Mathematical modeling of collective behavior in socio-economic and life sciences

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Author: Giovanni Naldi | Lorenzo Pareschi | Giuseppe Toscani

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Modeling and Simulation in Science, Engineering and Technology Series Editor Nicola Bellomo Politecnico di Torino Italy Advisory Editorial Board M. Avellaneda (Modeling in Economics) Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY 10012, USA

[email protected] K.J. Bathe (Solid Mechanics) Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139, USA

[email protected] P. Degond (Semiconductor and Transport Modeling) Mathematiques ´ pour l’Industrie et la Physique Université P. Sabatier Toulouse 3 118 Route de Narbonne 31062 Toulouse Cedex, France

[email protected] A. Deutsch (Complex Systems in the Life Sciences) Center for Information Services and High Performance Computing Technische Universität Dresden 01062 Dresden, Germany

[email protected] M.A. Herrero Garcia (Mathematical Methods) Departamento de Matematica Aplicada Universidad Complutense de Madrid Avenida Complutense s/n 28040 Madrid, Spain

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W. Kliemann (Stochastic Modeling) Department of Mathematics Iowa State University 400 Carver Hall, Ames, IA 50011, USA

[email protected] H.G. Othmer (Mathematical Biology) Department of Mathematics University of Minnesota 270A Vincent Hall Minneapolis, MN 55455, USA

[email protected] L. Preziosi (Industrial Mathematics) Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino, Italy

[email protected] V. Protopopescu (Competitive Systems, Epidemiology) CSMD Oak Ridge National Laboratory Oak Ridge, TN 37831-6363, USA

[email protected] K.R. Rajagopal (Multiphase Flows) Department of Mechanical Engineering Texas A&M University College Station, TX 77843, USA

[email protected] Y. Sone (Fluid Dynamics in Engineering Sciences) Professor Emeritus Kyoto University 230-133 Iwakura-Nagatani-cho Sakyo-ku Kyoto 606-0026, Japan

[email protected]. kyoto-u.ac.jp

For other titles published in this series, go to http://www.springer.com/series/4960

Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences

Giovanni Naldi Lorenzo Pareschi Giuseppe Toscani Editors

Birkhäuser Boston • Basel • Berlin

Editors Giovanni Naldi Università di Milano Dipartimento di Matematica Via Saldini, 50 20133 Milano Italy [email protected]

Giuseppe Toscani Università di Pavia Dipartimento di Matematica Via Ferrata, 1 27100 Pavia Italy [email protected]

Lorenzo Pareschi Università di Ferrara Dipartimento di Matematica Via Machiavelli, 35 44100 Ferrara Italy [email protected]

ISBN 978-0-8176-4945-6 e-ISBN 978-0-8176-4946-3 DOI 10.1007/978-0-8176-4946-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010930687 Mathematics Subject Classification (2010): 35B40, 60K35, 76P05, 82B21, 82C40, 91C20 c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Birkhäuser is part of Springer Science+Business Media (www.birkhauser-science.com)

Preface

The description of emerging collective behaviors and self–organization in a group of interacting individuals has gained increasing interest from various research communities in biology, engineering, physics, as well as sociology and economics. In the biological context, swarming behavior of bird flocks, fish schools, insects, bacteria, and people is a major research topic in behavioral ecology with applications to artificial intelligence. Likewise, emergent economic behaviors, such as distribution of wealth in a modern society and price formation dynamics, or challenging social phenomena such as the formation of choices and opinions are also problems in which the emergence of collective behaviors and universal equilibria has been shown. These behaviors occur widely in nature and are part of our daily lives. It is quite surprising (ever astonishing) to learn that they can be fruitfully described and understood by means of suitable mathematical tools. Mathematical modeling using partial differential equations, which touches core areas of physics and engineering, is in fact playing an increasing role in emerging fields such as the social, economic and life sciences. Mathematical efforts are gradually gaining strength in this multidisciplinary area. Typically, the underlying equations are highly nonlinear; in many cases, they are also vectorial systems and represent a challenge even for the most modern and sophisticated mathematical-analytical and mathematical-numerical techniques. Among others, multidimensional computations of complex multi-scale phenomena are now within reach. Sophisticated nonlinear analysis deepens our understanding of increasingly complex models. Computational results feed back into the modeling process, and provide insight into detailed mechanisms that often cannot be studied by real-life experiments. The novelty here is that important phenomena in seemingly different areas such as sociology, economics and biology can be described by closely related mathematical models. In this book we present selected research topics that can be regarded as new and challenging frontiers of applied mathematics. These topics have been chosen to elucidate the common methodological background underlining the main idea of this book: to identify similar modeling

VI

Preface

approaches, similar analytical and numerical techniques, for systems made out of a large number of “individuals” that show a “collective behavior,” and obtain from them “average” information. The expertise obtained from dealing with physical situations is considered as the basis for the modeling and simulation of problems for applications in the socio-economic and life sciences, as a newly emerging research field. In most of the selected contributions, the main idea is that the collective behaviors of a group composed of a sufficiently large number of individuals (agents) could be described using the laws of statistical mechanics as it happens in a physical system composed of many interacting particles. This opens a bridge between classical statistical physics and the socio-economic and life sciences. In particular, powerful methods borrowed from the kinetic theory of rarefied gases can be fruitfully employed to construct kinetic equations that describe the emergence of universal structures through their equilibria. The book is subdivided into three parts, and each part contains several chapters listed alphabetically by author. Part I deals with the microscopic and kinetic modeling of simple economies and financial markets. Some of this kinetic modeling is clearly rather ad hoc, but if one is willing to accept the analogies between trading agents and colliding particles, then various wellestablished methods from statistical physics and applied mathematics seem ready for application to the field of economics. Most notably, the numerous tools originally devised for the study of the energy distribution in a rarefied gas can now be used to analyze wealth and price distributions. Likewise, microscopic models of both social and political phenomena describing collective behaviors and self-organization in a society can be analyzed using these methods. These topics are the contents of Part II. Among others, the modeling of opinion formation and vote intention dynamics has attracted the interest of an increasing number of researchers in recent years. The last part of the book deals mainly with the collective self-driven motion of self-propelled particles such as flocking of birds, schooling of fishes, swarming of insects and bacteria, traffic and crowd movements. These coherent and synchronized structures are apparently produced without the active role of a leader in the grouping, and can be described according to similar concepts of statistical physics and applied mathematics. General population dynamics and modern human warfare models are also presented here. The idea of publishing a book to highlight these new emerging applications of mathematics occurred to us at the conclusion of a short series of lectures we organized in Vigevano in November 2008, with the support of the Municipality of Vigevano, the Center for Interuniversity Research in Land Economics founded by the Universities of Milano-Bicocca, Pavia and Ferrara (CRIET), the Advanced Applied Mathematical and Statistical Sciences Center of the University of Milan (ADAMSS) and the Center for Modeling Computing & Statistics of the University of Ferrara (CMCS). These lectures served as the springboard starting point for the present book, which includes several other contributions of distinguished researchers.

Preface

VII

We warmly hope this book will be of great interest to applied mathematicians, physicists, biologists, economists, and engineers involved in the modelling of complex socio-economic systems, and in aggregation and collective phenomena in general. Milano, Ferrara, Pavia December 2009

Giovanni Naldi Lorenzo Pareschi Giuseppe Toscani

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V Part I Economic modelling and financial markets Agent-based models of economic interactions Anirban Chakraborti and Guido Germano . . . . . . . . . . . . . . . . . . . . . . . . . . .

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On kinetic asset exchange models and beyond: microeconomic formulation, trade network, and all that Arnab Chattejee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Microscopic and kinetic models in financial markets Stephane Cordier, Dario Maldarella, Lorenzo Pareschi, and Cyrille Piatecki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 A mathematical theory for wealth distribution Bertram D¨ uring and Daniel Matthes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Tolstoy’s dream and the quest for statistical equilibrium in economics and the social sciences Ubaldo Garibaldi and Enrico Scalas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Part II Social modelling and opinion formation New perspectives in the equilibrium statistical mechanics approach to social and economic sciences Elena Agliari, Adriano Barra, Raffaella Burioni, and Pierluigi Contucci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Kinetic modelling of complex socio-economic systems Giulia Ajmone Marsan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

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Mathematics and physics applications in sociodynamics simulation: the case of opinion formation and diffusion Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi . . . . . . . . . . . . 203 Global dynamics in adaptive models of collective choice with social influence Gian-Italo Bischi and Ugo Merlone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Modelling opinion formation by means of kinetic equations Laurent Boudin and Francesco Salvarani . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Part III Human behavior and swarming On the modelling of vehicular traffic and crowds by kinetic theory of active particles Nicola Bellomo and Abdelghani Bellouquid . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Particle, kinetic, and hydrodynamic models of swarming José A. Carrillo, Massimo Fornasier, Giuseppe Toscani, and Francesco Vecil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints Emiliano Cristiani, Benedetto Piccoli, and Andrea Tosin . . . . . . . . . . . . . . 337 Statistical physics and modern human warfare Alex Dixon, Zhenyuan Zhao, Juan Camilo Bohorquez, Russell Denney, and Neil Johnson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Diffusive and nondiffusive population models Ansgar J¨ ungel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

Part I

Economic modelling and financial markets

Agent-based models of economic interactions Anirban Chakraborti1 and Guido Germano2,3 1 2 3

Laboratoire de Mathématiques Appliquées aux Systèmes, Ecole Centrale Paris, 92290 Chˆ atenay-Malabry, France, [email protected] Fachbereich 15 und WZMW, AG Computersimulation, Philipps-Universit¨ at Marburg, 35032 Marburg, Germany, [email protected] Dipartimento di Scienze Economiche e Metodi Quantitativi, Universit` a del Piemonte Orientale “Amedeo Avogadro,” Via Ettore Perrone 18, 28100 Novara, Italy, [email protected]

Summary. The interdisciplinary field of econophysics has enjoyed recently a surge of activities especially with numerous agent-based models, which have led to a substantial development of this field. We review three main application areas of agentbased models in econophysics: order books, distributions of wealth in conservative economies, and minority games.4

1 Introduction The term “complex system” was coined to cover a great variety of systems which include examples from natural sciences (physics, chemistry, biology, etc.) as well as social sciences (economics and sociology), especially those where the constituents are dissimilar (heterogeneous) and the interactions among them are not known particularly well. In this respect, a socio-economic system, as e.g. an economic market, is a perfect example of a complex system, since every constituent entity (agent) has different characteristics and behaves in a different way. The system is not just an assembly of many identical particles as we commonly encounter in physics, for example. Econophysicists have been particularly intrigued by a number of phenomena described by power laws in economic systems, for example the distribution of price changes and of individual income and wealth. All have a sort of an universal power-law tail. Fitting with power laws does not trigger any interest in most economists, but in does in physicists since many complex physical systems display a similar intermittent dynamics; for example, the avalanche dynamics in random magnets under a slowly varying external field and the progression of cracks in a slowly strained disordered material. The important thing about these physical examples is that the exogenous driving force is regular and steady, but 4

Portions of this text have been taken from Sec. IV of the review by A. C. et al. “Econophysics: Empirical facts and agent-based models”, arXiv:0909.1974. G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 1, c Springer Science+Business Media, LLC 2010

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4

Anirban Chakraborti and Guido Germano

the resulting endogenous dynamics is complex and critical. Econophysicists have thus proposed simple multi-agent models with heterogeneities, where the nature of the dynamics comes from collective effects. In these models, the individual entities or components have a relatively simple behavior, but interactions lead to new, emergent and cooperative phenomena. The whole complex system is fundamentally different from its subparts or elementary constituents. This has been summarized in the famous paradigma “more is different” [103]. As this intermittent behavior appears to be generic for physical systems with both heterogeneities and interactions, it leads to think that the dynamics of socio-economic systems may also have the same underlying mechanisms. In economic models there is usually a representative agent, who has unlimited foresight and capability of deliberation, “perfect rationality”, and uses the “utility maximization” principle to act economically, taking into account all potential future events with the correct probabilities. The “rational-agent” paradigm is also coupled with another reductionism: As there is a single way to act perfectly rationally, all agents should display exactly the same behavior. So one representative agent would be sufficient. Thus the typical format of current economic models is that of a single agent or firm maximizing its utility or profit with perfect foresight over a finite or infinite period. Instead, the agent-based models that have originated from simple statistical physics considerations have allowed one, for example, to go beyond the prototype theories with a “representative” agent in traditional economics. The recent failure of economists to anticipate the collapse of markets worldwide since 2007, has led over a short period of time voices even from within the field of economics itself, suggesting that new foundations for the discipline are required [104,105]. Some scientists have suggested that econophysics and in particular agent-based models may provide such an alternative theoretical framework for rebuilding economics [106]. Statistical physics, a branch of physics that combines the principles and procedures of statistics with the laws of both classical and quantum mechanics, and explains the measurable properties of macroscopic systems on the basis of the properties and behavior of their microscopic constituents, has turned out to be useful in the study of these diverse complex systems. In the following, we review three main fields where econophysicists have applied this approach through agent-based models to the description of particular aspects of economics: order books, the distribution of wealth, and minority games. More recently these and other sectoral models have led to the development of comprehensive agent-based models of a whole economy including a spatial structure and the major markets considered in quantitative macroeconomic modelling (consumer goods, investment goods, labour, credit and finance) [107, 108]; however, due to a lack of time, we could not include the latter in our review.

Agent-based models of economic interactions

5

2 Order books Much has been done in the past fifteen years by providing new models that can reproduce the “stylized facts,” that have often been left aside when modeling financial markets referring to them as “anomalous” characteristics, as if observations failed to comply with theory. These recent developments have been built on top of early attempts at modeling mechanisms of financial markets with agents. Stigler [1], who investigated some rules of the Security Exchange Commission (SEC), and Garman [2], who investigated double-auction microstructure, belong to those historical works. The first modern attempts at this type of models were perhaps made in the field of behavioral finance. Agent-based models in financial economics were built with numerous agents who can exchange shares of stocks according to exogenously defined utility functions reflecting their preferences and risk aversions. LeBaron [3] provides a recent review of that type of models. Although achieving some of their goals, these models suffer from many drawbacks: (a) they were complicated, and it was a difficult task to identify the role of their numerous parameters and their dependencies; (b) the chosen utility functions did not reflect what was observed on the mechanisms of financial markets. Simpler models were introduced in the last decade, implementing only well-identified and presumably more realistic behavior: Cont and Bouchaud [4] used noise traders that are subject to “herding,” i.e., form random clusters of traders sharing the same view on the market. The idea was used by Roberto et al. [5] too. A complementary approach was to characterize traders as fundamentalists, chartists, or noise traders. Lux and Marchesi [6] proposed an agent-based model in which these types of traders interact. In all these models, the price variation directly results from the excess demand: at each time step, all agents submit orders and the resulting price is computed. Therefore, everything is cleared at each time step and there is no order book structure to keep track of orders. One big step was made with models taking into account limit orders and keeping them in an order book once submitted and not yet executed. Chiarella and Iori [7] built an agent-based model where all traders submit orders depending on the three groups identified in the Lux and Marchesi model [6]: chartists, fundamentalists and noise traders. Submitted orders are then stored in a persistent order book. One of the first simple models with this feature was proposed by Bak, Paczuski and Shubik [8]. In this model, orders are particles moving along a price line, and each collision is a transaction. Due to numerous caveats in this model, the authors proposed in the same paper an extension with fundamentalist and noise traders in the spirit of the models mentioned previously. Maslov [9] went further in the modeling of trading mechanisms by taking into account fixed limit orders and market orders that trigger transactions, and the order book was here really simulated. This model was solved analytically by Slanina using a mean-field approximation [10]. Following this trend of modeling, the rational agent models in economics started to be replaced by models with a notion of flows: orders are not

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submitted any more by an agent following a strategic behavior, but are viewed as arriving in flows whose properties are to be determined by empirical observations on market mechanisms. Biais, Hilllion and Spatt [11] made a thorough empirical study of the order flows in the Paris Bourse a few years after its complete computerization. Market orders, limit orders, time of arrivals, and placement were studied. Challet and Stinchcombe [12] proposed a simple model of order flows: limit orders are deposited in the order book and can be removed if not executed, in a simple deposition–evaporation process. Bouchaud and Potters [13] used this type of model with empirical distribution as inputs. Preis et al. [14] attempted to add perturbations to the order flows in order to reproduce stylized facts. An empirical model was proposed by Mike and Farmer [15], where order placement and cancellation models are proposed and fitted on empirical data. Here, we review some of these models that we feel are representative of a specific trend of modeling; certainly our review is not exhaustive. 2.1 Order-driven market models Starting from the mid-1990s, physicists have propose order book models directly inspired from physics, where the analogy “order = particle” is emphasized. The Bak, Paczuski and Shubik model This is a simple model [8] where the authors consider a market with N noise traders able to exchange one share of stock at a time. Price p(t) at time t is constrained to be an integer (i.e., price is quoted in number of ticks) with an upper bound p¯: ∀t, p(t) ∈ {0, . . . , p¯}. Simulation is initiated at time 0 with half of the agents asking for one share of stock (buy orders, bid) with price pjb (0) ∈ {0, p¯/2},

j = 1, . . . , N/2,

(1)

and the other half offering one share of stock (sell orders, ask) with price pjs (0) ∈ {¯ p/2, p¯},

j = 1, . . . , N/2.

(2)

At each time step t, agents revise their offer by exactly one tick, with equal probability to go up or down. Therefore, at time t, each seller or buyer chooses his new price as pjs (t + 1) = pjs (t) ± 1

or pjb (t + 1) = pjb (t) ± 1.

(3)

A transaction occurs when there exists (i, j) ∈ {1, . . . , N/2}2 such that pib (t + 1) = pjs (t + 1). In such a case the orders are removed and the transaction price is recorded as the new price p(t). Once a transaction has been recorded, two orders are placed at the extreme positions on the grid:


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pib (t + 1) = 0 and pjs (t + 1) = p¯. As a consequence, the number of orders in the order book remains constant and equal to the number of agents. As pointed out by the authors, this process of simulation is similar to the reaction–diffusion model A + B → ∅ in Physics. In such a model, two types of particles are inserted at each side of a pipe of length p¯ and move randomly with steps of size 1. Each time two particles collide, they are annihilated and two new particles are inserted. Following this analogy, it thus can be shown that the variation Δp(t) of the price p(t) obeys Δp(t) ∼ t

1/4

1/2 t log . t0

(4)

Thus, at long time scales, the series of price increments simulated in this model exhibit a Hurst exponent H = 1/4. As for the stylized fact H ≈ 0.7, this subdiffusive behavior appears to be a step in the wrong direction compared to the random walk H = 1/2. Moreover, Slanina [16] pointed out that no fat tails are observed in the distribution of the returns of the model, but rather fits the empirical distribution with an exponential decay. Many more drawbacks of the model could be mentioned, the main one being that “moving” orders is highly unrealistic as for modeling an order book. However, we feel that such a model is interesting because of its simplicity and its representation of an order-driven market. The Maslov model Maslov [9] keeps the zero-intelligence structure of the earlier model [8] but adds more realistic features in the order placement and evolution of the market: (1) limit orders are submitted and stored in the model, without moving, (2) limit orders are submitted around the best quotes, and (3) market orders are submitted to trigger transactions. More precisely, at each time step, a trader is chosen to perform an action. In contrast to previous models, the number of traders is not fixed, but one trader enters the market at each time step. This trader can either submit a limit order with probability ql or submit a market order with probability 1 − ql . Once this choice is made, the order is a buy or sell order with equal probability. All orders have a one unit volume. As usual, we denote p(t) as the current price. In case the submitted order at time step t + 1 is a limit ask (respectively bid) order, it is placed in the book at price p(t) + Δ (respectively p(t) − Δ), Δ being a random variable uniformly distributed in ]0; ΔM = 4]. In case the submitted order at time step t + 1 is a market order, one order at the opposite best quote is removed and the price p(t + 1) is recorded. In order to prevent the number of orders in the order book from large increase, two mechanisms are proposed by the author: either keeping a fixed maximum number of orders (new limit orders are the discarded) or removing them after a fixed lifetime if they haven’t been executed. As for numerical simulations, results show that this model exhibits

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nongaussian heavy-tailed distributions. For a time scale δt = 1, the author fit the tails distribution with a power law with exponent 3.0, i.e., reasonable compared to empirical value. However, the Hurst exponent of the price series is still H = 1/4 with this model. This model brings interesting innovations in order book simulation: order book with (fixed) limit orders, market orders, necessity to cancel orders waiting too long in the order book. These features are of prime importance and open a new trend in order book modelling. The Challet and Stinchcombe model Challet and Stinchcombe [12] continue the work of Bak et al. [8] and Maslov [9] expliciting the analogy between dynamics of an order book and an infinite one-dimensional grid where particles of two types (ask and bid) are subject to three types of events: deposition (limit orders), annihilation (market orders), and evaporation (cancelation). Note that annihilation occurs when a particle is deposited on a site occupied by a particle of another type. Therefore, the model goes as follows: at each time step, a bid (or ask) order is deposited with probability λ at a price n(t) drawn according to a Gaussian distribution centered on the best ask a(t) (or best bid b(t)) and with variance depending linearly on the spread s(t) = a(t) − b(t): σ(t) = Ks(t) + C. If n(t) > a(t) (respectively n(t) < b(t)), then it is a market order: annihilation takes place and the price is recorded. Otherwise, it is a limit order and it is stored in the book. Finally, each limit order stored in the book has a probability δ to be canceled (evaporation). The series of price returns simulated with this model exhibit a Hurst exponent H = 1/4 for short time scales, and that tends to H = 1/2 for larger time scales. This behavior might be the consequence of the random evaporation process (which was not modeled in [9], where H = 1/4 for large time scales). Although some modifications of the process (more than one order per time step) seem to shorten the subdiffusive region, it is clear that no over-diffusive behavior is observed. 2.2 The market interactions models In all the models we have reviewed until now, flows of orders are treated as an independent processes. In this section we present some toy models implementing mechanisms that aim at bringing heterogeneity: Herding behavior on markets in [4], heterogenous agents in [6], and dynamic order submission in [14]. The Cont and Bouchaud model This model [4] considers a market with N agents trading a given stock with price p(t). At each time step, agents choose to buy or sell one unit of stock,


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i.e., their demand is φi (t) = ±1, i = 1, . . . , N with probability a or are idle with probability 1 − 2a. The price change is assumed to be linearly linked with N the excess demand D(t) = i=1 φi (t) with a factor λ measuring the liquidity of the market: N 1 p(t + 1) = p(t) + φi (t). (5) λ i=1 λ can also be interpreted as a market depth, i.e., the excess demand needed to move the price by one unit. In order to evaluate the distribution of stock returns from (5), we need to know the joint distribution of the individual demands (φi (t))1≤i≤N . As pointed out by the authors, if the distribution of the demand φi is independent and identically distributed with finite variance, then the Central Limit Theorem stands and the distribution of the price variation Δp(t) = p(t + 1) − p(t) will converge to a Gaussian distribution as N goes to infinity. The idea of the model is to model the diffusion of the information among traders by randomly linking their demand through clusters. At each time step, agents i and j can be linked with probability pij = p = c/N , c being a parameter measuring the degree of clustering among agents. Therefore, an agent is linked to an average number of (N − 1)p other traders. Once clusters are determined, the demand are forced to be identical among all members of a given cluster. Denoting nc (t) the number of cluster at a given time step t, Wk the size of the kth cluster, k = 1, . . . , nc (t) and φk = ±1 its investement decision, the price variation is then straightforwardly written: nc (t) 1 Δp(t) = Wk φk . λ

(6)

k=1

This modeling is a direct application to the field of finance of the random graph framework as studied in [17]. Ref. [18] previously suggested it in economics. Using these previous theoretical works, and assuming that the size of a cluster Wk and the decision taken by its members φk (t) are independent, the author are able to show that the distribution of the price variation at time t is the sum of nc (t) independent identically distributed random variables with heavy-tailed distributions: nc (t) 1 Δp(t) = Xk , λ

(7)

k=1

where the density f (x) of Xk = Wk φk is decaying as: f (x) ∼|x|→∞

−(c−1)|x| A W0 e . |x|5/2

(8)

Thus, this simple toy model exhibits fat tails in the distribution of prices variations, with a decay reasonably close to empirical data. Therefore, [4]

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shows that taking into account a naive mechanism of communication between agents (herding behavior) is able to drive the model out of the Gaussian convergence and produce non-trivial shapes of distributions of price returns. The Lux and Marchesi model Lux and Marchesi [6] proposed a model very much in line with agent-based models in behavioral finance, but where trading rules are kept simple enough so that they can be identified with a presumably realistic behavior of agents. This model considers a market with N agents that can be a part of two distinct groups of traders: nf traders are “fundamentalists,” who share an exogenous idea pf of the value of the current price p; and nc traders are “chartists,” who make assumptions on the price evolution based on the observed trend (mobile average). The total number of agents is constant, so that nf + nc = N at any time. At each time step, the price can be moved up or down with a fixed jump size of ±0.01 (a tick). The probability to go up or down is directly linked to the excess demand ED through a coefficient β. The demand of each group of agents is determined as follows: 1. Each fundamentalist trades a volume Vf accordingly (through a coefficient γ) to the deviation of the current price p from the perceived fundamental value pf : Vf = γ(pf − p). 2. Each chartist trades a constant volume Vc . Denoting n+ the number of optimistic (buyer) chartists and n− the number of pessimistic (seller) chartists, the excess demand by the whole group of chartists is written as (n+ − n− )Vc . Finally, assuming that there exists some noise traders on the market with random demand μ, the global excess demand is ED = (n+ − n− )Vc + nf γ(pf − p) + μ.

(9)

The probability that the price goes up (respectively down) is then defined to be the positive (respectively negative) part of βED. As in [4], the authors expect nontrivial features of the price series to results from herding behavior and transitions between groups of traders. Referring to Kirman’s work as well, a mimicking behavior among chartists is thus proposed. The nc chartists can change their view on the market (optimistic, pessimistic), their decision being based on a clustering process modelized buy an opinion index x = (n+ − n− )/(nc ) representing the weight of the majority. The probabilities π+ and π− to switch from one group to another are formally written: nc π± = v e±U U = α1 x + α2 p/v, (10) N where v is a constant, and α1 and α2 reflect, respectively, the weight of the majority’s opinion and the weight of the observed price in the chartists’ decision.


11

Finally, transitions between fundamentalists and chartists are also allowed, decided by comparison of expected returns. We do not reproduce here the explicit probabilities of transitions of the model (see [6] for details), as this would need to introduce more parameters that are not useful for the purpose of our review. They are formally similar to π± with of course a different function U . The authors present both theoretical and empirical results. First, they are able to derive approximate differential equation in continuous time governing mean values of the model, thus derive stationary solution for these mean values. Second, they provide simulation results and show that the distribution of returns generated by their model have excess kurtosis. Using a Hill estimator, they fit a power law to the fat tails of the distribution and observe exponents grossly ranging from 1.9 to 4.6. The authors also check hints for volatility clustering: absolute returns and squared returns exhibit a slow decay of autocorrelation, while raw returns do not. Hence, it appears that such a model can grossly fit some “stylized facts.” However, the number of parameters involved, as well as the quite obscure rules of transition among agents makes clear identification of sources of phenomena and calibration to market data difficult, if not intractable. The Preis model Preis [14] studies a model similar to those of Challet and Stinchcombe [12,19], adding a link between order placement and market trend as an attempt to obtain an over-diffusive behavior in the price series. Doing so, it is in line with some extensions of the previous models we have already mentioned. The simulation process goes similarly to the one with deposition, annihilation and evaporation events [12]. Price p(t) is stored as an integer (number of ticks). Every order has a volume of one unit of stock. As in previous presentations, a(t) (respectively b(t)) is the best ask (respectively best bid) at time t. First, at each time step, N liquidity providers submit limit orders at rate λ, i.e., on average λN limit orders are inserted in the order book per time step. The price of a new limit ask (respectively bid) orders is n(t) = b(t) + Δ (respectively n(t) = a(t)−Δ) where Δ is a random variable with an exponential distribution with parameter α. Second, N liquidity takers submit market orders at rate μ, with an equal probability q = 1/2 to buy or sell. Finally, each order staying in the order book is canceled with probability δ. Note that this formalism allows for a direct computation of the average number order of orders Ne in the order book if an equilibrium exists: μ 1 Ne = N λ −1 − . (11) δ δ As expected, this zero-intelligence framework neither produce fat tails in the distribution of (log-)returns nor an overdiffusive Hurst exponent. As in the previous similar models, the authors observe a underdiffusive behavior

12


for short time scales, and a Hurst exponent H = 1/2 for larger time scales. However, an interesting result appears when extending the basic model with random demand perturbation and dynamic limit order placement depth. The author propose that the probability q for a buy/sell market order evolves at each time step according to a feedback random walk: q(t + 1) =

q(t) − Δq q(t) + Δq

with probability q(t), with probability 1 − q(t),

(12)

where δq is a constant. This stochastic perturbation alone is not sufficient to produce nontrivial Hurst exponent. Thus, it is then combined with a dynamic placement order depth, i.e., the author propose that the parameter α vary during the simulation as follows: α(t) = α(0) (1 + C|q(t) − 1/2|)

(13)

This equation formalizes the assumption that in a market with an observed trend in prices, liquidity providers expect larger variations and submit limit orders at a wider depth in the order book. Although the assumption behind such a mechanism may not be empirically confirmed (e.g., the assumed symmetry) and should be further discussed, it is interesting enough that it directly provides fat tails in the log-return distributions and an over-diffusive Hurst exponent H ≈ 0.6 − 0.7 for medium time scales.

3 Wealth distributions The distribution of wealth or income, i.e., how richness is shared among the population of a given country and among different countries, is a topic of wide interest for over centuries. Of course wealth is a stock and income is a flow, however these two quantities and thus their distributions are related, as income is the derivative of wealth (a pun by N. Kalecki says that economics is the science of confusing stocks with flows.) From the point of view of the science of complex systems, wealth distributions represent an unique example of a quantitative outcome of a collective behavior which can be directly compared with the predictions of theoretical models and numerical experiments. Also, there is a basic interest in wealth distributions from the social point of view, in particular in their degree of inequality. It was first observed by Pareto [20] that in an economy the higher end of the distribution of income f (x) follows a power law, f (x) ∼ x−1−α ,

(14)

with α, now known as the Pareto exponent, estimated by Pareto to be α ≈ 3/2. For the last 100 years the value of α ≈ 3/2 seems to have changed little in


13

time and across the various capitalist economies. However, in 1931, Gibrat [21] clarified that, while Pareto’s law is valid only for the high-income range, for the middle-income range the income distribution is described by a log-normal probability density, 1 log2 (x/x0 ) f (x) ∼ √ , (15) exp − 2σ 2 x 2πσ 2 where log(x0 ) = log(x) is the mean value of the logarithmic variable √ and σ 2 = [log(x)−log(x0 )]2 the corresponding variance. The factor β = 1/ 2σ 2 , also know as an Gibrat index, measures the equity of the distribution. More recent empirical studies on income distribution have been carried out by physicists, e.g., those by Dragulescu and Yakovenko for UK and US [22,23], and Fujiwara et al. for Japan [24], for an overview see [25]. The distributions obtained have been shown to follow either the Gibbs or power-law types, depending on the range of wealth. Thus, one of the current challenges is to write down the “microscopic equation” which governs the dynamics of the evolution of wealth distributions, possibly predicting the observed shape of wealth distributions, including the exponential law at intermediate values of wealth as well as the century-old Pareto law. To this aim, several studies have been made to investigate the characteristics of the real income distribution and provide theoretical models or explanations, for a review see [25–27]. The model of Gibrat [21] mentioned earlier and other models formulated in terms of a Langevin equation for a single wealth variable, subjected to multiplicative noise [28–31], can lead to equilibrium wealth distributions with a power law tail, since they converge toward a log-normal distribution. However, the fit of real wealth distributions does not turn out to be as good as that obtained using, e.g., a Γ- or a β-distribution, in particular due to too large asymptotic variances [32]. Other models describe the wealth dynamics as a wealth flow due to exchanges between (pairs of) basic units. In this respect, such models are basically different from the class of models formulated in terms of a Langevin equation for a single wealth variable. For example, Levy and Solomon studied the generalized Lotka-Volterra equations in relation to power-law wealth distribution [33, 34]. Ispolatov et al. [35] studied random exchange models of wealth distributions. Other models describing wealth exchange have been formulated using matrix theory [36], the master equation [37–39], the Boltzmann equation [38, 40–45], or Markov chains [46– 48]. Here, we consider in greater detail a class of models usually referred to as kinetic wealth exchange models (KWEM), formulated through finite timedifference stochastic equations [32, 38, 49–59]. 3.1 Kinetic wealth exchange models We consider KWEMs, which are statistical models of a closed economy, in which N agents exchange a quantity x, defined as wealth in some models and

14


as money in others. As explained earlier, money can be interpreted as the unity of measure for all the goods that constitute the agents’ wealth, so that it is possible, in order to avoid confusion, to use only the term wealth. In these models the states of the agents are defined in terms of their wealth variables {xn }, n = 1, 2, . . . , N . The evolution of the system is carried out according to a “trading rule” between agents which, for obtaining the final equilibrium distribution, can be interpreted as the actual time evolution of the agent states as well as a Monte Carlo optimization. The algorithm is based on a simple update rule performed at each time step t, when two agents i and j are extracted randomly and an amount of wealth Δx is exchanged, xi = xi − Δx , xj = xj + Δx .

(16)

Notice that the quantity x is conserved during single transactions, xi + xj = xi + xj , where xi = xi (t) and xj = xj (t) are the agent wealths before, whereas xi = xi (t+1) and xj = xj (t+1) are the final ones after the transaction. Several rules have been studied for the model defined by (16). We refer to the Δx there defined in illustrating the various models. Exchange models without saving In a first version of the model, considered in early works by Bennati [60–62] and rediscovered independently by Dragulescu and Yakovenko [38], the money difference Δx in (16) is assumed to have a constant value, Δx = Δx0 .

(17)

This rule, together with the constraint that transactions can take place only if xi > 0 and xj > 0, provides an equilibrium exponential distribution, see the curve for λ = 0 in Fig. 1. Various other trading rules were studied by Dragulescu and Yakovenko [38], choosing Δx as a random fraction of the average money between the two agents, Δx = (xi + xj )/2, or of the average money of the whole system, Δx = x , where is a uniform random number ∈ (0, 1). In the reshuffling model, which can be obtained as a particular case from the model with saving introduced in [51] for a saving parameter λ = 0 (see below), the wealths of the two agents are reshuffled randomly, xi = (xi + xj ) , xj = (1 − )(xi + xj ) .

(18)

In this case the Δx appearing in the trading rule (16) is given by: Δx = (1 − )xi − xj .

(19)


15

All the models mentioned here, as well as some more complicated models [38], lead to a robust equilibrium Boltzmann distribution, f (x) = β exp(−βx) ,

(20)

with the effective temperature of the system equal to the average wealth, β −1 = x . This result is largely independent of the details of the models, e.g., the multiagent nature of the interaction, the initial conditions, and the random or consecutive order of extraction of the interacting agents. The Boltzmann distribution is sometimes referred to as an “unfair distribution,” since it is characterized by a majority of poor agents and a few rich agents (sue to the exponential tail). The exponential distribution is characterized by a Gini coefficient of 0.5. Exchange models with saving As a generalization and more realistic version of the basic exchange models without saving, a saving criterion can be introduced. A saving parameter 2.5

10

λ=0 λ = 0.2 λ = 0.5 λ = 0.7 λ = 0.9

2

λ=0 λ = 0.2 λ = 0.5 λ = 0.7 λ = 0.9

1 0.1 0.01 f (x)

f (x)

1.5 1

0.001 0.0001 1e-05

0.5

1e-06

0

0

0.5

1

1.5

x

2

2.5

3

3.5

1e-07

0

1

2

3

4

5 x

6

7

8

9

10

Fig. 1. Probability density for wealth x. The curve for λ = 0 is the Boltzmann function f (x) = x−1 exp(−x/x) for the basic model of Sect. 3.1. The other curves correspond to a global saving propensity λ > 0, see Sect. 3.1

0 < λ < 1 representing the fraction of wealth saved and not reshuffled, was introduced in the model introduced in [51]. In this model (CC) wealth flows simultaneously toward and from each agent during a single transaction, the dynamics being defined by the equations xi = λxi + (1 − λ)(xi + xj ) , xj = λxj + (1 − )(1 − λ)(xi + xj ) ,

(21)

or, equivalently, by a Δx in (16) given by: Δx = (1 − λ)[(1 − )xi − xj ] .

(22)

16


All these models lead to an equilibrium distribution qualitatively different from the purely exponential distribution of models without saving. In fact, there is now a mode xm > 0, in agreement with real data of wealth and income distributions [23,39,63–66]. Furthermore, the limit for small x is zero, i.e., P (x → 0) → 0, see the example in Fig. 1. The functional form of such a distributions has been conjectured to be a Gamma distribution, on the base of the excellent fitting provided to numerical data [32, 49, 67–70], f (x) = n x −1 γn (nx/ x ) = n(λ) ≡

1 n Γ (n) x

nx x

n−1

nx , exp − x

Dλ 3λ =1+ , 2 1−λ

(23) (24)

where γn (ξ) is the standard Γ -distribution. The analogy between models is more than qualitative. For instance, it has been found [70] that the only difference between the equilibrium solution above and that of the symmetrical model of Angle is in the effective dimension, which in the Angle model is nA (λ) = n(λ)/2, where n is given in Eq. (24). The ubiquitous presence of Γ -functions in the solutions of kinetic models (see also later heterogeneous models) suggests a close analogy with kinetic 100

1

Tλ/〈 x 〉

Dλ

0.8

10

0.6 0.4 0.2

1 0.01

0.1 λ

1

0

0

0.2

0.4

λ

0.6

0.8

1

Fig. 2. Effective dimension Dλ and temperature T as a function of the saving parameter λ

theory of gases. In fact, interpreting Dλ = 2n as an effective dimension, the variable x as kinetic energy, and introducing the effective temperature β −1 ≡ Tλ = x /2Dλ according to the equipartition theorem, (23) and (24) define the canonical distribution βγn (βx) for the kinetic energy of a gas in Dλ = 2n dimensions, see [69] for details. The analogy is illustrated in Table 1 and the dependence of Dλ = 2n and of β −1 = Tλ on the saving parameter λ are shown in Fig. 2. The exponential distribution is recovered as a special case, for n = 1. In the limit λ → 1, i.e., for n → ∞, the distribution f (x) above tends to a Dirac δ-function, as shown in [69] and qualitatively illustrated by the curves


17

Table 1. Analogy between kinetic the theory of gases and the kinetic exchange model of wealth Variable Units Interaction Dimension Temperature definition Reduced variable Equilibrium distribution

Kinetic model K (kinetic energy) N particles Collisions Integer D kB T = 2K/D ξ = K/kB T f (ξ) = γD/2 (ξ)

Economy model x (wealth) N agents Trades Real number Dλ Tλ = 2x/Dλ ξ = x/Tλ f (ξ) = γDλ /2 (ξ)

in Fig. 1. This shows that a large saving criterion leads to a final state in which economic agents tend to have similar amounts of money and, in the limit of λ → 1, exactly the same amount x . The equivalence between a kinetic wealth-exchange model with saving propensity λ ≥ 0 and an N -particle system in a space with dimension Dλ ≥ 2 is suggested by simple considerations about the kinetics of collision processes between two molecules. In one dimension, particles undergo head-on collisions in which the whole amount of kinetic energy can be exchanged. In a larger number of dimensions the two particles will not travel in general exactly along the same line, in opposite verses, and only a fraction of the energy can be exchanged. It can be shown that during a binary elastic collision in D dimensions only a fraction 1/D of the total kinetic energy is exchanged on an average for kinematic reasons, see [71] for details. The same 1/D dependence is in fact obtained inverting (24), which provides for the fraction of exchanged well 1 − λ = 6/(Dλ + 4). The basic models considered earlier, giving equilibrium wealth distributions with an exponential tail, interpolate well real data at small and intermediate values of wealth [22, 23, 32, 39, 63, 67, 72]. However, more realistic generalized models have been studied, in which agents are diversified by assigning different values of the saving parameter. Diversified saving parameters were introduced in Refs. [54, 73] by generalizing the model introduced in [51], xi = λi xi + [(1 − λi )xi + (1 − λj )xj ] , xj = λxj + (1 − )[(1 − λi )xi + (1 − λj )xj ] .

(25)

The surprising result is that if the parameters {λi } are suitably diversified, a power law appears in the equilibrium wealth distribution. In particular if λi are uniformly distributed in (0, 1) the wealth distribution exhibits a robust power-law tail, f (x) ∝ x−α−1 , (26) with the Pareto exponent α = 1 largely independent of the details of the λ-distribution, as reviewed by Chatterjee elsewhere in this book. This result is supported by theoretical considerations based on different approaches, such as mean field theory [74] or the Boltzmann equation [41, 58, 76, 77].

18


4 Minority games Minority games (MGs) [78] refer to multiagent models of financial markets according to the original basic formulation introduced by Challet and Zhang in 1997 [79] or later variants [80, 81, 109], most of which share the principal features that the models are repeated games and the agents have an inductive nature. 4.1 Basic minority game The basic minority game consists of an odd natural number N of agents, who choose between one of two options at each round of the game, using their own simple inductive strategies. The two options, denoted 0 and 1, could be for example buying or selling commodities or assets at a given time t. The agents who belong to the minority group win the game, and thus rewards are given to the strategies that predict the winning side. All the agents have access to a finite amount of public information, which is a common bit-string memory of the M most recent outcomes, composed of the winning sides in the past few rounds. Thus, the agents are said to exhibit “bounded rationality” [82]. Consider for example, memory M = 2; then there are P = 2M = 4 possible “history” bit strings: 00, 01, 10, and 11. A “strategy” consists of a response, i.e., 0 or 1, to each possible history bit strings; therefore, there are M G = 2P = 22 = 16 possible strategies which constitute the “strategy space.” At the beginning of the game, each agent randomly picks k strategies, and after the game, assigns one “virtual” point to a strategy which would have predicted the correct outcome. The actual performance r of the player is measured by the number of times the player wins, and the strategy, using which the player wins, gets a “real” point. Unlike most economics models which assume agents are “deductive” in nature, here a “trial-and-error” inductive thinking approach is implicitly implemented in process of decision-making when agents make their choices in the games. A record of the number of agents who have chosen a particular action, say, “selling” denoted by 1, A1 (t) as a function of time is kept. The fluctuations in the behavior of A1 (t) actually indicate the system’s total utility. For example, we can have a situation where only one player is in the minority and all the other players lose. The other extreme case is when (N − 1)/2 players are in the minority and (N + 1)/2 players lose. The total utility of the system is obviously greater for the latter case and from this perspective, the latter situation is more desirable. Therefore, the system is more efficient when there are smaller fluctuations around the mean than when the fluctuations are larger. For modeling purposes, minority games were meant to serve as a class of simple models which could produce some of the macroscopic features being observed in the real financial markets [81,83], usually termed as “stylized facts” which included the fat-tail price return distribution and volatility clustering [78, 80]. Despite the initial furore [84–86] they have, however, failed to capture or reproduce most important stylized facts. However, in the physicists’


19

community, minority games became an interesting and established class of complex disordered systems [83,87], yielding a large amount of deep physical insights [88, 89]. As in the BMG model a Hamiltonian function can be defined and analytic solutions can be developed in some regimes of the model, the model may be viewed with a more complete physical picture [90]. It is characterized by a clear two-phase structure with very different collective behaviors in the two phases, as in many known conventional physical systems [89, 91]. Savit et al. [89] first found that the macroscopic behavior of the system does not depend independently on the parameters N and M , but instead depends on the ratio 2M P = , (27) N N which serves as the most important control parameter in the game. The variance in the attendance ([92] or volatility σ 2 /N for different values of N and α≡

Asymmetric Phase

Symmetric Phase

σ2/N

100

N N N N N

10

1

= = = = =

51 101 251 501 1001

Worse-than-random Better-than-random

0.1 0.001

0.01

0.1

α

1

10

100

Fig. 3. The simulation results of the variance in attendance σ 2 /N as a function of the control parameter α = 2M /N for games with k = 2 strategies for each agent, ensemble averaged over 100 sample runs. Dotted line shows the value of volatility in random choice limit. Solid line shows the critical value of α = αc ≈ 0.3374. Resolution of the curve can be improved to shows σ 2 /N attains minimum at α ≈ αc . Reproduced from Yeung and Zhang [109]

M depends only on the ratio α. Figure 3 shows a plot of σ 2 /N against the control parameter α, where the data collapse of σ 2 /N for different values of N and M is clearly evident. The dotted line in Fig. 3 corresponds to the “coin-toss” limit (random choice or pure chance limit), in which agents play by simply making random decisions (by coin-tossing) at every rounds of the game. This value of σ 2 /N in coin-toss limit can be obtained by simply assuming a binomial distribution of the agents’ binary actions, with probability 0.5,

20


where σ 2 /N = 0.5(1 − 0.5) · 4 = 1. When α is small, the value of σ2 /N of the game is larger than the coin-toss limit which implies the collective behaviors of agents are worse than the random choices. In the early literature, it was popularly called as the worse-than-random regime. When α increases, the value of σ 2 /N decreases and enters a region where agents are performing better than the random choices, which was popularly called as the better-than-random regime. The value of σ 2 /N reaches a minimum value which is substantially smaller than the coin-toss limit. When α further increases, the value of σ 2 /N increases again and approaches the coin-toss limit. This allowed one to identify two phases in the minority game, as separated by the minimum value of σ 2 /N in the graph. The value of α where the rescaled volatility attended its minimum was denoted by αc , which represented the phase transition point; αc has been shown to have a value of 0.3374 . . . (for k = 2) by analytical calculations [90]. 4.2 A few variants of the basic minority game Challet generalized the basic minority game [79, 93] mentioned earlier to include the Darwinist selection: the worst player is replaced by a new one after some time steps, the new player is a “clone” of the best player, i.e., it inherits all the strategies but with corresponding virtual capitals reset to zero (analogous to a new born baby, though having all the predispositions from the parents, does not inherit their knowledge). To keep a certain diversity they introduced a mutation possibility in cloning. They allowed one of the strategies of the best player to be replaced by a new one. As strategies are not just recycled among the players any more, the whole strategy phase space is available for selection. They expected this population to be capable of “learning” because bad players are weeded out with time, and fighting is among the so-to-speak the “best” players. They observed that the learning emerged in time: Fluctuations are reduced and saturated, which implies the average gain for everybody is improved but never reaches the ideal limit. Li, Riolo, and Savit [94, 95] studied the minority game in the presence of evolution. In particular, they examined the behavior in games in which the dimension of the strategy space, m, is the same for all agents and fixed for all time. We find that for all values of m, not too large, evolution results in a substantial improvement in overall system performance. They also showed that after evolution, results obeyed a scaling relation among games played with different values of m and different numbers of agents, analogous to that found in the nonevolutionary, adaptive games. Best system performance still occurs, for a given number of agents, at mc , the same value of the dimension of the strategy space as in the nonevolutionary case, but system performance is now nearly an order of magnitude better than the nonevolutionary result. For m < mc , the system evolves to states in which average agent wealth is better than in the random choice game, despite (and in some sense because of) the persistence of maladaptive behavior by some agents. As m gets large, overall systems performance approaches that of the random choice game.


21

They continued the study of evolution in minority games by examining games [94, 95] in which agents with poorly performing strategies can trade in their strategies for new ones from a different strategy space. In the context of the games which meant allowing for strategies that use information from different numbers of time lags, m. They found, in all the games, that after evolution, wealth per agent is high for agents with strategies drawn from small strategy spaces (small m), and low for agents with strategies drawn from large strategy spaces (large m). In the game played with N agents, wealth per agent as a function of m was very nearly a step function. The transition is at m = mt , where mt mc − 1, and mc is the critical value of m at which N agents playing the game with a fixed strategy space (fixed m) have the best emergent coordination and the best utilization of resources. They also found that overall system-wide utilization of resources is independent of N . Furthermore, although overall system-wide utilization of resources after evolution varied somewhat depending on some other aspects of the evolutionary dynamics, in the best cases, utilization of resources was on the order of the best results achieved in evolutionary games with fixed strategy spaces. Sysi-Aho et al. [92, 96–98] presented a simple modification of the basic minority game where the players modify their strategies periodically after every time interval τ , depending on their performances: if a player finds that he is among the fraction n (where 0 < n < 1) who are the worst performing players, he adapts himself and modifies his strategies. They proposed that the agents use hybridized one-point genetic crossover mechanism (as shown in Fig. 4), inspired by genetic evolution in biology, to modify the strategies and replace the bad strategies. They studied the performances of the agents under different conditions and investigate how they adapt themselves in order to survive or be the best, by finding new strategies using the highly effective mechanism. They also studied the measure of total utility of the system U (xt ),

breaking point

si

sj

sk

sl

0

1

1

0

1

1

1

1

1

1

1

1

1

0

1

0 one−point crossover 1 1 0

1

1

1

1

1

1

1

1

1

1

1

1

parents

0

children

Fig. 4. Schematic diagram to illustrate the mechanism of one-point genetic crossover for producing new strategies. The strategies si and sj are the parents. We choose the breaking point randomly and through this one-point genetic crossover, the children sk and sl are produced and substitute the parents. Reproduced from [97]

22


which is the number of players in the minority group; the total utility of the system is maximum Umax as the highest number of players win is equal to (N − 1)/2. The system is more efficient when the deviations from the maximum total utility Umax are smaller, or in other words, the fluctuations in A1 (t) around the mean become smaller. If the parents are chosen randomly from the pool of strategies then the mechanism represents a “one-point genetic crossover” and if the parents are the best strategies then the mechanism represents a “hybridized genetic crossover.” The children may replace parents or two worst strategies and accordingly four different interesting cases arise: (a) one-point genetic crossover with parents “killed,” i.e., parents are replaced by the children, (b) onepoint genetic crossover with parents “saved,” i.e., the two worst strategies are replaced by the children but the parents are retained, (c) hybridized genetic crossover with parents “killed,” and (d) hybridized genetic crossover with parents “saved.” In order to determine which mechanism is the most efficient, we have made a comparative study of the four cases, mentioned earlier. We plot the attendance as a function of time for the different mechanisms in Fig. 5. 800

800

a

600

Attendance

Attendance

700

400 200

b

600 500 400 300 200

0 0

800

100 0

2000 4000 6000 8000 10000 Time

800

c

400 200 0 0

d

600 Attendance

Attendance

600

2000 4000 6000 8000 10000 Time

2000 4000 6000 8000 10000 Time

400 200 0 0

2000 4000 6000 8000 10000 Time

Fig. 5. Plots of the attendances by choosing parents randomly (a) and (b), and using the best parents in a player’s pool (c) and (d). In (a) and (c) case parents are replaced by children and in (b) and (d) case children replace the two worst strategies. Simulations have been done with N = 801, M = 6, k = 16, t = 40, n = 0.4, and T = 10, 000


23

It should be noted that the mechanism of evolution of strategies is considerably different from earlier attempts [94, 95, 99]. This is because in this mechanism the strategies are changed by the agents themselves and even though the strategy space evolves continuously, its size and dimensionality remain the same. In the optimal cases, it usually turns out that in the end the best players are those who use the hybridized mechanism, second best are those using the one-point mechanism, and the bad players those who do not adapt at all. In addition, it turns out that the competition among the players who adapt using the hybridized genetic crossover mechanism is severe. Because of the simplicity of these models [92, 96–98], a lot of freedom is found in modifying the models to make the situations more realistic and applicable to many real dynamical systems, and not only financial markets. Many details in the model can be fine-tuned to imitate the real markets or behavior of other complex systems. Many other sophisticated models based on these games can be setup and implemented, which show a great potential over the commonly adopted statistical techniques in analyses of the financial markets.

5 Final remarks In this review, we have tried to give a flavor of some agent-based models of complex socio-economic systems. The agent-based models of order books are a good example of interactions between ideas and methods usually linked to economics and finance (microstructure of markets, agent interaction) and physics (reaction–diffusion processes, deposition–evaporation process). As of today, existing models exhibit a trade-off between realism and calibration in its mechanisms and processes, and explanatory power of simple observed behaviors. In the former case, some of the “stylized facts” may be reproduced, but using empirical processes that may not be linked to any behavior observed on the market. In the latter case, these are only toy models that cannot be calibrated on data. Note that the mixing of many features, as is usually the case in behavioral finance, leads to untractable models where the sensitivity of the model to a particular parameter is hardly understandable. Toy model features explaining volatility clustering or market interactions are yet to be done. Implementing agents trading simultaneously several assets in a way that reproduces empirical observations on correlation and volatility clustering remains an open challenge. From the studies carried out using the simple kinetic wealth-exchange models, where the agents exchange randomly some wealth (similar to kinetic energy in physics), it emerges that it is possible to use them to model realistic wealth distributions and predict power-law tails. However, a general understanding of the dependence of the shape of the equilibrium distribution on the underlying mechanisms and parameters is still missing. In modeling heterogeneity and bounded rational behavior through minority games, besides rich intermittent collective behaviors, physicists became also

24


interested in the dynamics of games such as crowd–anticrowd movement of agents, periodic attractors, etc. [100–102]. In this way, minority games serve as a useful tool and provide a new direction for physicists in viewing and analyzing the underlying dynamics of complex evolving systems such as the financial markets. On the whole, we can conclude that agent-based models have, however, given an extra-dimension to the traditional approaches.

6 Acknowledgements We are grateful to F. Abergel, A. Chatterjee, E. Heinsalu, I. Muni Toke, M. Patriarca, and V. Yakovenko for critical discussions and to all the collaborators whose work has been described.

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On kinetic asset exchange models and beyond: microeconomic formulation, trade network, and all that Arnab Chattejee Condensed Matter and Statistical Physics Section, The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste I-34014, Italy, [email protected] Summary. We review the kinetic models of asset exchange and report the latest microeconomic formulations. We also discuss the role of topology and disorder in such models. The role of such models from a trading network perspective are also discussed.

1 Introduction The distribution of wealth among individuals in an economy has been an important area of research in economics, for more than a hundred years [8,19, 41, 50, 67]. The same is true for income distribution in any society. Detailed analysis of the income distribution [8, 19] so far indicate α m exp(−m/T ) for m < mc , P (m) ∼ (1) for m ≥ mc , m−(1+ν) where P denotes the number density of people with income or wealth m and α, ν denote exponents and T denotes a scaling factor. The power law in income and wealth distribution (for m ≥ mc ) is named after Pareto and the exponent ν is called the Pareto exponent. A historical account of Pareto’s data and that from recent sources can be found in [55]. The crossover point (mc ) is extracted from the crossover from a Gamma distribution form to the power law tail. One often fits the region below mc to a log-normal form log P (m) ∝ −(log m)2 . Although this form is often preferred by economists, we think that the other Gamma distribution form (1) fits better with the data, because of the remarkable fit with the Gibbs distribution in [27, 58, 65]. Considerable investigations revealed that the tail of the income distribution indeed follows the above mentioned behavior and the value of the Pareto exponent ν is generally seen to vary between 1 and 3 [4, 20, 23, 24, 28, 40, 60].

G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 2, c Springer Science+Business Media, LLC 2010

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Arnab Chattejee

It is also known that typically less than 10% of the population in any country possesses about 40% of the total wealth of that country and they follow the above law, while the rest of the low-income population, follow a different distribution which is debated to be either Gibbs [4, 10, 26, 28, 36, 40] or lognormal [20, 23]. The striking regularities observed in the income distribution for different countries, have led to several new attempts at explaining them on theoretical grounds. Much of it is from physicists’ modeling of economic behavior in analogy with large systems of interacting particles, as treated, e.g., in the kinetic theory of gases. According to physicists, the regular patterns observed in the income (and wealth) distribution may be indicative of a natural law for the statistical properties of a many-body dynamical system representing the entire set of economic interactions in a society, analogous to those previously derived for gases and liquids. By viewing the economy as a thermodynamic system, one can identify the income distribution with the distribution of energy among the particles in a gas. In particular, a class of kinetic exchange models have provided a simple mechanism for understanding the unequal accumulation of assets. Many of these models, while simple from the perspective of economics, has the benefit of coming to grips with the key factor in socioeconomic interactions that results in very different societies converging to similar forms of unequal distribution of resources (see [8, 19], which consists of a collection of large number of technical papers in this field). In recent years, apart from the analysis of these models, there has been efforts in understanding the revance of such models in the economics context. More precisely, some of these models are shown to emerge out of natural consequences in a trading economy. We discuss in detail such an approach in a following section (Sect. 3). Another approach to understand these models better is to test the consequences of putting them on an underlying disordered trade network where exchanges are not mean-field in nature compared to the basic proposed models (Sec. 4). Finally, we discuss the development of a trade network in a generalised preferential trading model (Sec. 5).

2 Gas-like models The study of pairwise money transfer and the resulting statistical distribution of money has almost no counterpart in modern economics. Econophysicists initiated a new direction here. The search theory of money [39] is somewhat related, but this work was largely influenced by [5] studying the dynamics of money. A probability distribution of money among the agents was only recently obtained numerically within the search-theoretical approach [49]. In analogy to two-particle collision process which results in a change in their individual kinetic energy or momenta, income exchange models may be defined using two-agent interactions: two randomly selected agents exchange

On kinetic asset exchange models

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money by some predefined mechanism. Assuming the exchange process does not depend on previous exchanges, the dynamics follows a Markovian process: mi (t) mi (t + 1) =M , (2) mj (t + 1) mj (t) where mi (t) is the income of (individual or corporate) agent i at time t and the collision matrix M defines the exchange mechanism. In this class of models, one considers a closed economic system where the total money M and number of agents N are fixed. This corresponds to no production or migration in the system where the only economic activity is confined to trading. In any trading, a pair of traders i and j exchange their money [10, 11, 26, 36], such that their total money is locally conserved and nobody ends up with negative money (mi (t) ≥ 0, i.e., debt not allowed): mi (t + 1) = mi (t) + Δm; mj (t + 1) = mj (t) − Δm.

(3)

Time (t) changes by one unit after each trading. Such conservative models are also studied in the economic literature [39,49]. The main reason for disregarding the presence of common commodities is that most of them are tangible assets or consumables and they disappear quickly, while money stays in the system. However, models that relax conservation laws can also produce a variety of interesting features [61]. The simplest model considers a random fraction of total money to be shared [26]. At steady-state (t → ∞) money follows a Gibbs distribution: P (m) = (1/T ) exp(−m/T ); T = M/N , independent of the initial distribution. This follows from the conservation of money and additivity of entropy: P (m1 )P (m2 ) = P (m1 + m2 ).

(4)

This result is quite robust and is independent of the topology of the (undirected) exchange space, be it regular lattice, fractal, or small-world [9]. The Boltzmann distribution was independently applied to social sciences by Mimkes [46, 47] using Lagrange principle of maximization with certain constraints. The exponential distribution of money was also found by the economist Shubik [57] using a Markov chain approach to strategic market games. If one allows for debt in such simple models, things look very different. From the point of view of individuals, debt can be viewed as negative money. As an agent borrows money from a bank, its cash balance M increases, but at the expense of cash obligation or debt D which is a negative money. Thus, the total money of the agent Mb = M −D remains the same. Thus, if the boundary condition mi ≥ 0 is relaxed, P (m) never stabilises and keeps spreading in a Gaussian manner toward m = +∞ and m = −∞. As total money is conserved, and some agents become richer in expense of others going to debt, so that M = Mb + D.

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Xi et al. [66] imposed a constraint on the total debt of all agents in the system. Banks set aside a fraction R of the money deposited into bank accounts, whereas the remaining 1 − R can be loaned further. If the initial amount of money in the system is Mb , then, with repeated loans and borrowing, the total amount of positive money available to the agents increases to M = Mb /R, 1/R is called the money-multiplier, and this extra money comes from the increase of the total debt in the system. The maximal total debt is D = Mb /R − Mb and is limited by the factor R. for maximal debt, the total amounts of positive (Mb /R) and negative (Mb (1 − R)/R) money circulate among the agents in the system. The distributions of positive and negative money are exponential with two different money temperatures T+ = Mb /RN and T− = Mb (1 − R)/RN , as confirmed by computer simulations [66]. Similar results were also observed elsewhere [32]. 101

2.5

100 λ0 = 0 λ0 = 0.1 λ0 = 0.6 λ0 = 0.9

10–2

1

–3

10

10–4 10–5 10–6

0.5

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–8

0

0.5

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1.5

2 m

2.5

3

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101 100 10–1 10–2 10–3 10–4 10–5 10–6 –2 –1 0 1 2 3 10 10 10 10 10 10 m

x–2

Q(m)

1.5

M/N = 1 N = 1000

10–1

P(m)

P(m)

2

10–2

10–1

100

101

102

103

m

Fig. 1. Left: Steady state money distribution P (m) for the model with uniform savings. The data shown are for different values of λ: 0, 0.1, 0.6, 0.9 for a system size N = 100. All data sets shown are for average money per agent M/N = 1. Right: Steady state money distribution P (m) for the distributed λ model with 0 ≤ λ < 1 for a system of N = 1, 000 agents. The x−2 is a guide to the observed power-law, with 1 + ν = 2. Here, the average money per agent M/N = 1

2.1 Models with savings Savings [56] is an essential ingredient in a trading market. A saving propensity factor λ was introduced in the random exchange model [11], where each trader at time t saves a fraction λ of its money mi (t) and trades randomly with the rest: mi (t + 1) = λmi (t) + ij [(1 − λ)(mi (t) + mj (t))] ,

(5)

mj (t + 1) = λmj (t) + (1 − ij ) [(1 − λ)(mi (t) + mj (t))] ,

(6)

ij being a random fraction, coming from the stochastic nature of the trading. Of course, in this model, no debt is allowed and hence individual money is non-negative.


35

10

1

10

0

10

–1

10

–2

10

–3

10

–4

10

–5

10

–6

10

–7

102

δ = − 0.8 δ = − 0.4 δ = +0.4 δ = +0.8 1+ν=2

δ = –0.5 δ = 1.0 δ = 2.0

101 100 10–1

P(m)

P(m)

In this model (CC model hereafter), the steady state distribution P (m) of money is decaying on both sides with the most-probable money per agent shifting away from m = 0 (for λ = 0) to M/N as λ → 1 [11]. This model has been understood to a certain extent [52,54] and argued to resemble a Gamma distribution [52]. But, the actual form of the distribution for this model still remains to be found out. It seems that a very similar model was proposed by Angle [2,3] several years back in sociology journals. The numerical simulation results of Angle’s model fit well to Gamma distributions. Different attempts using transport equation [22] and also a matrix formulation [37] have provided some insight into the nature of the models.

10–2

x–1.5

–3

10

10–4 N = 200, M/N = 1

10–5

N = 200, M/N = 1

x– 4.0

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–2

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–1

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0

m

1

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2

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102

m

Fig. 2. Left: Steady state money distribution P (m) in the model for N = 200 agents with λ distributed as ρ(λ) ∼ λδ with different values of δ. A guide to the power law with exponent 1 + ν = 2 is also provided. For all cases, the average money per agent M/N = 1. Right: Steady state money distribution P (m) in the model for N = 200 agents with λ distributed as ρ(λ) ∼ |1 − λ|δ with different values of δ. The distributions P (m) have power law tails P (m) ∼ m−(1+ν) , where the power law exponents 1 + ν approximately equal to 2 + δ indicated by the dotted straight lines. For all cases, the average money per agent M/N = 1

Empirical observations in homogeneous groups of individuals as in waged income of factory laborers in UK and USA [65] and data from population survey in USA among students of different school and colleges produce similar distributions [3]. This is a simple case where a homogeneous population (say, characterized by a unique value of λ) has been identified. In a real society or economy, saving λ is a very inhomogeneous parameter. The evolution of money in a corresponding trading model (CCM model hereafter) can be written as [16]: mi (t + 1) = λi mi (t) + ij [(1 − λi )mi (t) + (1 − λj )mj (t)] ,

(7)

mj (t + 1) = λj mj (t) + (1 − ij ) [(1 − λi )mi (t) + (1 − λj )mj (t)] .

(8)

The trading rules are same as CC model, except that λi and λj , the saving propensities of agents i and j, are different. The agents have fixed (over time)

36

Arnab Chattejee

saving propensities, distributed independently, randomly as Λ(λ), such that Λ(λ) is nonvanishing as λ → 1, λi value is quenched for each agent (λi are independent of trading or t). The actual asset distribution P (m) in such a model will depend on the form of Λ(λ), but for all of them the asymptotic form of the distribution will become Pareto-like: P (m) ∼ m−(1+ν) ; ν = 1 for m → ∞. This is valid for all such distributions, unless Λ(λ) ∝ (1 − λ)δ , when P (m) ∼ m−(2+δ) [9, 14, 16, 18, 48]. In the CCM model, agents with higher saving propensity tend to hold higher average wealth, which is justified by the fact that the saving propensity in the rich population is always high [31]. Analytical understanding of CCM model has been possible until now under certain approximations [17], and mean-field theory [48]. Unlike the earlier models with savings as a quenched disorder, one can also consider savings as an annealed variable and still derive a power law distribution in wealth [15]. 2.2 Average money at any saving propensity and the distribution Several numerical studies investigated [51, 53] the saving factor λ and the average money held by an agent whose savings factor is λ. This numerical study revealed that the product of this average money and the unsaved fraction remains constant, or in other words, the quantity m(λ)(1 − λ) = c,

(9)

where c is a constant. This key result has been justified using a rigorous analysis by Mohanty [38, 48]. We give later a simpler argument and proceed to derive the steady state distribution P (m) in its general form. In a mean field approach, one can calculate [48] the distribution for the ensemble average of money for the model with distributed savings. It is assumed that the distribution of money of a single agent over time is stationary, which means that the time averaged value of money of any agent remains unchanged independent of the initial value of money. Taking the ensemble average of all terms on both sides of (7), one can write: ⎡ ⎤ N 1 mi = λi mi + ⎣(1 − λi )mi + (1 − λj )mj ⎦ . (10) N j=1 It is assumed that any agent on the average, interacts with all others in the system. The last term on the right is replaced by the average over the agents. Writing N 1 (11) (1 − λ)m ≡ (1 − λj )mj N j=1 and as is assumed to be distributed randomly and uniformly in [0, 1], so that = 1/2, (10) reduces to: (1 − λi )mi = (1 − λ)m.


37

As the right side is free of any agent index, it suggests that this relation is true for any arbitrary agent, i.e., mi (1 − λi ) = constant, where λi is the saving factor of the ith agent (as in (9)) and what follows is: dλ ∝

dm . m2

(12)

An agent with a particular saving propensity factor λ, therefore, ends up with a characteristic average wealth m given by (9) such that one can in general relate the distributions of the two: P (m) dm = ρ(λ) dλ.

(13)

This, together with (9) and (10) gives [48]: P (m) = ρ(λ)

c ρ(1 − m ) dλ ∝ , dm m2

(14)

giving P (m) ∼ m−2 for large m for uniform distribution of savings factor λ, i.e., ν = 1; and ν = 1 + δ for ρ(λ) = (1 − λ)δ . This study, therefore, explains the origin of the universal (ν = 1) as well as the nonuniversal (ν = 1 + δ) Pareto exponent values in the distributed savings model, and shown in Fig. 2. 2.3 Rigorous treatments of the problems In recent years, several papers discuss these models at length and provide rigorous analysis of the models and related ones [21, 29, 30, 44, 45] using a variety of approaches like Fokker–Planck equations and generalised Boltzmann transport equations. Several issues regarding the structure and dynamics of such models are known by now.

3 A microeconomic formulation Recently, Chakrabarti and Chakrabarti [7] have put forward a microeconomic formulation of the above models, using the utility function as a guide to the behavior of agents in the economy. They consider an N -agent exchange economy, where each produces a single perishable commodity. Each of these goods is different from all other goods and money exists in this economy to facilitate transactions. These agents care for their future consumptions and hence they care about their savings in the current period as well. Each of these agents are endowed with an initial amount of money (the only type of nonperishable asset considered) which is assumed to be unity for every agent for simplicity. At each time step, two agents meet randomly to carry out transactions according to their utility maximization principle. It is also assumed that the agents have time dependent preference structure, i.e., the parameters of the utility function can vary over time [59].

38

Arnab Chattejee

It is assumed that agent 1 produces Q1 amount of commodity 1 only and agent 2 produces Q2 amount of commodity 2 only and the amounts of money they possess at time t are m1 (t) and m2 (t), respectively. Both of them will be willing to trade and buy the other’s goods by selling a fraction of their own productions and also with the money that they have. In general, at each time step there would be a net transfer of money from one agent to the other. The utility functions are defined as follows: For agent 1, U1 (x1 , x2 , m1 ) = αm 1 α2 m xα and for agent 2, U2 (y1 , y2 , m2 ) = y1α1 y2α2 mα where the argu1 x2 m1 2 ments in both of the utility functions are consumption of the first (i.e., x1 and y1 ) and second good (i.e., x2 and y2 ) and the amount of money they possess, respectively. For simplicity, they assume that the utility functions are of the Cobb–Douglas form with the sum of the powers normalized to 1, i.e., α1 + α2 + αm = 1 [43]. Let p1 and p2 be the commodity prices to be determined in the market. The budget constraints are defined as follows: For agent 1 the budget constraint is p1 x1 + p2 x2 + m1 ≤ M1 + p1 Q1 and similarly, for agent 2 the constraint is p1 y1 + p2 y2 + m2 ≤ M2 + p2 Q2 . This means that the amount that agent 1 can spend for consuming x1 and x2 added to the amount of money that he holds after trading at time (t + 1) (i.e., m1 ) cannot exceed the amount of money that he has at time t (i.e., M1 ) added to what he earns by selling the good he produces (i.e., Q1 ), and the same is true for agent 2. Subject to their respective budget constraints, the agents try to maximize their respective utilities and the price mechanism (invisible hand) works to clear the market for both goods (i.e., total demand equals total supply for both goods at the equilibrium prices), i.e., agent 1’s problem is to maximize his utility U1 (x1 , x2 , m1 ) subject to p1 × x1 + p2 × x2 + m1 = M1 + p1 × Q1 , and for agent 2, maximize U2 (y1 , y2 , m2 ) subject to p1 × y1 + p2 × y2 + m2 = M2 + p2 × Q2 . Those two maximization exercises are solved by Lagrange multipliers and applying the condition that the market remains in equilibrium, the competitive price vector (ˆ p1 , pˆ2 ) as pî = (αi /αm )(M1 +M2 )/Qi for i = 1, 2 has been derived. The outcomes of such a trading process are: (a) At optimal prices (ˆ p1 , pˆ2 ), m1 (t) + m2 (t) = m1 (t + 1) + m2 (t + 1), i.e., demand matches supply in all market at the market-determined price in equilibrium. Because money is also treated as a commodity in this framework, its demand (i.e., the total amount of money held by the two persons after trade) must equal what was supplied (i.e., the total amount of money held by them before trade). (b) Making a restrictive assumption that α1 in the utility function can vary randomly over time with αm remaining constant, it readily follows that α2 also varies randomly over time with the restriction that the sum of α1 and α2 is a constant (1 − αm ). Now, in the money demand equations derived from the above, if αm is substituted by λ and α1 /(α1 + α2 ) by , the money evolution equations are: m1 (t + 1) = λm1 (t) + (1 − λ)[m1 (t) + m2 (t)] (15) m2 (t + 1) = λm2 (t) + (1 − )(1 − λ)[m1 (t) + m2 (t)].


39

For a fixed value of λ, if α1 (or α2 ) is a random variable with uniform distribution over the domain [0, 1 − λ] then is also uniformly distributed over the domain [0, 1]. It may be noted that λ (i.e., αm in the utility function) is the savings propensity used in the CC model [11]. (c) For the limiting value of αm in the utility function (i.e., αm → 0 which implies λ → 0), the money transfer equation describing the random sharing of money without savings is retrieved, as used in the model of Dr˘agulescu and Yakovenko [26]. (d) These asset evolution equations under the microeconomic framework differ from a previous known work [59], where the asset evolution equation for the ith agent depends on its own assets.

3.1 Random exchange For random sharing of assets, the exchange equations look like: mi (t + 1) = (mi (t) + mj (t)) mi (t + 1) = (1 − )(mi (t) + mj (t)),

(16)

where ∈ [0, 1] and uniform (one can also consider ∈ [δ, 1 − δ], and results still hold good). One can show m =1. Also, in the steady state Δmi = Δ[(mi + mj )] = x2 − x2 , where x = [(mi + mj )]. Note that x = 1. Hence Δm = 2 m2i + m2j + 2mi mj − 1. (17) Using the fact that mi and mj are uncorrelated and Δ = 2 − 1/4, one gets 1 (2Δm + 4) − 1. (18) Δm = Δ + 4 Simplifying, one gets Δm =

4Δ

1 2 −2Δ

.

3.2 Exchange with savings In (5), one can show m = 1. The authors also calculate the variance operator as: 1 2 2 (Δm + 2) + λ(1 − λ)(Δm + 2) − 1. Δm = λ (Δm + 1) + 2(1 − λ) Δ + 4 Because ∈ [0, 1] and uniform, Δ = 1/12. Thus, one gets: Δm =

(1 − λ)2 . (1 − λ)(1 + 2λ)

(19)

40

Arnab Chattejee

Hence, if λ = 1, Δm = (1 − λ)/(1 + 2λ) as in [52]. Thus, Δm = 1 for λ = 0, which is the case for an exponential distribution and for 0 ≤ λ < 1, the distribution is approximated by: p(m) =

mα−1 e−βm Γ (α)β −α

with α = (1 + 2λ)/(1 − λ) and β = α as is conjectured in [52]. For λ → 1, by applying l’Hˆ opitals rule one gets Δm = 0, explaining why the steady state distribution tends to a delta function as the rate of savings, i.e., λ → 1 as widely observed in simulations [11, 17, 52]. 3.3 Global market Reference [7] also considers trading in a global market under the above microeconomic framework. Here the price mechanism does not work locally (matching the bipartite supply and demand), but, globally, and the market will also be cleared globally. Subject to budget constraint, each agent maximises its utility and allocates income accordingly between present and future consumptions, both the present and future consumptions are represented by the amount of money spent on the present and future consumptions. What the agents save for future consumption earns them interest income. Hence, the market grows over time. On the production side, they can invest in production and get their returns accordingly. Formally, there are N agents in the economy each taking part in production and consumption. A typical agent’s behavior at any step t is analyzed as follows: (1) Each agent has to maximize utility subject to his budget constraint. For simplicity, the utility function is assumed to be of Cobb–Douglas type. Briefly, at time t the ith agent’s problem is to maximize u(f, c) = f λ c(1−λ) subject to f /(1 + r) + c = m(t), where f is the amount of money kept for future consumption, c the amount of money to be used for current consumption, m(t) the amount of money holding at time t, and r is the interest rate prevailing in the market. This is a standard utility maximization problem and solving it by Lagrange multiplier, one gets the optimal allocation as c∗ = (1 − λ)m(t) and f ∗ = (1 + r)λm(t). (2) The ith agent invests (1 − λi )mi (t) in the market and produces an output vector yi (t) which he sells in the market at market-determined price vector pt , being same for everybody. By a perfect competition assumption, it follows that (1 − λi )mi (t) = p(t)yi (t). Roughly, the argument is: If l.h.s ≥r.h.s, then it is not optimal to produce because cost is higher than revenue. Again, if r.h.s ≥ l.h.s, then there exists a supernormal profit which attracts more agents to produce more. But that leads to a fall in price and hence the economy comes to the equilibrium


41

only when l.h.s = r.h.s. Summing up the above equation over all agents, one gets i (1 − λi )mi (t) = p(t) i yi (t). One can rewrite this as: M (t)V (t) = p(t)Y (t),

(20)

where M (t) is the total money in the system and V (t) is equivalent to the velocity of money at time t. It is evident that V (t) depends on the parameter of the utility functions λi for all agents. It may be noted that the derived equation is analogous to the Fisher equation of ‘quantity theory of money’ [42]. For an alternative interpretation of the Fisher equation in the context of CC type exchange models, see [63]. (3) In this closed economy, no money is neither created nor destroyed during the exchange process. After all trading are done, each agent has whatever they saved for future consumption and the interest income earned from it added to some fraction αi (t) of the total amount of money invested in production of current consumption. i.e., mi (t + 1) = (1 + r)λi mi (t) + αi (t) (1 − λi )mi (t) i

or, from (20), mi (t + 1) = (1 + r)λi mi (t) + αi (t)p(t)Y (t).

(21)

It is assumed that r = 0 and αi (t)p(t)Y (t) = (t). One gets the following reduced equation, mi (t + 1) = λi mi (t) + (t). (22) 3.4 Steady state distribution of money and price Each agent money follows the dynamics defined by (22), where (t) can be assumed to be a white noise. It can be easily shown that this process produces a Gamma function-like part with a power-law tail [6]. Equation (21) is its more general version. This is an autoregressive process of order 1 with (1+r) λi < 1 assuming that the last term is a white noise. Taking expectation for the whole expression [1 − (1 + r)λi ]mi = αi (t)p(t)Y (t). Denoting αi (t)p(t)Y (t) by a finite constant C, we rewrite the equation in terms of average money holding 1 C λ= 1− 1+r m which immediately shows that dλ ∝ dm/m2 . Because P (m)dm = ρ(λ)dλ where P (m) is the distribution of money and ρ(λ) is the distribution of λ, it follows

42

Arnab Chattejee

P (m) = ρ

1 1+r

C 1 1− . m m2

(23)

Thus, if λ is distributed uniformly, the distribution of money has a power-law feature with the exponent 2 (see also in [6, 14, 18]). Particularly, [6] provides examples of the emergence of gamma function-like behavior in money distribution for a variety of noise terms. Writing (20) without subscripts p=

V . Y /M

One observes that M is of the order of N , the number of agents, and Y is their total production. If we assume that both V and Y /M are distributed uniformly, one can show that the distribution of price is a power law: f (p) ∼ p−2 .

(24)

So, in the earlier model, price may also have power-law fluctuations. However, there is no clear evidence supporting the existence of a power law in commodity price fluctuation, although it has been verified in stock price fluctuations [62].

4 Models on directed networks The topology of exchange space in a real society is quite complicated. There are strong notions of directionality and sometimes, hierarchy in the underlying network, where money is preferentially transferred in certain direction that others, contributing in irreversible flow of money. A mean-field scenario (Sec. 2) does not include the constraints on the flow of money or wealth. A way to imitate this is to consider wealth exchange models on a directed network [1, 25, 64]. There have been previous attempts to obtain the same using the physics of networks [33–35]. We consider N agents, each sit on a separate node, connected to the rest N − 1 by directed links. The directionality of the links denote the direction of flow of wealth in this fully connected network. The directed network parametrized by p is constructed in the following way [13]: (a) There are no self-links, so that the adjacency matrix A [1, 25] has diagonal elements aii = 0 for all sites i. (b) For each matrix element aij , i = j, we call a random number r ∈ [0, 1]. aij = +1 if r < p and aij = −1 otherwise. Also aji = −aij . Thus, we have N (N − 1)/2 such calls of r. aij = +1 denotes a directed link from i to j and aij = −1 denotes a directed link from j to i. The link disorder at site i is ρi = N1−1 j aij × j denotes the sum over all N − 1 sites j linked to i. Thus, ρi = 1 is a node which has all links outgoing and ρi = −1 is a node for which all links are incoming. The parameter p has


43

a symmetry about 0.5 and the distribution R(ρi ) is also symmetric about 0, which is, in fact, a consequence of the conservation of the number of incoming and outgoing links. A network with p = 0.5 has the lowest degree of disorder, given by a narrow distribution R(ρ) of ρ, around ρ = 0 (see Fig. 3). This 8

p = 0.50 p = 0.30 p = 0.10 p = 0.05 p = 0.01

Probability, R(ρi)

7 6 5 4 3 2 1 0 -1

-0.8 -0.6 -0.4 -0.2 0 0.2 ri = Si aii /(N-1)

0.4

0.6

0.8

1

Fig. 3. The distribution R(ρi ) of the link disorder ρi for a directed network with different values of p = 0.50, 0.30, 0.10, 0.05, 0.01 for a system of N = 100 nodes, obtained by numerical simulation, averaged over 104 realizations

means that almost all nodes have more or less equal number of incoming and outgoing links. On the other extreme, p = 0.01 is a network which has a small but finite number of nodes where most links are incoming/outgoing, thus giving rise to a very wide distribution of link disorder R(ρ). The rules of exchange are: If aij = −1, mi (t + 1) = mi (t) + μj mj (t) mj (t + 1) = mj (t) − μj mj (t) else, aij = +1, mi (t + 1) = mi (t) − μi mi (t) mj (t + 1) = mj (t) + μi mi (t). 0 < μi < 1 is the ‘transfer fraction’ associated with agent i, and mi (t) is the money of agent i (or, money at node i) at time t. The total money in the system is conserved, no money is created or destroyed, defined by (25) and (25). If there is a link from j to i, the node i gains μj fraction of jth agent’s money. Otherwise, if there is a link from i to j, the node j gains μi fraction of ith agent’s money. In the Monte Carlo simulations, one assigns random amount of money to agents to start with, such that the average money M/N = 1. A pair of agents (nodes) are chosen at random, and depending on the directionality of the link between them (the sign of aij ), the relevant rule, is chosen. This

44

Arnab Chattejee

is repeated until a steady state is reached and the money distribution does not change in time. The distribution of money P (m) is obtained by averaging over several ensembles (different random initial distribution of money). This model is different from the CC and CCM models, but one can relate the transfer fraction μ analogous to λ in the CC and CCM models. 4.1 Model with uniform µ We first discuss the case of homogeneous agents, i.e., when all agents i have μi = μ. The μ = 0 limit is trivial, as the system does not have any dynamics. Figure 4a shows the steady state distribution P (m) of money m for μ = 0.1 for different values of network disorder p. In general, the distribution of money has a most probable value, which shifts monotonically from about 0.85 for p = 0.5 to 0 as p → 0. P (m) has an exponential tail, but a power-law region develops as p → 0 below the exponential cutoff, which fits approximately to m−1.5 . At p → 0, the condensation of wealth at the node(s) with strong disorder (ρ → 1) is apparent from the single, isolated data point at the mmax = M end (see inset of Fig. 4a, for p = 0.01). There is a strong finite size effect involved in this behavior. To emphasize this, we plot P (m) for p = 0.01 for N = 100, 500, and 1, 000 (inset of Fig. 4a). While N = 100 and N = 500 does show the isolated data point, it is absent for N = 1, 000. This also indicates that this behavior is absent for infinite systems for p → 0. For larger values of p, the distribution resembles Gamma distributions, as in the CC model. At μ = 0.5, the most-probable value of P (m) is always at 0 (see Fig. 4b). For weak disorder (p = 0.5), P (m) is exponential, but it shows a wider distribution as one goes to higher disorder (p → 0). The condensation of wealth at node(s) with high value of ρ (ρ → 1) is again apparent from the single, isolated data point at the mmax = M end (see Fig. 4b, for p = 0.05). For μ = 0.9, P (m) is always decaying, with a wide distribution upto mmax = M (see Fig. 4c). As like previous plots, the condensation of wealth at node(s) with high value of ρ is visible: see plot for p = 0.05 in Fig. 4c. A common feature for the curves for all values of p is that, P (m) exhibits log-periodic oscillations, while resembling roughly a power-law decay. Another important feature is that P (m) → N for m → 0, which indicates that money is distributed in a very small fraction of nodes, while most nodes have almost no money at a given instance. For a particular value of the network disorder p, the wealth distribution P (m) becomes more and more ‘fat tailed’ as μ is increased. This is in contrast to what is observed in CC model [11] where P (m) organizes to a narrower distribution as λ increases. 4.2 Model with distributed µ We now consider the case when agents i have different values of μi , which do not change in time. This is a case of heterogeneous agents where the heterogeneity can be viewed as a ‘quenched disorder.’ We consider a random uniform


b

2.5 0

P(m)

Probability, P(m)

2 1.5

10–2 10–4 –6

10

–8

10

1 0.5 0

c

0

0.5

1

p=0.01 N=100 N=500 N=1000

10–2 10–1 100 101 102 103 m p=0.50 p=0.40 m=0.10 p=0.30 N=100 p=0.20 p=0.10 1.5 2 money, m

102

3

100 –1

10

1

10

0

p = 0.50 p = 0.40 p = 0.30 p = 0.20 p = 0.10 p = 0.05

–1

10

–2

10

10–3 10–4

N = 100 m = 0.50

10–7 –2 10

10–1

100 money, m

102

10

N = 100 m = 0.90

10–5

100 10–1

102

m–2

10–2 10

101

p = 0.50 p = 0.40 p = 0.30 p = 0.20 p = 0.10 p = 0.05

101

–3

10–6 –2 10

10

10–6

d

10–2

10–4

102

10–5

3.5

p = 0.50 p = 0.40 p = 0.30 p = 0.20 p = 0.10 p = 0.05

101

Probability, P(m)

2.5

Probability, P(m)

m–1.5

10

Probability, P(m)

a

45

–3

10–4

N = 100 D(m) = 1

10–5

10–1

100 money, m

101

102

10–6 –2 10

10–1

100

101

102

money, m

Fig. 4. The steady state distribution P (m) of money m for directed networks characterized by different values of p. (a) For μ = 0.1, the inset shows P (m) for p = 0.01 for N = 100, 500, 1, 000, and the power law m−1.5 is also indicated. (b) For μ = 0.5 and (c) μ = 0.9. (d) shows the plots for uniform, random distributed μ, D(μ) = 1, and also a guide to the power law m−2 . The data are obtained by numerical simulation, for a system of N = 100 nodes, averaged over 103 realizations in the steady state and over 104 initial configurations. The average money M/N is 1

distribution of μ, i.e., D(μ) = 1 in 0 < μ < 1. This is the case analogous to the CCM model [16]. Figure 4d shows the plots of the money distribution P (m) for different values of network disorder p. All curves have a power-law tail [13], resembling a m−2 variation. However, the effect of topology of the underlying network are visible: For strong disorder in topology p = 0.05, condensation of wealth at node(s) with high value of ρ (ρ → 1) is also apparent from the single, isolated data point at the mmax = M end. Further investigations also indicate that the power-law exponent is similarly related to the distribution of ‘transfer fraction’ μ, as one observes in the CCM model [16, 48], i.e., P (m) ∼ m−2 for most distributions, while one can obtain P (m) ∼ m−(2+δ) if D(μ) ∝ (1 − μ)δ . For a particular value of the network disorder p, the wealth distribution P (m) becomes more and more ‘fat tailed’ as μ is increased. This is in contrast to what is observed in CC model [11] where P (m) organizes to a narrower distribution as λ increases.

46

Arnab Chattejee

5 Preferential transactions and weighted trade network Very recently, Chakraborty and Manna [12] proposed models for preferential transactions. The main idea being the fact that rich traders invest much more in trade and, hence, take part more frequently in the trading process. In the model, a pair of traders i and j are selected for trading with probabilities directly proportional to mi (t)α and mj (t)β , respectively. The trading rules are same as (7) and (8). The α = β = 0 case corresponds to the CCM model, the trading topology is a random graph (RG). But, when they are nonzero positive, rich traders have higher probability of getting selected in a trading process. However, in this case it takes a long time for all of the traders to take part in the exchange process.

DIMER

8 8

( , )

STAR

8

(0, )

Pareto Distribution

β RG

( ,0) 8

(0,0)

α

STAR

Fig. 5. The phase diagram in the (α, β) plane. The origin corresponds to the CCM model, while at corners (0, ∞) and (∞, 0), the richest trader participates in every transaction, hence the network is star-like. At the (∞, ∞) corner only the richest and the second richest traders trade, and the network essentially is a dimer

When α = β and finite, the resultant wealth distribution exhibits a powerlaw tail with ν = 1.00 apart from slight variations. This shows that the wealth distribution is robust with respect to the parameter values in the region, and the nonzero values of α and β only controls the frequency at which different traders take part in the trading process. When either of α or β is infinity and the other is zero, the richest trader is always selected while all others are selected with uniform probability. This corresponds to a star-like topology in the trade graph. It is observed that Pareto law still holds good. Now, when both (α, β) assume large values, the situation looks very different. In its limiting case, (∞, ∞), only the richest and the next richest trader take part in the trading process, i.e., the trade graph is merely a dimer. Thus, qualitatively, one can summarise the results in a schematic diagram (Fig. 5).


47

5.1 The trade network One can associate a network in this trading system: each trader being a node and a link appears between a pair when they trade for the first time. No further link is added between a pair even if they trade again. As more and more traders take part in the trading process, the number of links grow in the system. When α = β = 0 this growth is exactly same as a random graph, but it looks much different for α > 0, β > 0, because rich nodes preferentially enter into trading more often and get linked than the poor ones. The degree ki of a node i is the number of distinct traders with whom it traded. The dynamics is found to have two distinct time scales, T1 at which the network is a single component connected graph, and T2 at which the graph is a N -clique (each node is connected to all others). This paper studies the growth of the giant component sM (ρ, N ) which is the order parameter of this percolation problem with respect to the link density ρ = n/[N (N − 1)] in the network. A finite size scaling collapses the order parameter for different sizes: scaling ρ axis by a factor of N θ . The critical density of percolation transition ρc (N ) is defined to be that value of ρ for which sM (ρ, N ) = 1/2. The paper reports that ρc (N ) varies with N −θ . In general, the exponent θ(α) depends on α: for α ≤ 1/2, θ(α) = 1 while for α > 1/2, α decreases. For Erd˝ os–Renyi random graphs, θ = 1, and hence it indicates that this trade network is different from random graphs for α > 1/2. 5.2 Degree distribution The degree distribution shows interesting observations. The paper studies the average degree distribution P (k, N ) as a function of k at different system sizes N for different values of α, β. It is observed that almost the entire degree distribution obeys the usual finite-size scaling analysis and confirms the validity of the following scaling form: P (k, N ) ∝ N −ηk (α) G[k/N ζk (α) ],

(25)

where the scaling function G(y) has its usual forms G(y) ∼ y −γ(α) as y → 0 and G(y) approaches 0 very fast for y >> 1. This is satisfied only when γk (α) = ζk (α)/ηk (α), and exponents ηk (α) and ζk (α) fully characterize the scaling of P (k, N ). γk (α) is observed to decrease with α. 5.3 The weighted network Between an arbitrary pair of traders, a large number of bipartite trading takes place within a certain time T . The sum of the amounts δij invested in all trades between traders I and j within time T is defined as the total volume of trade wij = T δij . Here wij is known as the weight of the link (i, j). The probability distribution P (w, N ) of the link weights is calculated when

48

Arnab Chattejee

the average degree k reaches a specific preassigned value. When the trade networks is a N -clique graph, i.e., when each trader has traded with all other traders at least once, each node has same degree, i.e., P (k) = δ(k − (N − 1)) and k = N − 1. The distribution has a very long tail, and P (w, N ) ∝ w−γw with γw 2.52. The strength of a node si = j wij where j are all neighbors ki of i, is a measure of the total volume of trade handled by the ith node. Nodal strengths vary widely over different nodes. The strength distribution also follows a power-law decay P (s, N ) ∼ s−γs for N → ∞. Similarly, as (25). Often weighted networks have nonlinear strength-degree relations indicating the presence of nontrivial correlations, as in the airport networks and the international trade network. For a network where the link weights are randomly distributed, the s(k) grows linearly with k. However a nonlinear growth like s(k) ∼ k φ with φ > 1, exhibits the presence of nontrivial correlations. For this case, φ(α) increases with α. The paper also reports the variation of the mean wealth of a trader with its degree, and gets: x(k) ∼ k μ(α) , where μ(α) decreases with α.

6 Discussions and outlook In this extensive review, we address the behavior of wealth distribution focussing mostly on the toy models, which fall in the class of kinetic exchange models. We review the essential structure of these ‘gas-like’ models and present the current status of understanding from various viewpoints. A recent microeconomic formulation helps in understanding the relevance of such models in a real economic context and shows their relevance and natural consideration [7]. We also discuss how the structure of these models change when the topology of the interaction space gets modified, not everybody is able to perform exchanges with others at the same rate [13]. In this context, one can also introduce preferential transaction rules depending on the amount of money each agent holds at a particular time. This serves as a generalised framework under which one can expect a variety of trade structures, whose limiting cases reproduce the basic CCM model. One can look at the structure and dynamics of the underlying ‘trade network’ which shows interesting features which are different from a Erd˝os–Renyi random graph. The degree distribution and weighted network in this case also show nontrivial scaling behavior [12].

7 Acknowledgements The author is thankful to P. Bhattacharyya, B. K. Chakrabarti, S. S. Manna, S. Sinha, and R. B. Stinchcombe for collaborations. Useful discussions with A. S. Chakrabarti, A. Chakraborti, A. KarGupta, P. K. Mohanty, and V. Yakovenko are also acknowledged.


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Arnab Chattejee KarGupta, A.: Physica A 359, 634 (2006). KarGupta, A.: In: Ref. [8] (2006). Kiyotaki, N., Wright, R.: Am. Econ. Rev. 83, 63 (1993). Levy, M., Solomon, S.: Physica A 242, 90 (1997). Mandelbrot, B.B.: Int. Econ. Rev. 1, 79 (1960). Mankiw, N.G.: Macroeconomics, Worth Publishers, New York (2003). Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory, Oxford University Press, New York (1995). Matthes, D., Toscani, G.: J. Stat. Phys. 130, 1087 (2008). Matthes, D., Toscani, G.: Kinetic Relat. Models 1, 1 (2008). Mimkes, J.: J. Calorim. Therm. Anal. 60, 1055 (2000). Mimkes, J.: In: [19], pp. 61 (2005). Mohanty, P.K.: Phys. Rev. E 74, 011117 (2006). Molico, M.: Int. Econ. Rev. 47, 701 (2006). Pareto, V.: Cours d’economie Politique. F. Rouge, Lausanne (1897). Patriarca, M., Chakraborti, A., Germano, G.: Physica A 369, 723 (2006). Patriarca, M., Chakraborti, A., Kaski, K.: Phys. Rev. E 70, 016104 (2004). Patriarca, M., Chakraborti, A., Kaski, K., Germano, G.: In: Ref. [19], p. 93 (2005). Repetowicz, P., Hutzler, S., Richmond, P.: Physica A 356, 641 (2005). Richmond, P., Hutzler, S., Coelho, R., Repetowicz, P.: in Ref. [8] (2006). Samuelson, P.A.: Economics, Mc-Graw Hill Int., Auckland (1980). Shubik, M.: The Theory of Money and Financial Institutions, The MIT Press, Cambridge, Vol 2, 192 (1999). Silva, A.C., Yakovenko, V.M.: Europhys. Lett. 69, 304 (2005). Silver, J., Slud, E., Takamoto, K.: J. Econ. Theory 106, 417 (2002). Sinha, S: Physica A 359, 555 (2006). Slanina, F.: Phys. Rev. E 69, 046102 (2004). Sornette, D.: Why Stock Markets Crash, Princeton University Press, Princeton, New Jersey (2004). Wang, Y., Ding, N.: in ref. [19] p. 126 (2005). Wassermann, S., Faust, K.: Social network analysis, Cambridge University Press, Cambridge (1994). Willis, G., Mimkes, J.: arxiv:cond-mat/0406694 (2004). Xi, N., Ding, N., Wang, Y.: Physica A 357, 543 (2005). Yakovenko, V.M., Barkley Rosser, J.: Rev. Mod. Phys. 81, 1703 (2009).

Microscopic and kinetic models in financial markets Stephane Cordier1 , Dario Maldarella2 , Lorenzo Pareschi3, and Cyrille Piatecki4 1 2 3 4

Laboratoire MAPMO UMR 66128, University of Orléans and CNRS, Fédération Denis Poisson, 45067 Orléans, France, [email protected] Department of Mathematics and CMCS, University of Ferrara, Via Machiavelli 35 I-44100 Ferrara, Italy, [email protected] Department of Mathematics and CMCS, University of Ferrara, Via Machiavelli 35 I-44100 Ferrara, Italy, [email protected] Laboratoire d’Economie d’Orlans (LEO) UMR 6221, University of Orléans and CNRS, 45067 Orléans, France, [email protected]

Summary. We review different microscopic and kinetic models of financial markets which have been developed by economists, physicists, and mathematicians in the last years. We first give a summary of the microscopic models and then introduce the corresponding kinetic equations. Our selective review outlines the main ingredients of some influential models of multiagent dynamics in financial markets like Levy, Levy, and Solomon (Economics Letters, 45, 1994) and Lux and Marchesi (International Journal of Theoretical and Applied Finance, 3, 2000). The introduction of kinetic equations permits to study the asymptotic behavior of the wealth and the price distributions and to characterize the regimes of lognormal behavior and the ones with power-law tails.

1 Introduction Most speculative markets at national and international level share a number of stylized facts, like volatility clustering and fat tails of returns, for which a satisfactory explanation is still lacking in standard theories of financial markets [34]. Such stylized facts are now almost universally accepted among economists and physicists and it is now clear that financial markets dynamics give rise to some kind of universal scaling laws. Showing similarities with scaling laws for other systems with many interacting particles, a description of financial markets as multiagent interacting systems appeared to be a natural consequence [22, 28, 31, 39, 45]. This topic was pursued by quite a number of contributions appearing in both the physics and economics literature in recent years [1, 4, 5, 9, 12, 17, 21, 31, 37, 45]. This new research field borrows several methods and tools from classical statistical G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 3, c Springer Science+Business Media, LLC 2010

51

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Stephane Cordier et al.

mechanics, where complex behavior arises from relatively simple rules due to the interaction of a large number of components. Starting from the microscopic dynamics, kinetic models can be derived with the tools of classical kinetic theory of fluids [1, 3, 7–10, 17, 30, 32, 36, 40]. In contrast with microscopic dynamics, where behavior often can be studied only empirically through computer simulations, kinetic models based on PDEs allow us to derive analytically general information on the model and its asymptotic behavior. For example, the knowledge of the tails behavior for the distributions of returns is of primary importance, as it determines a posteriori whether the model can fit data of real financial markets [24, 41]. In this selective review we first give a summary of some influential models of multiagent dynamics in financial markets like Levy–Levy–Solomon [21] and Lux–Marchesi [28] and then introduce the corresponding kinetic equations [8, 30] and their main properties.

2 Microscopic models After the pioneering microscopic market models from economists like Nobel laureate Stigler [43], numerous microscopic models were published in the physics literature in the last two decades [2, 6, 20–22, 26–28]. Here we do not aim at a comprehensive review of microscopic models in finance we refer to [22,31,39,45] for a more detailed introduction to this topic. We mainly concentrate here on those models of speculative financial markets which are enough realistic to include some essential economic features like the notion of price, dividends, and interest rates, and that at the same time, thanks to their particular structure, admit an interpretation as a kinetic model. These include the models of Levy–Levy–Solomon (LLS) [21,22] and Lux–Marchesi (LM) [26–28]. From a mathematical viewpoint several of these models can be seen as generalizations of the the “Law of Proportionate Effect” introduced by Gibrat [13] according to which the expected value of the growth rate of a quantity is proportional to the current size of the quantity. It is well-known that this simple random multiplicative approach yields a lognormally distributed quantity whereas some seemingly trivial variations of the same process lead to power laws [33, 38]. Microscopic and kinetic models that can be considered as variations to a random multiplicative process are the Cordier–Pareschi–Toscani model [7], the Bouchad–Mézard model [1] and the generalized Lotka–Volterra model by Solomon et al. [15, 23, 29]. All these models, although able to reproduce fat tails, are concerned with the distribution of wealth and the price formation dynamic is not considered. Thus, we leave these wealth distribution models to another review in this book.

Microscopic and kinetic models in financial markets

53

2.1 The Levy–Levy–Solomon model The LLS model considers a set of financial agents i = 1, . . . , N who can create their own portfolio between two alternative investments: a stock and a bond. Let us denote by wi the wealth of agent i and by ni the number of stocks of the agent. Additionally, we use the notations S for the price of the stock and n for the total number of stocks. The wealth dynamic The essence of the dynamic is the choice of the agent’s portfolio. More precisely, at each time step each agent selects which fraction of wealth to invest in bonds and which fraction in stocks. They indicate with r the (constant) interest rate of bonds. The bond is assumed to be a risk-less asset yielding a return at the end of each time period. The stock is a risky asset with overall returns rate x composed of two elements: a capital gain or loss and the distribution of dividends. To simplify the description we omit the presence of dividends. Thus, if an agent has invested γi wi of its wealth in stocks and (1 − γi )wi of its wealth in bonds, at the next time step in the dynamic he will achieve the new wealth value: wi = (1 − γi )wi (1 + r) + γi wi (1 + x), (1) where the rate of return of the stock is given by: x=

S − S , S

(2)

and S is the new price of the stock. The dynamic now is based on the agent choice of the new fraction of wealth he wants to invest in stocks at the next stage. According to the standard theory of investment each investor is characterized by a utility function (of its wealth) U (w) that reflects the personal risk taking preference [16]. The optimal γi is the one that maximizes the expected value of U (w). Utility function and optimal investments Different models can be used for this (see [22, 45]), for example, maximizing a von Neumann-Morgenstern utility function with a constant risk aversion of the type w1−α U (w) = , (3) 1−α where α is the risk aversion parameter, or a logarithmic utility function U (w) = log(w).

(4)

As they do not know the future stock price S , the investors estimate the stock’s next period return distribution and find an optimal mix of the stock

54


and the bond that maximizes their expected utility E[U ]. In practice, for any hypothetical price S h , each investor finds the hypothetical optimal proportion γih (S h ) which maximizes his/her expected utility evaluated at wih (S h ) = (1 − γih )wi (1 + r) + γih wi (1 + x (S h )),

h

h

(5)

where x (S ) = (S − S )/S and S is estimated in some way. For example in [22] the investors expectations for x are based on extrapolating the past values thus originating a memory span for the trader. Once each investor decides on the hypothetical optimal proportion of wealth γih that he/she wishes to invest in stocks, one can derive the number of stocks nhi (S h ) he/she wishes to hold corresponding to each hypothetical stock price S h . Since the total number of shares in the market n, is fixed there is a particular value of the price S for which the sum of the nhi (S h ) equals n. This value S is the new market equilibrium price and the optimal proportion of wealth is γi = γih (S ). Market clearance and equilibrium price More precisely, following [22], each agent formulates a demand curve γ h (S h )wih (S h ) Sh characterizing the desired number of stocks as a function of the hypothetical stock price S h . This number of share demands is a monotonically decreasing function of the hypothetical price S h . As the total number of stocks nhi = nhi (S h ) =

n=

N

ni

(6)

i=1

is preserved, the new price of the stock at the next time level is given by the so-called market clearance condition. Thus, the new stock price S is the unique price at which the total demand equals the supply N

nhi (S ) = n.

(7)

i=1

This will fix the value w in (1) and the model can be advanced to the next time level. To make the model more realistic, typically a source of stochastic noise, which characterizes all factors causing the investor to deviate from his/her optimal portfolio, is introduced in the proportion of investments γi and in the rate of return of the stock x . As shown in [21, 22] the model is capable to provide realistic features of a stock market such as boom, crashes, and cycles (see Fig. 1). It is not clear, however, if such model is able to reproduce fat tails for the price and/or the wealth distribution. Numerical simulations seem to exclude this possibility [39]. In particular, in Sect. 3.1, we give a mathematical proof of self-similar lognormal behavior for the corresponding kinetic model [8].

Microscopic and kinetic models in financial markets 1.00

45

0.90

fraction of total wealth

50 40

Price

35 30 25 20 15 10 5 0 0

memory 256

0.80 0.70 0.60 0.50

memory 141

0.40 0.30 0.20 0.10

20

40

60

80 100 120 140 160 180 200

55

0.00 0

memory 10 10000

Days

20000

30000

time (days)

Fig. 1. Cyclic behavior in LLS model with periodic booms and crashes using only one type of traders and a logarithmic utility function (left). Fraction of the total wealth as a function of time in LLS model with three equal investor population with different memory span (right)

2.2 The Lux–Marchesi model The LM model considers the behavior of an ensemble of N speculators. These traders may adhere to chartist or fundamentalist practices. The number of chartists at any point of time will be denoted by NC , the number of fundamentalist is Nf , (Nc + Nf = N ). Furthermore, they distinguish two subgroups of chartists: those with an optimistic disposition and those who are pessimistic about the market’s development in the near future. The number of individual in these groups is denoted by N+ and N− , respectively, (N+ + N− = NC ). The dynamic of the model encapsulates the endogenous switching of agents between the groups defined earlier and the prices dynamics resulting from their market activities. So the entire dynamic is characterized by three elements, which we will discuss later. Chartists switching between optimistic and pessimistic An opinion index Y is introduced which is defined as the difference between optimistic and pessimistic chartists scaled by their total number Y =

N+ − N− , Y ∈ [−1, 1]. NC

(8)

Denoting the price change in continuous time by S˙ = dS/dt and following a convenient formalization for transition probabilities the probability of a formerly pessimistic individual to switch to the optimistic group (P−+ ) and vice versa (P+− ) within some small time interval Δt may be written as: P−+ = ν1 (NC /N ) exp(U1 ),

P+− = ν1 (NC /N ) exp(−U1 ),

(9)

56


where U1 = α1 y +

α2 S˙ , (Sν1 )

ν1 is a parameter for the frequency of revaluation of opinion, and α1 and α2 are parameters measuring the importance the individuals place on majority opinion and actual price trend in forming expectation about future price changes. Switching between chartist and fundamentalist strategy Chartists are assumed to buy (sell) a fixed number of units if they are optimistic (pessimistic). Fundamentalists on the other hand are assumed to buy (sell) if the actual market price is below (above) the fundamental price SF . These behavioral changes are modeled in the following way: agents meet individuals from the other groups, compare excess profits from both strategies and with a probability depending on the pay-off differential switch to the more successful strategy. Excess profits per unit (compared to alternative invest˙ ments) gained by chartist are given by (D + S)/S. These are composed of ˙ Dividnominal dividends (D) and capital gains due to the price change (S). ing by the actual market price gives the revenue per unit of the asset. Excess returns compared with other investment opportunities are computed by subtracting the average real return R received by the holders of other assets in our economy. In the case of fundamentalist traders excess profits per unit of the asset can be written as: k|(SF − S)/S|. As the gains of chartists are immediately realized whereas those claimed by fundamentalists occur only in the future (and depend on the uncertain time for reversal to the fundamental value) the latter are discounted by a factor k < 1. Neglecting the dividend term, the fundamentalists’ profits are justified by assuming that they correctly perceive the (long-term) real returns to equal the average return of the economy (i.e., D/SF = R) so that the only source of excess profits in their view is arbitrage when S = SF . According to above, transition probabilities for changes of strategies are formalized as follows: PF+ = ν2 (N+ /N ) exp(U2,1 ),

with

S˙ r+ ν2

U2,1 = α3

P+F = ν2 (NF /N ) exp(−U2,1 ),

(10)

SF − S 1 , − R − k S S

and PF− = ν2 (N− /N ) exp(U2,2 ),

with U2,2 = α3

R−

P−F = ν2 (NF /N ) exp(−U2,2 ),

S˙ r+ ν2

SF − S 1 . −k S S

(11)


57

Here ν2 is again a parameter for the frequency of the transition, while α3 is a measure of the pressure exerted by profit differentials. As transition are governed by some type of pair interaction, the probability for an individual to change strategy also depends on the number of individuals pursuing other strategies at that time. The Price formation process The dynamic of the price is explained by the presence of an auctioneer which react with respect to the excess demand by adjusting the price to the next higher (lower) possible value within the next time increment with a certain probability depending on the extent of the balance between demand and supply. Assuming, furthermore, that there are additional liquidity traders in the market whose excess demand is stochastic or that the value of excess demand (ED) is perceived with some imprecision by the auctioneer, a small noise term μ is added, and arrive at transition probabilities for an increase or decrease of the market price by a fixed amount ΔS P↑S = max{0, β(ED + μ)},

P↓S = min{β(ED + μ), 0}.

Where β is a parameter for the reaction speed of the auctioneer. Hence, if for example, the perceived excess demand is positive, an increase of the price toward the next elementary unit occurs with probability β(ED + μ)Δt within an infinitesimal time increment. Aggregate excess demand ED is composed of excess demand of chartists and fundamentalists ED = EDC +EDF . The former is EDC = (N+ − N− )tC because all chartists either buy or sell the same number tC of units. Fundamentalists’ excess demand is given by EDF = NF γ(SF − S)/S, depending on the deviation from the fundamental value, reaction strength γ, and the number of individuals behaving this way at that time, NF . This probabilistic rule for price adjustments is, in fact, equivalent to the traditional Walrasian adjustment scheme. It can be shown that the mean value dynamics of the price is governed by the simple differential equation [26, 28] dS = βED S = β ((N+ − N− )tC S + NF γ(SF − S)) . dt

(12)

Typically, in order to assure that none of the stylized facts of financial prices can be traced back to exogenous factors, one assumes that the log-changes of SF in time are Gaussian random variables. The overall results of this dynamics is easily understood by investigation of the properties of stationary states. Introducing the fraction of chartists Z = NC /N we have [26, 28]. Proposition 1. (a) The mean-value dynamics of Y , S, and Z possesses the following stationary solutions: (i) Y = 0, S = SF with arbitrary Z,

58


(ii) Y = 0, Z = 1 with arbitrary S, (iii) Z = 0, S = SF with arbitrary Y ; (b) No stationary states with both Y = 0 and S = SF exist. The equilibria of major interest are those depicted in item (i) of the first part of the proposition. These stationary states are characterized by a balanced disposition among chartists and the price equal to the fundamental value. Situations (ii) and (iii) denote the prevalence of one strategy over the other, as in case (ii) only chartists are present whereas in case (iii) only fundamentalists survive. It is remarkable that state (i) characterizes a stable point only under certain assumptions over the parameters otherwise it is unstable [26, 28]. In addition, numerical results show the emergence of deviation from normal behavior with presence of fat tails for the distribution of the time series of returns [26]. Figure 2 illustrates the interplay between the dynamics of relative price changes and the fraction of chartists Z among traders. An increase of the number of chartists leads to intermittent fluctuations. Note that, thanks to the presence of fundamentalists, the model incorporates self-stabilizing forces leading to a reduction of the number of chartists after a period of severe fluctuations. In fact, large deviations of the price from its fundamental value lead to high potential profits of the fundamentalist strategy which induces a certain number of agents to switch away from chartism. The dashed line in the bottom picture is the critical threshold for Z leading to the loss of the self-stabilizing forces and to the extinction of fundamentalists. As we will see in Sect. 3.2 using a related kinetic model we will determine an analogous of Proposition 1 and show how the presence of fundamentalists is essential in order to obtain fat tails in the price distribution.

3 Kinetic models Starting from the microscopic dynamics described earlier one can aim at deriving the corresponding kinetic or mesoscopic models using the toolbox of classical kinetic theory [1, 7–10, 17, 30, 32, 36, 40]. In contrast with microscopic dynamics, where the model behavior often can be studied only through computer simulations, kinetic models allow to derive analytically rigorous results on the model and its asymptotic behavior. In the sequel we will consider two different kind of kinetic models for financial markets introduced in [8, 30] which are strictly related to the LLS and the LM model, respectively. For standard Boltzmann-like models the determination of an explicit form of the asymptotic wealth/price distribution of the kinetic equation remains difficult and requires the use of suitable numerical methods. A complementary method to extract information on the tails is linked to the possibility to obtain particular asymptotics which maintain the characteristics of the solution to the original problem for large times. Following the analysis developed in [7], we shall prove that the Boltzmann models converge in a suitable asymptotic limit


59

Fig. 2. Time series of returns (top) and the fraction of chartists (bottom) from a typical simulation of the LM model

toward convection–diffusion equations of Fokker–Planck type. Other Fokker– Planck equations were obtained using different approaches in [1, 29, 42]. This permits to study the asymptotic behavior of the wealth and the price distributions and to characterize the regimes of lognormal behavior and the ones with power law tails. 3.1 A kinetic model for a speculative market In this model we derive a mesoscopic description of the behavior of a simple financial market where a population of homogeneous agents can create their own portfolio between two investment alternatives: a stock and a bond. The model has been introduced in [8] and is closely related to the LLS microscopic model in finance described before [21]. Kinetic setting We define f = f (w, t), w ∈ R+ , t > 0 the distribution of wealth w, which represents the probability for an agent to have a wealth w. Let μ = μ(S) be a given monotonically nonincreasing function of the price S ≥ 0 such that 0 < μ(0) < 1. This function represents the optimal demand curve of an

60

Stephane Cordier et al. 1 μ(S)

0.8

_

Sn/w(t)

0.6 0.4 0.2 equilibrium price

0

S

Fig. 3. Example of equilibrium price

agent and we assume it is a given function characterizing the optimal agent behavior on the market. Here we restrict to the case of one single type of agent for simplicity. We assume that at time t the percentage of wealth invested is of the form γ(ξ) = μ(S) + ξ, where ξ is a random variable in [−z, z], and z = min{−μ(S), 1 − μ(S)} is distributed according to some probability density Φ(μ(S), ξ) with zero mean and variance ζ 2 . This probability density characterizes the individual strategy of an agent around the optimal choice μ(S). We assume Φ to be independent of the wealth of the agent. In the sequel because the total number of agents remains constant we assume that f (w, t) has been normalized so that ∞ f (w, t)dw = 1. 0

Note that given f (w, t), because γ and w are independent random variables, at each time t, the price S(t) satisfies: S(t) =

1 1 1 E[γw] = E[γ]E[w] = μ(S(t))w(t), ¯ n n n

(13)

where E[X] denotes the mathematical expectation of the random variable X, E[γ] = μ(S(t)) and the mean wealth w(t) ¯ is given by: ∞ w(t) ¯ = E[w] = f (w, t)wdw. (14) 0

The microscopic evolution of the wealth of an investor will depend on the future price S and the percentage γ of wealth, and is given by: w (S , γ, η) = (1 − γ)w(1 + r) + γw(1 + x(S , η)),

(15)


61

where the expected rate of return of stocks is given by: x(S , η) =

S − S + D + η . S

(16)

In the above relation, D ≥ 0 represents a constant dividend paid by the company and η is a random variable distributed according to Θ(η) with zero mean and variance σ 2 , which takes into account fluctuations due to price uncertainty and dividends [14, 22]. We assume η to take values in [−d, d] with 0 < d ≤ S + D so that w ≥ 0 and thus negative wealths are not allowed in the model. Note that (16) requires estimation of the future price S , which is unknown. The dynamic is then determined by the agent’s new fraction of wealth invested in stocks, γ (ξ ) = μ(S )+ξ , where ξ is a random variable in [−z , z ] and z = min{μ(S ), 1 − μ(S )} is distributed according to Φ(μ(S ), ξ ). We have the demand-supply relation S =

1 E[γ w ], n

(17)

which permits us to write the following equation for the future price S = Now

1 1 E[γ ]E[w ] = μ(S )E[w ]. n n

w (S , γ, η) = w(1 + r) + γw(x(S , η) − r),

thus

E[w ] = w(t)(1 ¯ + r) + μ(S)w(t) ¯

S − S + D −r . S

(18)

(19)

(20)

Using (13) in (18) we can eliminate the dependence on the mean wealth and write S =

μ(S ) (1 − μ(S))μ(S ) (1 + r)S + D. (1 − μ(S ))μ(S) 1 − μ(S )

(21)

Equation (21) determines implicitly the future value of the stock price. Let us set 1 − μ(S) g(S) = S. μ(S) Then, the future price is given by: g(S ) = g(S)(1 + r) + D for a given S. Note that dμ(S) S 1 − μ(S) dg(S) =− > 0, + 2 dS dS μ(S) μ(S)

62


so the function g(S) is strictly increasing with respect to S. This guarantees the existence of a unique solution S = g −1 (g(S)(1 + r) + D) > S.

(22)

Moreover, if r = 0 and D = 0, the unique solution is S = S and the price remains unchanged in time. For the average stock return, we have x ¯(S ) − r =

μ(S )D (μ(S ) − μ(S))(1 + r) + , (1 − μ(S ))μ(S) S(1 − μ(S ))

(23)

where

S − S + D . (24) S Note that the average stock return is above the bonds rate r only if the (negative) rate of variation of the investments is above a certain threshold x ¯(S ) = E[x(S , η)] =

D μ(S ) − μ(S) S≥− . μ(S)μ(S ) (1 + r)

The kinetic equation and its property The microscopic dynamics of agents originate the following linear kinetic equation for the evolution of the wealth distribution d z ∂f (w) 1 = f ( w) − β(w → w )f (w) dξ dη, (25) β( w → w) ∂t j(ξ, η) −d −z where the dependence of time has been omitted for notation simplicity. The first part of the integral on the right hand side takes into account all possible gains of the test wealth w coming from a pretrading wealth w. The function β( w → w) gives the probability per unit time of this process. Thus, w is obtained simply by inverting the dynamics to get

w=

w , j(ξ, η, t)

j(ξ, η, t) = 1 + r + γ(ξ)(x(S , η) − r),

(26)

where the value S is given as the unique fixed point of (18). The presence of the term j in the integral is needed in order to preserve the total number of agents d ∞ f (w, t)dw = 0. dt 0 The second part of the integral on the right hand side of (25) is a negative term that takes into account all possible losses of wealth w as a consequence


63

of the direct dynamic (15), the rate of this process now being β(w → w ). In our case, the kernel β takes the form: β(w → w ) = Φ(μ(S), ξ)Θ(η).

(27)

Note that (25) in weak form takes the simpler form d ∞ f (w, t)φ(w)dw dt 0

∞

D

z

= 0

−D

−z

(28) Φ(μ(S), ξ)Θ(η)f (w, t)(φ(w ) − φ(w))dξ dη dw.

From (28) follows the conservation of the total number of investors if φ(w) = 1. The choice φ(w) = w is of particular interest because it gives the time evolution of the average wealth which characterizes the price behavior. The mean wealth is not conserved in the model because we have ∞ d ∞ S −S+D −r f (w, t)w dw = r + μ(S) f (w, t)w dw. dt 0 S 0 Note that because the sign of the right hand side is non-negative, the mean wealth is nondecreasing in time. In particular, we can rewrite the equation as: d w(t) ¯ = ((1 − μ(S))r + μ(S)¯ x(S )) w(t). ¯ dt

(29)

From this we get the equation for the price d μ(S(t)) S(t) = ((1 − μ(S(t)))r + μ(S(t))¯ x(S (t))) S(t), dt μ(S(t)) − μ(S(t))S(t) ˙ where S is given by (21) and μ(S) ˙ =

dμ(S) ≤ 0. dS

Now since from (23) it follows by the monotonicity of μ that def

x¯(S ) ≤ M :=r +

D , S(0)(1 − μ(S(0)))

using (29) we have the bound w(t) ¯ ≤ w(0) ¯ exp (M t) . From (13) we obtain immediately S(0) S(t) ≤ exp (M t) , μ(S(t)) μ(S(0))

(30)

64


which gives S(t) ≤ S(0) exp (M t) .

(31)

Analogous bounds to (30) for moments of higher order can be obtained in a similar way [8]. If we denote with M0 the space of all probability measures on R+ and with

Mp =

Ψ ∈ M0 :

R+

|ϑ|p Ψ (ϑ) dϑ < +∞, p ≥ 0 ,

(32)

the space of all Borel probability measures of finite momentum of order p, equipped with the topology of the weak convergence of the measures, we can summarize the above bounds in the following [8]. Theorem 1. Let the probability density f0 ∈ Mp , where p = 2 + δ for some δ > 0. Then the average wealth is increasing exponentially with time following (30). As a consequence, if μ is a nonincreasing function of S, the price does not grow more than exponentially as in (31). Similarly, higher order moments do not increase more than exponentially. Fokker–Planck asymptotics and wealth distribution Unfortunately, in general it is difficult to study in detail the large time behavior of the system. A standard technique in kinetic theory is to use asymptotic analysis to derive simplified models whose behavior is easier to analyze. Here, following the analysis in [7, 36] and inspired by similar asymptotic limits for inelastic gases [11,40], we consider the limit of large times in which the market originates a very small exchange of wealth (small rates of return r and x). In order to study the asymptotic behavior of the distribution function f (w, t), we set τ = rt,

f˜(w, τ ) = f (w, t),

˜ ) = S(t), S(τ

˜ = μ(S), μ ˜ (S)

which implies that f˜(w, τ ) satisfies the weak form of the kinetic equation ∞ d f˜(w, τ )φ(w)dw dτ 0 (33) 1 ∞ d z ˜ ˜ Φ(˜ μ(S), ξ)Θ(η)f (w, τ )(φ(w ) − φ(w))dξ dη dw = r 0 −d −z and consider a second-order Taylor expansion of φ around w, 1 φ(w ) − φ(w) = w(r + γ(x(S , η) − r))φ (w) + w2 (r + γ(x(S , η) − r))2 φ (w), ˜ 2


65

where, for some 0 ≤ ϑ ≤ 1, w ˜ = ϑw + (1 − ϑ)w. Now we insert this expansion into the right-end-side, and compute the limit of very small values of the constant rate r. In order for such a limit to make sense and preserve the characteristics of the model, we must assume that σ2 = ν, r→0 r lim

lim

r→0

D = λ. r

(34)

Note that the above limits in (21) imply, immediately, that ˜ lim S˜ = S.

(35)

r→0

Now, omitting the details of the computations, sending r → 0 under these assumptions, we obtain the weak form: ∞ d f˜(w, τ )φ(w)dw dτ 0 ∞ ˜ ˜ − 1) + 1 λ μ ˜(S)(κ( S) ˜ ˜ f˜(w, τ )wφ (w) dw = 1+μ ˜(S) (κ(S) − 1) + ˜ 1−μ ˜(S) S˜ 0 ˜ 2 + ζ 2) ∞ μ(S) 1 (˜ ν f˜(w, τ )w2 φ (w) dw, + 2 S˜2 0 with ˜ def := 0 < κ(S)

˜ ˜ μ ˜(S)(1 −μ ˜(S)) ≤ 1, ˜ −μ ˜ − S˜μ ˜ μ ˜(S)(1 ˜ (S)) ˜˙ (S)

˜ μ(S) ˜ = d˜ μ ˜˙ (S) ≤ 0. dS˜

(36)

This corresponds to the Fokker–Planck equation ∂ 1 ∂ 2˜ ∂ ˜ ˜ f = −A(τ )wf + B(τ ) w f , ∂τ ∂w 2 ∂w with

˜ ˜ − 1) + 1 λ μ ˜ (S)(κ( S) ˜ ˜ A(τ ) = 1 + μ ˜(S) (κ(S) − 1) + ˜ 1−μ ˜(S) S˜ B(τ ) =

˜ 2 + ζ2) (˜ μ(S) ν. S˜2

(37)

(38) (39)

Furthermore, the following theorem can be stated [8]. Theorem 2. Let the probability density f0 ∈ Mp , where p = 2 + δ for some δ > 0. Then, as r → 0, σ → 0, and D → 0 in such a way that σ2 = νr and D = λr, the weak solution to the Boltzmann equation (28) for the scaled density f˜r (w, τ ) = f (v, t) with τ = rt converges, up to extraction of a subsequence, to a probability density f˜(w, τ ). This density is a weak solution of the Fokker–Planck equation (37).

66


In order to search for self-similar solutions, we consider the scaling 1 f˜(w, τ ) = g˜(χ, τ ), w

χ = log(w).

(40)

Simple computations show that g˜(χ, τ ) satisfies the linear convection–diffusion equation ∂ B(τ ) ∂ B(τ ) ∂ 2 g˜(χ, τ ) = − A(τ ) g˜(χ, τ ) + g˜(χ, τ ), ∂τ 2 ∂χ 2 ∂χ2 which admits the self-similar solution (see [25] for example) (χ + b(τ )/2 − a(τ ))2 1 , exp − g˜(χ, τ ) = 2b(τ ) (2b(τ )π)1/2

where a(τ ) =

0

τ

A(s) ds + C1 ,

b(τ ) = 0

(41)

τ

B(s) ds + C2 .

Reverting to the original variables, we obtain the lognormal asymptotic behavior of the model, 1 (log(w) + b(τ )/2 − a(τ ))2 f˜(w, τ ) = . (42) exp − 2b(τ ) w(2b(τ )π)1/2 The constants C1 = a(0) and C2 = b(0) can be determined from the initial data at t = 0. If we denote by w(0) ¯ and e¯(0) the initial values of the first two central moments, we get e¯(0) C1 = log(w(0)), . ¯ C2 = log 2 (w(0)) ¯ Numerical simulations In this section we report the results of different numerical simulations for the CCP model. In all the numerical simulations we consider N = 1, 000 agents and n = 10, 000 shares. Initially, each investor has a total wealth of 1, 000 composed of 10 shares, at a value of 50 per share, and 500 in bonds. The random variables ξ and η are assumed distributed according to truncated normal distributions so that negative wealth values are avoided (no borrowing and no short selling). In the first test we compare the results obtained with the Monte Carlo simulation of the kinetic model to a direct solution of the price equation (3.1). In the second test case we consider the time-averaged Monte Carlo asymptotic behavior of the kinetic model and compare its numerical self-similar solution with the explicit one computed in the last section using the Fokker–Planck model.


67

Test 1 In the first test we take a riskless interest rate r = 0.01 and an average dividend growth rate D = 0.015 and assume that the agents simply follow a constant investments rule, μ(·) = 0.5. We report the results after 400 stock market iterations with ξ and η/S(0) distributed with standard deviation 0.2 and 0.3, respectively. In Fig. 4 (left) we report the simulated price behavior together with the evolution computed from (3.1). The fraction of investments in time during the Monte Carlo simulation fluctuates around its optimal value and is given in Fig. 4 (right). 0.55

3000

0.54 2500

0.53 0.52

2000

μ(t)

S(t)

0.51 1500

0.5 0.49

1000

0.48 0.47

500

0.46 0 0

50

100

150

200

t

250

300

350

400

0.45

0

50

100

150

200

250

300

350

400

t

Fig. 4. Exponential growth of the price S(t) in time (left) and fluctuations of the corresponding fraction of investments in time (right) in a Monte Carlo simulation

Test 2 In the second test case we consider the asymptotic limit of the Boltzmann model and compare its numerical self-similar solution with the explicit one computed in the last section using the Fokker–Planck model. To this end, we consider the self-similar scaling (40) and compute the solution for the values r = 0.001, D = 0.0015 with ξ and η/S(0) distributed with standard deviation 0.05. We report the numerical solution for a constant value of μ = 0.5 at different times t = 50, 200, and 500 in Fig. 5. 3.2 A kinetic model for multiple agents interactions In this model we describe a simple financial market characterized by a single stock or good and an interplay between two different traders populations, chartists and fundamentalists, which determine the price’s dynamic of such stock (good). The model has been introduced in [30] inspired by the Lux– Marchesi model [28]. The aim was to introduce a kinetic description both for the behavior of the microscopic agents and for the price, and then to exploit the tools given by kinetic theory to get more insight about the way the microscopic dynamic of each trading agent can influence the evolution of the price, and be responsible of the appearance of ‘stylized’ fact like ‘fat tails’ and lognormal’ behavior.

68

Stephane Cordier et al. –3

2.5

x 10

10 10

2 10 10

f(w,t)

f(w,t)

1.5

10

−2

−3

−4

−5

−6

1 10

0.5

10 10

0 0

500

1000

1500

2000

2500

−7

−8

−9

3000

w

3

4

10

10 w

Fig. 5. Distribution function f (w, t) at t = 50, 200, and 500 in standard scale (left) and log-log scale (right). The continuous line is the lognormal Fokker–Planck solution

Kinetic setting Similarly to Lux and Marchesi model, the starting point is a population of two different kinds of traders, chartists and fundamentalists. Chartists are characterized by their number density ρC and the investment propensity (or opinion index) y of a single agent whereas fundamentalists appear only through their number density ρF . The value ρ = ρF + ρC is invariant in time so that the total number of agents remains constant. In the sequel we will assume for simplicity ρ = 1. Dynamic of investment propensity among chartists Let us define f (y), y ∈ [−1, 1], the distribution function of chartists with investment propensity y. Positive values of y represent buyers, negative values characterize sellers and close to y = 0 we have undecided agents. Clearly 1 ρC (t) = f (y, t) dy. −1

Moreover, we define the mean investment propensity 1 1 f (y, t)y dy. Y (t) = ρC (t) −1

(43)

˙ For a given price S(t) and price derivative S(t) = dS(t)/dt the microscopic dynamic of the investment propensity of chartists is characterized by the following binary interactions (y, y∗ ) → (y , y∗ ) with ˙ S(t) + DC (y)η, y = (1 − α1 H(y) − α2 )y + α1 H(y)y∗ + α2 Φ S(t) ˙ S(t) + DC (y∗ )η∗ . y∗ = (1 − α1 H(y∗ ) − α2 )y∗ + α1 H(y∗ )y + α2 Φ S(t)

Microscopic and kinetic models in financial markets 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

D(y)

H(y)

0.6 0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 −1

69

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y

0 −1

−0.8 −0.6 −0.4 −0.2

0 y

0.2

0.4

0.6

0.8

1

Fig. 6. Typical examples of herding function H(y) (left) and diffusion function DC (y) (right)

Here α1 ∈ [0, 1] and α2 ∈ [0, 1], with α1 + α2 ≤ 1, measure the importance the individuals place on others opinions and actual price trend in forming expectations about future price changes. The random variables η and η∗ are assumed distributed accordingly to Θ(η) with zero mean and variance σ 2 and measure individual deviations to the average behavior. The function H(y) ∈ [0, 1] is taken symmetric on the interval I, and characterizes the herding behavior whereas DC (y) defines the diffusive behavior, and is also taken symmetric on I. Simple examples of herding function and diffusion function are given by: H(y) = a + b(1 − |y|),

DC (y) = (1 − y 2 )γ ,

with 0 ≤ a + b ≤ 1, a ≥ 0, b > 0, γ > 0. Other choices are of course possible, note that in order to preserve the bounds for y it is essential that D(y) vanishes in y = ±1. Both functions take into account that extremal positions suffer less herding and fluctuations. For b = 0, H(y) is constant and no herding effect is present and the mean investment propensity is preserved when the market influence is neglected (α2 = 0) as in a model of opinion [44]. A remarkable feature of the above relations is the presence of the nor˙ malized value function Φ(S(t)/S(t)) in [−1, 1] in the sense of Kahneman and Tversky [18, 19] that models the reaction of individuals toward potential gain and losses in the market [18]. This permits to introduce behavioral aspects in the market dynamic and take into account the influence of psychology on the behavior of financial practitioners. The value function is defined on deviations from a reference point, which will be assumed usually equal to zero, but can be considered also positive or negative, and is normally concave for gains (implying risk aversion), commonly convex for losses (risk seeking) and is generally steeper for losses than for gains (loss aversion). Let us ignore for the moment the price evolution. The earlier binary interaction gives the following kinetic equation for the time evolution of chartists

70

Stephane Cordier et al. 1 0.8 0.6

Φ(S’(t)/S(t))

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

S’(t)/S(t)

˙ Fig. 7. An example of value function Φ(S(t)/S(t))

∂f = QC (f, f ), ∂t

(44)

where interaction operator QC can be conveniently written in weak form as: 1 QC ϕ(y) dy = BC (y, y∗ )f (y)f (y∗ )(ϕ(y ) − ϕ(y))dη dη∗ dy∗ dy, −1

[−1,1]2

R2

where ϕ is a test function and the transition rate has the form: BC (y, y∗ ) = Θ(η)Θ(η∗ )χ(|y | ≤ 1)χ(|y∗ | ≤ 1), with χ(·) the indicator function. Note that the mass density of chartists ρC (t) is an invariant for the interaction, (ϕ ≡ 1). Strategy exchange chartists–fundamentalists In addition to the change of investment propensity due to a balance between herding behavior and the price followers nature of chartists the model includes the possibility that an agent change its strategy from chartist to fundamentalists and vice versa. Fundamentalists are modeled as in Lux and Marchesi [28]. Agents meet individual from the other group, compare excess profits from both strategies and with a probability depending on the pay-off differential switch to the more successful strategy. When a chartist and a fundamentalist meet they characterize the success of a given strategy trough the profits earned by comparing ˙ +D SF − S(t) S(t) . XC (y, t) = ψ(y) − R , XF (t) = k S(t) S(t)


71

Here ψ(y) ∈ [−1, 1] has the same sign of y and takes into account the change of sign in the profits accordingly to the actual behavior of the agent in the market which rely on his investment propensity y. The simplest choice is ψ(y) = sign(y). The value D is the nominal dividend and R the average real return of the market, such that R = D/SF , i.e., evaluated at its fundamental value SF in a state of stable price S˙ = 0 the asset yield the same returns of other investments, or equivalently XC = XF = 0. The discount factor k < 1 is justified by the observation that XF is an expected gain realized only after reversal to the fundamental value. A chartist characterized by an investment propensity y and a fundamentalist meets each other, and after comparing their strategies, they exchange their strategies with a rate given by a suitable monotone function B(·) ≥ 0. More precisely a chartist switches to fundamentalist with a rate B(XF − XC ) and a fundamentalist switches to chartist at a rate B(XC − XF ). A possible choice for the rate function is for example B(x) = ex . For chartists we define the following linear strategy exchange operator QFC (fC ) = μρF (t)fC (y)(BFC (XC − XF ) − BFC (XF − XC )), where μ > 0 measures the frequency of the exchange rates. Taking into account strategy exchange we have the chartists–fundamentalists model ⎧ ∂fC ⎪ ⎪ ⎨ ∂t = QC (fC , fC ) + μρF (t)fC (y)(BFC (XC − XF ) − BFC (XF − XC )) 1 ∂ρF ⎪ ⎪ ⎩ = μρF (t) fC (y)(BFC (XF − XC ) − BFC (XC − XF )) dy. ∂t −1 (45) It is immediate to verify that the total number density ρC + ρF is conserved in time. Price evolution Finally, we introduce the probability density V (s, t) of a given price s at time t. The real market price S(t) is defined as the mean value: ∞ S(t) = V (s, t)s ds. 0

Following Lux and Marchesi [28] the microscopic dynamic of the price is given by: s = s + β(ρC tC Y (t)s + ρF γ(SF − s)) + ηs, where the parameters β, represent the price speed evaluation, η is a random variable with zero mean and variance ζ 2 , distributed accordingly to Ψ (η). In the above relation chartists either buy or sell the same number tC of

72


units and γ is the reaction strength of fundamentalists to deviations from the fundamental value. Thus, the chartists–fundamentalists system of (45) is complemented with the equation for the price distribution ∂V = QV (V ), ∂t

(46)

where the operator QV in weak form reads ∞ ∞ QV ϕ(s) ds = BV (s)V (s)(ϕ(s ) − ϕ(s))dη ds 0

0

R

with the transition rate BV (s) = Ψ (η)χ(s ≥ 0). Note that the expected value for the stock price satisfies the same differential equation (12) as in[26, 28] dS(t) = βρC tC Y (t)S(t) + βρF γ(SF − S(t)). dt

(47)

Booms, crashes, and macroscopic stationary states In order to study the macroscopic steady states of the system let us start by observing that equilibrium states for the price satisfy ρC tC Y S + ρF γ(SF − S) = 0 and fall in one of the following categories (1) ρF = 0, (2) (3)

ρF = 0, ρF = 0,

Y = 0, S = 0,

S=

ρF γSF − ρC tC Y , ρF γ

ρF γSF − ρC tC Y ≥ 0.

S arbitrary, Y arbitrary.

At equilibrium we require ρF , ρC , and Y to be constants. In order for the number densities to be constants we require QFC = 0. For ρF = 0 and ρC = 0, thanks to monotonicity of BFC , we have XC = XF or equivalently S = SF . Note that QFC vanishes also when ρF = 0 or ρC = 0. These considerations reduce the set of possible equilibrium configurations to (1) (2) (3)

ρF = 0, ρF = 0, ρF = 0,

S = SF , Y = 0, Y = 0, S arbitrary, S = 0, Y arbitrary.

Finally, we consider the requirements for Y to be constant. In the case QFC = 0 the first moment equation reads: 1 d Y (t) = − α1 H(y)yf (y)dy − α2 ρC Y (t) dt −1 1 ˙ S(t) , H(y)f (y)dy + α2 ρC Φ + α1 Y (t) S(t) −1


73

which gives the steady state condition −α1

1

−1

H(y)yf (y)dy − α2 ρC Y + α1 Y

1

−1

H(y)f (y)dy + α2 ρC Φ

˙ S(t) S(t)

.

The study of the above equation for general H is quite complicated [30], however in the simple case of H constant reduces to ˙ S(t) α2 ρC Φ − Y = 0. S(t) Now using the fact that ˙ S(t) (SF − S(t)) = βρC tC Y (t) + βρF γ , S(t) S(t) we have Proposition 2. The system of equations (45)–(46) in the case of H constant admits the following possible equilibrium configurations (i) ρF = 0, (ii) ρF = 0, (iii) ρF = 0,

S = SF , Y = 0, Φ(0) = 0, Y = 0, Φ(0) = 0, S arbitrary, Y = Y∗ , with Y∗ = Φ(βtC Y∗ ), S = 0.

Note that if the reference point for the value function Φ(0) = 0 configuration (i) and (ii) are not possible for a constant H. This is in good agreement with the fact that an emotional perception of the market from the chartists acts as a source of instability for the market itself. In contrast configuration (iii), corresponding to a market crash, can be achieved also for Φ(0) = 0. The existence of a unique fixed point Y∗ has to be guaranteed by the choice of Φ, β, and tC . Of course if the reference point is set to zero, Φ(0) = 0, we have Y∗ = 0. It is easy to verify that these possible equilibrium configurations includes the ones in the original Lux–Marchesi model (see Proposition 1). In addition to the earlier equilibrium configurations the model admits several other possible asymptotic behavior in the form of booms and cycles. Some of the fundamental features of the model are summarized in the following. • Chartists alone (ρF = 0, ρC = 1) influence the price through their mean propensity to invest Y (t) and at the same time the price trend influences ˙ their mean propensity through the value function Φ(S(t)/S(t)), because ˙ S(t)/S(t) = βY (t)tC . Thus, except for the particular shape of the value function, if the mean propensity is initially (sufficiently) positive then it will continue to grow together with the price and the opposite occurs if it is initially (sufficiently) negative.

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The market goes toward a boom (exponential grow of the price) or a crash (exponential decay of the price) with S(0)e−βtC ≤ S(t) ≤ S(0)eβtC , and agents tend to concentrate in y = 1 and y = −1, respectively, depending on the choices of H and Φ. This is in good agreement with the price followers nature of chartists. • Fundamentalists alone (ρF = 1, ρC = 0) influence the price through their expectation of the fundamental price. So their effect is to drive the price toward the fundamental price. For a constant fundamental price SF the equilibrium state reached is characterized by S = SF and the trend is exponential. • The presence of fundamentalists acts in contrast to the chartists pressure toward market booms or crashes. If their number is large enough they are capable to drive the price toward the fundamental value otherwise the chartists dynamic may dominate. In addition to booms and crashes, we have now the possibility of price cycles/oscillations around the fundamental value. Fokker–Plank limit and kinetic asymptotic behavior Now we consider what happens at the kinetic scale. Due to the extreme difficulty to get detailed information on the asymptotic behavior of the kinetic coupled system, we will recover for both distribution functions f , and V , simplified Fokker–Planck models which preserve the main features of the original kinetic model. To keep notations simple, as we are mostly interested in the study of the equilibrium states we ignore the presence of the terms describing the change of strategy. However, they can be easily included in the scaling described later [30]. For this purpose we introduce a time scaling parameter ξ and define τ = ξt,

f˜(y, τ ) = f (y, t),

V˜ (s, τ ) = V (s, t).

To preserve the chartists dynamic in the limit, we must require that α1 = α˜1 , α1 ,ξ→0 ξ lim

α2 = α˜2 , α2 ,ξ→0 ξ lim

σ2 = λ, σ,ξ→0 ξ lim

where λ is a positive constant. Similarly for the price dynamic, we assume β ˜ = β, β,ξ→0 ξ lim

ζ2 = ν. ζ,ξ→0 ξ lim

Performing similar computations as in Sect. 3.1 and omitting the details we recover the following Fokker–Plank system [30]


⎧ ⎪ ∂ f˜ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂τ

∂ ∂y λρC ∂ 2 [(DC 2 (y))f˜], = ⎪ 2 ⎪ 2 ∂y ⎪ ⎪ 2 ⎪ ⎪ ⎩ ∂ V˜ + ∂ β˜ ρ Y˜ t s + ρ γ(S˜ − s) V˜ = ν ∂ 2˜ s V . C C F F ∂τ ∂s 2 ∂s2 +

75

S˜˙ ˜ ρC α˜1 H(y)(Y − y) + ρC α˜2 Φ −y f˜ S˜ (48)

For notation simplicity in the sequel we will omit the tildes in the variables f , V , Y , and S. If we now take DC (y) = 1 − y 2 , and H(y) = 1 we can compute explicitly the equilibrium state for chartists with a constant mean investment propensity Y = Y∗ as: (α˜1 +α˜2 )

f ∞ (y) = C0 (1 + y)−2+Y∗ 2λ (1 − y)−2−Y∗ (1 − Y∗ y)(α˜1 + α˜2 ) , exp − λ(1 − y 2 )

(α˜1 +α˜2 ) 2λ

(49)

where C0 = C0 (Y∗ , λ/(α˜1 + α˜2 )) is such that the mass of f ∞ is equal to ρC . Other choices of the diffusion function originate different steady states (see [44]). Observe that, in the case Y∗ = 0, the distribution is not symmetric and in the chartist population a predominant behavior arise. Otherwise when the reference point of the value function is set to zero we have a symmetric distribution with two peaks and mean value zero, and the macroscopic state of indecision is given, microscopically, by a polarization of the chartist population among two opposite kind of behaviors (see Fig. 8). In order to study the asymptotic behavior for the price we must distinguish between the case ρF = 0 and ρF = 0. Let us consider first the situation in which ρF = 0 (or equivalently ρC = 1). For this purpose, we introduce the scaling V (s, τ ) =

1 v(χ, τ ), s

χ = log(s).

It is straightforward to show that v(χ, τ ) satisfies the following linear convection diffusion equation ν 2 ∂ ˜ tC ∂ v(χ, τ ) + ν ∂ v(χ, τ ). v(χ, τ ) = − βY ∂τ 2 ∂χ 2 ∂χ2 which admits the self-similar solution [8, 30] v(χ, τ ) =

1 1

(2 log(E(τ )/S(τ )2 )π) 2

exp −

(χ + log(

E(τ )/S(τ )) − log(S(τ )))2 , 2 log(E(τ )/S(τ )2 )

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Stephane Cordier et al. 1.4

80 Y*=0

Y*=0

1.2

Y*=0.2

70

Y*=0.2

Y*=− 0.2 60

Y*=−0.2

1

50

f(y)

S(t)

0.8 40

0.6 30 0.4

20

0.2 0 −1

10

−0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

0.8

1

0 0

10

20

y

30

40

50

60

70

80

90

100

t

Fig. 8. Equilibrium distribution function of the chartist investment propensity for different values of Y∗ = 0, 0.2, and − 0.2 (left) and corresponding behavior of the price S (right). Exact solutions with ρC = 1, β = 0.1, tC = 1, λ/(α˜1 + α˜2 ) = 1, and f (y, 0) = f ∞ (y)

with E(τ ) =

∞

V (s, τ )s2 ds.

0

Then reverting to the original variables it gives the lognormal behavior (log(s E(τ )/S(τ )2 )2 1 V (s, τ ) = . (50) − 1 exp 2 log(E(τ )/S(τ )2 ) s(2 log(E(τ )/S(τ )2 )π) 2 Note that E(τ ) satisfy the differential equation dE ˜ tC + ν)E(τ ). = (2βY dτ Thus, for a steady state characterized by (ii) in Proposition 2 we have S(τ ) = S0 , Y = 0, and E(τ ) = eντ E0 . Besides the above equilibrium state (50) characterizes also the self-similar behavior of the price distribution in the case of booms and crashes, when the price S(τ ) grows arbitrary or decays to zero. In particular in the limit S(τ ) → 0, point (iii) in Proposition 2, the distribution function V (s, τ ) concentrates near zero. Finally, we consider the microscopic behavior of the model where both ρC = 0 and ρF = 0. Recall now the Fokker–Plank equation for the price in (48) and consider the stationary case (i) in Proposition 2. The Fokker–Planck equation in such case reads ∂ ∂ ν ∂2 2 V + [(ρF γ(SF − s)) V ] = s V . ∂τ ∂s 2 ∂s2


77

In this case the steady state can be computed as [1, 7] V ∞ (s) = C1 (μ)

1 − (μ−1)SF s e , s1+μ

(51)

where μ = 1 + 2ρF γ/ν and C1 (μ) = ((μ − 1)SF )μ /Γ (μ) with Γ (·) being the usual Gamma function. Therefore, the stationary state is described by a Gamma-like distribution with Pareto power law tails [35]. Remark 3.1. The presence of fundamentalists is then essential in order to obtain fat tails in the price distribution. Their presence force the price to approach the mean value SF in a way similar to the redistribution of wealth in the models proposed in [1, 7]. This feature seems to be essential for the development of power law behaviors. The stationary state for the price (51) has in fact the same structure of the stationary states for the wealth in [1, 7]. Numerical tests We considered a Monte Carlo simulation of the kinetic system using N = 50, 000 chartists agents and no averages. In order to simulate the kinetic behavior of the price, we use a set of NS = 50000 samples which can be though as possible realizations of the random variable s denoting the price. As at the initial time the stock price S0 is supposed to be known, all samples are initialized at the same value initially. In all our computations we take the value function r ⎧ ⎨ x−R0 , L > x > R0 ; L−R0 l Φ(x) = ⎩ − R0 −x , −L < x ≤ R , 0 R0 +L where x ∈ [−L, L], R0 is the reference point and 0 < l ≤ r < 1. For example, we choose r = 1/2 and l = 1/4. Test 1 In the first test we consider the case with ρf = 0, i.e., only chartists in the model. We computed the equilibrium distribution for Φ(0) = 0 for the investment propensity. We take β = 0.1, tC = 1, a constant herding function H(y) = 1 and the coefficients α1 = α2 = 0.01. The initial data for the chartists is perfectly symmetric with Y = 0, so the price remains constant S = S0 with S0 = 10. A particular care is required in the simulation to keep Y = 0 as the equilibrium point is unstable and as soon as Y = 0 the results deviate toward a market boom or crash. After T = 1, 500 iteration the solution for the investment propensity has reached a stationary state and is plotted together with the solution of the Fokker Plank-limit in Fig. 9. In the same figure we report also the computed solution for the price distribution and the self-similar lognormal solution of the corresponding Fokker–Plank equation.

78

Stephane Cordier et al. 0.8

0.12

0.7 0.1

0.6 0.08

V(s,t)

f(y,t)

0.5 0.4

0.06

0.3 0.04

0.2 0.02

0.1 0 −1 −0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

0.8

0

1

0

5

10

15

y

20

25

30

35

40

s

Fig. 9. Equilibrium distribution function of the chartist investment propensity, with Φ(0) = 0 (left) and log-normal distribution for the price (right) at t = 1, 500. The continuous line is the solution of the corresponding Fokker–Plank equations

Test 2 In the second test case we considered the most interesting situation with the presence of fundamentalists, i.e., both chartists and fundamentalists interact in the stock market. We compute an equilibrium situation where ρf = ρc = 1/2 and the price is stationary at the fundamental value SF = 20. We take β = 0.1, tc = 1, γ = 1.3, α1 = α2 = 0.01 and report the result of the simulation for the price distribution at the stationary state. In Fig. 10 we show the price distribution together with the steady state of the corresponding Fokker–Planck equation. The emergence of a power law is clear also for the Boltzmann model.

10

0.06

10

0.05

10

0.04

V(s,t)

V(s,t)

0.07

10

−1

−2

−3

−4

0.03 10

−5

0.02 10

0.01 0

0

50

s

100

150

10

−6

−7

10

1

2

s

10

Fig. 10. Stationary price distribution for the price with ρF = ρC = 0.5. Figure on the right is in log–log scale. The continuous line is the Fokker–Planck solution


79

References 1. Bouchaud, J.P., Mézard, M., Wealth condensation in a simple model of economy, Physica A, 282, (2000), 536. 2. Caldarelli, G., Marsili, M., Zhang, Y.C., Europhysics Letters 40, (1997), 476. 3. Cercignani, C., Illner, R., Pulvirenti, M., The mathematical theory of dilute gases. Springer, New-York (1994) 4. Chakraborti, A., Chakrabarti, B.K., Statistical mechanics of money: how saving propensity affects its distributions, The European Physical Journal B, 17, (2000), 167. 5. Chatterjee, A., Chakrabarti, B.K., Manna, S.S., Pareto law in a kinetic model of market with random saving propensity, Physica A, 335, (2004), 155–163. 6. Cont, B., Bouchaud, J.P., Herd behaviour and aggregate fluctuations in financial markets. Macroeconomic Dynamics, 4, (2000), 170–196. 7. Cordier, S., Pareschi L., Toscani G., On a kinetic model for a simple market economy. Journal of Statistical Physics, 120, (2005), 253–277. 8. Cordier, S., Pareschi L., Piatecki C., Mesoscopic modelling of financial markets. Journal of Statistical Physics, 134, (2009), 161–184. 9. Drˇ agulescu, A., Yakovenko, V.M., Statistical mechanics of money, The European Physical Journal B, 17, (2000), 723–729. 10. D¨ uring, B., Toscani, G., Hydrodynamics from kinetic models of conservative economies, Physica A, 384, (2007), 493–506. 11. Ernst, M.H., Brito, R., Scaling solutions of inelastic Boltzmann equation with over-populated high energy tails. Journal of Statistical Physics, 109, (2002), 407–432. 12. Gabaix, X., Gopikrishnan, P., Plerou, V., Stanley, H.E., A Theory of Power-Law Distributions in Financial Market Fluctuations, Nature 423, (2003), 267–270. 13. Gibrat, R., Les inè galitès èconomiques, Librairie du Recueil Sirey, Paris, (1931). 14. Hill, I., Taylor, R., Recent trends in dividends payments and share buy-backs, Economic Trends, 567, (2001), 42–44. 15. Huang, Z.F., Solomon, S., Stochastic multiplicative processes for financial markets. Physica A, 306, (2002), 412–422. 16. Ingersoll, J.E. Jr., Theory of Financial Decision Making, Rowman and Littlefield Publishers, Inc., (1987). 17. Ispolatov, S., Krapivsky, P.L., Redner, S., Wealth distributions in asset exchange models, The European Physical Journal B, 2, (1998), 267–276. 18. Kahneman, D., Tversky, A., Prospect theory: an analysis of decision under risk, Econometrica, (1979), 263–292. 19. Kahneman, D., Tversky, A., Choices, Values, and Frames. Cambridge University Press, Cambridge (2000). 20. Kim, G., Markowitz, H.M., Investment rules, margin and market volatility. Journal of Portfolio Management, 16, (1989), 45–52. 21. Levy, M., Levy, H., Solomon, S., A microscopic model of the stock market: Cycles, booms and crashes, Economics Letters, 45, (1994), 103–111. 22. Levy, M., Levy, H., Solomon, S., Microscopic simulation of financial markets: from investor behaviour to market phoenomena, Academic Press, London (2000). 23. Levy M., Solomon, S., Dynamical explanation for the emergence of power law in a stock market. International Journal of Modern Physics C, 7, (1996), 65–72. 24. Levy, M., Solomon, S., New evidence for the power law distribution of wealth, Physica A, 242, 90, (1997).

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25. Liron, N., Rubinstein, J., Calculating the fundamental solution to linear convection-diffusion problems, SIAM Journal on Applied Mathematics 44, (1984), 493–511. 26. Lux, T., The socio-economic dynamics of speculative markets: interacting agents, chaos, and the fat tails of return distributions, Journal of Economic Behavior & Organization, 33, (1998), 143–165. 27. Lux, T., Marchesi, M., Scaling and criticality in a stochastich multi-agent model of a financial market. Nature, 397 No. 11, (1999), 498–500. 28. Lux, T., Marchesi, M., Volatility clustering in financial markets: a microscopic simulation of interacting agents, International Journal of Theoretical and Applied Finance, 3, (2000), 675–702 29. Malcai, O., Biham, O., Solomon, S., Richmond, P., Theoretical analysis and simulations of the generalized Lotka-Volterra model, Physical Review E, 66, (2002), 031102. 30. Maldarella, D., Pareschi L., Kinetic models for socio-economic dynamics of speculative markets. CMCS Report, (2009). 31. Mantegna, R.N., Stanley, H.E., An Introduction to Econophysics Correlations and Complexity in Finance, Cambridge University Press, Cambridge (2000). 32. Matthes, D., Toscani, G., On steady distributions of kinetic models of conservative economies, Journal of Statistical Physics, 130, (2008), 1087–1117. 33. Mitzenmacher, M., A Brief history of generative models for power law and lognormal distributions, Internet Mathematics, 1, No. 2, (2004), 226–251. 34. Pagan, A., The econometrics of financial markets, Journal of Empirical Finance 3, (1996), 15–102. 35. Pareto, V., Cours d’Economie Politique, Lausanne and Paris, (1897). 36. Pareschi, L., Toscani, G., Self-similarity and power-like tails in nonconservative kinetic models, Journal of Statistical Physics, 124, (2006), 747–779. 37. Plerou, V., Gopikrishnan, P., Stanley, H.E., Two-phase behaviour of financial markets, Nature, 421, (2003), 130. 38. Redner, S., Random multiplicative processes: an elementary tutorial, American Journal of Physics, (1990). 39. Samanidou, E., Zschischang, E., Stauffer, D., Lux, T., Microscopic models of financial markets, Economic working papers, 15, Christian-Albrechts-University of Kiel, Department of Economics, (2006). 40. Slanina, F., Inelastically scattering particles and wealth distribution in an open economy, Physical Review E 69, (2004), 046102. 41. Solomon, S., Richmond, P., Power laws of wealth, market order volumes and market returns, Physica A 299, (2001), 188–197. 42. Solomon, S., Stochastic Lotka-Volterra systems of competing auto-catalytic agents lead generically to truncated Pareto power wealth distribution, truncated Levy distribution of market returns, clustered volatility, booms and crashes, Computational Finance 97, eds. A-P. N. Refenes, A.N. Burgess, J.E. Moody (Kluwer Academic Publishers, Dordrecht (Hingham, MA), 1998). 43. Stigler, G.J., Public regulation of the securities market. Journal of Business, 37, (1964), 117–142. 44. Toscani, G., Kinetic Models of opinion formation, Communications in Mathematical Sciences, (2006). 45. Voit, J., The Statistical Mechanics of Financial Markets, Springer, Berlin, (2005).

A mathematical theory for wealth distribution Bertram D¨ uring1 and Daniel Matthes2 1 2

Institut f¨ ur Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria, [email protected] Institut f¨ ur Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria, [email protected]

Dedicated to Prof. Giuseppe Toscani, on the occasion of his 60th birthday Summary. We review a qualitative mathematical theory of kinetic models for wealth distribution in simple market economies. This theory is a unified approach that covers a wide class of such models which have been proposed in the recent literature on econophysics. Based on the analysis of the underlying homogeneous Boltzmann equation, a qualitative description of the evolution of wealth in the largetime regime is obtained. In particular, the most important features of the steady wealth distribution are classified, namely the fatness of the Pareto tail and the tails’ dynamical stability. Most of the applied methods are borrowed from the kinetic theory of rarefied gases. A concise description of the moment hierarchy and suitable metrics for probability measures are employed as key tools.

1 Introduction Kinetic models for wealth and income distributions successfully use methods from statistical mechanics to model the behavior of a large number of interacting individuals or agents in a closed economic system. In the rapidly growing field of econophysics, kinetic market models are presently of particular interest, see, e.g., the various contributions in the recent books [5,13,18,47,48]. The basic paradigm, dating back to the works of Mandelbrot [36], is that the laws of statistical mechanics govern the behavior of a huge number of interacting individuals just as well as that of colliding particles in a gas. The classical theory for homogeneous gases is adapted to the economic framework in the following way: molecules and their velocities are replaced by agents and their wealth, and instead of binary collisions, one considers trades among two agents. The specific model is then defined by trade rules on the microscopic level, which specify how wealth is exchanged in trades. Such rules are usually derived from plausible assumptions in an ad hoc manner. The output of the model are the macroscopic statistics of the wealth distribution in G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 4, c Springer Science+Business Media, LLC 2010

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Bertram D¨ uring and Daniel Matthes

the society. It is commonly accepted that the wealth distribution approaches a stationary profile for large times, and that the latter exhibits a Pareto tail [44]. Such overpopulated tails are a manifestation of the existence of an upper class of very rich agents, i.e., an indication of social inequality. A variety of models have been proposed and studied in view of the relation between parameters in the microscopic rules and the resulting macroscopic statistics, however most often only by numerical simulations. Two basic mechanisms drive the system on the microscopic level. First, a mechanism for saving, that ensures that agents exchange only a certain fraction of their wealth in a single trade event. Second, random effects are frequently incorporated to account for the risks involved in trading. Depending on the specific choice of these two mechanisms, and in particular on the balance between saving propensity and risk, these systems may or may not produce wealth distributions with the desired Pareto tail. We review a unified mathematical treatment, which applies to a large class of recently proposed models. We will focus our attention on trade interactions which are conservative, either microscopically or in the statistical mean. In this situation, the mean wealth in the model Boltzmann equation is preserved, and one expects the formation of a stationary profile. However, our mathematical framework and the essential results also carry over to various cases in which the mean wealth is not preserved. The main difference is that then the longtime behavior of the wealth distribution is not described by the convergence toward a stationary, but rather to a self-similar profile. In the class of conservative trades, the focus is on models with risky investments, originally introduced by Cordier, Pareschi, and Toscani [22], and on variants of the model designed by Chakraborti and Chakrabarti [12]. The results reviewed here are essentially based on [22, 28–30, 37, 38, 43]. The mathematical rigorous approach that we review is complementary to the numerous theoretical and numerical studies that can be found in the recent physics literature on the subject. In particular, the analysis is entirely based on the spatially homogeneous Boltzmann equation associated to the microscopic trade rules of the respective model. Thus, agents on the market are treated as a continuum, just like molecules in classical gas dynamics. Not only does this approach constitute the most natural generalization of the classical ideas to econophysics, but also it clarifies that certain peculiar observations made in ensembles of finitely many agents and in numerical experiments (like the apparent creation of steady distributions of infinite average wealth, e.g., [15–17]) are genuine finite size effects. In absence of explicit formulas for the large-time distributions of wealth, the analysis based on the Boltzmann description reveals itself as a powerful instrument to provide relations that allow to calculate characteristic features, like the Pareto index of the large-time wealth distribution, directly from the model parameters. In addition, apart from the shape of the steady states, the steady solutions of the underlying kinetic equations can be investigated by estimating the speed of relaxation of transient solutions to stationarity.

A mathematical theory for wealth distribution

83

The 1-Wasserstein metric is used to estimate the distance between the wealth distribution at finite times, and the steady state. We emphasize that the developed analytical techniques easily generalize to a broader class of conservative economic games, specifically to those in which the wealth distribution evolves according to some Kac-like equation. Indeed, the techniques have been applied, adapted, and extended to cover, e.g., wealth distribution among traders with dynamically adapting saving propensities [26], among different groups of traders [27], or in societies with taxation [50]. Further extensions of the theory lead to models for asset prices [21], for opinion and choice formation [20, 25, 49] and so on. Also, the mathematical aspects of the underlying Boltzmann equations, like propagation of regularity, have been further investigated in the current mathematical literature, see, e.g., [4, 10, 39]. As a closing remark for this introduction, a comment on the relevance of the presented models is in place. Clearly, they are – as often in applied mathematics – very simplified approximations of the complex reality. The exchange of wealth in a society is governed by a multitude of rational and irrational factors, and thus follows more difficult rules (if any rules at all) than the ones discussed here. Nonetheless, the results obtained for the simple models at hand do provide some insight into the relation between the trade behavior of individuals on the microscopic level, and the shape of the wealth distribution on the macroscopic level. One of the most important conclusions is that a suitable combination of the two basic ingredients “individual saving” and “market risk” already suffices to produce a wide spectrum of possible equilibria that have the shape of realistic wealth distributions. Also, we emphasize that it is easy to develop more complex models on top of the ones presented here, because the kinetic description provides a lot of flexibility for the definition of the exchange rules.

2 Kinetic wealth distribution models and mathematical tools 2.1 Wealth distributions In a closed ensemble of agents (i.e., a market), the wealth distribution f (t; w) refers to the relative density of agents with wealth w at time t ≥ 0. Debts are excluded in the models considered here, i.e., f (t; w) = 0 for w < 0, but concentration in w = 0 is allowed. The first moment of f (t; w) yields the average wealth per agent, M1 (t) = wf (t; w) dw. (1) R+

In the models under consideration, the density f (t; w) stabilizes at some stationary wealth curve f∞ (w) in the large-time limit t → ∞. The central notion

84


in the theory of wealth distributions is that of the celebrated Pareto index α ≥ 1. This number describes the size of the rich upper class in the considered ensemble of agents. Roughly, the smaller α is, the more of the total wealth is concentrated in the hands of a small group of individuals. The stationary curve f∞ (w) satisfies the Pareto law [44] with index α, provided that f∞ decays like an inverse power function for large w, f∞ (w) ∝ w−(α+1)

as w → +∞.

More precisely, f∞ has Pareto index α ∈ [1, +∞) if the moments ws f∞ (w) dw Ms :=

(2)

(3)

R+

are finite for all positive s < α, and infinite for s > α. If all Ms are finite (e.g., for a Gamma distribution), then f∞ is said to possess a slim tail. According to empirical data from ancient Egypt until today [15, 18], the wealth distribution among the population in a capitalistic country follows the Pareto law, with an index α ranging between 1.5 and 2.5. Slim tails are typical for societies with a highly equal distribution of wealth. Intuitively, one may think of socialist countries. Surprisingly, the mathematical description of the stationary wealth curve f∞ attracted the interest of mathematicians many years before Mandelbrot’s works [36]. A description of this curve by means of a generalized Gamma distribution is due to Amoroso [1] and D’Addario [23]. If one assumes for f∞ a unit mean, the Amoroso distribution reads: (α − 1)α exp − α−1 w fα (w) = , α > 1. (4) Γ (α) w1+α Note that this stationary distribution exhibits a Pareto power law tail of order α for large w’s. 2.2 One-dimensional Boltzmann models Here we consider a class of models in which agents are indistinguishable. Then, an agent’s “state” at any instant of time t ≥ 0 is completely characterized by his current wealth w ≥ 0. When two agents encounter in a trade, their pretrade wealths v, w change into the post-trade wealths v ∗ , w∗ according to the rule v ∗ = p1 v + q1 w, w∗ = q2 v + p2 w. (5) The interaction coefficients pi and qi are non-negative random variables. While q1 denotes the fraction of the second agent’s wealth transferred to the first agent, the difference p1 − q2 is the relative gain (or loss) of wealth of the first agent due to market risks. We assume that pi and qi have fixed laws, which are independent of v and w, and of time.


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In one-dimensional models, the wealth distribution f (t; w) of the ensemble coincides with agent density and satisfies the associated spatially homogeneous Boltzmann equation, ∂t f + f = Q+ (f, f ), (6) on the real half line, w ≥ 0. The collisional gain operator Q+ acts on test functions ϕ(w) as: Q+ (f, f )[ϕ] := ϕ(w)Q+ f, f (w) dw R+ 1 ϕ(v ∗ ) + ϕ(w∗ )f (v)f (w) dv dw, (7) = 2 R2+ with · denoting the expectation with respect to the random coefficients pi and qi in (5). The large-time behavior of the density is heavily dependent of the evolution of the average wealth M (t) := M1 (t) = wf (t; w) dw. (8) R+

Conservative models are such that the average wealth of the society is conserved with time, M (t) = M , and we will generally assume that the value of M to be finite. In terms of the interaction coefficients, this is equivalent to p1 + q2 = p2 + q1 = 1. Nonconservative models are such that M (t) is not conserved with time. We will restrict ourselves to the case in which p1 + q2 = p2 + q1 = 1, so that the average wealth is exponentially increasing or decreasing M (t) = M (0)e(p1 +q2 −1)t .

(9)

From the point of view of its kinetic classification, the Boltzmann equation (6) belongs to the Maxwell type. In the Boltzmann equation for Maxwell molecules, in fact, the collision frequency is independent of the relative velocity [7], and the loss term in the collision operator is linear. This introduces a great simplification, that allows to use most of the well established techniques developed for the three-dimensional spatially homogeneous Boltzmann equation for Maxwell molecules in the field of wealth redistribution. In the following we will restrict ourselves to the conservative case. However, we note that the main results carry over to the nonconservative case. In this case one needs to study self-similar solutions and rescale the solutions so that average wealth is conserved as time evolves. A more extensive presentation of these results can be found in [30]. 2.3 Saving propensity as additional variable Arguing that agents are not indistinguishable in reality, but have personal trading preferences, Chatterjee et al. [16] introduced the concept of quenched

86


saving propensity. In this Chatterjee-Chakrabarti-Manna (CCM) model, λ is not a global quantity, but characterizes the agents. The current “state” of an agent is consequently described by two numbers, his wealth w > 0 and his personal saving propensity λ ∈ (0, 1). We shall only discuss the case where λ does not change with time. Trade rules which allow the agents to adapt their saving strategy in time (“annealed saving”) have been investigated [14, 16], but seemingly do not exhibit genuinely novel effects. The configuration of the kinetic system is described by the extended density function f (t; λ, w). The wealth distribution h(t; w) is recovered from the density f (t; λ, w) as marginal, 1 h(t; w) = f (t; λ, w) dλ, (10) 0

but is no longer sufficient to characterize the configuration completely. The other marginal yields the time-independent density of saving propensities, f (t; λ, w) dw. (11) χ(λ) = R+

Clearly, χ(λ) is determined by the initial condition f (0; λ, w), and should be considered as defining parameter of the model. The collision rules are similar to the one-dimensional model, taking into account the individual characteristics: two agents with pretrade wealth v, w and saving propensities λ, μ, respectively, exchange wealth according to v ∗ = λv + [(1 − λ)v + (1 − μ)w], ∗

w = μw + (1 − )[(1 − λ)v + (1 − μ)w].

(12) (13)

Clearly, wealth is strictly conserved, v ∗ + w∗ = v + w, so the mean wealth M is constant in time. The Boltzmann equation (6) is now posed on a twodimensional domain, (λ, w) ∈ (0, 1) × (0, ∞). The collisional gain operator Q+ satisfies 1 ϕ(v ∗ )f (λ, v)f (μ, w) dv dw dμ (14) Q+ (f, f )[ϕ](λ) = R2+

0

after integration against a regular test function ϕ(w). For simplicity, we assume that is symmetric around 1/2. Another two-dimensional model of Fokker–Planck type has been recently described in [27]. The idea is to generalize a Fokker–Planck system derived in [27] for groups of traders where each group has a common saving propensity to the case in which the trading rate is randomly distributed on the interval (0, 1), with distribution Γ (λ), where Γ (λ) = g0 (λ, v) dv, 0 < λ < 1, R+


87

is the λ-marginal of the initial density of wealth. Introduce the notation 1 Φ(λ) dλ. Φ(λ)λ = 0

The Fokker–Planck equation with a continuous varying trading rate reads ∂g(λ, v) μ ∂2 2 = v ρ(τ ; λ)λ g(λ, v) ∂τ 2 ∂v 2 ∂ ((λvρ(τ ; λ)λ − λM (τ ; λ)λ ) g(λ, v)) . (15) + ∂v Taking into account that the total mass is preserved, ρ(λ)λ ≡ 1, while the distribution Γ (τ ; λ) = g(τ ; λ, v) dv R+

does not depend on time, i.e., Γ (τ ; λ) = Γ (λ), (15) simplifies to μ ∂2 2 ∂g(λ, v) ∂ = (λv − λM (λ)λ ) g(λ, v) . v g(λ, v) + 2 ∂τ 2 ∂v ∂v

(16)

The analytical study of the behavior of the solution to (15) would certainly deserve attention. The main difference between the CCM model and the present one is related to the fact that the Fokker–Planck equation (16) is obtained from the CPT model with risky components. A detailed study of the Fokker– Planck equation (16) is forthcoming.

2.4 Wasserstein and Fourier based distances Since Monte Carlo simulations produce distributions of point masses instead of smooth curves, a good notion of distance between measures is important to quantify the convergence of numerical results to the continuous limit. In most of our applications, we will consider probability distributions possessing finite moments of some order r > 1. Accordingly, for given constants c > 0 and r > 1, define Mc,s as the set of (Borel) probability measures on R+ satisfying wf (w) dw = c, wr f (w) dw < ∞. (17) R+

R+

Among other distances, the Wasserstein distance (of order one) of two density functions f1 (w), f2 (w) is an extremely useful instrument. This distance is given by: F1 (v) − F2 (v) dv, W[f1 , f2 ] := (18) R+

88


where the Fi denote the distribution functions, ∞ fi (w) dw (i = 1, 2). Fi (v) =

(19)

v

Equivalently, the Wasserstein distance is defined as the infimum of the costs for transportation [51], |v − w| dπ(v, w). (20) W[f1 , f2 ] := inf π∈Π

Here Π is the collection of all measures in the plane R2 with marginal densities f1 and f2 , respectively. The infimum is in fact a minimum, and is realized by some optimal transport plan πopt . Convergence of densities f (t; w) to a limit f∞ (w) in the Wasserstein distance is equivalent to the weak convergence f (t; w)dw f∞ (w)dw in the sense of measures, and convergence of the first moments. Note that definition (20) is a particular case (p = 1) of the general expression of the Wasserstein distance of order p > 0, Wp [f1 , f2 ]p := inf |v − w|p dπ(v, w). (21) π∈Π

There is an intimate relation of Wasserstein to Fourier metrics [32], defined by: ds [f1 , f2 ] = sup[|ξ|−s |fˆ1 (ξ) − fˆ2 (ξ)|], ξ

s > 0,

(22)

where fˆ(t; ξ) is the Fourier transform of f (t; x), f (t; ξ) = e−iξv f (t; v) dv. R+

Note that the distance (22) is finite for some s > 1 if the distribution functions have the same moments up to [s], where [s] denotes as usual the entire part of s. The interested reader can have an almost complete picture of the key properties of these metrics by looking at the notes [11]. There, however, mostly the case of the Wasserstein distance of order two (assuming finite second moment of the occurring densities) is considered. In the economic framework, where the typical case is p = 1, for s > 1, the Wasserstein and Fourier distance are related [28] by: W[f1 , f2 ] ≤ C(ds [f1 , f2 ])−(s−1)/s(2s−1) .

(23)

Below, we sketch the proof of this result. Lemma 1. Assume that two probability densities f and g have first moment equal to one, and some moment of order s ∈ (1, 2] bounded. Then there exists a constant C > 0, depending only on s and the values of the sth moments of f and g, such that


W[f, g] ≤ C(ds [f, g])

s−1 s(2s−1)

.

89

(24)

Conversely, one has d1 [f, g] ≤ W[f, g],

(25)

even if no moments of f and g above the first are bounded. Proof. To prove (24), we adapt the proof of Theorem 2.21 in [11], corresponding to s = 2. Define v s f (v) dv,

M = max R+

v s g(v) dv

.

R+

Starting from the definition of the Wasserstein distance in (18), we estimate F (v) − G(v) dv W[f, g] = R+

≤

0

R

F (v) − G(v) dv + R1−s

≤ R1/2

R+

F (v) − G(v)2 dv

∞

R

1/2

v s−1 F (v) − G(v) dv +R1−s

(26)

v s−1 F (v) − G(v) dv,

∞ R

where the parameter R = R(t) > 0 is specified later. By Parseval’s identity, 2 F − G (v)2 dv = (F (iξ)−1 fˆ(ξ) − gˆ(ξ) 2 dξ − G)(ξ) dξ = R+

R

≤ (ds [f, g])2

|ξ| 0. The constant C1 depends only on s > 1. This gives a bound on the first term in (26). We estimate the second term, integrating by parts: ∞ ∞ v s−1 F (v) − G(v) dv ≤ v s−1 F (v) + G(v) dv R R ∞ 1 ∞ s = v f (v) + g(v) dv+ v s F (v) + G(v) s R R 2M + lim rs F (r) + G(r) . ≤ r→+∞ s The last expression is easily estimated by Chebyshev’s inequality, i.e., ∞

v s f (v) dv = 0, lim rs F (r) ≤ lim rs Pf v > r ≤ lim r→∞

r→∞

r→∞

r

90


since the sth moment of f is finite. In summary, (26) yields 1/2

W[f, g] ≤ C1 R1/2 (ds [f, g])1/(2s) + 2s−1 M R1−s . Optimizing this over R yields the desired inequality (24). The other inequality (25) is derived from the alternative definition (18) of W[f, g], with πopt being the optimal transport plan

−1 −ivξ −iwξ d1 [f, g] = sup |ξ| e f (v) dv − e g(w) dw ξ =0

≤ sup |ξ|−1 ξ =0

R+

R2+

R+

−ivξ e − e−iwξ dπopt (v, w)

|1 − ei(v−w)ξ | |v − w| dπopt (v, w) |v − w||ξ| R2+ ξ =0

|1 − eix | W[f, g]. = sup |x| x∈R

≤

sup

In view of the elementary inequality |1 − exp(ix)| ≤ |x| for x ∈ R, this yields the claim (25). Examples. Two Dirac distributions have Wasserstein distance W[δx , δy ] = |x − y|. Likewise, d1 [δx , δy ] = |x − y|, but notice that ds [δx , δy ] = +∞ for s > 1 unless x = y. More generally, a density f1 (v) and its translate f2 (v) = f1 (v−z) have Wasserstein distance W[f1 , f2 ] = |z| and Fourier distance d1 [f1 , f2 ] = |z|. To indicate that the Wasserstein and the Fourier based distances provide a more sensible notion of “closeness” than, e.g., the classical Lp -distances, consider densities f1 and f2 that are supported in small intervals [x − , x + ] and [y − , y + ], respectively. Then f1 − f2 L1 = 2 as soon as |x − y| > 2. 2.5 Other Fourier based distances One of the weak points of the Fourier based distance (22) is that, for a given s such that 1 < s < 2, it is not known if the space of probability measures Mc,s with metric ds is complete or not. This unpleasant fact is discussed in [11], together with a possible remedy. A further metric, however, can be introduced, which does not have the same problem, while it possesses most of the properties of the metric ds . This metric has been introduced in [3] to characterize fixed points of convex sums of random variables with a small number of moments. For s ∈ (1, 2), Ds [f1 , f2 ] = |ξ|−(s+1) |fˆ1 (ξ) − fˆ2 (ξ)| dξ, s > 0. (27) As proven in [3], (Mc,s , Ds ) is complete. A proof of the analogous of Lemma 1 would be desirable.


Let fμ (w) =

1 w μ f ( μ ).

91

Then, the metric (27) is such that Ds [fμ , gμ ] = μs Ds [f, g].

(28)

The scaling property (28), which holds also for the metric ds , is at the basis of most of the applications of Fourier based metrics to kinetic models.

3 Results for the one-dimensional models First, we shall give an overview on the available analytical results for onedimensional models, and indicate the derivation of these results on an intuitive, nonrigorous level. The differences between pointwise conservative and conservative-in-the-mean models are discussed. Subsequently, some mathematical details and proofs are provided in Sect. 3.4. 3.1 Pareto tail of the wealth distribution We introduce the characteristic function S(s) =

2 1 s pi + qis − 1, 2 i=1

(29)

which is convex in s > 0, with S(0) = 1. Also, S(1) = 0 because of the conservation property (8). The results from [28, 37] imply the following. Unless S(s) ≥ 0 for all s > 0, any solution f (t; w) tends to a steady wealth distribution P∞ (w) = f∞ (w), which depends on the initial wealth distribution only through the conserved mean wealth M > 0. Moreover, exactly one of the following is true: (PT) if S(α) = 0 for some α > 1, then P∞ (w) has a Pareto tail of index α; (ST) if S(s) < 0 for all s > 1, then P∞ (w) has a slim tail; (DD) if S(α) = 0 for some 0 < α < 1, then P∞ (w) = δ0 (w), a Dirac Delta at w = 0. To derive these results, one studies the evolution equation for the moments ws f (t; w) dw, (30) Ms (t) := R+

which is obtained by integration of (6) against ϕ(w) = ws , d Ms = Q+ [ϕ] − Ms . dt

(31)

Using an elementary inequality for x, y ≥ 0, s ≥ 1, xs + y s ≤ (x + y)s ≤ xs + y s + 2s−1 (xy s−1 + xs−1 y),

(32)

92


in (7), one calculates for the right-hand side of (31) S(s)Ms ≤ Q+ [ϕ] − Ms ≤ S(s)Ms + 2s−2

2 i=1

pi qis−1 + ps−1 qi M Ms1−1/s . (33) i

Solving (31) with (33), one finds that either Ms (t) remains bounded for all times when S(s) < 0 or it diverges like exp[tS(s)] when S(s) > 0, respectively. In case (PT), exactly the moments Ms (t) with s > α blow up as t → ∞, giving rise to a Pareto tail of index α. We emphasize that f (t; w) possesses finite moments of all orders at any finite time. The Pareto tail forms in the limit t → ∞. In case (ST), all moments converge to limits Ms (t) → Ms∗ , so the tail is slim. One can obtain additional information on the stationary wealth distribution P∞ (w) from the recursion relation for the principal moments, −S(s)Ms∗ =

s−1 2

1 s ∗ pki qis−k Mk∗ Ms−k , 2 k i=1

s = 2, 3, . . .

(34)

k=1

The latter is obtained by the integration of (6) against ϕ(w) = ws in the steady state ∂t f = 0. In case (DD), all moments Ms (t) with s > 1 blow up. The underlying process is a separation of wealth as time increases: while more and more agents become extremely poor, fewer and fewer agents possess essentially the entire wealth of the society. In terms of f (t; w), one observes an accumulation in the pauper region 0 ≤ w 1, while the density rapidly spreads into the region w 1. The expanding support of f (t; w) is balanced by a decrease in magnitude, because the average wealth is fixed. This induces a pointwise convergence f (t; w) → 0 for all w > 0. Such a condensation of wealth has been observed and described in several contexts [8, 9, 19, 35] before. An illustration of the solution’s behavior in the (DD) case is provided by the “Winner takes all” dynamics, with rules v ∗ = v + w,

w∗ = 0.

(35)

In each trade, the second agent loses all of his wealth to the first agent. The solution for the initial condition f (0; w) = exp(−w) is explicit, f (t; w) =

2 2 2 t exp − w + δ0 (w). 2+t 2+t 2+t

(36)

Note that the average wealth is conserved at all finite times t ≥ 0, so that limt→∞ M1 (t) = M1 (0), but f∞ = δ0 has vanishing average wealth. We have simulated the “Winner takes all” model (35) using direct Monte Carlo simulations. As time evolves, all agents but one become pauper and give rise to a Dirac Delta at w = 0. We run 100 simulations for systems consisting of N = 100, 1,000, and 10,000 agents, respectively. Figure 1 displays the


93

simulation-averaged fraction of the population with zero wealth. This fraction of pauper agents grows linearly until a saturation effect becomes visible. The blow up figure shows the improving approximation of the theoretically predicted rate for growing system size. 0.992

90

80

70

N=100 N=1000 N=10000 Analytical

60

50 100

101

102

103

Agents with zero wealth (%)

Agents with zero wealth (%)

100

0.991 0.99 0.989 0.988 0.987

0.985 0.984 190

104

N=100 N=1000 N=10000 Analytical

0.986

Time (steps)

200

210

Time (steps)

Fig. 1. “Winner takes all” model: Evolution of the fraction of agents with zero wealth (left) and blow up (right)

3.2 Pointwise conservative models The first explicit description of a binary wealth exchange model dates back to Angle [2] (although the intimate relation to statistical mechanics was only described about one decade later [24, 35]): in each binary interaction, winner and loser are randomly chosen, and the loser yields a random fraction of his wealth to the winner. From here, Chakraborti and Chakrabarti [12] developed the class of strictly conservative exchange models, which preserve the total wealth in each individual trade, v ∗ + w∗ = v + w.

(37)

In its most basic version, the microscopic interaction is determined by one single parameter λ ∈ (0, 1), which is the global saving propensity. In interactions, each agent keeps the corresponding fraction of his pretrade wealth, while the rest (1 − λ)(v + w) is equally shared among the two trade partners, 1 v ∗ = λv + (1 − λ)(v + w), 2

1 w∗ = λw + (1 − λ)(v + w). 2

(38)

In result, all agents become equally rich eventually. Indeed, the stochastic variance of f (t; w) satisfies d 1 2 2 (w − M ) f (t; w) dw = − (1 − λ ) (w − M )2 f (t; w) dw. (39) dt R+ 2 R+

94


The steady state f∞ (w) = δM (w) is a Dirac Delta concentrated at the mean wealth, and is approached at the exponential rate (1 − λ2 )/2. More interesting, nondeterministic variants of the model have been proposed, where the amount (1−λ)(v +w) is not equally shared, but in a stochastic way: v ∗ = λv + (1 − λ)(v + w),

w∗ = λw + (1 − )(1 − λ)(v + w),

(40)

with a random variable ∈ (0, 1). Independent of the particular choice of , the characteristic function 1 S(s) = [λ+(1−λ)]s +[1−(1−λ)]s +[s +(1−)s ](1−λ)s −1 (41) 2 is negative for all s > 1, hence case (ST) applies. Though the steady state f∞ is no longer explicit – for approximations see [42, 46] – one concludes that its tail is slim. In conclusion, no matter how sophisticated the trade mechanism is chosen, one-dimensional, strictly conservative trades always lead to narrow, “socialistic” distributions of wealth. 3.3 Conservative in the mean models Cordier et al. [22] have introduced the CPT model, which breaks with the paradigm of strict conservation. The idea is that wealth changes hands for a specific reason: one agent intends to invest his wealth in some asset, property, etc. in possession of his trade partner. Typically, such investments bear some risk, and either provide the buyer with some additional wealth or lead to the loss of wealth in a nondeterministic way. An easy realization of this idea [37] consists in coupling the previously discussed rules (38) with some risky investment that yields an immediate gain or loss proportional to the current wealth of the investing agent, v∗ =

1 + λ 2

1−λ + η1 v + w, 2

w∗ =

1 + λ 2

1−λ + η2 w + v. 2

(42)

The coefficients η1 , η2 are random parameters, which are independent of v and w, and distributed so that always v ∗ , w∗ ≥ 0, i.e., η1 , η2 ≥ −λ. Unless these random variables are centered, i.e., η1 = η2 = 0, it is immediately seen that the mean wealth is not preserved, but it increases or decreases exponentially (see the computations in [22]. For centered ηi , v ∗ + w∗ = (1 + η1 )v + (1 + η2 )w = v + w,

(43)

implying conservation of the average wealth. Various specific choices for the ηi have been discussed [37]. The easiest one leading to interesting results is ηi = ±μ, where each sign comes with probability 1/2. The factor μ ∈ (0, λ) should be understood as the intrinsic risk of the market: it quantifies the


95

fraction of wealth agents who are willing to gamble on. Figure 2 displays the various regimes for the steady state f∞ in dependence of λ and μ, which follow from numerical evaluation of s 1 + λ s 1 − λ s 1 1 + λ −μ + +μ S(s) = + − 1. (44) 2 2 2 2 Zone I is forbidden by the constraint μ < λ. In zone II, corresponding to low market risk, the wealth distribution shows again “socialistic” behavior m 1

Zone IV Condensation

0.75

Zone I Not allowed Zone III Pareto Tails

0.5

0.25

Zone II Slim Tails 0.25

0.5

0.75

1

l

Fig. 2. Regimes for the formation of Pareto tails

with slim tails. Increasing the risk, one falls into “capitalistic” zone III, where the wealth distribution displays the desired Pareto tail. A minimum of saving (λ > 1/2) is necessary for this passage; this is expected because if wealth is spent too quickly after earning, agents cannot accumulate enough to become rich. Inside zone III, the Pareto index α decreases from +∞ at the border with zone II to unity at the border to zone IV. Finally, in zone IV, the steady wealth distribution is a Delta in zero. Both risk and saving propensity are so high that a marginal number of individuals manages to monopolize all of the society’s wealth. In the long-time limit, these few agents become infinitely rich, leaving all other agents truly pauper. 3.4 Mathematical details We will now give some details about proofs. One of the main tools is the use of the Fourier transform. This idea, which goes back to the seminal work of Bobylev [6, 7], is well-suited to treat collision kernels of Maxwellian type. In particular, the Fourier representation is particularly adapted to the use of various Fourier metrics. An auxiliary tool is the study of the evolution of moments.

96


Evolution of Fourier metrics According to the collision rule (5), the transformed kernel reads fˆ, fˆ (ξ) = 1 f(p1 ξ)f(q1 ξ) + f(p2 ξ)f(q2 ξ) − f(ξ)f(0). Q 2

(45)

Assuming the initial distribution of wealth in MM,r , with r > 1, the initial conditions turn into f0 (0) = 1

and f0 (0) = iM.

Hence, the Boltzmann equation (6) can be rewritten as: 1 ∂ f(t; ξ) + f(t; ξ) = f(p 1 ξ)f (q1 ξ) + f (p2 ξ)f (q2 ξ) = f (pi ξ)f (qi ξ) + . (46) ∂t 2 Details about existence of solutions to (46) can be found in [37]. Let f1 and f2 be two solutions of the kinetic equation (46), corresponding to initial values f1,0 and f2,0 in MM,r , with r > 1, and denote by f1 , f2 their Fourier transforms. Let s ≥ 1 be such that ds (f1,0 , f2,0 ) is finite. Then f1 (pi ξ)f1 (qi ξ) − f2 (pi ξ)f2 (qi ξ) + ∂ f1 (ξ) − f2 (ξ) f1 (ξ) − f2 (ξ) + = . ∂t |ξ|s |ξ|s |ξ|s (47) Now, because |f1 (t; ξ)| ≤ 1 and |f2 (t; ξ)| ≤ 1 for all ξ ∈ R, we obtain f1 (pi ξ)f1 (qi ξ) − f2 (pi ξ)f2 (qi ξ)+ |ξ|s f1 (qi ξ) − f2 (qi ξ) s f1 (pi ξ) − f2 (pi ξ) s pi ≤ |f1 (pi ξ)| + |f2 (qi ξ)| qi |qi ξ|s |pi ξ|s + + f1 (ξ) − f2 (ξ) s pi + qis + . ≤ sup |ξ|s ξ In terms of the auxiliary quantity h(t; ξ) =

f1 (ξ) − f2 (ξ) , |ξ|s

the preceding computation shows that ∂h s s ∂t + h ≤ pi + qi + h ∞ . Gronwall’s lemma yields at once that h(t) ∞ ≤ exp (psi + qis + − 1)t h0 ∞ .

(48)


97

This introduces the quantity S(s), defined in (29), into the game. As h(t) ∞ = ds [f1 (t), f2 (t)], we obtain from (48) ds [f1 (t), f2 (t)] ≤ exp {S(s) · t} ds [f1,0 , f2,0 ].

(49)

In particular, if S(s) is negative, then the ds -distance of f1 and f2 decays exponentially in time. We remark that, thanks to the scaling property (28), the same result holds for the metric Ds . Thus, Ds [f1 (t), f2 (t)] ≤ exp {S(s) · t} Ds [f1,0 , f2,0 ].

(50)

Theorem 4.1. [37] Let f1 (t) and f2 (t) be two solutions of the Boltzmann equation (6), corresponding to initial values f1,0 and f2,0 in MM,r , r > 1. Let s ≥ 1 be such that ds [f1,0 , f2,0 ] is finite. Then, for all times t ≥ 0, (49) and (50) hold. In particular, if S(s) is negative then the Fourier based distances of f1 and f2 decay exponentially in time. Putting f1,0 = f2,0 = f0 in (49), and using s = 1 yields. Corollary 4.2 If f0 is a non-negative density in MM,r , r > 1. Then there exists a unique weak solution f (t) of the Boltzmann equation with f (0) = f0 . Evolution of moments In Theorem 4.1 about the large-time behavior of solutions to (6), the essential quantity S has been introduced. Below, we prove that the values S(s) also control the asymptotic behavior of moments. In fact, if S(s) is negative for some s > 0, then the sth moment of the solution, Ms (t) = v s f (t; v) dv, R+

remains bounded for all times. On the other hand, if S(s) is positive for some s > 1, then Ms (t) diverges exponentially fast as t → ∞. We exploit this information to prove decay properties to the steady state. To start with, we note that conservation of the total wealth allows to conclude that at least all moments of order s ≤ 1 remain uniformly bounded. In fact, by H¨ older’s inequality, R+

v s f (v) dv ≤

vf (v) dv R+

s ·

R+

1−s f (v) dv

= M1s ,

Now, let s > 1 and suppose that the initial density f0 (v) satisfies Ms (0) = v s f0 (v) dv < ∞. R+

0 < s < 1.

(51)

98


Then, putting φ(v) = v s in the weak form (7), we obtain d dt

1 v f (t; v) dv = 2 R+ s

2 R2+

(pi v + qi w) − v − w f (v)f (w) dv dw . s

s

s

i=1

+

(52) In the following, we establish upper and lower bounds for the right-hand side of (52). These bounds rely on the following elementary inequality. Lemma 2. For arbitrary non-negative real numbers a and b, and s > 1, as + bs + θs (as−1 b + abs−1 ) ≤ (a + b)s ≤ as + bs + Θs (as−1 b + abs−1 ), (53) ⎧ ⎪ (s > 3) ⎨s s (2 ≤ s ≤ 3) s−3 with θs = 2 s (2 ≤ s ≤ 3) and Θs = s−3 ⎪ 2 s (otherwise) ⎩ 0 (1 < s < 2) Remark. An investigation of the limit behavior as a 0 and b > 0 shows that θs = 0 for 1 < s < 2 cannot be improved in general. Proof. By homogeneity, it suffices to prove the inequality for a + b = 1. Define for s > 1, φ(s) := as + bs + sab. A calculation yields φ(2) = φ(3) = 1, independently of a and b = 1 − a. Furthermore, φ is convex in s since φ

(s) = as ln2 a + bs ln2 b ≥ 0. Hence φ(s) ≤ 1 if and only if 2 ≤ s ≤ 3. Observe that as−2 + bs−2 is concave w.r.t. a = 1 − b ∈ (0, 1) for 2 ≤ s ≤ 3, and convex for all other s > 1; the expression attains its extremal value 23−s at a = b = 12 . Hence ≤ 23−s ab (2 ≤ s ≤ 3) s−1 s−1 s−2 s−2 a b + ab = ab(a +b ) ≥ 23−s ab (otherwise) Thus we obtain, for 2 ≤ s ≤ 3 as + bs + 2s−3 s(as−1 b + abs−1 ) ≤ φ(s) ≤ 1 = (a + b)s , and with reversed inequalities for 1 < s < 2 or s > 3. Now let s > 1 be fixed and consider for a ∈ [0, 1] fs (a) := as + bs + s(as−1 b + abs−1 ), with b = 1 − a. Observe that fs (0) = fs (1) = 1, and furthermore fs (a) = s(s − 1)(as−2 b − abs−2 ),


99

so that fs has a ¯ = 1/2 as the only critical point in (0, 1). For s < 3, one has fs (¯ a) = (s + 1)21−s > 1, so a ¯ is a maximum point and hence fs (a) ≥ 1 for a ∈ [0, 1]; for s > 3, it is a minimum point and fs (a) ≤ 1. Consequently, for s ≤ 3, as + bs + s(as−1 b + abs−1 ) = fs (a) ≥ 1 = (a + b)s . The reversed inequality holds for s ≥ 3. Using the upper bound in (53), estimate (pi v + qi w)s ≤ psi v s + qis ws + Θs (ps−1 qi v s−1 w + pi qis−1 vws−1 ) i under the integral in (52), leading to s d 1 Ms (t) ≤ (p1 + ps2 − 1)v s + (q1s + q2s − 1)ws f (v)f (w) dv dw dt 2 R2+ s−1 Θs s−1 (p1 q1 + ps−1 w + 2 q2 )v 2 R2+ s−1 s−1 s−1 + (p1 q1 + p2 q2 )vw f (v)f (w) dv dw = S(s)Ms (t) + Θs ps−1 qi + pi qis−1 + · M1 (t) · i

ws−1 f (w) dw.

R+

(54) Recall that the total wealth vf (v) dv = M1 is conserved in time. Further, by H¨ older’s inequality, it follows that R+

ws−1 f (w) dw ≤

ws f (w) dw

1− 1s .

R+

Hence, we obtain: d Ms (t) ≤ S(s)Ms (t) + Θs K(s)M1 · dt

v s f (v) dv

1− 1s ,

(55)

R+

where K(s) := ps−1 qi + pi qis−1 + ≤ psi + qis + = S(s) + 1. i

(56)

In particular, if S(s) is a finite number, then so is K(s). In this case, the sth moment grows at most exponentially, with rate S(s), if it was finite initially. Moreover, if S(s) < 0, then the sth moment remains uniformly bounded for

100


all times. In fact, an upper bound on the sth moment is determined by the solution of the associated ordinary differential equation 1

y˙ = S · y + Θs K(s)M1 y 1− s

with initial condition y(0) = Ms (0). The solution is explicitly given by: !t " #s 1/s Θs K(s)M1 y(t) = Ms (0) exp tS(s) exp S(s) − 1 + . S(s) s Notice that the first term in the square bracket vanishes for t → ∞ if S(s) < 0, so that the limiting value depends on the initial condition f0 only through the total wealth M1 . By the same reasoning as earlier, we construct a bound from below on the time-derivative of the integral. For this, we use the lower bound given in the elementary inequality (53). Replacing the respective expressions under the integral, we obtain s d 1 Ms (t) ≥ p1 + ps2 − 1v s + q1s + q2s − 1ws f (v)f (w) dv dw dt 2 R2+ θs s−1 (ps−1 q1 + ps−1 w + 1 2 q2 )v 2 R2+ s−1 s−1 s−1 f (v)f (w) dv dw + (p1 q1 + p2 q2 )vw = S(s)Ms (t) + θs K(s)M1 · ws−1 f (w) dw. R+

We use Hölder’s inequality to estimate w R+

s−1

f (w) dw ≥

s−1 wf (w) dw

R+

= M1s−1 .

By Gronwall’s inequality, a lower bound is given by: θs · K(s) · M1s exp t · S(s) − 1 . Ms (t) ≥ Ms (0) · exp t · S(s) + S(s) We conclude that if S(s) > 0, then the moment Ms diverges exponentially in time. In the special case that S(s) = 0, similar but simpler arguments give that the corresponding moment remains either bounded (iff K(s) = 0) or diverges, but only at the algebraic rate ts . Finally, if S(s) = +∞, an easy argument shows that the sth moment of f (t; v) must be infinite for all positive times. Theorem 4.3. [37] Consider a solution f to the Boltzmann equation (6). Let s > 1 be such that Ms (0) = R+ v s f0 (v) dv < ∞.


101

1. If 0 < S(s) < +∞, then the sth moment diverges exponentially fast as t → ∞: Ms (0) +

θs K(s) · M1s + o(1) S(s) #s Θs K(s) Ms (t) ≤ Ms1/s (0) + · M1 + o(1). ≤ exp{t · S(s)} S(s)

(57)

2. If S(s) < 0, then the sth moment remains uniformly bounded as t → ∞:

s θs K(s) Θs K(s) · M1s + o(1) ≤ Ms (t) ≤ · M1s + o(1). (58) |S(s)| |S(s)| 3. If S(s) = 0, then the sth moment either remains bounded or diverges at an algebraic rate:

s

s θs K(s) Θs K(s) · M1s + o(1) ≤ t−s · Ms (t) ≤ · M1s + o(1). (59) s s 4. Finally, if S(s) = +∞, then the sth moment is infinite for all t > 0.

Existence and tails of the steady state The analysis of the previous sections shows that the long-time behavior of solutions is essentially determined by the quantity S. For this reason, let us investigate this function in further detail. First recall that for an arbitrary non-negative number p, the exponential s → ps is convex in s > 0. Hence S(s), which is the average of convex functions, is convex on its domain. By the dominated convergence theorem, S(s) is welldefined at least for 0 < s ≤ 1, but possibly S(s) = +∞ for all s > s∞ > 1. Because S(1) = 0, convexity leaves only three possibilities for the behavior of S: 1. S(s) is non-negative for all s > 0. 2. S(s) is negative for some s ∈ (0, 1), and positive for all s > 1. 3. S(s) is negative for all 1 < s < s¯, and positive for all s < 1 and all s > s¯; here s¯ = +∞, S(¯ s) = 0, or S(s) = +∞ for s > s¯. If S is differentiable at s = 1, then the first case corresponds to S (1) = 0, the second to S (1) > 0, and the last to S (1) < 0. These four cases are now discussed in detail. In the first case, no information about the existence of a long-time limit can be extracted. In the second case, r := −S(s) > 0 for some s ∈ (0, 1). Observe that f∗ = δ0 , corresponding to a mass concentrated in v = 0, trivially constitutes a

102


stationary solution of the Boltzmann equation. Recall that the initial condition f0 is a probability density of finite first moment. As s < 1, it follows K := ds (f0 , f∗ ) < ∞. By the contraction estimate, ds (f (t), f∗ ) ≤ K exp(−rt) → 0 as t → ∞. Thus, the solution f (t) converges weakly to f∗ . It is worthwhile to observe that, by Theorem 4.1, all moments Ms (t) with s > 1 diverge as t → ∞. So, although each f (t) for t > 0 has the same (positive and finite) first moment as f0 , one cannot invoke Prokhorov’s theorem to conclude that also the weak limit f∗ has positive first moment. The third case is the most interesting one. Choose some s ∈ (1, 2) with s < s¯; then r := −S(s) > 0. Assume that the initial datum of f possesses a moment of order S > s. In view of the completeness of (MM,s , Ds ) when s ∈ (1, 2) [3], the existence of the long-time limit can be concluded directly from the contractivity of the kinetic equation in Ds -norm (cf. Theorem 4.1). The same result can be achieved by means of the metric ds . In fact, f (t; v) has the Cauchy property in ds ; notice that ds [f (t), f0 ] is always finite because s < 2 and the first moment (mean wealth) is conserved under evolution. Moreover, as we required s < s¯, there exists a s with s < s < min(¯ s, S); by Theorem 4.3, the moment of order s remains uniformly bounded. It follows that f (t) converges in ds to a limit distribution f∞ (v), which is normalized and has the same first moment as f (t). This convergence implies that f∞ is a steady state for the kinetic equation (6). Indeed, denote by f∞ (t) the solution to (6) with initial datum f∞ , then Theorem 4.1 gives: ds [f∞ (t), f∞ ] ≤ ds [f∞ (t), f (t + T )] + ds [f (t + T ), f∞ ] ≤ e−rt ds [f∞ , f (T )] + ds [f (t + T ), f∞ ]. The last expression can be made arbitrarily small by choosing T large enough, so that f∞ (t) = f∞ for all t ≥ 0. In fact, f∞ is the only steady state with the

respective value of the first moment; for if f∞ is another steady state with the

same first moment, then ds [f∞ , f∞ ] is finite, and so, invoking Theorem 4.1 again,

ds [f∞ , f∞ ] ≤ e−r ds [f∞ , f∞ ],

. Finally, consider a solution f which has arbitrary which forces f∞ = f∞ moments bounded initially. Theorem 4.3 gives a time-uniform bound for moments of order less than s¯. As convergence f (t) → f∞ in ds implies weak* convergence of the associated measures, it follows that f∞ has finite moments of all orders less than s¯. On the other hand, no moment of order larger than s¯ can be finite. To see this, simply apply Theorem 4.3 to the steady state solution f∞ (t) ≡ f∞ to derive a contradiction. We summarize the results of this section.

Theorem 4.4. [37] Let f (t; v) be the (unique) weak solution of the Boltzmann equation (6), which has initially finite moments up to order S > 0. Further


103

assume that S(s) < 0 for some s ∈ (0, S). Then f converges exponentially fast in Ds (respectively, in ds ) to a steady state f∞ , Ds [f (t), f∞ ] ≤ Ds [f0 , f∞ ] exp {−|S(s)|t} .

(60)

If s < 1, then f∞ is a Dirac distribution centered at v = 0, and there are no other steady states. If s > 1, then f∞ has mean wealth equal to M , and it is the only steady state with this mean wealth. Moreover, if S(s ) < 0 exactly for 1 < s < s¯, possibly with s¯ = +∞, then f∞ has finite moments of all orders less than s¯, while moments of order larger than s¯ are infinite. 3.5 Numerical results To verify the analytical results for the relaxation behavior numerically, we resort to Monte Carlo simulations. Numerical experiments by means of Monte Carlo methods have been recently done in [28, 29] for both the CPT and the CCM models. In these rather basic simulations, known as direct simulation Monte Carlo (DSMC) method or Bird’s scheme, pairs of agents are randomly and nonexclusively selected for binary collisions, and exchange wealth according to the respective trade rules. One time step corresponds to N/2 such interactions, with N denoting the number of agents. In all experiments, every agent possesses unit wealth initially. The state of the kinetic system at time t > 0 is characterized by the N wealth values w1 (t), . . . , wN (t). The densities for the current wealth f (N ) (t; w) (N ) and the steady state f∞ are each a collection of scaled Dirac Deltas at positions wi . The associated distribution functions are build of a sequence of rectangles, F (N ) (t; w) = #{agents with wealth wi (t) > w}/N, (N )

and, respectively, for F∞ (w). The goal is to monitor the convergence of the wealth distribution f (N ) (t; w) (N ) to the approximate steady state f∞ (w) over time in terms of the Wassersteinone-distance. This amounts to computing the area between the two distribu(N ) tion functions F (N ) (t; w) and F∞ (w). The relaxation behavior of the CPT model (42) is investigated when the random variables η1 , η2 attain values ±μ with probability 1/2 each. According to the analytical results, the shape of the steady state can be determined from Fig. 2. Results are reported for zones II and III. Recall that zone I is forbidden by the constraint |μ| < λ, whereas parameters in zone IV lead to wealth condensation (without convergence in Wasserstein metrics). For zones II and III simulations are performed for systems consisting of N = 500, 5,000, and 50,000 agents, respectively. The relaxation in the CPT model occurs exponentially fast. Though the system has virtually reached equilibrium after less than 102 time steps, simulations are performed for 104 time steps. In order to obtain a smooth result,

104


the wealth distribution is averaged over another 103 time steps. The resulting (N ) reference state P∞ is used in place of the (unknown) steady wealth curve. For zones II and III a risk index of μ = 0.1, and a saving propensity of λ ≡ 0.7 for zone II, and λ ≡ 0.95 for zone III, respectively, are chosen. The nontrivial root of S(s) in (29) is s¯ ≈ 12.91 in the latter case. For each choice of N and each pair (μ, λ), averages over 100 simulations have been made. Figure 3 shows the decay of the Wasserstein-one-distance of the wealth distribution to the approximate steady state over time. In both zones, we observe exponential decay. The reason for the residual Wasserstein distance of order 10−2 lies in the statistical nature of this model, which never reaches equilibrium in finite-size systems, due to persistent thermal fluctuations. Note that before these fluctuations become dominant, relaxation is extremely rapid. The exponential rate is independent of the number of agents N . 0

10

N=500 N=5000 N=50000 −1

10

−2

10

10−3 0

2

4

6

8

10

12

Wasserstein distance


100

N=500 N=5000 N=50000 −1

10

−2

10

10−3 0

Time (steps)

10

20

30

Time (steps)

Fig. 3. CPT model: Decay of the Wasserstein distance to the steady state in zones II (left) and III (right)

4 Results in the two-dimensional case 4.1 Pareto tail of the wealth distribution Due to its two-dimensionality, the CCM model behaves very different from the strictly conservative model (40). In particular, h∞ (w) may possess a Pareto tail. In analogy to S(s) from (29), define the function 1 χ(λ) Q(r) := dλ, (61) (1 − λ)r 0 which determines the properties of the steady wealth distribution h∞ (w) as follows [38]: (PT’) if Q(1) < +∞, and α ∈ [1, +∞) is the infimum of r for which Q(r) = +∞, then h∞ (w) has a Pareto tail of index α; (ST’) if Q(r) < +∞ for all r ≥ 1, then h∞ (w) has a slim tail; (DD’) if Q(1) = +∞, then h∞ (w) = δ0 (w).


105

To derive these results, it is useful to think of the global wealth distribution h∞ (w) as superposition of λ-specific steady wealth distributions f∞ (λ, w)/ χ(λ), i.e., the wealth distributions of all agents with a certain personal saving propensity λ. The individual λ-specific distributions are conjectured [16, 45] to resemble the wealth distributions associated to the one-dimensional model (40), but their features are so far unknown. However, they are conveniently analyzed in terms of the λ-specific moments 1 ∗ ˆ Ms (λ) = ws f∞ (λ, w) dw. (62) χ(λ) R+ Integration of the stationary Boltzmann equation f∞ (λ, w) = Q+ (f∞ , f∞ )

(63)

against ϕ(w) = ws for a non-negative integer s gives ˆ ∗ (λ) = 1 M s χ(λ)

R2+

1

0

s [λ+(1−λ)]v+(1−μ)w f∞ (λ, v)f∞ (μ, w) dμ dv dw

After simplifications, ˆ ∗ (λ) (1 − λ)φs (λ)M s

1 s−1 s ∗ ˆ s∗ (λ) ˆ s−k s−k [λ + (1 − λ)]k M (1 − μ)s−k M (μ)χ(μ) dμ, = k 0 k=0

(64) where φs (λ) is a polynomial with no roots in [0, 1]. The λ-specific steady wealth distributions have slim tails, and moments of arbitrary order can be calculated recursively from (64). From ˆ 0∗ (λ) ≡ 1, M

ˆ 1∗ (λ) = M (1 − λ)−1 , M Q(1)

(65)

it follows inductively that ˆ s∗ (λ) = rs (λ)(1 − λ)−s , M

(66)

and rs (λ) is a continuous, strictly positive function for 0 ≤ λ ≤ 1. By Jensen’s inequality, formula (66) extends from integers s to all real numbers s ≥ 1. In conclusion, the total momentum 1 1 χ(λ) ∗ ∗ ˆ Ms = dλ (67) Ms (λ) dλ ∝ (1 − λ)s 0 0 is finite exactly if Q(s) is finite.

106


Q(1) = +∞ would imply infinite average wealth per agent in the steady wealth distribution by formula (67). This clearly contradicts the conservation of the mean wealth at finite times. In reality, the first moment vanishes, and h∞ is a Dirac distribution; see Sect. 4.3. We emphasize this fact since a noticeable number of theoretical and numerical studies has been devoted to the calculation of h∞ for uniformly distributed λ, i.e., χ(λ) ≡ 1, where clearly Q(1) = +∞. In the corresponding experiments [14, 16, 17, 41, 45] with finite ensembles of N agents, an almost perfect Pareto tail h∞ (w) = CN w−2 of index α = 1 has been observed over a wide range wN < w < WN . However, the “true” tail of h∞ (w) – for w WN – is slim. As the systems size N increases, also WN ∝ N increases and CN ∝ 1/ ln N → 0. In fact, one proves [38] weak convergence of h∞ (w) to δ0 (w) in the thermodynamic limit N → ∞. 4.2 Rates of relaxation: Pareto tail The discussion of relaxation is more involved than in one dimension, and we restrict our attention to the deterministic CCM model, ≡ 1/2, in the case (PT’) of Pareto tails of index α > 1. In fact, it is believed [17] that the randomness introduced by has little effect on the large-time behavior of the kinetic system. The stationary state of the deterministic CCM model is characterized by the complete stop of wealth exchange. This is very different from the steady states for the one-dimensional models, where the macroscopic wealth distribution is stationary despite the fact that wealth is exchanged on the microscopic level. Stationarity in (12) and (13) is achieved precisely if v(1 − λ) = w(1 − μ) for arbitrary agents with wealth v, w and saving propensities λ, μ, respectively. Correspondingly, the particle density concentrates in the plane on the curve K∞ = { (λ, w) | (1 − λ)w = M/Q(1)}, (68) and the steady wealth distribution is explicitly given by Mohanty’s formula [40], M M h∞ (w) = 2 χ 1 − , (69) w w with the convention that χ(λ) = 0 for λ < 0. The conjectured [15, 41] time scale for relaxation of solutions is t−(α−1) , lim

t→∞

ln W[h(t; w), h∞ (w)] = α − 1. − ln t

(70)

It has been proven [28] for all α > 1 that the limit in (70) is at most α − 1, i.e., relaxation cannot occur on a faster time scale. The complete statement (70), however, was made rigorous only for 1 < α < 2 so far [38].


107

The key tool for the analysis is the equation for the λ-specific mean wealth, 1 1−μ ˆ d ˆ 1−λ ˆ (71) M1 (t; λ) = − M1 (t; λ) + M1 (t; μ)χ(μ) dμ. dt 2 2 0 Intuitively, the slow algebraic relaxation is explained by the temporal behavior ˆ 1 (t; λ) grows at of the richest agents. By (71), the λ-specific average wealth M most linearly in time, ˆ 1 (t; λ) ≤ t + M ˆ 1 (0; λ). M (72) Thus, the tail of the wealth curve h(t; w) becomes slim for w t. The cost of transportation in (20) to “fill up” the fat tail h∞ (w) ∝ w−(α+1) is approximately given by: ∞ ∞ w h∞ (w) dw ∝ w−α dw ∝ t−(α−1) . (73) t

t

That equilibration works no slower than this (at least for 1 < α < 2) follows from a detailed analysis of the relaxation process. In [38], it has been proven that 1 M ˆ (74) χ(λ) dλ ∝ t−(α−1) M1 (t; λ) − λQ(1) 0 by relating (71) to the radiative transfer equation [31]. Moreover, the λ-specific variance ˆ 1 (t; λ)2 ˆ 2 (t; λ) − M (75) Vˆ (t; λ) = M was shown to satisfy 0

1

(1 − λ)2 Vˆ (t; λ)χ(λ) dλ ∝ t−α

(76)

provided 1 < α < 2. Combination of (74) and (76) leads to (70). Moreover, relaxation may be decomposed into two processes. The first ˆ 1 (t; λ); i.e., all is concentration of agents at the λ-specific mean wealth M agents with the same saving propensity become approximately equally rich. According to (76), this process happens on a time scale t−α/2 . Second, the localized mean values tend toward their respective terminal values M/λQ(1). Thus, agents of the same saving propensity simultaneously “adjust” their wealth. By (74), the respective time scale is t−(α−1) , which is indeed slower than the first provided α < 2. 4.3 Rates of relaxation: Dirac delta Finally, the deterministic CCM model is considered with a density χ(λ) where limλ0 χ(λ) > 0, e.g., χ ≡ 1 on [0, 1]. Clearly, Q(1) = +∞. An analysis of (71) provides [38] for λ < 1 the estimate c ˆ 1 (t; λ) ≤ C ≤ ln t · M 1−λ 1−λ

(t > Tλ ),

(77)

108


with 0 < c < C < +∞, and Tλ → +∞ as λ → 1. Convergence of h(t; w) to ˆ 1 (t; λ) tends to zero for a Delta in w = 0 is a direct consequence, because M each 0 ≤ λ < 1 as t → ∞. Estimate (77) has a direct interpretation. Agents of very high saving propensity λ ≈ 1 drain all wealth out of the remaining society as follows. At intermediate times t 1, agents equilibrate in microscopic trades so that the product (1 − λ)w becomes approximately a global constant m(t). Agents with low saving propensity λ < 1 − m(t)/t indeed satisfy w ≈ m(t)/(1 − λ). Agents with higher saving propensity, however, are in general far from this (apparent) equilibrium; their target wealth m(t)/(1 − λ) is very large, whereas their actual wealth is bounded by t on the average. Correspondingly, a “Pareto region” of the shape h(t; w) ≈ χ(1)m(t)w−2 forms over a range 1 w ≤ t, whereas the tail of h(t; w) for w t is slim. The average wealth per agent contained in the Pareto region amounts to t wh(t; w) dw ≈ χ(1)m(t) ln t. (78) 1

By conservation of the average wealth, the global constant m(t) tends to zero logarithmically in t and gives rise to (77). 4.4 Numerical simulations The CCM model is expected to relax at an algebraic rate (70). As simulations indeed take much longer to reach equilibrium than in the case of CPT, the numerical experiments are carried out for about 105 time steps, and then the wealth distribution is averaged over another 104 time steps. Again, this reference state is used in place of the (unknown) steady wealth curve. The saving propensities for the agents are assigned at the beginning of each run and are kept fixed during this simulation. Agents are assigned the propensities 1/2.5 λj = 1−ωj , where the ωj ∈ (0, 1) are realizations of a uniformly distributed random variable. Simulations are performed for the deterministic situation ≡ 1/2 as well as for uniformly distributed ∈ (0.4, 0.6). In both situations, computations are carried out for systems consisting of N = 500, 5, 000, and 50, 000 agents, respectively. The steady state reached in one simulation is typically nonsmooth, and smoothness is only achieved by averaging over different simulations. However, in contrast to the CPT model, the steady states for CCM do depend on the initial conditions, namely through the particular realization of the distribution of saving propensities λ1 , . . . , λN among the agents. Consequently, there are two possibilities to calculate the relaxation rates. One can monitor either the convergence of the wealth distributions in one run to the steady distribution corresponding to that specific realization of the saving propensities, or the convergence of the transient distributions, obtained from averaging over several simulations, to the single smooth steady state that results from averaging the simulation-specific steady states.

A mathematical theory for wealth distribution 0

N=500 N=5000 N=50000 −5

10

10

−10

10−15

0.5

1

1.5

2

Time (steps)

2.5

10−1.2



10

109

N=500 N=5000 N=50000

−1.3

10

10−1.4 10−1.5 10−1.6

3 x 104

200

400

600

800

1000 1200 1400

Time (steps)

Fig. 4. CCM model: Decay of the averaged Wasserstein distance to the steady states for ε ≡ 1/2 and for ε ∈ (0.4, 0.6) uniformly distributed 100

N=500 N=5000 N=50000 10−1

10−2

10−3 0.0

0.5

1.0

Time (steps)

1.5

x 104



100

N=500 N=5000 N=50000

10−1

10−2 0.0

0.5

1.0

Time (steps)

1.5

x 104

Fig. 5. CCM model: Decay of the Wasserstein distance to the averaged steady state for ε ≡ 1/2 (left) and for ε ∈ (0.4, 0.6) uniformly distributed (right)

Figure 4 shows the evolution of the Wasserstein-one-distance of the wealth distributions to the individual steady states, both in the purely deterministic setting ≡ 1/2 (left), and for uniformly distributed ∈ (0.4, 0.6). (The curves in the figures represent averages of the Wasserstein distances calculated in the individual simulations.) In comparison, the distance of the simulationaveraged wealth distributions to the single (averaged) steady state is display in Fig. 5. The average has been taken over 100 simulation results. Again, results are shown for ≡ 1/2 (left), and for uniformly distributed ∈ (0.4, 0.6), respectively. Some words are in order to explain the results. The almost perfect exponential (instead of algebraic) decay displayed in Fig. 4 obviously originates from the finite size of the system. The exponential rates decrease as the system size N increases. In the theoretical limit N → ∞, one expects subexponential relaxation as predicted by the theory. We stress that, in contrast, the exponential decay rate for the CPT model in Fig. 3 is independent of the system size.

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5 Conclusions We have reviewed a unified mathematical approach to study the qualitative behavior of conservative kinetic models for the distribution of wealth in simple market economies. We have focussed on two types of money conservation: pointwise, and in the mean only. The risky market approach (CPT) by Cordier et al. [22] belongs to this last class, while the model with quenched saving propensities (CCM) by Chatterjee et al. [16] is obtained from pointwise conservative trades. Both models constitute refinements of the idea developed by Angle [2]. For CPT, randomness – related to the unknown outcome of risky investments – plays the pivotal rôle. In contrast to Angle’s original model, the market risk is defined in a way that breaks the strict conservation of wealth in microscopic trades and replaces it by conservation in the statistical mean. The founding idea of CCM is to incorporate individual trading preferences by assigning personal saving propensities to the agents. Our main result is that for suitable choices of the respective model parameters, both approaches produce realistic Pareto tails in the wealth distribution. While solutions of the CPT model are attracted exponentially fast by the unique steady state, the relaxation in CCM is algebraic. Moreover, for unsuitable parameter choices, the wealth distribution converges weakly toward a singular limit, and the total wealth is lost in that limit despite the fact that it is conserved at finite times. Another important conclusion is that one must be careful when interpreting numerical simulation results. The simulated ensembles in kinetic Monte Carlo experiments are necessarily of finite size, and the qualitative features of finite-size systems differ in essential points from those proven for the continuous limit. Most remarkably, the finite-size CCM model exhibits nontrivial steady states with (apparent) Pareto tail in situations where the continuous model produces a Dirac distribution. Also, the typical time scale for relaxation in the deterministic CCM model changes from exponential convergence (finite size) to algebraic convergence (continuous). It is arguable which kind of approach (finite size or continuous) provides the better approximation to reality. However, it is important to notice that the predictions are qualitatively different.

Acknowledgements Bertram D¨ uring is supported by the Deutsche Forschungsgemeinschaft, grant JU 359/6 (Forschergruppe 518). Daniel Matthes is supported by the Deutsche Forschungsgemeinschaft, grant JU 359/7.


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Tolstoy’s dream and the quest for statistical equilibrium in economics and the social sciences Ubaldo Garibaldi1 and Enrico Scalas2 1 2

IMEM-CNR and Universit` a di Genova, Dipartimento di Fisica, via Dodecaneso 33, 16146 Genova, Italy, [email protected] Universit` a del Piemonte Orientale, Dipartimento di Scienze e Tecnologie Avanzate, viale T. Michel 11, 15121 Alessandria, Italy, [email protected]

Summary. The meaning of the notion of statistical equilibrium in economics is discussed as well as its relevance for economic theory. A simple agent-based model of taxation and redistribution is presented. Its invariant equilibrium distribution is the generalized P´ olya sampling distribution. It turns out that the expected wealth distribution is the dichotomous P´ olya whose continuous limit is the Beta distribution and whose appropriate thermodynamic limit is the Gamma distribution, often found in describing empirical data. The shape parameter of the Gamma distribution is the inverse of the wealth preferential attachment α−1 .

1 Tolstoy’s dream The agent-based approach to Economics and the Social Sciences is becoming more and more popular among scholars interested in going beyond mainstream analyses [14]. This approach is trying to reconcile methodological individualism [13] with the existence of emergent phenomena in social systems [2]. Some of these concepts may appear brand new, but, at least, they can be traced back to the philosophical and scientific discussions taking place in the XIXth Century. The basic idea is that there is an analogy between human societies where many individuals interact and gases where many atoms or molecules interact. Indeed, as discussed by Hacking in The Taming of Chance [8], Boltzmann himself used this analogy in order to justify the atomic hypothesis. This idea was pervasive in XIXth Century thinkers. We like to think of Tolstoy’s novel War and Peace [15] as an early agent-based simulation. The author explores the behaviour and interactions of his 580 characters during the Napoleonic invasion of Russia. More specifically, the second epilogue of the


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novel reveals Tolstoy’s theoretical interests and his model of human history. Let Tolstoy directly speak: Speaking of the interaction of heat and electricity and of atoms, we cannot say why this occurs, and we say that it is so because it is inconceivable otherwise, because it must be so and that it is a law. The same applies to historical events. Why war and revolution occur we do not know. We only know that to produce the one or the other action, people combine in a certain formation in which they all take part, and we say that this is so because it is unthinkable otherwise, or in other words that it is a law. Therefore, in the XIXth Century, the analogy on which current agent-based simulations are grounded was so popular that it found its way through literature. Unfortunately for Economics, the mechanical analogy was used in its static version and the concept of statistical equilibrium remained unknown to most economists troughout all the XXth Century and up to now.

2 Statistical equilibrium in economics 2.1 What is the common notion of equilibrium in economics? The concept of equilibrium referred to in General Equilibrium Theory is taken from Physics. It coincides with mechanical equilibrium. When looking for mechanical equilibrium one minimizes a potential function subject to boundary conditions, in order to find equilibrium positions; when looking for standard (micro)economic equilibrium, one maximizes a utility function subject to budget constraints (this is the consumer side, in other words, demand) and maximizes the profit subject to cost constraints (this is producer side, in other words, supply); then one equates supply and demand, and finds equilibrium quantities and prices. In both cases, the mathematical tool is optimization with constraints using the method of Lagrange multipliers. Walras and Pareto explicitly inspired their pioneering work on General Equilibrium Theory to Physics and mechanical equilibrium. This was made clear by Ingrao and Israel [9].

2.2 What is statistical equilibrium? Statistical equilibrium is another notion of equilibrium in Physics. It was defined by Maxwell and Boltzmann in their early work on the theory of gases, trying to reconcile mechanics and thermodynamics. In order to better understand this notion, it is useful to make use of a Markovianist approach to

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statistical equilibrium as discussed by Oliver Penrose (the brother of Roger Penrose) in his 1970 book [10]. By the way, a similar approach was promoted by Richard von Mises (the brother of Ludwig von Mises) in a book reprinted in 1945 (actually the book was written by R. von Mises before World War II) [16]. A finite Markov chain is a stochastic process defined as a sequence of random variables X1 , . . . , Xn on the same probability space that assume values in a finite set S, known as the state space. For a Markov chain, the predictive probability P(Xn = xn |Xn−1 = xn−1 , . . . , X1 = x1 ) has the following simple form: P(Xn = xn |Xn−1 = xn−1 , . . . , X1 = x1 ) = P(Xn = xn |Xn−1 = xn−1 ).

(1)

In other words, the predictive probability does not depend on all the past states, but on the last state occupied by the chain. As a consequence of the multiplication theorem, one gets that the finite-dimensional distribution P(X1 = x1 , . . . , Xn = xn ) is given by: P(X0 = x0 , . . . , Xn = xn ) = P(Xn = xn |Xn−1 = xn−1 ) · · · P(X1 = x1 |X0 = x0 )P(X0 = x0 ).

(2)

As a consequence of Kolmogorov’s representation theorem, this means that a Markov chain is fully characterized by the knowledge of the functions P(Xm = xm |Xm−1 = xm−1 ), also known as transition probabilities and P(X0 = x0 ), also known as initial probability distributions. If the transition probabilities do not depend on the index m but only on the initial and on the final state, then the Markov chain is called homogeneous. In the following, only homogeneous Markov chains will be considered. For the sake of simplicity, it is useful to introduce the notation P (x, y) = P(Xm = y|Xm−1 = x)

(3)

for the transition probability and p(x) = P(X0 = x)

(4)

for the initial probability distribution. Note that P (x, y) is nothing else than a matrix in the finite case under scrutiny, with the property that P (x, y) = 1; (5) y∈S

in other words the rows of the matrix sum up to 1 as a consequence of the addition axiom. Such matrices are called stochastic matrices (to be distinguished from random matrices which are matrices with random entries). Note

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that the initial distribution can be written as a row vector, so that one can obtain the marginal distribution of the random variable Xn as: p(x)P (n) (x, y), (6) P(Xn = y) = x∈S

where P (n) (x, y) represents the (x, y) entry of the n-step transition matrix. Now, assume there is a distribution π(x) satisfying the equation: π(y) = π(x)P (x, y), (7) x∈S

then π(x) is called a stationary distribution or invariant measure. If at time step t the chain is described by P(Xt = x) = π(x), then from (7), it follows that P(Xt+1 = x) = π(x) = P(Xt = x); in other words, the distribution does not change as time goes by. Note that the states are jumping from one to another one, but the probability of finding the system in a specific state does not change. This is exactly the idea of statistical equilibrium put forward by Ludwig Boltzmann. However, more can and should be said. First of all, the stationary distribution may not exist; secondly, the chain usually starts from a specific state, so that the initial distribution is a vector full of 0’s and with a single 1 in the initial state. The latter state of affairs can be represented by a Kronecker delta π(x) = δ(x, x0 ), where x0 is the specific initial state. This is not a stationary distribution and the convergence of the chain to the stationary distribution is not granted at all. Fortunately, it turns out that under some rather mild conditions: • The stationary distribution exists and it is unique; • The chain always converges to the stationary distribution irrespective of its initial distribution. It is indeed sufficient to consider a finite chain that is irreducible and aperiodic. A chain is irreducible if all the states are persistent; this is equivalent to claim that any state can be reached from any other state with finite probability in a finite number of steps. The chain is aperiodic if for any x, one has that P (s) (x, x) > 0 for s > s0 (x); in other words, after a possible transitory period, the probability of return is positive. All these conditions essentially mean that the s-step matrix P(s) no longer has zero entries after a sufficient number of steps. If the finite Markov chain is irreducible and aperiodic, then it has a unique invariant distribution π(x) and lim P (n) (x, y) = π(y)

n→∞

(8)

irrespective of the initial state x. This means that, after a transient period, the distribution of chain states reaches a stationary distribution, which can then be interpreted as an equilibrium distribution in the statistical sense.

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2.3 Why and where statistical equilibrium may be useful in economics? There are several possible domains of application of the concept of statistical equilibrium in Economics. Incidentally, note that many agent-based models used in Economic theory are intrinsically Markov chains (or Markovian processes). Therefore, the ideas discussed earlier naturally apply. Up to now, we have used these concepts: • To discuss some toy models for the distribution of wealth (not of income!) as in Scalas et al. (2006) [11] and in Garibaldi et al. (2007) [6]. • To generalize a sectoral productivity model originally due to Aoki and Yoshikawa [1], in Scalas and Garibaldi (2009) [12]. In [6, 11, 12], we promote the use of a finitary approach to combinatorial stochastic processes. This approach is the subject of a forthcoming book [7] and will be illustrated by an example in the next section.

3 An example: the taxation-redistribution game 3.1 Basic descriptions Consider a system of n coins to be divided into g agents. There are three levels of description for the system. • (individual descriptions) Let the integers from 1 to n denote the coins and the integers from 1 to g denote the agents. Let us introduce the variables V1 , . . . , Vn whose values are given by the integers between 1 and g; by Vi = j, we mean the the ith coin belongs to the jth agent. • (frequency or occupation descriptions) If the names (or labels) of the coins are irrelevant, it is possible to use the variables Y1 , . . . , Yg where Yi = ni is the number of coins in the pocket of the ith agent. In symbols, one can write Yi = #{Vj = i, j = 1, . . . , n}. If the vector Y n = (n1 , . . . , ng ) = g denotes a particular frequency description, one has i=1 ni = n. • (frequency of frequencies or partitions) For k = 1, . . . , n, the variables defined by Zk = #{Yi = k, i = 1, . . . , g} give the number of agents with k coins. If the vector Z = z = (z0 , . . . , zn ) denotes a particular partition, it n n must satisfy the two constraints k=0 zk = g and k=1 kzk = n. Example (n = 3 objects (coins) into g = 2 categories) • There are eight individual descriptions: (1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (2, 2, 1), (2, 1, 2), (1, 2, 2), (2, 2, 2). • There are four occupation vectors: (3, 0) corresponding to (1, 1, 1); (2, 1) corresponding to (1, 1, 2), (1, 2, 1) and (2, 1, 1); (1, 2) corresponding to (2, 2, 1), (2, 1, 2) and (1, 2, 2); (0, 3) corresponding to (2, 2, 2).

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• There are two partition vectors: (1, 0, 0, 1) corresponding to (3, 0) and (0, 3); (0, 1, 1, 0) corresponding to (1, 2) and (2, 1). The three basic descriptions define possible constituents of the sample space for the individual descriptions. Note that: • For each occupation vector n = (n1 , . . . , ng ) there are g

n!

i=1

ni !

(9)

corresponding individual descriptions; • For each partition vector z = (z0 , . . . , zn ) there are g! n

i=0 zi !

(10)

corresponding occupation vectors; • The total number of individual descriptions is g n ; • The total number of occupation vectors is (g + n − 1)!/[n!(g − 1)!]; • For the total number of partition vectors, a closed formula is not available. 3.2 Taxation (destruction) and redistribution (creation) In this section, a stylized probabilistic model for taxation and redistribution will be introduced, based on [6]. A taxation is a step in which a coin is randomly taken out of n coins and a redistribution is a step in which the coin is given back to one of the g agents. A taxation move is equivalent to a destruction/annihilation and a redistribution move to a creation [3–5]. This model is conservative as the numbers of agents g and of coins n do not change in time. Moreover, it only includes so-called unary moves. If the initial state is given by n = (n1 , . . . , ni , . . . , nj , . . . , ng ), the final state is nji = (n1 , . . . , ni − 1, . . . , nj + 1, . . . , ng ), after taxation and redistribution. Note that indebtedness is not possible. If a coin is randomly selected out of n coins, the probability of selecting a coin belonging to agent i is ni /n. Therefore, in this model, agents are taxed proportionally to their wealth measured in terms of the number of coins in their pockets. The redistribution step is crucial as it can favour agents with many coins (a rich get richer mechanism, also known as preferential attachment) or agents with few coins (a taxation scheme leading to equality). This can be done by assuming that the probability of giving the coin taken from agent i to agent j is proportional to wj + nj , where nj is the number of coins in the pocket of agent j and wj is a suitable weight. Depending on the choice of wj , one can obtain different equilibrium situations. Based on the previous considerations, it is assumed that the transition probability is: P(nji |n) =

ni wj + nj − δi,j , n w+n−1

(11)

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g

where w = i=1 wi and the Kronecker symbol δi,j takes into account the case i = j. If the condition wj = 0 is satisfied, then also agents without coins can receive them. If all the agents are equivalent, one has wj = a, uniformly and w = ga = θ, so that (11) becomes P(nji |n) =

ni a + nj − δi,j . n θ+n−1

(12)

3.3 Statistical equilibrium From (7), one can see that the invariant distribution is the left eigenvector corresponding to eigenvalue 1 for the matrix of transition probabilities. However, the direct diagonalization of (11) is cumbersome. In this case, it is easier to use detailed balance. If a probability p(n) can be found satisfying detailed balance, then this is an invariant distribution! In our case, if i = j, the direct flux is given by: n i a + nj p(n)P(nji |n) = p(n) (13) n θ+n−1 whereas the inverse flux is given by: p(nji )P(n|nji ) = p(nji )

nj + 1 a + ni − 1 . n θ+n−1

(14)

Equating the two fluxes, we get p(n) p(nji )

=

n j + 1 a + ni − 1 . ni a + nj

(15)

The g-variate P´ olya distribution discussed in the Appendix satisfies (15), so that, eventually, we get the invariant distribution for the taxation-redistribution model (it is the case α1 = α2 = · · · = αg = a) p(n) =

g n! a[ni ] . θ[n] i=1 ni !

(16)

Moreover, a little thought should convince the reader that the Markov chain defined by (12) is irreducible and aperiodic. Therefore, the invariant distribution (16) is unique and it is also the equilibrium distribution. Three important particular cases of (16) are: • For a = 1 p(n) =

−1 n+g−1 ; n

this is the uniform distribution on all occupation vectors n;

(17)

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• For |a| → ∞

n!

1 ; n g n ! i i=1

p(n) = g

(18)

this coincides with the multinomial distribution and corresponds to the uniform distribution on the individual descriptions; • For a = −1 −1 g p(n) = ; (19) n this is again the uniform distribution on the restricted support of all occupation vectors n with ni = 0, 1. The case a = 1 coincides with the so-called Bose-Einstein distribution, the case |a| → ∞ with the so-called Maxwell-Boltzmann distribution, and the case a = −1 leads to the so-called Fermi-Dirac distribution. As discussed in the Appendix, these three remarkable cases correspond to three urn models. The Bose-Einstein distribution is related to the P´ olya urn, the MaxwellBoltzmann distribution to the Bernoullian urn and the Fermi-Dirac distribution to the hypergeometric urn. However, in this model, the parameter a needs not be confined to the three values discussed earlier and it can assume any real positive value and any negative integer value. Moreover, in our stylized model, the redistribution policy is characterized by the value of the parameter a. If a is small and positive, one has that rich agents become richer, but for a → ∞ the redistribution policy becomes random: any agent has the same probability of receiving the coin. Eventually, the case a < 0 favours poor agents, but |a| is the maximum allowed wealth for each agent. 3.4 Wealth (coin) distribution As discussed in the Appendix, agents’ exchangeability lead to a simple relationship between the joint probability distribution of partitions and the probability of a given occupation vector. One has that g! P(Z = z) = n

i=0 zi !

g! P(Y = n) = n

i=0 zi !

g a[ni ] [n] j=1 ni ! j=1 θ

g

n!

g a[ni ] g!n! , zi [n] i=0 zi !(i!) j=1 θ

= n

(20)

where, as discussed in Sect. 3.1, zi is the number of agents with i coins. Now, both (16) and (20) are multivariate distributions. In order to get a univariate distribution, to be compared with empirical data, we consider the marginal distribution that describes a single agent. Given that all the agents are characterized by the same weight a, we can focus on the behaviour of the random variable Y = Y1 representing the number of coins of agent 1. Starting from Yt = k, one can define the following transition probabilities

Tolstoy’s dream

w(k, k + 1) = P(Yt+1 = k + 1|Yt = k) =

n−k a+k , n θ+n−1

123

(21)

meaning that a coin is randomly selected among the other n−k coins belonging to the other g − 1 agents and given to agent 1 according to the weight a and to the number of coins k, w(k, k − 1) = P(Yt+1 = k − 1|Yt = k) =

k θ−a+n−k , n θ+n−1

(22)

meaning that a coin is randomly removed from agent 1 and redistributed to one of the other agents according to the weight θ − a and the number of coins n − k, and w(k, k) = P(Yt+1 = k|Yt = k) = 1 − w(k, k + 1) − w(k, k − 1),

(23)

meaning that agent 1 is not affected by the move taking place at step t + 1. These equations define a birth–death Markov chain corresponding to a random walk with semi-reflecting barriers. This chain represents the wealth dynamics of a single agent interacting with a thermal bath consisting of the other g − 1 agents. Indeed, the invariant (and equilibrium) distribution of the birth-death chain can be directly obtained marginalizing (16). This leads to the dichotomous P´ olya distribution (see the Appendix): P(Y = k) = pk =

a[k] (θ − a)[n−k] n! . k!(n − k)! θ[n]

(24)

Equation (24) can be compared with the behaviour of the agent as time goes by. As a consequence of the ergodic theorem for irreducible chains, it follows that #{Ys = k, s = 0, . . . , t} lim = pk , (25) t→∞ t where pk is given by (24). In other words, the marginal equilibrium probability is also the long-run limit of the hitting time relative frequency. These consideration are important, in order to identify the probabilistic objects to be compared to empirical (or to simulated) data. The same procedure can be used for the wealth distribution z. The random (k) variable Zk counts the number of agents with k coins. Denoting by IYj = IYj =k the indicator function of the event {Yj = k}, the random variable Zk can also be written as follows (k)

(k)

(k)

Zk = IY1 + IY2 + . . . + IYg ;

(26)

Therefore, we find that E(Zk ) =

g j=1

(k)

E(IYj ) =

g j=1

P(Yj = k),

(27)

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where P(Yj = k) is the marginal distributions for the jth agent. As a consequence of the equivalence of all agents, from (24) and (27), one gets that E(Zk ) = gP(Y = k) = g

a[k] (θ − a)[n−k] n! . k!(n − k)! θ[n]

(28)

Equation (28) gives the first moment of the probability function on all possible wealth distributions (20) for the taxation-redistribution model. The thermodynamic limit for (24) when n 1, g 1 and n/g = aχ leads to the negative binomial distribution as an approximation of the dichotomous Pólya distribution (see the Appendix) P

TL

a[k] (Y = k) = NegBin(k|a, χ) = k!

1 1+χ

a

χ 1+χ

k .

(29)

On the other side, for α > 0, the continuous limit for the wealth distribution is (see the Appendix) fB (x) =

Γ (θ) xa−1 (1 − x)θ−a−1 , Γ (a)Γ (θ − a)

(30)

where x = k/n is the continuous variable corresponding to the normalized wealth of the first agent (0 ≤ x ≤ 1) and fB (x) is its Beta probability density function. The thermodynamic limit of (30) leads to the Gamma(x|a, u) density pTL (x) =

x u−a a−1 x , exp − Γ (a) u

(31)

and w = n/g, whose obvious meaning is the expectation for the wealth of the selected agent, which stays constant when the continuous thermostat becomes infinite. (See the Appendix). 3.5 Block taxation and the convergence to equilibrium Consider the case in which taxation is made in the following way: instead of drawing a single coin from an agent at each step, m ≤ n coins are randomly taken from various agents and then redistributed with the mechanism described earlier, that is with a probability proportional to the actual number of coins and to an a priori weight. If n = (n1 , . . . , ng ) is the initial occupation g vector, m = (m1 , . . . , mg )(with i=1 mi = m) is the taxation vector, and g m = (m1 , . . . , mg ) (with i=1 mi = m) is the redistribution vector, we can also write n = n − m + m . (32) The block taxation-redistribution model still has (16) as its equilibrium distribution, as the block step is equivalent to m steps of the original taxationredistribution model. However, the convergence rate to equilibrium is faster.

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The marginal analysis for the block taxation-redistribution model in terms of a birth–death Markov chain is more cumbersome than for the original model because, now, the difference |ΔY | can vary from 0 to m. In any case, given that (24) always gives the equilibrium distribution, this means that (see the Appendix) a n E(Y ) = n = , (33) θ g and aθ−aθ+n ng−1θ+n Var(Y ) = n = . (34) θ θ θ+1 g g θ+1 We can write Yt+1 = Yt − Dt+1 + Ct+1 , (35) where Dt+1 is the random taxation for the given agent and Ct+1 is the random redistribution to the given agent. The expected value of Dt+1 under the condition Yt = k is k (36) E(Dt+1 |Yt = k) = m ; n this result is valid as m coins are taken at random out of the n coins and the probability of removing a coin from the first agent is k/n under the given condition. Moreover, if Yt = k and Dt+1 = d, we get that the probability of giving a coin back to agent 1 is (a+ k − d)/(θ + n− m), so that, after averaging over Dt+1 |Yt = k, we have k a+k−m n . (37) E(Ct+1 |Yt = k) = m θ+n−m The expected value of Yt+1 − Yt conditioned on Yt = k can be found from the expectation of (35) and using (36) and (37). This yields: E(Yt+1 − Yt |Yt = k) = −

a mθ k−n . n(θ + n − m) θ

(38)

The following remarks on (38) are possible: 1. If k is different from its equilibrium value nα/θ = n/g, then the expected motion E(Yt+1 |Yt = k) points towards n/g; 2. If k = na/θ then the chain is first-order stationary. If one begins with n/g, then one always gets E(Yt+1 − Yt |Yt = k) = 0; 3. r = mθ/(n(θ + n − m)) is the intensity of the restoring force. The inverse of r, gives the order of magnitude for the number of transitions needed to reach the equilibrium value. 4. If m = n, meaning that all the coins are taken and then redistributed, the new state has no memory of the previous one and statistical equilibrium is reached in a single step (r−1 = 1)! This result casts some light on the meaning of statistical equilibrium: not a state, rather a timeless and memoryless probability distribution on all accessible states.

126


Before concluding this section, it is interesting to discuss the case θ < 0 in detail. In this case the marginal equilibrium distribution becomes the hypergeometric one: |a| |θ − a| k n−k P(Y = k) = , (39) |θ| n with a = θ/g and θ negative integers. The range of k is (0, 1, . . . , min(|a|, n)). The states with ni > |a| are transient and they do not appear any longer as times goes by. If, for instance, |a| = 10n/g (ten times the average wealth), one has that |θ| = 10n and r = 10m/(10n − n + m) (10m)/(9n). If m n, this is not so far from the independent redistribution case, where r = m/n. On the contrary, in the extreme case |a| = n/g, the occupation vector n = (n/g, . . . , n/g) is obtained with probability 1. If an initial state containing individuals richer than |a| is considered, that is if one considers (38) for k > |a|, then E(Dt+1 |Yt = k) is still mk/n but E(Ct+1 |Yt = k, Dt+1 = d) = 0 unless k − d < |a|. More precisely, one has ⎧ k ⎪ ⎪ ⎪ k ⎨ |a| − k + m n m if k − m ≤ |a| (40) E(Ct+1 |Yt = k) = |θ| − n + m n ⎪ ⎪ k ⎪ ⎩ 0 if k − m > |a| n If the percent taxation is f = m/n, then one gets E (Yt+1 − Yt |Yt = k) fθ n − kt − if k(1 − f ) ≤ |a| = θ − n(1 − f ) g ⎩ = −k(1 − f ) if k(1 − f ) > |a| ⎧ ⎨

(41)

As k(1 − f ) is the expected value of Y after taxation, even if the agent is initially richer than |a| he/she can participate to redistribution when the percentage of taxation is high enough.

Appendix: the P´ olya distribution Finite (n-step) stochastic processes The sequence of individual random variables V1 , . . . , Vn is an n-step stochastic process. It is completely determined by the knowledge of all the finite dimensional distributions of the kind: pV1 ,...,Vm (v1 , . . . , vm ) = P(V1 = v1 , . . . , Vm = vm ),

(42)

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where 1 ≤ m ≤ n. The finite dimensional distributions are subject to Kolmogorov’s compatibility conditions pV1 ,...,Vm (v1 , . . . , vm ) = P(V1 = v1 , . . . , Vm = vm ) = P(Vi1 = vi1 , . . . , Vim = vim ) = pVi1 ,...,Vim (vi1 , . . . , vim ),

(43)

where i1 , . . . , im is any of the m! permutations of the indices, and g

pV1 ,...,Vm−1 (v1 , . . . , vm−1 ) =

pV1 ,...,Vm (v1 , . . . , vm−1 , vm ).

(44)

vm =1

Finite dimensional distributions can be conveniently characterized in terms of predictive probabilities. Indeed, as a consequence of the multiplication theorem, one has P(V1 = v1 , . . . , Vm = vm ) = P(V1 = v1 )P(V2 = v2 |V1 = v1 ) · · · · · · P(Vm = vm |V1 = v1 , . . . , Vm−1 = vm−1 )

(45)

and Kolmogorov’s compatibility conditions are automatically satisfied if predictive probabilities satisfy the basic probability axioms. Exchangeable processes An exchangeable process is characterized by additional symmetry conditions on the finite dimensional distributions pV1 ,...,Vm (v1 , . . . , vm ) = P(V1 = v1 , . . . , Vm = vm ) = P(Vi1 = v1 , . . . , Vim = vm ) = pVi1 ,...,Vim (v1 , . . . , vm ),

(46)

where i1 , . . . , im is any of the m! permutations of the indices, Note that condition (46) differs from condition (43). For an exchangeable process, the probability of an individual sequence V(m) = v(m) = (V1 = v1 , . . . , Vm = vm ) only depends on the occupation vector of the sequence m = (m1 , . . . , mg ) with g m = m. This leads to: i i=1 P(V(m) = v(m) ) =

m! i=1 mi !

g

−1 P(Y = m)

(47)

as a consequence of (9). The P´ olya process The Pólya process is an exchangeable process characterized by the predictive probability P(Vm+1 = j|V1 = v1 , . . . , Vm = vm ) =

αj + mj , α+m

(48)

128


where mj is the number of times in which category j has been observed up to step m, α = (α1 , . . . , αg ) is a vector of parameters and α = gi=1 αi . If the new parameters pj = αj /α are introduced, (48) becomes P(Vm = j|V1 = v1 , . . . , Vm = vm ) =

αpj + mj . α+m

(49)

pj = P(V1 = j) is the a priori probability of category j and (49) is nothing else than a mixture between initial or a priori probabilities and the observed frequencies. As a consequence of (48), and of exchangeability (see (47)), one gets the following finite dimensional distributions P(V(m) = v(m) ) =

m! i=1 mi !

g

−1 Polya(m|m; α),

(50)

where the multivariate generalized P´ olya sampling distribution is given by: Polya(m|m; α) =

g [m ] m! αi i , α[m] i=1 mi !

(51)

where x[n] = x(x + 1) · · · (x + n − 1) is the rising factorial. The Pólya process encompasses the following remarkable cases: • The multivariate hypergeometric process for integer αj < 0, ∀j ∈ {1, . . . , g} with the constraints mj ≤ |αj | and m ≤ α. In this case |αj | represents the initial number of marbles of colour j in an urn from which they are randomly drawn without replacement; this process is not extendible to infinity and ends after n = |α| steps; • The multinomial process in the limit |α| → ∞ and |αj | → ∞, with pj = αj /α constant. In this case pj represents the probability of drawing a marble of colour j with replacement from and urn; this process can be exteded to infinity; • The P´ olya urn process for integer αj > 0, ∀j ∈ {1, 2, . . . , g}. In this case αj is the initial number of marbles of colour j in an urn. They are randomly drawn and replaced with another ball of the same kind. Also this process is indefinitely extendible. Marginal distributions The marginal distributions for the g-variate generalized Pólya distribution can be easily derived from the predictive probability given by (48). Consider the probability P(Vm+1 ∈ A|V1 = v1 , . . . , Vm = vm ), where the set A is a set of categories A = {j1 , . . . , jr }. This new predictive probability is given by: P(Vm+1 ∈ A|V(m) ) =

r i=1

P(Vm+1 = ji |V(m) ),

(52)

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where, as usual, V(m) = (V1 = v1 , . . . , Vm = vm ) summarizes the evidence. In the P´ olya case, P(Vm+1 = j|V(m) ) is a linear function of both the weights and the occupation numbers; therefore, one gets: αA + mA j∈A αj + j∈A mj (m) P(Vm+1 ∈ A|V ) = = , (53) α+m α+m where α = j αj , αA = j∈A αj and mA = j∈A mj . As a direct consequence of (53), the marginal distributions of the g-variate generalized Pólya distribution are given by the dichotomous P´ olya distribution of weights αi and α − αi , where i is the category with respect to which the marginalization is performed. In other words, one gets that Polya(m|m, α) = Polya(mi , m − mi ; αi , α − αi ) mj , j=i [mi ]

=

αi m! mi !(m − mi )!

(α − αi )[m−mi ] . α[m]

(54)

Moments of the P´ olya distribution Consider the evidence vector V(m) = (V1 = v1 , . . . , Vm = vm ). In the general case of g categories, it is natural to introduce the indicator function IXi =j (ω) = m (j) (j) (j) (j) Ii , and define Sm = i=1 Ii . Therefore, the random variable Sm gives the number of successes for the jth category out of m observations or trials and (j) (k) (k) (k) (k) Sm = mj . One can determine E(Ii ) and E(Ii Ij ) and derive E(Sm ) as (k)

(k)

(k)

well as Var(Sm ). As for the expected value, one has that E(Ii ) = 1·P(Ii = (k) (k) 1) + 0 · P(Ii = 0) = P(Ii = 1) coinciding with the marginal probability of success, that is the probability of observing category k at the ith step. (k) From (48), in the absence of any evidence, one has P(Ii = 1) = P(Xi = (k) k) = αk /α = pk . Therefore, the random variables Ii are equidistributed and m (k) (k) (k) exchangeable, and E(Sm ) = i=1 E(Ii ) = mE(I1 ), yielding (k) E(Sm ) = mpk . (k)

(55) (k)

(k)

As for the variance Var(Sm ), the covariance matrix of I1 , . . . , Im is needed. (k) (k) (k) Because of the exchangeability of I1 , . . . , Im , the moment E[(Ii )2 ] is the (k) (k) same for all i, and E(Ii Ij ) is the same for all i, j, with i = j. Note that (k)

(k)

(Ii )2 = Ii

(k)

and this means that E[(Ii )2 ] = pk ; it follows that (k)

(k)

(k)

Var(Ii ) = E[(Ii )2 ] − E2 (Ii ) = pk (1 − pk )

(56)

one can show that (k) (k)

E(Ii Ij ) = P(Xi = k, Xj = k);

(57)

130


now, from exchangeability, from (57), and from (48), one gets (k) (k)

E(Ii Ij ) = P(Xi = k, Xj = k) = P(X1 = k, X2 = k) αk + 1 (k) (k) . = E(I1 I2 ) = P(X1 = k)P(X2 = k|X1 = k) = pk α+1

(58)

Therefore, the covariance matrix is given by: (k)

(k)

(k)

(k)

Cov(Ii , Ij ) = Cov(I1 , I2 ) α − αk (k) (k) (k) (k) . = E(I1 I2 ) − E(I1 )E(I2 ) = pk α(α + 1)

(59)

(k)

The variance of the sum Sm follows from (56) and (59) (k)

(k)

(k)

(k) ) = mVar(I1 )+m(m−1)Cov(I1 , I2 ) = mpk (1−pk ) Var(Sm

α+m . (60) α+1

Thermodynamic limit Let α1 denote the weight of the chosen category and let α − α1 denote the weight of the thermostat. The thermodynamic limit is n, α 1 with χ = n/α. Consider that (α − α1 )[n−k] (α − α1 )(α − α1 + 1) · · · α(α + 1) · · · (α − α1 + n − k − 1) = [n] α(α + 1) · · · (α + n − 1) α (α − α1 )(α − α1 + 1) · · · (α − 1) . (61) (α − α1 + n − k) · · · (α + n − 1) 1 The numerator contains the product α (α − i) αα1 , whereas at the α1 +ki=1 denominator, one has the product i=1 (α + n − i) (α + n)α1 +k and the ratio is approximated by: αα1 ; (62) (α + n)α1 +k =

therefore, we eventually get P(k|n; α1 , α) NegBin(k|α1 , χ) α1 k [k] 1 χ α , k = 0, 1, 2, . . . ; = P(k|α1 , χ) = 1 k! 1+χ 1+χ

(63)

this distribution is called negative binomial distribution; the geometric distribution is a particular case of (63) in which α1 = 1. If α1 is an integer number, the usual interpretation of the negative binomial random variable is the description of the (discrete) waiting time of (i.e., the number of failures before)

Tolstoy’s dream

131

the first α1 th success in a Bernoullian process with parameter p = 1/(1 + χ). The moments of the negative binomial distribution can be obtained from the corresponding moments of the Polya(m1 , m − m1 ; α1 , α − α1 ) in the limit n, α 1, with χ = n/α yielding: E(Y1 = k) = n Var(Y1 = k) = n

α1 → α1 χ, α

(64)

α1 α − α1 α + n → α1 χ(1 + χ). α α α+1

(65)

Note that if α1 is an integer, k can be interpreted as the sum of α1 independent geometric variables. Continuous limit Consider the multivariate generalized P´ olya distribution given by (51). Noting that Γ (m + α) α[m] = (66) Γ (α) (51) can be re-written as: g Γ (α) Γ (mi + αi ) m! . mi ! i=1 Γ (αi ) Γ (m + α) i=1

Polya(m|m; α) = g

(67)

The variables xi = mi /m satisfy the following constraint g i=1

xi =

g mi i=1

m

= 1;

(68)

moreover, ∀i ∈ {1, . . . , g}, we have that 0 ≤ xi ≤ 1. If one considers the continuous limit in which m → ∞, mi → ∞ with constant xi = mi /m for all the categories i, one finds that Γ (mi + αi ) Γ (mi + αi ) i −1 = mα i mi ! Γ (mi + 1) replacing (69) for any mi and for m in (67) leads to g αi −1 Γ (α) i=1 mi Polya(m|m; α) g mα−1 i=1 Γ (αi ) g g Γ( αi ) αi −1 1 = g i=1 xi · g−1 . m Γ (α ) i i=1 i=1

(69)

(70)

132


Equation (70) can be interpreted as follows; based on the exchangeability of the variables Yi = mi , the probability of the variables Xi = Yi /m of assuming values X1 = x1 , . . . Xn = xn with xi = mi /m is g g Γ ( i=1 αi ) αi −1 1 P(X1 = x1 , . . . , Xn = xn ) g xi · g−1 m Γ (α ) i i=1 i=1 g g Γ( αi ) αi −1 g i=1 xi dx1 · · · dxg−1 , i=1 Γ (αi ) i=1

(71)

where the relationship becomes exact in the continuous limit. In fact, the ratio 1/m can be interpreted as Δxi because Δmi = 1 and xi = mi /m. The function g g Γ ( i=1 αi ) αi −1 p(x1 , . . . , xg ; α1 , . . . αg ) = p(x; α) = g xi (72) i=1 Γ (αi ) i=1 g defined on the simplex i=1 xi = 1 and 0 ≤ xi ≤ 1 for all the i ∈ {1, . . . g} is the probability density function for the so-called Dirichlet distribution. Let X ∼ Dir(x; α) denote the fact that the random vector X is distributed according to the Dirichlet distribution. As a consequence of the P´ olya marginalization property (53), we obtain the so-called aggregation property of the Dirichlet distribution: let X1 , . . . , Xg be a sequence of random variables with values on the simplex gi=1 xi with 0 ≤ xi ≤ 1, ∀i ∈ {1, . . . g} whose distribution is Dir(x1 , . . . , xi , . . . , xi+k , . . . , xg ; α1 , . . . , αi , . . . , αi+k , . . . , αg ), then i+k the new sequence X1 , . . . , XA = j=i Xj , . . . Xg is distributed according to i+k i+k Dir(x1 , . . . , xA = j=i xj , . . . , xg ; α1 , . . . , αA = j=i αj , . . . , αg ). Thanks to the aggregation property, we can find the marginal distribution of the Dirichlet distribution, whose probability density function is nothing else than the Beta distribution. If X1 , . . . , Xg ∼ Dir(x1 , . . . , xg ; α1 , . . . , αg ) then Xi ∼ Beta(xi ; αi , α − αi ).

(73)

Starting from the probability density function Beta(x; a, b). p(x) =

Γ (a + b) a−1 x (1 − x)b−1 . Γ (a)Γ (b)

(74)

and defining y = Ax, then we get f (y) =

Γ (a + b) 1 y a−1 y b−1 1− , Γ (a)Γ (b) A A A

(75)

with y ∈ [0, A]. While x is the fraction of wealth belonging to the selected agent, now y represents his absolute wealth, being A the total wealth. In the

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133

limit A → ∞, b → ∞, A/b = w/a = u constant, the Beta density can be approximated by the Gamma(y|a, u) density given by: g(y) =

y u−a a−1 y . exp − Γ (a) u

(76)

The meaning of w is the expected value of the wealth of the selected agent, which stays constant when the continuous thermostat becomes infinite.

References 1. Aoki M., Yoshikawa H.: Reconstructing Macroeconomics. A Perspective from Statistical Physics and Combinatorial Stochastic Processes. Cambridge University Press, Cambridge, UK (2007) 2. Bunge M.: Systemism: the alternative to individualism and holism. Journal of Socio-Economics, 29, 147–157 (2000) 3. Costantini D., Garibaldi U.: A Probabilistic Foundation of Elementary Particle Statistics. Part I. Stud. Hist. Phil. Mod. Phys., 28, 483–506 (1997) 4. Costantini D., Garibaldi U.: A Probabilistic Foundation of Elementary Particle Statistics. Part II. Stud. Hist. Phil. Mod. Phys., 29, 37–59 (1998) 5. Costantini D., Garibaldi U.: The Ehrenfest Fleas: From Model to Theory. Synthese, 139, 107–142 (2004) 6. Garibaldi U., Scalas E., Viarengo P.: Statistical equilibrium in simple exchange games II. The redistribution game. Eur. Phys. J. B, 60, 241–246 (2007) 7. Garibaldi U., Scalas E.: Finitary Probabilistic Methods in Econophysics. Cambridge University Press, Cambridge UK (2010) 8. Hacking I.: The Taming of Chance. Cambridge University Press, Cambridge UK (1990) 9. Ingrao B., Israel G.: The Invisible Hand. Economic Equilibrium in the History of Science. MIT Press, Cambridge MA (1990) 10. Penrose O.: Foundations of Statistical Mechanics: A Deductive Treatment. Dover, New York (1970) 11. Scalas E., Garibaldi U., Donadio S.: Statistical equilibrium in simple exchange games I. Methods of solution and application to the Bennati-DragulescuYakovenko (BDY) game. Eur. Phys. J. B, 53, 267–272 (2006) 12. Scalas E., Garibaldi U.: A Dynamic Probabilistic Version of the AokiYoshikawa Sectoral Productivity Model. Economics, The Open-Access, Open-Assessment E-J., 3, 2009–15 http://www.economics-ejournal.org/economics/ journalarticles/2009--15 (2009) 13. Schumpeter J.: Das Wesen und der Hauptinhalt der theoretischen National¨ okonomie. Duncker und Humblot, M¨ unchen and Leipzig (1908) 14. Tesfatsion L., Judd K.L. (eds.): Handbook of Computational Economics, 2. Agent-Based Computational Economics. North Holland, Amsterdam (2006) 15. Tolstoy L.N.: War and Peace, http://www.gutenberg.org/dirs/etext01/wrnpc11.txt (1869) 16. von Mises R.: Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik. Rosenberg, New York (1945)

Part II

Social modelling and opinion formation

New perspectives in the equilibrium statistical mechanics approach to social and economic sciences Elena Agliari1 , Adriano Barra2 , Raffaella Burioni3 , and Pierluigi Contucci4 1 2 3 4

Dipartimento di Fisica, Universit` a di Parma, Italy, [email protected] Dipartimento di Fisica, Sapienza Universit` a di Roma and Dipartimento di Matematica, Universit` a di Bologna, Italy, [email protected] Dipartimento di Fisica, Universit` a di Parma and INFN, Gruppo Collegato di Parma, Italy, [email protected] Dipartimento di Matematica, Universit` a di Bologna, Italy, [email protected]

Summary. In this chapter we review some recent development in the mathematical modeling of quantitative sociology by means of statistical mechanics. After a short pedagogical introduction to static and dynamic properties of many body systems, we develop a theory for particle (agents) interactions on random graph. Our approach is based on describing a social network as a graph whose nodes represent agents and links between two of them stand for a reciprocal interaction. Each agent has to choose among a dichotomic option (i.e., agree or disagree) with respect to a given matter and he is driven by external influences (as media) and peer to peer interactions. These mimic the imitative behavior of the collectivity and may possibly be zero if the two nodes are disconnected. For this scenario we work out both the dynamics and, given the validity of the detailed balance, the corresponding equilibria (statics). Once the two-body theory is completely explored, we analyze, on the same random graph, a diffusive strategic dynamics with pairwise interactions, where detailed balance constraint is relaxed. The dynamic encodes some relevant processes which are expected to play a crucial role in the approach to equilibrium in social systems, i.e., diffusion of information and strategic choices. We observe numerically that such a dynamics reaches a well defined steady state that fulfills a shift property: the critical interaction strength for the canonical phase transition is higher with respect to the expected equilibrium one previously obtained with detailed balanced dynamical evolution. Finally, we show how the stationary states of this kind of dynamics can be described by statistical mechanics equilibria of a diluted p-spin model, for a suitable noninteger real p > 2. Several implications from a sociological perspective are discussed together with some general outlooks.


137

138

Elena Agliari, Adriano Barra, Raffaella Burioni, and Pierluigi Contucci

1 Introduction Born as a microscopic foundation of thermodynamics, statistical mechanics provides nowadays a flexible approach to several scientific problems whose depth and wideness increases continuously. In the last decades, in fact, statistical mechanics has invaded fields as diverse as spin glasses [26], neural networks [7], protein folding [22], immunological memory [28], and also made some attempt to describe social networks [16], theoretical economy [17], and urban planning [14]. In this paper we study statistical mechanics of imitative diluted systems, paying particular attention to its applications in social sciences. After a review of the statistical mechanics methodology, we introduce the tools, both analytical and numerical, used for the investigation of many-body problems. We apply such a machinery to study at first the global behavior of a large amount of dichotomic agents (i.e., able to answer only “yes/no” to a given question) whose decision making is driven both by a uniform external influence (as the media) and by pairwise imitative interactions among agent themselves. In general, the agents making up a community are not all contemporarily in contact with each other, namely, the network representing the social structure is not fully connected but rather randomly diluted, hence mirroring acquaintances or family relationships. Even though refined models as small-worlds graphs have recently been proposed, a standard one for such a network is provided by the famous Erd¨ os– Renyi graph. The model turns out to be nontrivial as, tuning the degree of connectivity and/or the strength of interaction, a cooperative state among the agents appears. Moreover, we show the existence of a region of the parameter space which is more convenient for the global behavior of the society, i.e., it corresponds to a minimum in the free energy. As social systems do not need to obey Maxwell–Boltzmann distribution, detailed balance is not strictly required for their evolution, so, after having explored the “canonical” 2-body model in full detail, we introduce a more realistic description of its free temporal evolution by adopting a diffusive strategic dynamics [3,4]. This dynamic takes into account two crucial aspects, which are expected to be effective in the temporal evolution of social systems, i.e., diffusion of information and strategic choices. We implement it on the same Erdös–Renyi graph and study its equilibria. Each agent is selected through a diffusive rule, and a flip in its dichotomic status is not weighted “a la Glauber” [7] but rather according to a strategic rule which produces the maximum energy gain. We stress that this operation involves more-than-twobody effective interactions, as the chosen agent interacts both with the first selected one as well as with its nearest neighbors, as a whole. This dynamics is shown to relax to a well defined steady state, where all the properties of stationarity are recovered [19], however the strength of the interactions at the critical line is lower, of a few percent, than the expected. The whole scenario suggests a “latent” many-body coupling influence, encoded into the particular

Statistical mechanics approach to social and economic sciences

139

rule for selecting the agents. This is also corroborated by further numerical analysis. As a consequence, we work out analytically a theory for the randomly diluted p-spin model so to fit an effective p ∈ R, which turns out to be p = 2.15, in order to match the numerical data available by the dynamics. This result has implication both in market trends, as well as in quantitative sociology, where the effective interactions always play an important role in decision making [18, 25]. The paper is organized as follows: In Sect. 2, for the sake of completeness, stochastic dynamics for discrete many body problems is outlined, Sect. 3 deals with definitions and introduction to their equilibrium via statistical mechanics. In Sect. 4, instead, the model we study is solved in full details with the aim of presenting both a scenario for these decision makers on random graphs as well as a general mathematical method which can be extended by the reader to other models. In Sect. 5 our alternative dynamics is introduced and shortly discussed; then further numerical investigations toward a better understanding of a p > 2 behavior are presented. In Sect. 6 the randomly diluted p-spin model is defined and exploited in all details, both analytically (within the cavity field framework) and numerically (within a Monte Carlo approach). Full agreement is found among the two methods. At the end, the last section is left for conclusions: the effective interaction is found and its implications analyzed. Furthermore, even though the paper is written within a theoretical physics approach, remarks concerning the application to quantitative sociology are scattered throughout the work. In particular, in the conclusion, a toy application of the outlined theory to trades in markets is shown.

2 A brief introduction to many-body dynamics In this section we introduce the fundamental principles of stochastic dynamics used to simulate the relaxation to equilibrium of the systems we are interested in. Even though for discrete systems two kinds of dynamics are available, parallel and sequential, we are going to deepen only the latter as is the one we will implement thought the chapter. Although the topic is well known (see, e.g., [23, 24]), for the sake of completeness and in order to offer to the reader a practical approach to these models, we present the underlying theory. 2.1 The model Let us consider an ensemble of N agents labeled as i = 1, .., N . Each agent has two possible choices, with respect to a given situation, which are encoded into a variable σi = ±1, say σi = +1 is “agreement” and vice versa for −1. Each agent experiences an external influence (for example by media) which is taken into account by the one-body coupling H1 (σ; θ)

140


H1 (σ; θ) = −

N

σi (t)θi (t),

(1)

i=1

where θi (t) is the stimulus acting on the ith agent at a given time t and σi (t) is the opinion of the ith agent at the same time. The interactions among the other agents are encoded into the H0 (σ; J) term as follows: N N H0 (σ; J) = − Jij (α)σi (t)σj (t). (2) i

j>i

For the moment there is no need to introduce explicitly the dilution of the underlying random network as the scheme applies in full generality and may be a benchmark for future development by the reader himself. We only stress that Jij (α) is quenched, i.e., does not evolve with time, and can be thought of as a symmetric adjacency matrix in such a way that a zero entry Jij = 0 means that the agents i and j are not in contact with each other, vice versa for Jij = 1 N there is a link between them. The ratio of connections i,j Jij /N 2 is tuned by a parameter α, such that for α → ∞ the graph recovers the fully connected one, while for α = 0 the graph is completely disconnected. Overall, the Hamiltonian defining the model is the sum of the two contributes, namely (with a little abuse of notation, thinking at J as (J, θ)) H(σ; J) = H0 (σ; J) + H1 (σ; θ). With the signs as they are here, a positive value of Jij makes the relevant spins want to line up together, i.e., to share the same opinion, and each spin also wants to be aligned with the corresponding external field. The investigation of the properties displayed by systems described by this kind of Hamiltonian are both analytical and numerical. The former relies on series expansions, field theoretical methods, cavity, and replica approaches. The latter are mainly based on Monte Carlo simulations where we directly simulate the temporal evolution of the system in such a way that an expectation value is calculated as a time average over the states that the system passes through. However, it must be underlined that the Hamiltonian contains no dynamical information, hence we have to choose a dynamic for our simulation, namely a rule for changing from one state to another during the simulation, which results in each state appearing with exactly the probability appropriate to it. Several possibility have been introduced in the past, ranging from deterministic, e.g., Q2R dynamics, to stochastic, e.g., Glauber algorithm and Wolff algorithm. 2.2 Transition rates and Markov process In statistical mechanics, Maxwell–Boltzmann statistics (hereafter simply “Boltzmann statistics”) describes the statistical distribution of material particles over various energy states in thermal equilibrium, when the temperature is high enough and density is low enough to render quantum effects negligible.


141

Now, given the generic configuration σ = {σi }i=1,...,N according to Boltzmann statistics, the value of a thermodynamic observable X(β; J) is given by: −βH(σ;J) {σi } X(σ; J)e , (3) X(β; J) = X(σ; J)β = −βH(σ;J) {σi } e where β represents the inverse of the temperature (that sometimes we call “noise”), i.e., β ≡ (kB T )−1 , being kB the Boltzmann constant, and the brackets are implicitly defined by the r.h.s. of (3). Monte Carlo techniques [27] work by choosing a subset of states S˜ at random from some probability distribution pσ which we specify. Our best estimate of the quantity X(σ; J) is then given by the so-called estimator X(σ; J)β,S˜ : −1 −βH(σ;J) ˜ pσ X(σ; J)e {σ∈S} X(σ; J)β,S˜ = . (4) −1 −βH(σ;J) ˜ pσ e {σ∈S} ˜ The estimator has the property that, as the number of sampled states |S| increases, it becomes a more and more accurate estimate of X(β; J), and, as ˜ → ∞ we have X(σ; J) ˜ = X(σ; J)β . |S| β,S The choice made for pσ is based on the following argument: when in equilibrium, the system is not sampling all states in S with equal probability, but according to the Boltzmann probability distribution. Hence, the strategy is this: instead of selecting the subspace S˜ in such a way that every state of the system is as likely to gets chosen as every other, we select them so that the (eq) probability that a particular state s is chosen is pσ = pσ = Z −1 e−βH(σ) , Z being a proper normalization factor by now. The estimator then simplifies into a simple arithmetic average X(σ; J)β,S˜ =

1 X(σ; J). ˜ |S| ˜

(5)

σ∈S

This definition for X(σ; J)β,S˜ works much better than the one we would obtain from a uniform distribution for pσ , especially when the system is spending the majority of its time in a small number of states. Indeed, the latter will be precisely the states sampled most often, and the relative frequency with which we select them will correspond to the amount of time the real system would spend in those states. Therefore, we need to generate an appropriate random set of states, accord(eq) ing to the Boltzmann weight pσ . In general, Monte Carlo schemes rely on Markov processes as the generating engine for the set of states to be used. Let us introduce a (normalized) transition probability W [σ, σ ] for any pair σ, σ of configurations in the phase space S. Such a set of transition probabilities, together with the specification of an initial configuration, allows to construct a Markov chain of configurations, S˜τ = (σ1 , σ2 , ..., στ ). The Markov process is chosen in such a way that, when it is run for long enough, starting from

142


any state of the system, it will eventually produce a succession of states which appear according to the canonical distribution. In order to achieve this, two conditions are sufficient: the condition of ergodicity and of detailed balance. The former is the requirement that it must be possible, for the Markov process, to reach any state of the system from any other state, if it is run for long enough. Otherwise stated, ∀ σ, σ , ∃ t : W t [σ, σ ] is non-null, where W t [σ, σ ] just represents the probability of reaching σ from σ in t steps. The ergodic condition is also consistent with the fact that, in the Boltzmann distribution, every state σ appears with nonzero probability. On the other hand, notice that this condition does not require that W (σ, σ ) = 0, ∀ σ, σ . Conversely, the detailed balance condition ensures that, in the limit τ → ∞, a given configuration σ appears in the Markov chain S˜τ just ac(eq) cording to the probability distribution pσ . The detailed balance condition requires that the system is in equilibrium (the rate of transitions into and out of any state must be equal) and that no limit cycles are present. As a result, the detailed balance condition can be stated as: (eq)

p(eq) σ W [σ, σ ] = pσ W [σ , σ],

(6)

where the l.h.s. represents the overall rate at which transitions from σ to σ occur in the system, while the r.h.s. is the overall rate for the reverse transition. This condition makes the system exhibit time-reversal symmetry at each move, and it provides a sufficient (but not necessary) condition ensuring that the application of these transition probabilities leads the system to an equilibrium distribution irrespective of the initial state. Now, as we wish the equilib(eq) rium distribution to be the Boltzmann one, we choose pσ = Z −1 e−βH(σ;J) , obtaining W [σ, σ ] = W [σ , σ]e−β[H(σ ;J)−H(σ;J)] . (7) The constraints introduced so far still leave a good deal of freedom over the definition of the transition probabilities. Indeed, the choice of a proper transition probability to apply to the system under study is crucial. In fact there is no kinetic information in the Hamiltonian given by (1) and by (2), as it only contains information about spin orientation and the spatial distribution of lattice sites. It is the transition probability which provides a dynamics, i.e., a rule according to which the system evolves. In the following, we will be especially interested in the so called single-spinflip dynamics, which means that the states involved in the transition only differ for the value of a single spin variable. More precisely, in this kind of dynamics, given the configuration σ, at each time step a single agent i ∈ [1, ..., N ], is randomly chosen among the N and updated with probability W [σ, Fi σ] to give rise to the the configuration Fi σ, where the N spin-flip operators Fi is defined as Fi σ ≡ Fi {σ1 , ..., σi , ..., σN } = {σ1 , ..., −σi , ..., σN }. The Metropolis and the Glauber dynamics [27] are examples of stochastic single-spin-flip dynamics. In particular, for the latter one has for the transition rates


−1 W [σ; Fi σ] = 1 + exp(βΔi H(σ; J)) ,

143

(8)

where Δi H(σ; J) = H(Fi σ; J) − H(σ; J).

3 Equilibrium behavior In synthesis, thermodynamics describes all the macroscopic features of the system and statistical mechanics allows to obtain such a macroscopic description starting from its microscopic foundation, that is, obtaining the global society behavior by studying the single agent based dynamics, and then, using Probability Theory, for averaging over the ensemble with the weight en(eq) coded by the equilibrium distribution pσ . This scenario is fully derivable when both the internal energy density of the system e(β, α) and the entropy density s(β, α) are explicitly obtained (we are going to introduce such quantities hereafter). Then, the two prescription of minimizing the energy e(β, α) (minimum energy principle) and maximizing entropy s(β, α) (second law of thermodynamics) give the full macroscopic behavior of the system, expressed via suitably averages of its microscopic element dynamics. To fulfil this task the free energy f (β, α) = e(β, α) − β −1 s(β, α) turns out to be useful because, as it is straightforward to check, minimizing this quantity corresponds to both maximizing entropy and minimizing energy (at the given temperature), furthermore, and this is the key bridge, there is a deep relation among it and the (eq) equilibrium measure pσ , in fact p(eq) ∝ exp(−βH(σ; J)), σ

−1 log exp(−βHN (σ; J)), N →∞ βN σ

f (β, α) = lim fN (β, α) = lim N →∞

e(β, α) = lim eN (β, α) = lim −∂β (βfN (β, α)), N →∞ N →∞ s(β, α) = lim sN (β, α) = lim fN (β, α) + β −1 ∂β (βfN (β, α)) . N →∞

N →∞

(9) (10) (11)

So, once explicitly obtained the free energy, equilibrium behavior is solved and the whole works once the equilibrium probability distribution is known, provided we use the Hamiltonian H(σ; J). Before proceeding in this derivation, we need some preliminary definitions: At first, in the following, it will be convenient to deal with the pressure A(β, α), defined as: A(β, α) = lim AN (β, α) = −β lim fN (β, α); N →∞

N →∞

(12)

we stress that often we are going to consider results in the “thermodynamic limit” N → ∞: such procedure allows us to use implicitly several theorem of convergence of random variables from Probability Theory and, when the

144


number of agents is large enough, the agreement among results at finite N and results in the thermodynamic limit is excellent, as usually the two differ by factors O(N −1 ) or at worse O(N −1/2 ). Let us now further introduce the partition function defined as: ZN (β, α) = e−βHN (σ,J) = p(eq) (13) σ . σN

σ

As we do not want a sample-dependent theory, using E for the average over the quenched variables (i.e., the connectivity), the quenched pressure can be written as: 1 AN (β, α) = E ln ZN (β, α), N the Boltzmann state is given by: 1 g(σ; J)e−βHN (σ;J) , ZN (β, α) σ

ω(g(σ, J)) =

(14)

N

with its replicated form on s replicas defined as: ω (s) (g(σ (s) ; J)) Ω(g(σ; J)) =

(15)

s

and the total average g as: g = E[Ω(g(σ; J))].

(16)

Let us introduce further, as order parameters of the theory, the multioverlaps N 1 (1) (n) q1...n = σ ...σi , (17) N i=1 i

with a particular attention to the magnetization m = q1 = (1/N ) N i=1 σi and N 1 2 to the two replica overlap q12 = (1/N ) i=1 σi σi . It is important to stress that the magnetization, which plays the role of the principal order parameter (able to recognize the different macroscopic phases displayed by the system), accounts for the averaged opinion into the social network, such that if m = 0 there is no net preference in global decision, while for m → 1 there is a sharp preference toward the “yes” state and vice versa for m → −1. Analogously, q accounts for similarity among two different “replicas” of the system (two independent realization of the adjacency matrix). It is easy to check that when β → 0 (or the interaction strength, that is always coupled with the noise), details of the Hamiltonian are unfelt by the agents, which will be on average one half up (yes) and one half down (no), giving a null net contribution to m.


145

At the contrary when the system is able to experience the rules encoded into the Hamiltonian it is easy to see that −

∂fN (β, a) ∂fN (β, a) = σi = 0, − = σi σj = 0. ∂θi ∂Jij

(18)

Averaging over the whole space of choices, we get the macroscopic response of the system in terms of the magnetization m. In the thermodynamic limit, further, self-averaging for this order parameter is expected to hold, which is expressed via lim m2N = lim mN 2 ,

N →∞

N →∞

that means that the mean value of the order parameter is not affected by the details of the microscopic structure in the N → ∞ limit (it is an expression of the Central Limit Theorem in this framework). From a purely thermodynamical viewpoint the equilibrium behavior (the phase diagram) is fundamental because it gives both the phase diagram and the critical scenario, so to say, the regions in the space of the tunable parameters β, α where the model displays a paramagnetic (independent agent viewpoint) or ferromagnetic (collective agent viewpoint) behavior, by which global decision on the whole society cannot leave aside. To obtain a clear picture of the equilibrium of the social network, we use standard techniques of statistical mechanics for positive valued interactions, namely the smooth cavity field technique [8]. For simplicity, as conceptually this does not change the picture, we deal with the simpler case θi = θ ∀i ∈ [1, N ].

4 Equilibrium statistical mechanics of the “2-body” model The “2-body” model has a long history in physics, having particular importance in interaction theories. In fact, from one side, (apart historical problems dealing with the deterministic dynamical evolution of the 3-body problem), for a long time the interaction in physics were thought of as particle scattering processes and in these events the probability of a more than 2-bodies instantaneous interaction were effectively negligible. From the other side, the structure of the 2-body energy is quadratic in its variables and this encodes several information: firstly, from a probabilistic viewpoint, the Maxwell–Boltzmann probability distribution assumes the form of a Gaussian, then, the forces (the derivatives of the energy with respect to its variable) are linear, such that superposition principle and linear response theory do hold and TLC is respected. However, as we will see later, neither of these properties of physical

146


systems need to be strictly obeyed in social theories and the interest to p > 2 body interactions will arise. In this section we consider diluted systems and systematically develop the interpolating cavity field method [8] and use it to sketch the derivation of a free energy expansion: the higher the order of the expansion, the deeper we could go beyond the ergodic (agent independent) region. Within this framework we perform a detailed analysis of the scaling of magnetization (and susceptibility) at the critical line. The critical exponents turn out to be the same expected for a fully connected system. Then, we perform extensive Monte Carlo (MC) simulations for different graph sizes and bond concentrations and we compare results with theory. Indeed, also numerically, we provide evidence that the universality class of this diluted Ising model is independent of dilution. In fact the critical exponents we measured are consistent with those pertaining to the Curie–Weiss model, in agreement with analytical results. The critical line is also well reproduced. The section is organized as follows: after a detailed and technical introduction of the model, in Sect. (4.1) we introduce the cavity field technique, which constitutes the framework we are going to use in Sect. (4.2) to investigate the free energy of the system at general values of temperature and dilution. Section (4.3) deals with the criticality of the model; there we find the critical line and the critical behavior of the main order parameter, i.e., magnetization. Section (4.4) is devoted to numerical investigations, especially focused on criticality. 4.1 Interpolating with the cavity field In this section, after a refined introduction of the model, we introduce further the cavity field technique on the line of [8]. Given N points and families {iν , jν } of i.i.d random variables uniformly distributed on these points, the (random) Hamiltonian of the diluted Curie– Weiss model is defined on Ising N -spin configurations σ = (σ1 , . . . , σN ) through P αN HN (σ, α) = − σiν σjν , (19) ν=1

where Pζ is a Poisson random variable with mean ζ and α > 1/2 is the connectivity. The expectation with respect to all the quenched random variables defined so far will be denoted by E, and is given by the composition of the Poissonian average with the uniform one performed over the families {iν } E[·] = EP Ei [·] =

1,N ∞ e−αN (αN )k k=0

k!N p

[·].

(20)

i1ν ....ip ν

As they will be useful in our derivation, it is worth stressing the following properties of the Poisson distribution: Let us consider a function g : N → R,


147

and a Poisson variable k with mean αN , whose expectation is denoted by E. It is easy to verify that E[kg(k)] = αN E[g(k − 1)] ∂αN E[g(k)] 2 ∂(αN )2 E[g(k)]

(21)

= E[g(k + 1) − g(k)] = E[g(k + 2) − 2g(k + 1) + g(k)].

(22) (23)

Turning to the cavity method, it works by expressing the Hamiltonian of a system made of N + 1 spins through the Hamiltonian of N spins by scaling the connectivity degree α and neglecting vanishing terms in N as follows Pα(N +1)

HN +1 (α) = −

∼

σiν σjν

ν=1

−

P αN ˜

σiν σjν −

ν=1

P2α ˜

σiν σN +1

(24)

ν=1

such that we can use the more compact expression ˆ N (˜ α) + H α)σN +1 HN +1 (α) ∼ HN (˜ with α ˜=

N N →∞ α −→ α, N +1

ˆ N (˜ H α) = −

(25)

P2α ˜

σiν .

(26)

ν=1

So we see that we can express the Hamiltonian of N + 1 particles via the one of N particles, paying two prices: the first is a rescaling in the connectivity (vanishing in the thermodynamic limit), the second being an added term, which will be encoded, at the level of the thermodynamics, by a suitably cavity function as follows: let us introduce an interpolating parameter t ∈ [0, 1] and the cavity function ΨN (˜ α, t) given by: α, β; t) Ψ (˜ α, β; t) = lim ΨN (˜ N →∞

= lim E ln

N →∞

{σ}

PαN ˜

eβ

ν=1

β σe Z (˜ N,t α, β) . = lim E ln N →∞ ZN (˜ α, β)

σiν σjν +β

PαN ˜ ν=1

P2αt ˜ ν=1

σiν

σiν σjν

(27)

The three terms appearing in the decomposition (25) give rise to the structure of the following theorem, which we prove by assuming the existence of the thermodynamic limit. (Actually we still do not have a rigorous proof of the existence of the thermodynamic limit but we will provide strong numerical evidences in Sect. 4.4). Theorem 6.1. In the N → ∞ limit, the free energy per spin is allowed to assume the following representation A(α, β) = ln 2 − α

∂A(α, β) + Ψ (α, β; t = 1). ∂α

(28)

148


Proof Consider the N + 1 spin partition function ZN +1 (α, β) and let us split it as suggested by (25) ˆ N (α)σ ˜ H ˜ N +1 ZN +1 (α, β) = e−βHN +1 (α) ∼ e−βHN (α)−β (29) =

σN +1

e

β

σN +1 PαN ˜ ν=1

σiν σjν +β

P2α ˜

ν=1

σN +1 σiν σN +1

=2

eβ

PαN ˜ ν=1

σiν σjν +β

P2α ˜

ν=1

σiν

,

σN

where the factor two appears because of the sum over the hidden σN +1 variable. Defining a perturbed Boltzmann state ω ˜ (and its replica product ˜=ω Ω ˜ × ···× ω ˜ ) as: −βHN (α) ˜ {σ } g(σ)e ˜ , Ω(g(σ)) = ω ˜ (i) (g(σ (i) )), ω ˜ (g(σ)) = N −βH ( α) ˜ N e {σN } i where the tilde takes into account the shift in the connectivity α → α ˜ and multiplying and dividing the r.h.s. of (29) by ZN (˜ α, β), we obtain α, β)˜ ω (eβ ZN +1 (α, β) = 2ZN (˜

P2α ˜

ν=1

).

(30)

Taking now the logarithm of both sides of (30), applying the average E and subtracting the quantity [ ln ZN +1 (˜ α, β)], we get α, β) ZN (˜ +ΨN (˜ α, β)] = ln 2+E ln α, β; t = 1) E[ln ZN +1 (α, β)]−E[ln ZN +1 (˜ ZN +1 (˜ α, β) (31) in the large N limit the l.h.s. of (31) becomes α, β)] E[ln ZN +1 (α, β)] − E[ln ZN +1 (˜ ∂ 1 ∂ [ ln ZN +1 (α, β)] = (α − α ˜ ) E[ln ZN +1 (α, β)] = α ∂α N + 1 ∂α ∂AN +1 (α, β) =α ∂α

(32)

then by considering the thermodynamic limit the thesis follows. Hence, we can express the free energy via an energy-like term and the cavity function. While it is well known how to deal with the energy-like [1, 20, 21], the same cannot be stated for the cavity function, and we want to develop its expansion via suitably chosen overlap monomials in a spirit close to the stochastic stability [6, 15, 29], such that, at the end, we will not have the analytical solution for the free energy in the whole (α, β) plane, but we will manage its expansion above and immediately below the critical line. To see how the machinery works, let us start by giving some definitions and proving some simple theorems:


149

Definition 1. We define the t-dependent Boltzmann state ω ˜ t as: ω ˜ t (g(σ)) =

PαN P2αt 1 ˜ ˜ g(σ)eβ ν=1 σiν σjν +β ν=1 σiν , ZN,t(α, β)

(33)

{σ}

where ZN,t (α, β) extends the classical partition function in the same spirit of the numerator of (33). As we will often deal with several overlap monomials let us divide them among two big categories: Definition 2. We can split the class of monomials of the order parameters in two families: • We define “filled” or equivalently “stochastically stable” all the overlap 2 monomials built by an even number of the same replicas (i.e., q12 , m2 , and q12 q34 q1234 ). • We define “fillable” or equivalently “saturable” all the overlap monomials which are not stochastically stable (i.e., q12 , m, and q12 q34 ). We are going to show three theorems that will play a guiding role for our expansion: as this approach has been deeply developed in similar contexts (as fully connected Ising model [9] or fully connected spin glasses [8] or diluted spin glasses [21], which are the boundary models of this subject) we will not show all the details of the proofs, but we sketch them as they are really intuitive. The interested reader can deepen this point by looking at the original works. Theorem 6.2. For large N , setting t = 1 we have n ω ˜ N,t (σi1 σi2 ...σin ) = ω ˜ N +1 (σi1 σi2 ...σin σN +1 ) + O

1 N

(34)

such that in the thermodynamic limit, if t = 1, the Boltzmann average of a fillable multioverlap monomial turns out to be the Boltzmann average of the corresponding filled multioverlap monomial. Theorem 6.3. Let Q2n be a fillable monomial of the overlaps (this means that there exists a multioverlap q2n such that q2n Q2n is filled). We have lim lim Q2n t = q2n Q2n

N →∞ t→1

(35)

2 , and q12 q34 t → (examples: for N → ∞ we get m1 t → m21 , q12 t → q12 q12 q34 q1234 ).

Theorem 6.4. In the N → ∞ limit the averages · of the filled monomials are t-independent in β average. For the proofs of these theorems we refer to [1].

150


It is now immediate to understand that the effect of Theorem (6.2) on a fillable overlap monomial is to multiply it by its missing part to be filled (Theorem (6.3)), while it has no effect if the overlap monomial is already filled 2n (Theorem (6.4)) because of the Ising spins (i.e., σN +1 ≡ 1 ∀n ∈ N). Now the plan is as follows: We calculate the t-streaming of the Ψ function in order to derive it and then integrate it back once we have been able to express it as an expansion in power series of t with stochastically stable overlaps as coefficients. At the end we free the perturbed Boltzmann measure by setting t = 1 and in the thermodynamic limit we will have the expansion holding with the correct statistical mechanics weight. P2αt ∂Ψ (˜ α, β, t) ∂ ˜ = E[ln ω ˜ (eβ ν=1 σiν )] ∂t ∂t

= 2α ˜ E[ln ω ˜ (eβ

P2αt ˜ ν=1

σiν +βσi0

)]−2α ˜E[ln ω ˜ (eβ

(36) P2αt ˜ ν=1

σiν

)] = 2α ˜ E[ln ω ˜ t (eβσi0 )]

and by the equality eβσi0 = cosh β + σi0 sinh β, we can write the r.h.s. of (36) as: ∂Ψ (˜ α, β, t) = 2α ˜ E[ln ω ˜ t (cosh β + σi0 sinh β)] ∂t = 2α ˜ log cosh β − 2α ˜ E[ln(1 + ω ˜ t (σi0 )θ)]. We can expand the function log(1 + ω ˜ t θ) in powers of θ = Tanh(β), obtaining ∞ (−1)n n ∂Ψ (˜ α, t) = 2α ˜ ln cosh β − 2α ˜ θ q1,...,n t . ∂t n n=1

(37)

We learn by looking at (37) that the derivative of the cavity function is built by nonstochastically stable overlap monomials, and their averages do depend on t making their t-integration nontrivial (we stress that all the fillable terms are zero when evaluated at t = 0 due to the gauge invariance of the model). We can escape this constraint by iterating them again and again (and then integrating them back too) because their derivative, systematically, will develop stochastically stable terms, which turn out to be independent by the interpolating parameter and their integration is straightforwardly polynomial. To this task we introduce the following Proposition 6.1. Let Fs be a function of s replicas. Then the following streaming equation holds s ∂Fs t,α˜ s+1 a = 2α ˜θ (38) Fs σi0 t,α˜ − sFs σi0 t,α˜ ∂t a=1 1,s s s(s + 1) s+1 s+2 2 a b a s+1 + 2α ˜θ Fs σi0 σi0 t,α˜ Fs σi0 σi0 t,α˜ − s Fs σi0 σi0 t,α˜ + 2! a=1 a Ts it starts increasing, since media action is expired. When Ts = 2 opinion 4 begins to appear with opinion 5. This phenomenon is more evident for Ts > 3. Notice that this is much different from the case with no media action. When dc = 4 the phenomenon is the same as before, just more evident: for Ts = 1 opinion 3 and 4 survive, then for Ts = 2 just opinion 4 survives and for Ts ≥ 3 opinion 4 survives together with opinion 5. See Fig. 4. Even in this case the differences with the case with no media are absolutely evident. Opinion 3 almost never survives and all the population is shifted toward opinion 5 depending on the critical distance dc . In particular, it seems that for low values of dc , namely dc = 0, 1, 2 this strategy is not very efficient because opinion 1 is still surviving. While for dc = 3, 4 the strategy is really efficient, since, for a sufficient time exposition to media action, almost all the population is captured by opinion 5 or 4. This is reasonable since, in this case, the strategy acts on intermediate opinions, that are the ones which play an important role when dc is high, if instead dc is low, intermediate opinions tend to be attracted by extreme ones and the strategy has no power.

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5.3 Strategy 3 Strategy 3 is such that opinion 1,2,3,4 move, respectively, to opinion 2,3,4,5 and opinion 5 remains the same. This strategy is the more uniform one in the sense that all the opinions are affected by media action. When dc = 0, 1, opinions 1 and 5 survive even if opinion 5 is much more intense. In this case media action response is highly evident and progressive, depending on Ts : the more Ts , the more opinion 5 has influence and opinion 1 decreases, as shown in Fig. 5. 1 0.8 0.6 0.4 0.2

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When dc = 2 the scenario is different: for Ts ≤ 4, opinion 5 is very influent, even if surviving with opinion 1 and 4 for small values of t. For Ts ≥ 4, opinion 5 is the unique to survive with a certain intensity. See Fig. 6.

Kinetic modelling of complex socio-economic systems

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Fig. 6. Strategy 3 – Evolution of the components of f1 , f2 , or f3 in the case of dc = 2, Ts = 1

When dc = 3, the surviving opinions are 3 and 4 or 4 and 5 depending on initial conditions, for Ts = 1. When Ts = 2, opinion 3 is no longer surviving and, for Ts ≥ 6, opinion 5 is the unique to survive. If dc = 4 results are almost the same but just for Ts ≥ 4 opinion 5 is the one which survives, with almost intensity 1, this means that almost all the population is influenced. See Figs. 7 and 8. 1 0.8 0.6 0.4 0.2

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This strategy is very efficient, especially for dc = 3, 4, even with a low value of Ts ; however, it is possible to see a very good dependence from media action even for dc = 0, 1, 2. With a sufficient value of Ts the goal of media is definitely reached, since almost all the population is captured by the desired media opinion. As far as strategy 1 is concerned, we notice that Ts plays an important role for low values of dc , while, regarding strategy 2, Ts plays an important role

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for high values of dc : this is explained by the fact that strategy 1 is very efficient in those cases in which extreme opinions are fundamental in the dynamics, while strategy 2 is efficient in those cases in which the intermediate opinions are important to determine the evolution. This means that these two strategy can be more or less efficient depending on the degree of radicalization of the initial population: if the population is not incline to compromise, strategy 1 is better, while if it is, strategy 2 works well. Anyway the most efficient results are obtained with strategy 3. In this case the dependence on Ts is strongly evident and, no matter which is dc , there is always a time Ts , which guarantees to influence all the population towards opinion 5. This is even more evident for high values of dc , when the dynamic is more free, since, due to people interactions, all the population can be easily oriented to the direction settled by media. This is definitely the most efficient strategy. Summarizing, it seems, from simulations, that concentrating upon all opinions always leads to a winning outcome, while the choice to influence not all the opinions can lead to a favorable equilibrium, depending on the initial predisposition of the population.

6 Conclusions This chapter focused on the mathematical modelling of complex socioeconomic phenomena, where individual human behaviors play a crucial role in determining the emerging collective dynamics. Several types of socio-economic systems are such that the collective dynamics are determined by complex individual interactions, so that the sum of the individual behaviors does not straightforwardly lead to the description of the collective dynamics. Stochastic games define the output of the interactions when the input states are given. Personal behaviors generate not only deviations from the most likely


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inputs, but may possibly change due to environmental conditions, for instance the onset of panic situations. The interest in complex socio-economic systems has witnessed a remarkable increase in recent years, thanks to the growing awareness of the fact that many systems related to human behaviors cannot be successfully modelled by traditional methods: new investigations leading to new methodologies are called for. This chapter aims at providing a preliminary contribution in this direction, keeping in mind that further research is required in order to develop a robust mathematical theory. Much remains to be done. We now provide our personal view of the most urgent further developments of the work reported in this chapter. The modelling approach proposed in this chapter is based on the concept of functional subsystem, considered as an aggregation of groups of interest expressing a common socio-economic function. Therefore, if the general context of the mathematical model changes, the characterization and the size of the functional subsystems may also need to change. In principle, the function expressed by a subsystem can be described as a vector with several components; however, the complexity inherent in modelling interactions suggests to refine the identification of the subsystem by an additional decomposition, in order to obtain that each subsystem expresses one scalar function only. Our modelling approach is based on the assumption that the number of functional subsystems is constant in time. However, the aggregation or fragmentation of agents into subsystems may possibly occur in many circumstances during the system dynamics. Including these possibilities in our mathematical models can add a new dimension to the global system dynamics. However, it must be observed that this goal cannot be pursued by a straightforward generalization of the approach proposed in this chapter, since a further development of the mathematical structure cannot describe aggregation or fragmentation events, unless properly modified. The proposed modelling approach accounts for interactions that do not depend on the geometry of the system, and specifically on the geographic position of the interacting subsystems. This simplification can be considered adequate when communications among agents occur through delocalized devices: for instance the media, the telephone, the Internet, and similar. On the contrary, communications among agents may be constrained by networks that organize and select the dialogue between and among functional subsystems. Like in the previous case, accounting for locality of interactions requires a further development of the mathematical modelling framework, to incorporate network structures. Accounting for this aspect in the mathematical modelling framework may allows the investigation of interesting new aspects, like the influence of the network structure and its degree of clustering on the overall system dynamics.

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References 1. Ajmone Marsan G., Bellomo N., and Egidi M., Towards a Mathematical Theory of Complex Socio-Economical Systems by Functional Subsystems Representation, Kinetic and Related Models, 1, 249–278, (2008). 2. Ajmone Marsan G., On the Modelling and Simulation of the Competition for a Secession under Media Influence by Active Particles Methods and Functional Subsystems Decomposition, Computers and Mathematics with Applications, 57, 710–728, (2009). 3. Ajmone Marsan G., New Paradigms Towards the Modelling of Complex Systems in Behavioral Economics, submitted to Mathematical and Computer Modelling, Mathematical and Computer Modelling, 50, 584–597, (2009). 4. Alesina A. and Spolaore E., Conflict, Defense Spending and the Number of Nations, European Economic Review, 50, 91–120, (2006). 5. Aleskerov F., Kalyagin V. and Pogorelskiy K., Actual Voting Power of the IMF Members Based on Their Political-Economic Integration, Mathematical and Computer Modelling, 48, 1554–1569, (2008). 6. Ansolabehere S. and Leblanc W., A Spatial Model of the Relationship Between Seats and Votes, Mathematical and Computer Modelling, 48, 1409–1420, (2008). 7. Arlotti L., Bellomo N. and De Angelis E., Generalized Kinetic (Boltzmann) Models: Mathematical Structures and Applications, Mathematical Models and Methods in Applied Sciences, 12, 567–591, (2002). 8. Arthur W. B., Durlauf S. N. and Lane D., Eds., The Economy as an Evolving Complex System II, Addison and Wesley, New York, (1997). 9. Axelrod R., The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration, Princeton University Press: Princeton, N.J., (1997). 10. Baum M. A., Soft News and Foreign Policy: How Expanding the Audience Changes the Policies, Japanese Journal of Political Science, 8, 115–145, (2007). 11. Bellomo N., Modelling Complex Living Systems - A Kinetic Theory and Stochastic Game Approach Birkh¨ auser, Boston, (2008). 12. Bellomo N., Bianca C. and Delitala M., Complexity Analysis and Mathematical Tools Towards the Modelling of Living Systems, Physics of Life Reviews 6, 144–175, (2009). 13. Bertotti M. L. and Delitala M., From Discrete Kinetic and Stochastic Game Theory to Modelling Complex Systems in Applied Sciences, Mathematical Models and Methods in Applied Sciences, 14, 1061–1084, (2004). 14. Bertotti M. L. and Delitala M., Conservation Laws and Asymptotic Behavior of a Model of Social Dynamics, Nonlinear Analysis RWA, 9, 183–196, (2008). 15. Bertotti M. L. and Delitala M., On the Existence of Limit Cycles in Opinion Formation Processes Under Time Periodic Influence of Persuaders, Mathematical Models and Methods in Applied Sciences, 18, 913–934, (2008). 16. Camerer C. F., Loewenstein G., and Rabin M., Advances in Behavioral Economics, Princeton University Press: Princeton, NJ (2003). 17. Camerer C. F., Behavioral Game Theory: Experiments in Strategic Interaction, Princeton University Press: Princeton, NJ (2003). 18. Camerer C. F. and Fehr E., When Does ‘Economic Man’ Dominate Social Behavior?, Science, 311, 47–52, (2006). 19. Epstein M. and Axtell R., Growing Artificial Societies: Social Science from the Bottom Up, MIT Press/Brookings Institution: Cambridge, MA (1996).


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20. Esser F. and Pfetsch B., Comparing Political Communication. Theories, Cases, and Challenger, Cambridge University Press: Cambridge, U.K., (2004). 21. Hartwell H. L., Hopfield J. J., Leibner S., and Murray A. W., From Molecular to Modular Cell Biology, Nature, 402, 47–52, (1999). 22. Iyengar S., The Accessibility Bias in Politics: Television News and Public Opinion, International Journal of Public Opinion, 2, 1–15, (1990). 23. Kirman A. and Zimmermann J., Eds. Economics with Heterogeneous Interacting Agents, Lecture Notes in Economics and Mathematical Systems, 503, Springer, Berlin, (2001). 24. Lumby C. and Probyn E., Remote Control. New Media, New Ethics, Cambridge University Press, Cambridge, U. K., (2003). 25. Rabin M., A Perspective on Psychology and Economics, European Economic Review, 46, 657–685, (2002). 26. Rubinstein A., Modeling Bounded Rationality, Cambridge University Press: Cambridge (1998). 27. Simon H. A., Theories of Decision-Making in Economics and Behavioral Science, The American Economic Review, 49, 253–283, (1959). 28. Simon H. A., Models of Bounded Rationality, Vols. 1 and 2., MIT Press: Cambridge, MA (1982). 29. Simon H. A., Models of Bounded Rationality, Vols. 3., MIT Press: Cambridge, MA (1997).

Mathematics and physics applications in sociodynamics simulation: the case of opinion formation and diffusion Giacomo Aletti1 , Ahmad K. Naimzada2 , and Giovanni Naldi3 1 2

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Department of Mathematics and ADAMSS Center, Universit` a degli studi di Milano, via Saldini 50, 20133 Milano, Italy, [email protected] Department of Quantitative methods for Business Economics, Universit` a degli studi di Milano Bicocca, Piazza dell’Ateneo Nuovo 1, 20126 Milano, Italy, [email protected] Department of Mathematics and ADAMSS Center, Universit` a degli studi di Milano, via Saldini 50, 20133 Milano, Italy, [email protected]

Summary. In this chapter, we briefly review some opinion dynamics models starting from the classical Schelling model and other agent-based modelling examples. We consider both discrete and continuous models and we briefly describe different approaches: discrete dynamical systems and agent-based models, partial differential equations based models, kinetic framework. We also synthesized some comparisons between different methods with the main references in order to further analysis and remarks.

1 Opinion dynamics Opinion dynamic models describe the process of opinion formation in groups of individuals: as the opinion behavior emerges, evolves, spreads, erodes, or disappears. Here we provide a brief overview of models and simulation tools for the opinion dynamics. Most people hold and exchange opinions about a lot of topics, from politics and sports to health, new products and the lives of others. These opinions can be either the result of serious reflection or as is often the case when information is hard to process or obtain, formed through interactions with others that hold views on given issues. The modelling of the opinion dynamics try to understand when the opinion formation leads to consensus, polarization or fragmentation within an interacting group. Opinion dynamics is one of the most widespread topics of Sociophysics (application of methods from physics to human relations) and Sociodynamics (the attempt to build up a modelling strategy allowing in principle of an integrative quantitative description of dynamic macro-phenomena in the society). Sociophysics covers numerous topics of social sciences and addresses many different problems G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 8, c Springer Science+Business Media, LLC 2010

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including social networks, language evolution, population dynamics, epidemic spreading, terrorism, voting, and coalition formation. Among them the study of opinion dynamics has become a main stream of research. In fact, the public opinion is nowadays a feature of central importance in modern societies making the understanding of its underlining mechanisms a major challenge. The dynamics of agreement/disagreement among individuals is complex, because the individuals are. Physicists working on opinion dynamics aim at defining the opinion states of a population, and the elementary processes that determine transitions between such states. The main question is whether this is possible and whether this approach can shed new light on the process of opinion formation. Computer simulations play an important role in the study of social dynamics as they parallel more traditional approaches of theoretical physics and mathematical modelling, where a system is described in terms of a set of equations. One of the most successful methodologies used in social dynamics is agent-based modelling [1, 2]. In agent-based modelling (ABM), a system is modelled as a collection of autonomous decision-making entities called agents. Each agent individually assesses its situation and makes decisions on the basis of a set of rules. Agents may execute various behaviors appropriate for the system they represent. Repetitive interactions between agents are a feature of agent-based modelling, which relies on the power of computers to explore dynamics out of the reach of pure mathematical methods. At the simplest level, an agent-based model consists of a system of agents and the relationships between them. Even a simple agent-based model can exhibit complex behavior patterns and provide valuable information about the dynamics of the real-world system that it emulates. In addition, agents may be capable of evolving, allowing unanticipated behaviors to emerge. Sophisticated ABM sometimes incorporates neural networks, evolutionary algorithms, or other learning techniques to allow realistic learning and adaptation. ABM is a mindset more than a technology: it consists of describing a system from the perspective of its constituent units. The description of emerging collective behaviors and self-organization in multiagent interactions has gained increasing interest from various research communities in biology, ecology, robotics, and control theory, as well as sociology and economics. In the biological context, the emergent behavior of bird flocks, fish schools, or bacteria aggregations, among others, is a major research topic in population and behavioral biology and ecology [3–11]. Likewise, the coordination and cooperation among multiple mobile agents (robots or sensors) have been playing central roles in sensor networking, with broad applications in environmental control [12]. Emergent economic behaviors, such as distribution of wealth in a modern society [13–17], or the formation of choices and opinions [18–21], are also challenging problems studied in recent years in which emergence of universal equilibria is shown. Also, the development of a common language in primitive societies is yet another example of a coherent collective behavior emerging within a complex system [22, 23].

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The social world that we observe reflects a lot of interdependent processes, with macro level structures of organizations, communities, and societies both emerging from and constraining the micro level interactions of individuals. Many social phenomena, such as the spread of epidemics, or the dissolution of organizations, are inherently time varying and depend on interactions between entities within a social system. Understanding the link between microlevel interactions and macrolevel dynamics promises to have profound impact on how human societies, organizations, and nations might be structured and how related policy decisions should be made. As we mentioned earlier, an increasing number of scientists are using mathematical and computational models to elucidate theoretical problems in social dynamics, often by applying general theories or methods that are well developed in the natural and physical sciences with a view to gaining insight into the underlying generative processes or the dynamic consequences of social relationships. It may be surprising, but the application of concepts from the natural sciences to social sciences is at least 25 centuries old. In fact, the Greek philosopher Empedocle stated that humans are like liquids: some mix easily like wine and water, and others refuse to mix. The discovery of quantitative laws in the collective properties of a large number of people was one of the pushing factors for the development of statistics. Many scientists and philosophers called for some quantitative understanding on how such precise regularities arise. Among many others, Hobbes, Laplace, Comte, Stuart Mill shared this line of thought. More recently, Majorana [24, 25] in the 1940s suggested to apply quantum-mechanical uncertainty to socio-economic questions. Weidlich [26] studied similar questions since 1971. In the same year, Thomas Schelling (Nobel Prize for Economics in 2005) published [27] his highly acclaimed model for for urban segregation in the first issue of Journal of Mathematical Sociology. Moreover, in the same issue of the journal Sakoda [28] presented a closely related work whose basic design was already present in his unpublished dissertation of 1949. In [33] Galam gave a personal testimony of Sociophysics going back to his 1982 publication. Other references may be found in the books of Arnopoulos [34] and Schweitzer [35], while some review articles can be found for example in [36–38]. 1.1 Schelling model We present here a significant and historical segregation model proposed by Schelling in 1971 [27] for the study of ethnic segregation in the United States. Schelling supposed that people had a threshold of tolerance of other ethnic groups based on the neighbored people. If, for instance, the threshold of tolerance was 40%, people were content to stay in the place they live provided that at least four in ten of their neighbors were from the same ethnic group. If this were not so, they would try to move to another neighborhood in which at least 40% were of their own group. The conventional assumption is that ethnic segregation in the USA is at least partly due to the fact that whites

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are prejudiced and have a tolerance threshold of over 50%. They therefore moved out of urban neighborhoods that had a majority of blacks, leaving the neighborhood with a still higher proportion of black people and, thus, accelerating the tendency towards complete segregation. Schellings point was that tolerance thresholds much lower than 50% could lead to the same result. Even a threshold as low as 30% could result in almost complete segregation. Thus, although people might be quite content with being in the minority in a neighborhood, so long as they demanded that some small proportion of their neighbors were of the same ethnic group as themselves, segregation could emerge. The original Schelling’s model was very simple. Take a large chessboard, and place a certain number of black and white counters on the board, leaving some free places. A counter prefers to be on a square where a certain fixed percentage of the counters in his Moore neighborhood (his eight nearest neighbors) are of its own color to the opposite situation. From the counters who wish to migrate one is chosen at random and moves to a preferred location. This model, when simulated, yields complete segregation even though people’s preferences for being with their own color are not strong. In Fig. 1 we show some simulations by using different threshold of tolerance, each chessboard refers to the steady state when there are no more unhappy. Schelling’s result has become famous precisely because the preferences of individuals for

Fig. 1. The role of preferences on the end pattern of segregation that emerges in the Schelling’s model


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segregation were not particularly strong. Note that the Schelling result is of interest to economists because it illustrates the emergence of an aggregate phenomenon that is not directly foreseen from the individual behavior. The Schelling model is based on the idea that an individual agent makes decisions based on his preferences or utility function. Then the agent’s satisfaction is equivalent to the energy stored in him. An increase in happiness is a decrease in internal energy. An agent, therefore, wants to minimize his energy, which is generated either by taking some action or through the interaction with his environment. The Schelling model assumes that the agent’s utility depends on his local environment and that he moves if the utility falls below a certain threshold. Such a process is easily simulated through the twodimensional Ising model [29–31]. In the simplest case, as black and white, we have two Ising spin orientations. In this Ising model, each site i on a square lattice carries a variable Si with value +1 or −1. For each pair (i, k) of nearest neighbors produces an energy contribution Eik equal to Eik = −JSi Sk with some proportionality constant J. The total energy E is proportional to the total unhappiness is the sum of this pair energy over all neighbor pairs of the lattice Eik . In statistical physics, different distributions of the spins Si are realized with a probability proportional to e(−E/KB T ) , where T is the absolute temperature and KB the Boltzmann constant. We point out that the relevant quantity is the ratio (KB T /J). The Metropolis kinetics of the system is simulated by flipping a spin if and only if a random number between 0 and 1 is smaller than the probability e(−ΔE/(KB T )) , where ΔE is the variation of the total energy. When the Glauber kinetics [32] is used the flipping probability is e(−ΔE/(KB T )) /(1 + e(−ΔE/(KB T )) ). For Glauber or Metropolis, after very long times (measured by the number of sweeps through the lattice) one of the two possibilities dominates at the end which depend on the temperature T . If T is not larger than the critical temperature Tc , for Kawasaki dynamics the fraction of black sites remains constant, and we get two large domains. For higher temperatures above Tc only small clusters and no large domains are formed. In this Ising model, two neighboring spins have due to their interaction JSi Sk a higher probability to belong to the same group than to belong to the two different groups. Schelling’s model avoided probabilistic rules and thus counted neighbors Si = ±1 at zero temperature. As Schelling moved only one person at a time, and made no exchange of two people simultaneously as in Kawasaki kinetics, he introduced a large fraction of empty residences. Thus at each step, one unhappy person or family moves into the closest vacancy where life would be happy. For a study of the model from this physical point of view and a link between Schellings socio-economic model of segregation and the physics of clustering we refer to [39] and [40]. There are several variants on Schelling’s original model modifying the form of the utility function used by Schelling, the size of neighborhoods, the rules for moving, and the amount of unoccupied space. Many of these fail to give large domains: only small clusters are seen. In many real cities, there are huge cluster which extends over many square kilometers. Thus, Schelling’s model

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does not give always the desired results. In general, the critique social systems models are too simplified is most of the times very well grounded. However, in most situations qualitative (and even some quantitative) properties of large scale phenomena can be simulated and reproduced. A discussion about the application of self-organising segregation and migration models can be found in [41–43].

2 Binary opinion dynamics and beyond Scientists working on opinion dynamics aim at defining the opinion states of a population, and the processes that determine transitions between such states. In a mathematical model, opinion has to be a variable, or a set of variables. This may appear too reductive, thinking about the complexity of a person and of each individual position. Rightly, Humans do not like to be treated like a number, and indeed the human brain is much more complex than a set of variables. But one can observe that the value of an opinion or a decision could be represented by a numerical vector. As example binary opinions and binary choices are frequent in the everyday life: Windows/Linux, buying/selling, Coca Cola/Pespi Cola, etc. Moreover, appear to act on the opinion some kind of social forces even though they may appear of little relevance to any important decisions. For example we can consider the following simple experiment, first performed four decades ago and reproduced many times since. Stand on a busy street corner and look up at the sky. The crowd will part around you, indifferent to whatever it is you may be looking at. Now enlist the help of a friend to stand beside you and also look skyward. Soon, others will stop and gaze up as well. A similar thing would happen if you and your friend boarded an empty elevator and faced the rear wall. As more passengers boarded many would face the back wall too. Then a kind of peer pressure seems to appear. 2.1 The Sznajd model There are several models used in modelling consensus formation and we tried to name a few historical introduction listed earlier. Here we consider in more detail a recent Sznajd model only as an example just to illustrate some issues and problems in the simulation of the opinion formation. The general framework of the model is as follows: at any given time a distribution of opinions exists in a given population; at each time step some group of individuals interact and as a result, one or several opinions are shifted (generally towards some consensus among the group); as a result of these interactions, some form of pattern of opinions is observed. An important aspect always present in social dynamics is topology which is the structure of the interaction network describing who is interacting with whom, how frequently and with which intensity. In the Sznajd model [20, 46], as in the traditional statistical physics and in the Ising model, the geometry is a one-dimensional regular lattice.

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Agents are, thus, supposed to sit on vertices (nodes) of the lattice, and each agent can interact with the two adjacent agents. More generally, it is possible to consider a d-dimensional lattice of N d sites carries opinion variables Si , i = 1, 2, . . . , N d . The lattice may be square (four nearest neighbours), triangular (six nearest neighbours), many other choices are also possible. The opinion variables could be scalar or vector real variables. Usually, they are scalar variables and are integers between P1 and P2 or take values in a finite set of opinions {P1 , P2 , . . . , Pk }. The individual opinions Si in the Sznajd model are represented in our model by Ising spins (“yes” or “no,” −1 or 1), that is to say the opinions are binary opinions. Such an approach is not new but it had been used earlier (see for example [33] or [47]). At each time step, each Si is calculated from a rule according the spin dynamics [20]: • A pair of opinions (spins) Si , Si+1 is chosen to change their nearest neighbors, i.e., the opinions (spins) Si−1 and Si+1 .

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• If Si Si+1 = 1 (the opinions are equal) then Si−1 = Si and Si + 2 = Si (social validation). • If Si Si+1 = −1 then Si−1 = Si+1 and Si+2 = Si . The model was originally called USDF after the trade union maxim “United we Stand, Divided we Fall,” in other words: If you see two or more people sharing an opinion on certain issue, you are tempted to join. The crucial difference of the model compared to other Ising-type models is that information flows outward. The dynamic rules of the model lead to two possible steady states: the complete consensus (ferromagnetic state) in which all Si = 1 or all Si = −1; the stalemate (antiferromagnetic state) in which 50% of opinions Si = 1 and 50% are equal to −1. However, the last 50 − 50 state is realized in a very special way: every member of the community disagrees with his nearest neighbor. Even if the Ising model with only next nearest neighbor interaction has such kind of fixed point (ferro and antiferromagnetic state) it is unrealistic as a social state. Then, new dynamic rules were proposed [46], for example in the case of the disagreements, Si Si+1 = −1, then a new update is computed: Si−1 = Si and Si+2 = Si+1 . Also a reasonable rule is to assume that the individual keeps its opinion in such a case, rather than opposing its neighbor. That is if Si = −Si+1 , nothing happens at that time step, the system is not altered. A global index which corresponds to the general opinion of the society is the magnetization m(t) at the time t defined as: (number of opinions 1) − (number of opinions − 1) 1 m(t) = = Si , N N i where N is the number of people in the community. In the Fig. 2 we show the behavior of the opinions in a community and the evolution of the magnetization in the case of the original Sznajd model and for the modified model with the different active update rule. In [20] Sznajd-Weron and Sznajd have investigated some statistical properties of the magnetization function m(t) and some interesting behavior of the opinion changes of one particular individual. In particular they observed that if an individual changes the opinion at time t, he (she) will probably change it also at time t + 1. Moreover, if τ denotes the time needed by an individual to change the opinion its distribution P (τ ) follows a power law with an exponent −3/2. It is well known that the changes of opinion in a community are not only determined by the individual contacts between neighbors but also the social impact has a role. It is possible to introduce a noise p, some kind of social temperature, which is the probability that an individual, instead of following the dynamic rules, will make a random decision. In [20] a computational study is performed to study the influence on the distribution P (τ ) of the parameter p. Sznajd model has been modified and applied in marketing, finance, and politics (see for example [38]). In agreement with social theory assigning four nearest neighbors to an individual rather than only two is a more realistic simplification of a real society. So in search for a more realistic model, two dimensional version of Sznajd


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model was proposed by Stauffer et al. [48] for two-dimensional regular lattice. In accordance with the dimensionality of the model, six different versions were proposed. Here, we focus on two of the most realistic of the six: • A 2 × 2 square of four neighbors are chosen at random. If all four opinions of the square are the same then all eight neighbors to this square follow the same value. Otherwise, the system is not altered, the neighbors are unchanged. • A bond is chosen at random. If the two opinions forming the bond are the same, then all six neighbors to this bond follow the same value. Otherwise, the system is not altered, the neighbors are unchanged. In Fig. 3 we show one example of the behavior of the two-dimensional Sznajd model following the two rules described earlier, simulations were performed on a N × N square lattice, where individuals were placed on the lattice as spins in the two dimensional Ising model. Periodic boundary conditions are used in both directions. As for the original Sznajd model simulation, random sequential updating was used as in Monte-Carlo simulation. In [48] the emergence of a phase transition phenomena with respect to the size of the system was studied: this is the only important difference between the onedimensional model and the two-dimensional model. A natural extension of the binary Sznajd model developed in order to fit the experimental observation in some elections is to allow more than two opinions for each individual. So, if there are M candidates running for the post then each individual can have M possible opinions. This is an example of a many-opinion modification to the Sznajd model, and also equivalent to extending the Ising model to M -state Potts model [49]. Other steps toward more realism is to investigate the model on a complex network with different topology, adding noise or diffusion (see for examples [46]). In a short note [50] Ochrombel suggested a drastic simplification of the Sznajd model. In the Ochrombel version it is not necessary to have a cluster of identical opinions to change the neighbors. Any individual is capable to convince her neighbors to select the same opinion. We choose an agent i at random. Then, choose j randomly among neighbors and set Sj = Si . In the case of a fully connected network the Ochrombel simplification is equivalent to voter model, whose dynamical properties were studied, e.g., in [51]. In this last case for binary opinions the state is described by the magnetisation variable m(t) only. It can be found in the thermodynamic limit N → ∞ that the probability density Pm (m, t) of the magnetization at time t evolves according to the partial differential equation [52] ∂ ∂2 Pm (m, t) = (1 − m2 )Pm (m, t) . ∂t ∂m2

(1)

For the original binary Sznajd model but in the case of a fullyconnected network a different time scaling is needed in order to get sensible thermodynamic

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Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi 0

0

50

50

100

0

50 t=1

100

100

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50

50

100

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50 t=10

100

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50 t=5

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50 t=50

100

0.52 0.515 0.51 m(t)

0.505 0.5 0.495 0.49 0.485 0.48 0

10

20

30

40

50

t

Fig. 3. The time evolution for a two-dimensional Sznajd model as proposed by Stauffer et al. [48] and the behavior of the corresponding magnetization function m(t)

limit [52]. Then the equation for the probability density is the following Fokker–Planck equation ∂ ∂ Pm (m, t) = − (1 − m2 )mPm (m, t) . ∂t ∂m

(2)

The dynamics is those of a pure deterministic drift and no diffusion ever smears out the evolving probability packet. The possibility to have the Fokker– Planck equation provides a great tool for the theoretical study of the opinion dynamics model.


213

One issue of interest in the opinion dynamics concerns the importance of the binary assumption: what would happen if opinion were a continuous variable such as the worthiness of a choice or some belief about the adjustment of a control parameter? Continuous opinions invalidate some of the concepts adopted in models with discrete choices so they require a different framework. The initial state is usually a population of N agents with randomly assigned opinions, represented by real numbers within some interval. The opinion clusters could be one (consensus), two (polarization), or more (fragmentation). In principle, each agent can interact with every other agent. In practice, there is a real discussion only if the opinions of the people involved are sufficiently close to each other. This realistic aspect of human communications is called bounded confidence. Usually, it is expressed by introducing a real number ε, the uncertainty or tolerance, such that an agent, with opinion x, only interacts with those of its peers whose opinion lies in the interval ]x − ε, x + ε[. As example consider the model introduced by Deffuant et al. [53]. There are, again, N individuals with opinions Si . In each update step, two of them, say, i and j, are chosen randomly. Then, we check to see whether their opinions differ less than (or equally to) the confidence bound ε . In the positive case, their opinions are slightly shifted towards each other, Si (t + 1) = (1 − μ)Si (t) + μSj (t), Sj (t + 1) = (1 − μ)Sj (t) + μSi (t), for |Si (t) − Sj (t)| ≤ ε, where ε is a parameter. For very large numbers of individuals, N → ∞, the dynamics can be expressed in terms of the continuous distribution of opinions P (s, t) which satisfies an equation with a fairly complex behavior [54].

3 Agents based model and discrete dynamical systems In [45], the authors consider a micro-scale approach. The population is composed by n agents. At each discrete time t = 0, 1, . . . the opinion of the ith agent is given by the quantity xi (t) and therefore the opinion profile at time t is given by the vector x(t) = (x1 (t), . . . , xn (t)). The ith agent takes into account the opinion of the jth agent by means of the weight aij (t, x(t)). Accordingly, the opinion formation evolves in the following way: xi (t + 1) = ai1 (t, x(t))x1 (t) + ai2 (t, x(t))x2 (t) + · · · + ain (t, x(t))xn (t) or, in matrix form, the general model is given by: x(t + 1) = A(t, x(t))x(t). The only assumption on the matrix A is that it is a stochastic matrix (each row sums to 1). Obviously, further assumptions on the weight matrix A lead to different studied models. The review article [45] presents models which can be analytically studied and others that are computationally faced. The consensum property is given when all the opinion converge to the same opinion

214


that depends only on the initial opinion profile: given x(0), there exists c = c(x(0)) such that xi (t) → c for any 1 ≤ i ≤ n. When x(t) converges to a vector with different components, we face an opinion fragmentation. The first part of [45] is devoted to a review of the linear models, while the second part refers to some computational results on some nonlinear models. 3.1 Linear models If A(t, x(t)) ≡ A, (3) reduces to the classical model of fixed weights x(t + 1) = Ax(t). A generalization of the previous model is given by Friedkin and Johnsen in [44] x(t + 1) = Gx(0) + (I − G)Ax(t), where G is a diagonal matrix containing the degree of adhesion to the initial opinion: G = diag(g1 , g2 , . . . , gn ). Obviously, when G ≡ 0, (3.1) reduces to (3.1). Note that in (3.1), x(t) = V (t)x(0), where V (t) = M t +

t−1

Mi G

with M = (I − G)A.

i=0

Analytical conditions on the matrix A ensure limit properties as consensum or fragmentation. As an example, if A is irreducible, limt x(t) = (I − M )−1 Gx(0) when G = 0. Similar results, even if less sharp, may be found with the last model presented in the review part of [45]. This model is still linear but timevariant (or, in the theory of Markov chains, inhomogeneous), that is x(t + 1) = A(t)x(t), where the entries of matrix A(t), i.e., the weights, are dependent on time only. The time variant model (3.1) portrays, e.g., the so called “hardening of positions” where agents put in the course of time more and more weight on their own opinion and less weight on the opinion of others. 3.2 Nonlinear models The models proposed in [45] depend on opinions itself and hence the models are nonlinear. Analytical insights are not so easy to obtain. Therefore, computer simulations were used to investigate these models. An agent i takes only those agents j into account whose opinions differ from his own not more than a certain confidence level. More precisely, if xi (t + 1) = |I(i, x(t))|−1 xk (t), where k

I(i, x(t)) = #{1 ≤ k ≤ n : l ≤ xk − xi ≤ r },


215

the confidence level for each agent is the interval [l , r ]. We have the nonlinear model x(t + 1) = A(x(t))x(t), where aij (x) =

1 . #{1 ≤ k ≤ n : l ≤ xk − xi ≤ r }

Asymptotic configurations are numerically studied both in the symmetric case (i.e., l = r ), and in the asymmetric one.

4 The kinetic approach The goal of the forthcoming kinetic model of opinion formation, is to describe the evolution of the distribution of opinions in a society by means of microscopic interactions among individuals which exchange information. Opinion is represented as a continuous variable w ∈ I, with, as example, I = [−1, 1]. The extremes w = ±1 represent extreme opinions. Toscani has recently studied [55], by means of kinetic collision-like models, the distribution of opinion among individuals in a simple, homogeneous society. This model is based on binary interactions. When two individuals with preinteraction opinion v and w meet, then their posttrade opinions (postcollisional) v ∗ and w∗ are given by: v ∗ = v − γP (|v|)(v − w) + ηv D(|v|) w∗ = w − γP (|w|)(w − v) + ηw D(|w|) where the coefficient γ ∈ (0, 1/2) is a given constant, while ηv and ηw are random variables with the same distribution with variance σ 2 and zero mean. The constant γ measures the compromise propensity while the variance σ 2 is related to the modification of opinion due to diffusion. The functions 0 ≤ P (·) ≤ 1 and 0 ≤ D(·) ≤ 1 describe the local relevance of the compromise and diffusion for a given opinion. In absence of the diffusion contribution, η = 0, if P (·) is not constant the total momentum is not conserved and it can increase or decrease depending on the opinions before the interaction. While if P is assumed constant, the interaction correspond to a granular gas like interaction [56]. Let f (w, t) denote the distribution of opinion w ∈ I at time t ≥ 0. Standard methods [57] of kinetic theory allow to describe the time evolution of f as a balance between bilinear gain and loss of opinion terms ∂f 1 = β f (v )f (w ) − βf (v)f (w) dwdηv dηw , (3) ∂t J B2 I where (v , w ) are the preinteraction opinions that generate the couple (v, w) of opinions after the interaction, J is the Jacobian of the transformation of (v, w) into (v , w ), while the kernels β and β are related to the details of the binary interaction, finally B is the space of the random variables η. Following

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Toscani [55] different Fokker–Planck equations can be obtained by considering different limits and simplifications. In the following section we consider only one example. For the case of a kinetic model for the the evolution of the opinion in a closed group with respect to a choice between multiple options, we refer to [58]. In some sense intermediate between the Kinetic approach and agent-based model we can also consider an alternative approach based on active Brownian particles which interact via a communication field (see [59] and references therein). This scalar field considers the spatial distribution of the individual opinions; further, it has a certain lifetime, reflecting a collective memory effect, and it can spread out in the community, modelling the transfer of information. Also the active particle approach [60] leads to the derivation of evolution equations for a one-particle distribution function over the microscopic state. Two types of equations, to be regarded as a general mathematical framework for deriving the models, are derived corresponding to short- and long-range interactions. The active particle approach gives a general mathematical framework in order to link microscopic and macroscopic descriptions of the dynamics. 4.1 An example of nonlinear continuous models of opinion formation In [61], the large-time behavior of solutions of the equation ∂f ∂

=γ (1 − x2 )(x − m(t))f ∂t ∂x

(4)

is studied, where the unknown f (x, t) is a time-dependent probability density which may represent the density of opinion in a community of agents. This opinion varies between the two extremal opinions represented by ±1, so that x ∈ [−1, 1]. The constant γ is linked to the spreading (γ = −1) or to the concentration (γ = +1) of opinions. In (4) m(t) represents the mean value of f (·, t), m(t) =

x f (x, t) dx,

(5)

[−1,1]

and its presence introduces a nonlinear effect into its evolution. Equation (4) describes the evolution of a probability density which represents the density of opinions in a community. For all values of the constant γ, the time-evolution driven by this equations leads the density toward a equilibrium state that is described in terms of two Dirac masses (γ = −1) or to a unique Dirac mass (γ = 1), see [61]. The convergence result toward equilibrium holds in weak-measure sense. A suitable way of treating convergence results of (4) is based on a rewriting of this equation in terms of pseudo-inverse functions. In fact, let F (x) denote the probability distribution induced by the density f (x), F (x) = f (y) dy. (6) (−∞,x]


217

As F (·) is not decreasing, we can define its pseudo inverse function (also called quantile function) by setting, for ρ ∈ (0, 1), X F (ρ) = inf{x : F (x) ≥ ρ}. Equation (4) for f (x, t) takes a simple form if written in terms of its pseudo inverse X(ρ, t). [61, Theorem 3.1] shows in fact that the evolution equation for X(ρ, t) reads ∂X(ρ, t) = −γ (X(ρ, t) − m(t)) (1 − X 2 (ρ, t)), (7) ∂t 1 where now ρ ∈ (0, 1) and m(t) = 0 X(ρ, t) dρ. Existence and uniqueness of solutions is studied. Then results on their large-time behavior are derived. First, it is noted that the initial masses in +1 (called p+1 ), and in −1 (called p−1 ) remain unchanged in time. For what concerns concentration (γ = 1), the steady state is characterized by the distribution p−1 δ−1 + (1 − p1 − p−1 )δa + p1 δ+1 . Obviously, the mean value m∞ = p1 −p−1 +a(1−p1 −p−1 ) characterize it. In [61], it is proved that if (1 − p1 )(1 − p−1 ) < 1 (i.e., if there are masses in ±1 at time t = 0) then, m∞ = p1 − p−1 . Otherwise, if log (1 + X(ρ, 0))/(1 − X(ρ, 0)) ∈ L1 (0, 1) then m∞ =

exp {T (0)} − 1 , exp {T (0)} + 1

(8)

1 where T (0) = 0 log 1+X(ρ,0) 1−X(ρ,0) dρ. The spreading case is difficult to handle analytically due to the nonlinearity present through the term m(t). Therefore, in [61], numerical methods are discussed to capture the behavior for large t. In Fig. 4 we show the evolution of numerical and quantile solution for a benchmark case as reported in [61]. 1 0.8

f(x;0) concentration T = 2 spreading T = 2

0.6 0.4

f(x;·)

X(½;·)

0.2 exact initial numerical

0 −0.2 −0.4 −0.6 −0.8

−1

−0.5

0

x

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−1 0

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1

Fig. 4. A benchmark case from [61]: evolution of density function (left hand side figure) and comparison between analytical and numerical solution for the quantile function (right hand side figure)

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5 New opportunities: an opinion Different computational techniques are useful for developing and adapting social dynamics models. Differential Equations Systems of both ordinary and partial differential equations are useful tools for developing models at the aggregate level, but are more limited when considering the interdependent behavior of individuals in a population. Agent-based methods are particularly useful for studying the microlevel behavior of individual objects in a system. The interactions between agents may be stochastic according to defined interaction and choice probabilities, and may be subject to exogenous shocks. Given the limited applicability of systems dynamics models using differential equations to populations of heterogeneous and interdependent agents, agentbased models may be used in conjunction with such systemlevel models or by coupling agents at different resolutions to serve as a basis for developing multilevel models. Markov chains and similar probabilistic models have a wide spectrum of applicability in social and economic process even if the independence assumptions that they make limit their modeling power. General theoretical frameworks provide the conceptual underpinnings for a variety of modeling techniques. Examples include Interacting Particle Systems and game theory. The fundamental complexity of the dynamics of social phenomena and the complications associated with representing agents who engage in rational, responsive, and realistic interactions present challenges. We point out only few opportunities and the list is certainly not exhaustive. • In order to study the effectiveness of possible interventions in social and economical problems, agent based methods must be simulated numerous times to compare alternative intervention scenarios, which is a limitation of method. Results of agent based methods are not optimal solutions, but rather scenarios with various assumptions. Thus, one possible improvement may affect the possibility of developing control-theoretic approaches for agent-based methods, which could be applied in studying interventions. • More realistic topologies must be taken into account in the opinion formation models. Some models has already been developed for suitable networks as scale-free networks but a more general stability analysis with respect to topology changes is needed. • Kinetic approach for a nonhomogeneous society must be considered (for example for hierarchical organization). • More effective validation methods for models in social sciences including methodological goals, criteria for evaluating fit, and model interpretation and use of error.

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Global dynamics in adaptive models of collective choice with social influence Gian-Italo Bischi1 and Ugo Merlone2 1 2

DEMQ (Dipartimento di Economia e Metodi Quantitativi), Universit` a di Urbino “Carlo Bo”, via Saffi n. 42, I-61029 Urbino, Italy, [email protected] Statistics and Applied Mathematics Department, Universit` a di Torino, Corso Unione Sovietica 218/bis, I-10134 Torino, Italy, [email protected]

Summary. In this chapter we present a unified approach for modelling the diffusion of alternative choices within a population of individuals in the presence of social externalities, starting from two particular discrete-time dynamic models – Galam’s model of rumors spreading [10] and a formalization of Schelling’s binary choices [7]. We describe some peculiar properties of discrete-time (or event-driven) dynamic processes and we show how some long-run (asymptotic) outcomes emerging from repeated short time decisions can be seen as emerging properties, sometimes unexpected, or difficult to be forecasted.

1 Introduction A classical theme in the mathematical modelling of social systems is the description of how the collective behavior affects the individual choices and, vice versa, how repeated choices of interacting individuals give rise, in the long run, to the emergence of collective behaviors and social structures. In other words, the mutual dependencies between “Micromotives and Macrobehavior” [25] or between “Individual Strategy and Social Structure” [27]. In this chapter we present two models describing the diffusion of alternative choices within a population of individuals in the presence of social externalities. The first model we consider is the Galam’s model of rumors spreading, as published in [10]. The diffusion of a given opinion by word-of-mouth mechanisms can be considered as a particular case of individual decisions (believing or not in a given opinion) affected by the opinion prevailing in the society. Several mathematical models have been used in sociological, mathematical, and physical literature (see, e.g., [12,15,26]). Starting from the pioneering works by Galam (see, e.g., [13]) opinion dynamics has become one of the main streams of sociophysics, an expanding field with hundreds of papers published in leading physical (and nonphysical) journals. It emerged in the 70s of last century and since then the number of topics covered has been increasing, see, e.g., [12] for a review. G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 9, c Springer Science+Business Media, LLC 2010

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Gian-Italo Bischi and Ugo Merlone

The second model we present is a mathematical formalization of a pioneering qualitative model proposed in [24], where a class of binary choice games with externalities is considered to model how individual choices are influenced by social externalities. In this famous seminal paper Schelling assumes that each agent’s payoff depends only on the number of agents who choose one way or the other and not on their identities; he provides qualitative explanations of several every-day life situations, as well as a general framework that includes some well known game-theoretic situations, such as the n-players prisoner’s dilemma or the minority games. Following this approach, in [7] an explicit discrete-time dynamic model is proposed; there a population of bounded rationality players is assumed to be engaged in a game where they repeatedly choose between two strategies through an adaptive adjustment process. This allowed the authors to study the effects on the dynamic behavior of different kinds of payoff functions that represent social externalities, as well as the qualitative changes of the asymptotic dynamics induced by variations of the main parameters of the model. The adaptive process by which agents switch their decisions depends on the difference observed between their own payoffs and those associated with the opposite choice in the previous turn; the switching intensity is modulated by a parameter λ representing the speed of reaction of agents: small values of such λ imply more inertia while, on the contrary, larger values of λ imply more reactive agents. This one-dimensional discrete dynamic model gives rise to different asymptotic dynamics, including convergence to steady states, periodic and even chaotic oscillations, as well as particular the structure of basins of attraction when several coexisting attractors are present. This latter effect is obtained in the case of non-monotonic payoff functions, a quite interesting situation in sociological applications (as explained by [24] and [15]) that in our mathematical framework leads to a dynamical system represented by an iterated noninvertible map. Moreover, in the limiting case of impulsive agents, represented by λ → ∞, a discontinuous dynamical system is obtained. In this case, border collision bifurcations cause the creation and destruction of periodic attractors as some parameters are varied. We finally illustrate a recent unified dynamic model, proposed in [8], which embeds these two models into a general discrete-time dynamic model for studying individual interactions in variously sized groups. In fact, while Galam’s model of rumors spreading considers a majority rule for interactions in several groups, Schelling’s framework considers individuals interacting in one large group, with payoff functions that describe how collective choices influence individual preferences. The general model proposed in [8] incorporates these two approaches and allows one to analyze how the social dynamics may differ depending on the size of the group they are taking place in. The chapter is organized as follows. In Sect. 2 Galam’s model of rumors is described, in Sect. 3 the formalization of Schelling’s model of binary choices, as given in [7], is summarized, together with the main results obtained in the case of monotonic and nonmonotonic payoff functions. In Sect. 4 the same model

Global dynamics in collective choices

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is considered in the limiting case of impulsive agents; some of the results concerning the global dynamic properties of piecewise linear discontinuous maps with the related problem of border collision bifurcations are provided. Finally, in Sect. 5, the unified model proposed by [8] is described; Sect. 6 concludes.

2 Galam’s model The model Galam proposes in [10] explains the so called Pentagon French hoax, according to which, on September the 11th, no plane crashed on the Pentagon. In this model individuals try to choose an opinion (true or false) on this rumor on the basis of repeated discussions in social gatherings. At each iteration, small groups of people get together and within each group they line up with a consensual opinion in which everyone agrees with the majority inside the group. The process is formalized as follows. The probability to be sitting at a group of size i is denoted by ai , i = 1, 2, . . . , L; obviously the constraint L

ai = 1

(1)

i=1

holds. In Galam’s model, the inclusion of one-person groups makes the assumption “everyone gathering simultaneously” realistic. At each social meeting, given the social spaces, individuals distribute among them according to probabilities ai Consider a N person population and assume two possible opinions, denoted3 by “A” and “B”; assume that at time t everyone is holding an opinion, i.e., NA (t) individuals are believing to opinion A and NB (t) persons are sharing the opinion B; it holds NA (t) + NB (t) = N . It is immediate to compute the probabilities to hold on A or B, respectively NA (t) and PB (t) = 1 − PA (t) . (2) N From this initial configuration, people discuss the issue at each social meeting; each cycle of multisize discussions is marked by an unitary time increment. The opinion modification process is the following. In any group of size k with j agents sharing opinion A and (k − j) sharing opinion B, if j > k/2 then all k members adopt opinion A; vice versa, if j < k/2, then everybody adopts opinion B. Finally, in the symmetric case j = k/2 the outcome is determined assuming a bias in favor of one of the two opinions. For a generic group of size k, the majority rule dynamics is formalized as: PA (t) =

PAk (t + 1) =

k

j

k−j

Cjk PA (t) (1 − PA (t))

,

(3)

j= k 2 +1 3

Actually, in [10] opinions are denoted by + and −; the notation we use here is to uniform the notation across all the models.

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where k/2 + 1 indicates the Integer Part of k/2 + 1 and Cjk = k!/(k − j)!j! are binomial coefficients. Considering as initial index of the sum k/2 + 1, models the bias in favor of opinion B in case of a local doubt is obtained. As a matter of fact, when k is odd the sum starts considering the minimum majority, while when k is even the sum starts at k/2 + 1. Aggregating all groups of size k = 1, ..., L, the overall updating process becomes L PA (t + 1) = ak PAk (t + 1) (4) k=1

with

PAk

(t + 1) given by (3), that is, the dynamics of PA is PA (t + 1) =

L

ak

k=1

k

j

k−j

Cjk PA (t) (1 − PA (t))

.

(5)

j= k 2 +1

A single step of the opinion dynamics can be illustrated as the example provided in Fig. 1. Assume there are three individuals among which two have opinion A and one opinion B; they can gather at a 2-size and a 1-size groups. For each individual the probability of sitting at a size 2 is 2/3 and the probability of sitting at the size 1 group 1/3.

Fig. 1. A one-step opinion dynamics. First stage, people sharing the two opinions are moving around. Grey have A opinion while black have opinion B. No discussion is occurring with 2 grey and 1 black. Second stage right, people is partitioned in groups of various sizes from one to two and no change of opinion occurs. Third stage, within each group consensus has been reached. As a result, they are now 1 grey and 2 black. Last stage, people are again moving around with no discussion

In this case, Galam’s formalization provides PA2 (1) =

2

4 Cj2 PA (0)j (1 − PA (0))2−j = C22 PA (0)2 (1 − PA (0))2−2 = 9 j= 22 +1 (6)


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and 1

2 Cj1 PA (0)j (1 − PA (0))1−j = C11 PA (0)1 (1 − PA (0))1−1 = 3 j= 12 +1 (7) And finally 2 14 24 12 PA (1) = + = . (8) ak PAk (1) = 39 33 27 PA1 (1) =

k=1

In the course of time, the same people keeps meeting in different groups in the same cluster configuration of size groups. The process and (5) is iterated to follow the time evolution of PA . This model, explains how the propagation of “absurd” rumors from initial tiny minorities may be explained by the bias driven by the tie effect. In particular, in [11] provides an analysis of how group shared belief may favor one opinion against the other. Finally, it must be observed that Galam’s model assumes implicitly that the number of agents is large. This is evident when considering the example discussed in Fig. 1, and labeling the agents as illustrated in Fig. 2. Here, the three individuals are identified as I1 , I2 , and I3 ; they have respective opinions A, A, and B, therefore, the probability PA (0) = 2/3. They can gather at a 2-size and a 1-size groups. For each individual the probability of sitting at a size 2 is 2/3 and the probability of sitting at the size 1 group 1/3. All the possible evolutions depend on the gathering configurations yet the final probabilities are, respectively, 2/3, 1/3, and 1/3, therefore, we have 1 2 1 1 4 12 11 11 PA (1) = + + = + + = 33 33 33 3 3 3 3 9 which is different from the result predicted by (8). By simple computations it is easy to find that when individuals are six, four with opinion A and two with opinion B, it holds PA (1) = 22/45. When the number of agents is large the model becomes more accurate; when the number of agents is small, [9] provides a different formula, yet, according to the same authors, the formula addresses this issue only partially.

3 A mathematical formalization of Schelling model In [24], a simple model which analyzes how individual choices are influenced by social interactions (social externalities), is proposed. In this model, agents face binary choices and are assumed to interact impersonally, i.e., each agent’s payoff depends only on the number of agents who choose one way or the other

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Fig. 2. A one-step opinion dynamics, when individuals follow different path. Each path has probability 1/3

and not on their identities. This model provides a qualitative explanation of a wealth of every-day life situations, and is general enough to include several games, such as the minority game and the well known n-players prisoner’s dilemma.


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A formalization of Schelling’s approach is proposed in [7]; there, a population of players is assumed to be engaged in a game where they have to choose between two strategies, A and B, respectively.4 The interaction is repeated over time in a discrete-time dynamic model. A continuum of agents is normalized to the interval [0, 1], and the real variable x ∈ [0, 1] denotes the fraction of players choosing strategy A. The payoffs associated to strategies A and B are function of x, say A : [0, 1] → R, B : [0, 1] → R; they are denoted by A(x) and B(x), respectively. As binary choices are considered, when fraction x is playing A, then fraction 1 − x is playing B. Therefore, x = 0 means that the whole population is playing B and x = 1 means that all the agents are playing A. Agents are homogeneous and myopic, that is, each of them is only interested to increase its own next period payoff. The dynamic adjustment is modeled assuming that x increases whenever A(x) > B(x) whereas it decreases when the opposite inequality holds. In this process all the agents update their binary choice at each time period t = 0, 1, 2, . . ., and xt represents the fraction of those playing strategy A at time period t. At time (t + 1), xt becomes common knowledge, therefore each agent is able to compute (or observe) payoffs A (xt ) and B (xt ). If A(xt ) > B (xt ) then a fraction δA of the (1 − xt ) agents that are playing B will switch to strategy A in the following turn; similarly, if A(xt ) < B (xt ) then a fraction δB of the xt players that are playing A will switch to strategy B. Formally, at any time period t, agents decide their action for period t + 1 according to: xt+1 = f (xt ) =

xt + δA g [λ (A (xt ) − B (xt ))] (1 − xt ) if A (xt ) ≥ B (xt ) xt − δB g [λ (B (xt ) − A (xt ))] xt if A (xt ) < B (xt ) , (9)

where • g : R+ → [0, 1] is a continuous and increasing function such that g(0) = 0 and limz→∞ g(z) = 1; it modulates how the difference between the previous turn payoffs affects the fraction of switching agents. • δA , δB ∈ [0, 1] represent the fraction of agents switching to A and B, respectively. When δA = δB , the propensity to switch to either strategies is the same. On the contrary, δA = δB represents a form of bias; in fact, given any payoff difference |A (x) − B (x)| > 0, δA > δB implies that when A(x) > B(x) switching from choice B to choice A is favored over switching from A to B when A(x) < B(x). • λ is a positive real number that represents the switching intensity (or speed of reaction) of agents as a consequence of the difference between payoffs. Small values of λ imply more inertia, i.e., anchoring attitude, of the actors involved, while, on the contrary, larger values of λ can be interpreted in terms of impulsiveness, see Sect. 4. 4

In [24] and [7] the two choices are denoted, respectively, R and L.

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The dynamics of model (9) has been examined in [7]. In particular, when the payoff functions have a single internal intersection point x∗ , depending on which choice is preferred on the left and right neighborhood of x∗ , the following propositions can be proved. Proposition 1. Assume that A : [0, 1] → R and B : [0, 1] → R are continuous functions such that • A(0) < B(0) • A(1) > B(1) • there exists unique x∗ ∈ (0, 1) such that A(x∗ ) = B(x∗ ), then dynamical system (9) has three fixed points, x = 0, x = x∗ , and x = 1, where x∗ is unstable and constitutes the boundary that separates the basins of attraction of the stable fixed points 0 and 1. All the dynamics generated by (9) converge to one of the two stable fixed points monotonically, decreasing if x0 < x∗ , increasing if x0 > x∗ . This proposition mathematically formalizes the qualitative results Schelling provides in [24]. By contrast, in the other case, the behavior is different; the discrete-time setting may give rise to oscillating behavior. Proposition 2. If A : [0, 1] → R and B : [0, 1] → R are continuous functions such that • A(0) > B(0) • A(1) < B(1) • there exists unique x∗ ∈ (0, 1) such that A(x∗ ) = B(x∗ ), then the dynamical system (9) has only one fixed point at x = x∗ , which is stable if f− (x∗ ) > −1 and f+ (x∗ ) > −1, and is unstable (in the sense of Lyapunov) if at least one of these two slopes is smaller than −1. Both slopes decrease as λ or δA or δB increase, i.e., if the propensity to switch to the opposite choice increases. This result differs from the description given in [24], where oscillations are ruled out as continuous time is implicitly assumed. Instead, in a discrete-time setting, overshooting (or overreaction) phenomena may occur for sufficiently large values of λ. Schelling [24] also describes interesting cases in which the payoff functions are nonmonotonic; in this case there may be more than one intersection and also two or more interior equilibria may exist. Several examples are discussed in [24]; more recently, [7] provides an analysis of the resistance to antibiotics phenomena in terms of payoff functions with two intersection points. Nonmonotonic payoff functions may lead to the existence of more than one intersection, i.e., two or more interior equilibria may exist, see Fig. 3 and the dynamic scenery may become more complicated.


231

Fig. 3. Payoff functions with two intersections, and the relative map obtained with g(·) = 2/π arctan(·), δA = δB = 0.5, λ = 6

Some interesting examples are discussed in [24]; among the others we mention the one about the local use of insecticides. While everyone benefits the use of insecticides by the others, the value of insecticides gets dissipated unless some neighbors use insecticides too. When others use it moderately it becomes cost-effective but when almost everybody uses there are not enough bugs to spray it, and they become cost-ineffective. The importance of nonmonotonic payoff functions is also highlighted in [15]. In this case, the results of the analysis can be formalized as Proposition 3. If R : [0, 1] → R and L : [0, 1] → R are continuous functions such that • A(0) < B(0) • A(1) < B(1) • there exist two points x∗1 < x∗2 both in (0, 1) such that A(x∗i ) = B(x∗i ), i = 1, 2 then dynamical system (9) has three fixed points x = 0, x = x∗1 , and x = x∗2 , where 0 is always stable, x∗1 is always unstable, and x∗2 may be stable or unstable. When x∗2 is unstable, then a cyclic (periodic or chaotic) attractor S(x∗2 ) ⊆ [f (cMax ), cMax ] exists around it and is bounded inside the trapping set [f (cMax ), cMax ], provided that f (cMax ) > x∗1 . The unstable fixed point x∗1 is both the upper boundary of the immediate basin of the stable fixed point 0, and the lower boundary of the immediate basin of x∗2 (or S(x∗2 ) if it exists); furthermore, if (1 − δL ) x∗2 > x∗1 then as λ increases non-connected portions of the basins are created.

232


The proof of this result goes beyond the scope of this chapter; it is reported in [7]. Intuitively, this can be explained observing that the map f is noninvertible, i.e., there exist at least a pair of distinct points that are mapped into the same point, see Fig. 5. Following the notation introduced in [20], we denote by Zk the subset of points, in the range of f , that have k preimages. In the particular case of the map f represented in Fig. 4, Z1 = [0, cmin), Z2 = (cmin , cMax ], and Z0 = (cMax, 1), where cmin and cMax , respectively, represent the relative minimum and maximum values. In addition, as it concerns the unstable fixed point x∗1 (located on the boundary that separates the two basins) it can be observed that x∗1 < cmin ; as a consequence x∗1 ∈ Z1 and the point itself is its unique preimage as in Proposition 1. This is the reason why x∗1 is the unique point that forms the boundary separating the two basins of attraction. However, any parameter variation such that cmin becomes lower than x∗1 as illustrated in Fig. 5a, brings x∗1 in region Z2 . ∗(−1) Therefore, there exists more than one preimage, say x1 , belonging to the ∗(−1)

basin boundary as well. Therefore, any initial condition x0 ∈ x1

,1

is

(0, x∗1 )

and then converges to the fixed point x = 0. In first mapped below other words, the basin of the stable fixed point x = 0 (everybody is choosing ∗(−1) ∗ L) is now B (0) = (0, x1 ) ∪ x1 , 1 , i.e., a nonconnected set, with the ∗(−1) ∗ ∗ “nested” inside. We recall that the widest com“hole” B (x2 ) = x1 , x1 ponent of the basin that contains the attractor is called immediate basin of the attractor.

Fig. 4. Map f obtained with the same parameter values as Fig. 3 but λ = 10; the trajectory starting from x0 = 0.9 converges to x∗2

For this kind of map there exists a value λ such that the topological structure of the basins exhibits a qualitative change; this value is characterized by


233

the contact cmin = x∗1 between a critical point (relative minimum value) and an unstable fixed point. This global bifurcation leads to a counterintuitive behavior of the system. In fact, as the initial fraction of the populations of players choosing strategy R (the initial condition x0 ) increases from 0 to 1, we first move from the basin of the lower equilibrium x = 0 into the basin of the upper one x∗2 , to finally re-enter into the basin of attraction of 0; that is, while when many players initially choose R the process will evolve toward a final equilibrium such that a large fraction of population chooses R, on the contrary when even more players initially choose R then nobody will end up playing R in the long run. The situation may become even more involved when the position of the minimum is shifted horizontally so that global shape of the map f implies a new zone Z3 . This is illustrated in Fig. 5b, where parameters are δR = δL = 0.4, λ = 40. In this case x∗1 ∈ Z3 , actually as λ increases a global bifurcation occurs: from cmin > x∗1 to cmin = x∗1 where the contact bifurcation occurs and, finally, cmin < x∗1 as depicted. At this stage, there exist three distinct preimages of the boundary point x∗1 : x∗1 itself and two more preimages denoted ∗(−1),1 ∗(−1),2 by x1 and x1 in the same figure. The result is that so that both basins consist of two disjoint portions: ∗(−1),1 ∗(−1),2 B (0) = (0, x∗1 ) ∪ x1 , x1 and

∗(−1),1 ∗(−1),2 B (A(x∗2 )) = x∗1 , x1 ∪ x1 ,1 .

In Fig. 6 three different trajectories can be observed; they are generated by initial conditions x0 = 0.8, x0 = 0.91, and x0 = 0.95, respectively (of course, any x0 < x∗1 generates a trajectory converging to 0). We also notice that, for this set of parameters, the larger fixed point x∗2 is not stable as around it there may exist a chaotic or high-period periodic attractor. However, we can observe that the occurrence of the global bifurcation that changes the topological structure of the attractors is not influenced by the kind of coexisting attractors. Furthermore, the global analysis of the dynamic properties of the model reveals the occurrence of a global bifurcation that causes the transition from connected to nonconnected basins of attraction. This implies that several basin boundaries are suddenly created; they may be seen as a possible mathematical description of an extreme form of path dependence, observed in social systems, which is responsible of irreversible transitions from one equilibrium to another (and distant) one as final outcome. The choice of a discrete time scale, allows the occurrence of overshooting and cyclic phenomena in social systems. In particular, with monotonic payoff functions, the model proposed in this paper allowed us to study the occurrence of oscillatory time series (periodic or chaotic). As it is well known,

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Fig. 5. Map f with different parameter values. (a) The parameters are the same as Fig. 4; the trajectory starting from x0 = 0.9 converges to x∗2 . (b) δR = δL = 0.4, λ = 40. In this case x∗1 ∈ Z3

Fig. 6. Three trajectories for the map illustrated in Fig. 5 (b) with respective initial condition x0 = 0.8 (a), x0 = 0.91 (b), and x0 = 0.95 (c)

discrete-time adaptive processes may lead to oscillations, often related to overshooting effects that are quite common in the presence of emotional human decisions.

4 The case of impulsive agents The effects of impulsivity are examined in [5] as the parameter λ increases, up to the limiting case obtained as λ → +∞. In fact, according to the Clinical Psychology literature [23], impulsiveness can be separated in different components such as lack of planning and acting on the spur of the moment. This case is equivalent to consider g (·) = 1, i.e., the switching rate only depends on the sign of the difference between payoffs, no matter how much


235

they differ. The dynamical system assumes then the form of the following discontinuous map ⎧ ⎨ δA + (1 − δA ) xt if B (xt ) < A (xt ) if B (xt ) = A (xt ) xt+1 = f∞ (xt ) = xt (10) ⎩ (1 − δB )xt if B (xt ) > A (xt ) . Any interior intersection between the payoff curves A (x) and B (x) generates a point of discontinuity. In the following, we summarize the results of the analysis of 1-discontinuity and 2-discontinuity cases. 4.1 Piecewise linear maps with one discontinuity In [5] the case of impulsive agents with payoff functions A(x) and B(x) with one and only one internal intersection x∗ ∈ (0, 1) has been examined. The family of iterated maps f : [0, 1] → [0, 1] we analyze in the case of impulsive agents can be expressed either ⎧ if x < d ⎨ (1 − δA )x if x = d x = T1 (x) = x (11) ⎩ (1 − δB )x + δB if x > d or

⎧ ⎨ (1 − δA )x + δA if x < d if x = d x = T2 (x) = x ⎩ (1 − δB )x if x > d

(12)

according to the situations described in Propositions 1 and 2, respectively. The parameter d ∈ (0, 1) represents the discontinuity point located at the interior equilibrium, i.e., d = x∗ , and, as usual, the parameters δA , δB are subject to the constraints 0 ≤ δA ≤ 1, 0 ≤ δB ≤ 1. It is worth noticing that the value of the map in the discontinuity point, x = d, is not important for the analysis which follows, therefore it will often be omitted. The study of the dynamic properties of iterated piecewise linear maps with one or more discontinuity points has been rising increasing interest in recent years, as witnessed by the high number of papers and books devoted to this topic, both in the mathematical literature (references to this huge literature are provided in [5] and [6]). The bifurcations involved in discontinuous maps are often described in terms of the so called border-collision bifurcations, that can be defined as due to contacts between an invariant set of a map with the border of its region of definition. The term border-collision bifurcation was introduced for the first time in [22] and it is now widely used in this context. However the study and description of such bifurcations was started several years before by Leonov in [16] and [17], who described several bifurcations of that kind

236


and gave a recursive relation to find the analytic expression of the sequence of bifurcations occurring in a one-dimensional piecewise linear map with one discontinuity point. His results are also described and used by Mira in [18] and [19]. By applying the methods suggested in [16] and [17], see also [18] and [19], to the map T2 , it is possible to give the analytical equation of the bifurcation curves where stable periodic cycles are created or destroyed. The boundaries that separate two adjacent periodicity regions in the space of parameters are also called periodicity tongues. They are characterized by the occurrence of a border-collision, involving the contact between a periodic point of the cycles existing inside the regions and the discontinuity point. To better formalize and explain our results it is suitable to label the two components of our map x = T2 (x) as follows: ⎧ ⎨ TL (x) = mL x + (1 − mL ) if x < d x = T2 (x) = (13) ⎩ if x > d, TR (x) = mR x where mL = (1 − δA ) and mR = (1 − δB ). First of all, notice that all the possible cycles of the map T2 of period k > 1 are always stable. In fact, the stability of a k-cycle is given by the slope (or eigenvalue) of the function T2k = T2 ◦ ... ◦ T2 (k times) in the periodic points of the cycle, which are fixed points for the map T2k , so that, considering a cycle with p points on the left side of the discontinuity and (k − p) on the (k−p) right side, the eigenvalue is given by mpL mR which, in our assumptions, is always positive and less than 1. In [5], the study of the conditions for the existence of the periodic cycles are limited to the analysis of the bifurcation curves of the so-called “principal tongues,”or “main tongues” [4] or “tongues of first degree” [16–19], which are the cycles of period k having one point on one side of the discontinuity point and (k − 1) points on the other side (for any integer k > 1). Let us begin with the conditions to determine the existence of a cycle of period k having one point on the left side L and (k − 1) points on the right side R. The condition (i.e., the bifurcation) that marks its creation is that the discontinuity point x = d is a periodic point to which we apply, in the sequence, the maps TL , TR , ..., TR . For example, the condition for the creation of a 3-cycle, i.e., k = 3, is TR ◦ TR ◦ TL (d) = d. Then the k-cycle with periodic points x1 , ..., xk , numbered with the first point on the left side, satisfies x2 = TL (x1 ), x3 = TR (x2 ), ..., x1 = TR (xk ), and this cycle ends to exist when the last point (xk ) merges with the discontinuity point, that is, xk−1 = d which may be stated as the point x = d is a periodic point to which we apply, in the sequence, the maps TR , TL , TR , ..., TR . The closing condition related with the 3-cycle is, TR ◦ TL ◦ TR (d) = d. Notice that both these conditions express the occurrence of a border collision bifurcation, being related to a contact between a periodic point and the boundary (or border) of the region of differentiability of the


237

corresponding branch of the map. In general, for a cycle of period k > 1, the equation of one boundary of the corresponding region of periodicity is: (k−1)

mL = mLi =

−d

mR

(14)

(k−1)

(1 − d)mR

while the other boundary, i.e., the closure of the periodicity tongue of the same cycle, is given by: (k−2)

mL = mLf =

mR

−d (k−2)

(1 − mR d)mR

.

(15)

The proof of these two equations is reported in [5]. Thus the k-cycle exists for mk−2 > d and mL in the range R mLi ≤ mL ≤ mLf

(16)

and the periodic points of the k-cycle, say (x∗1 , x∗2 , ..., x∗k ) where x∗1 < d and x∗i > d for i > 1, can be obtained explicitly as: x∗1 = x∗2 x∗3 x∗4 ... x∗k

= = =

(k−1)

mR

(1−mL ) (k−1)

1−mL mR TL (x∗1 ) = TR (x∗2 ) = TR (x∗3 ) =

mL x∗1 + 1 − mL mR (mL x∗1 + 1 − mL ) m2R (mL x∗1 + 1 − mL ) (k−2)

= TR (x∗k−1 ) = mR

(17)

(mL x∗1 + 1 − mL ).

An illustration of the periodicity tongues together with their boundaries is provided in Fig. 7.

Fig. 7. (a) The bifurcation curves obtained by the analytical expressions calculated for the map T , with the discontinuity point d=0.8, for cycles of periods 2, . . . , 15. (b) The tongues of periodicity relative to case considered in (a)

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4.2 Piecewise linear maps with two discontinuities As discussed in Sect. 3, when the payoff functions are nonmonotonic there may be more then one intersection point. In [6] the case of two internal intersection has been examined. In this case the family of iterated maps T : [0, 1] → [0, 1] has the form: ⎧ if x < d1 or x > d2 ⎨ (1 − δB )x if x = d x = T (x) = x ⎩ (1 − δA )x + δA if d1 < x < d2 with parameters subject to the following constraints: 0 < d1 < d2 < 1 0 < δA < 1, 0 < δB < 1. It is easy to realize that the origin is always an attractor; it may be the only one or it may coexist with an attracting cycle of period k > 1, according to the values of the four parameters d1 , d2 , δA , and δB . Also in this case it is possible to prove the existence of stable periodic cycles in analytically defined regions of the space of the parameters, as well as the analytic conditions for their creation or destruction through border collision bifurcations (see [1–4]). In the case of coexisting attractors, it is also important to bound the sets of initial conditions that generate trajectories converging to either one or to the other, i.e., the respective basins of attraction. Indeed, we only have the two following possibilities: 1. Each of the two basins consists of a single interval, separated by the discontinuity point d1 ; as a consequence, the basin of the origin is the first interval: B(O) = [0, d1 [ while the other points converge to the attracting cycle: B(C) = ]d1 , 1]; 2. Each of the two basins consists of two or more intervals, separated by the two discontinuity points d1 and d2 and their preimages. The immediate basin of the origin is clearly the segment [0, d1 [ so that the whole basin is given by this segment and all its preimages: B(0) = ∪j≥0 T −j ([0, d1 [), while the complementary region in [0, 1] gives the basin of the cycle: B(C) = [0, 1]\B(0). In [6], the authors provide the analytic description of the bifurcations occurring in a piecewise linear map T : [0, 1] → [0, 1] formed by three portions with two different slopes separated by two discontinuity points 0 ≤ d1 < d2 ≤ 1. The problem is approached as the generalization of a simpler map with only one discontinuity. They show how both the bifurcation diagram and the analytic expression of the periodicity tongues of first degree maintain some important aspects of the map with only one discontinuity point; also the effects, on the structure of the border collision bifurcation curves – induced


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Fig. 8. (a) The tongues of periodicity, in the parameter plane (δA , δB ), for cycles of periods 2, . . . , 15 when d1 = 0.2 and d2 = 0.7. (b) The bifurcation curves relative to the tongues depicted in (a) obtained by the analytical expressions calculated for the map T

by the introduction of the second discontinuity point – are examined. An illustration is provided in Fig. 8. The methods followed to obtain the analytic expressions are quite general and can be easily generalized to cases with different linear functions, several discontinuities, and with slopes different from the ones considered in the model studied in this chapter. Furthermore, differently from the case of only one discontinuity, when considering two discontinuities, cases of multistability can be obtained, i.e., the coexistence of a stable fixed point and a stable periodic cycle, each with its own basin of attraction. These basins may be either connected intervals, separated by a discontinuity point, or nonconnected sets formed by the union of several disjoint intervals, separated by a discontinuity point and its preimages of any rank. Finally, the discontinuous map – which can be interpreted as a model of the social behavior of impulsive agents – can be seen as the limit case of the continuous map, that models agents with some degree of inertia in making their choices. It is possible to show how the bifurcation curves of the limiting case, characterized by periodic cycles only, can be obtained from those of the continuous model, that also exhibit chaotic behavior, with a high value of the parameter λ.

5 A general model with different switching propensity In [8] the two models described in Sects. 2 and 3 are merged into a single one which includes each of them as a particular case. Furthermore, this general model allows us to describe other situations which cannot be studied by any of the original models described above.

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For each group of size k, define two payoff functions Ak (j) and Bk (j), where j ∈ 0, 1, . . . , k is the number of people in the group choosing A. As in Galam’s model, at each time t individuals gather in group and, after the interaction, leave the group with an updated preference which depends on the outcome of the interaction, in the sense that in each group a fraction of the agents with lower payoff will switch to the choice that gives higher payoff. This dynamic adjustment can be formalized as follows. At each time t, consider a size k group where 0 ≤ jt ≤ k agents are choosing A; after the social gathering, as a result of the intragroup interaction, individuals may switch opinion. That is, at time t + 1, the number of individuals choosing A will be: jt+1 = hk (jt , δA , δB ) ⎧ ⎪ ⎪ jt + δA (k − jt ) = kδA + (1 − δA ) jt if Ak (jt ) > Bk (jt ) ⎪ ⎪ (18) ⎨ if Ak (jt ) = Bk (jt ) = jt ⎪ ⎪ ⎪ ⎪ ⎩ jt − δB jt = jt (1 − δB ) if Ak (jt ) < Bk (jt ) , where δA ,δB ∈ [0, 1] represent the probability according to which agents may switch to A and B, respectively, depending on to the greater payoff observed. Furthermore, since in each group the number of agents is integer, it makes little sense to consider fraction of agents who are switching choices, therefore we have introduced integer parts in the right hand side of (18) are introduced to consider integer number of agents. In formulation (18) the Integer part has been considered; this indicates the behavioral momentum of agents, that is, how difficult is for agents to switch choices when facing inferior payoffs. Behavioral momentum is a well known psychological construct which has been examined for example in [14]. Finally, in this model the switching mechanism has no bias in the sense of [10].5 In order to obtain a bias toward B it is sufficient to consider, for example, ⎧ ⎨ kδA + (1 − δA ) jt if Ak (jt ) > Bk (jt ) jt+1 = hk,B (jt , δA , δB ) = (19) ⎩ jt (1 − δB ) if Ak (jt ) ≤ Bk (jt ) . Symmetrically, it is immediate to obtain a switching function biased toward A. In this formulation, such a bias – which is related to how local uncertainty Ak (jt ) = Bk (jt ) is resolved- is combined to the bias, related to the asymmetric switching propensity δA = δB as described in Sect. 3.

5

Recall that in Sect. 2, in the case of local doubt the choice was biased toward B.


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As in [10], it is possible to obtain the dynamics of the probability of choosing A: PA (t + 1) =

L

ak

k=1

k j=0

j

k−j

Cjk PA (t) (1 − PA (t))

hk (j, δA , δB ) , k

(20)

where the last term is the relative number of agents choosing option A in a group of size k. This model assumes that individuals are impulsive in the sense described in [5]. However, it is possible to model inertia in the agents’ reaction, as in [7], by introducing a modulating function g as described in Sect. 3. In [8] by the unified model, different situations besides the original ones proposed in [10] and [24] have been considered. Two of them are particularly interesting. The first one considers the case of a single large group. This situation is important from the theoretical point of view as it represents the case in which all the groups merge into one. Furthermore, it illustrates how the size of the group and the payoffs affect the dynamics. In fact, in the case of the majority rule [10], it takes just one iteration for the population to reach unanimity on the majority’s opinion, and the killing point for this model coincides with the floor (i.e., the Integer Part) of N/2; by contrast, when considering different payoffs the dynamics may become quite interesting. In fact, recall that, given the necessity to maintain an integer number of agent in each group, the fraction of agents switching choices must be rounded. In [8] we consider floor, where x defines the largest integer n ≤ x; ceiling, where x defines the smallest integer n ≥ x. Finally, we consider nearest integer, where x, defines the integer closest to x; as usual, to avoid ambiguity we adopt the convention according which half-integer are always rounded to even numbers. A situation similar to the one presented in [10] is analyzed and it is possible to observe that, considering different rounding functions, the dynamics can have a different evolution from the one predicted by Galam’s original model. The second example consists of payoff functions with – using Schelling’s terminology, see [24] – both contingent internality and contingent externality. In this case it is possible to observe that an A choice benefits those who choose B, and a B benefits choice those who choose A. Among the vivid real life examples provided in [24], we quote6 one situation related to traffic congestion after a blizzard: “... let A be staying home and B using the car right after a blizzard. The radio announcer gives dire warnings and urges everybody to stay home. Many do, and those who drive are pleasantly surprised by how empty the roads are; if the others had known, they would surely have driven. If they had, they would all be at the lower left extremity of the B curve.” ([24], page 405). This kind of games has been examined in [21] as compounded dispersion game. The dynamics in dispersion games when agent 6

Recall that we use, respectively, A and B instead of R and L.

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are allowed in finite size groups can be quite interesting depending on which rounding function is considered. In particular in [8] the payoff functions are assumed to be Ak (j) := j Bk (jt: ) := k − j with switching propensities δA = 0.3 and δB = 0.9. Also it is assumed that, when payoff are identical, agents have a bias toward A, that is those who have played B will switch choice with propensity δA . Finally, the probabilities to be sitting at the different sizes groups are defined as follows: a2 = 0.1, a3 = 0.1, ak = 0.8, where k is fixed as any value greater than 3 all other groups have probability 0. The dynamics of choices as k varies in the range [4, 276] have been examined; considering the different rounding functions the differences in the dynamics are stark, yet these differences are to be interpreted not as numerical artifacts rather as the result of the different biases implicitly modeled by the rounding functions.

6 Conclusion In this chapter we have described some discrete-time dynamic models used to represent adaptive mechanisms through which individuals perform repeated binary choices in the presence of social externalities. The long-run (asymptotic) outcomes emerging from repeated short time decisions can be seen as an emerging property, sometimes unexpected, or difficult to be forecasted. Moreover, when there are several coexisting attractors each with its own basin of attraction, i.e., in the presence of so called multistability, the adaptive dynamics can be seen as an equilibrium-selection device. In this case the long-run outcome becomes path dependent, and historical accidents may play a crucial role in the selection of the social emerging behavior. This path dependence can also be seen as an evolutionary approach to the explanation of collective behavior as the result of repeated individual and myopic choices. Starting from two particular discrete-time dynamic models – Galam’s model of rumors spreading [10] and a formalization of Schelling’s binary choices [7] – we have described some peculiar properties of discrete-time (or event-driven) dynamic processes. In particular, from a mathematical point of view, the kind of global dynamic analysis of the discrete time dynamics illustrated in this chapter, obtained through a continuous dialogue among analytic, geometric and numerical methods, is based on the properties of noninvertible one-dimensional maps, see, e.g., [20]. Moreover, in the limiting case of impulsive agents, the global dynamic properties of piecewise linear discontinuous maps allowed us to illustrate some results concerning border collision bifurcations, a topic that has been recently been at the center of a flourishing


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stream of literature, see, e.g., [1–4]. In other words, the models described in this chapter allowed us to get some insight into the two kinds of complexity, related to complex attracting sets and complex structure of the basins of attraction. Both these kinds of complexity are related to the presence of overshooting phenomena, typical of discrete-time adaptive systems, see, e.g., [21]. However, overshooting should not be seen as an artificial effect or a distortion of reality due to discrete time scale. In fact, as stressed in [25], overshooting and over-reaction arise quite naturally in social systems, due to emotional attitude, excess of prudence, or lack of information. The extreme form of agents’ impulsivity, represented by the limiting case of a switching intensity that tends to infinity, i.e., actors that decide to switch the strategy choice even when the discrepancy between the payoffs observed in the previous period is extremely small, may even be interpreted as the automatic change of an electrical or mechanical device that changes its state according to a measured difference between two indexes of performance. Finally, the general model described at the end of the chapter, constitutes a generalization and a synthesis of two models of social interaction of Galam and Schelling, respectively. It may be further used to analyze other situations, such as some of those considered in [24] for a single population that can be extended to small groups. Other aspects of this model that can be investigated concern, for example, the role of impulsiveness in small groups, and how some social parameters, such as agents’ impulsiveness and population size, affect the dynamics. In particular, it would be interesting to investigate what makes a group large and how its dynamic behavior is different from that of a small one.

References 1. Avrutin V., Schanz M.: Multi-parametric bifurcations in a scalar piecewise-linear map, Nonlinearity, 19, 531–552 (2006) 2. Avrutin V., Schanz M., Banerjee S.: Multi-parametric bifurcations in a piecewise-linear discontinuous map, Nonlinearity, 19, 1875–1906 (2006) 3. Banerjee S., Grebogi C.: Border-collision bifurcations in two-dimensional piecewise smooth maps, Phys. Rev. E, 59(4), 4052–4061, (1999) 4. Banerjee S., Karthik M.S., Yuan G., Yorke J.A.: Bifurcations in OneDimensional Piecewise Smooth Maps - Theory and Applications in Switching Circuits, IEEE Trans. Circuits Syst.-I: Fund. Theory Appl. 47(3), 389–394 (2000) 5. Bischi G.I., Gardini L., Merlone U.: Inpulsivity in binary choices and the emergence of periodicity, Disc. Dyn. Nat. Soc., Vol 2009, Article ID 407913, 22 pages, doi:10.1155/2009/407913 (2009) 6. Bischi G.I., Gardini L., Merlone U.: Periodic cycles and bifurcation curves for one-dimensional maps with two discontinuities, J. Dyn. Sys. Geom. Theor., 7, 101–124 (2009) 7. Bischi G.I., Merlone U.: Global dynamics in binary choice models with social influence, J. Math. Soc. 33(4), 277–302 (2009)

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8. Bischi G.I., Merlone U.: Binary choiches in small and large groups: A unified model, Phys. A, doi:10.1016/j.physa.2009.10.010 (2009) 9. Ellero A., Fasano G., Sorato A.: A modified Galam’s model for word-of-mouth information exchange. Physica A, 388, 3901–3910 (2009) 10. Galam S.: Modelling rumors: the no plane pentagon french hoax case. Physica A, 320, 571–580 (2003) 11. Galam S.: Heterogeneous beliefs, segregation, and extremism in the making of public opinions, Phys. Rev. E, 71, 046123-1-5 (2005) 12. Galam S., Sociophysics: A review of Galam models, Int. J. Modern Phys. C, 19, 409–440 (2008) 13. Galam S., Chopard B., Masselot A., Droz M.: “Competing Species Dynamics”, The Eur. Phys. J. B, 4, 529–531 (1998) 14. Goltz S.M.: Can’t Stop on a Dime, J. Org. Behav. Manag., 19(1), 37–63 (1999) 15. Granovetter M.: Threshold models of collective behavior, Am. J. Soc., 83(6), 1420–1443 (1978) 16. Leonov N.N.: Map of the line onto itself, Radiofish, 3(3), 942–956 (1959) 17. Leonov N.N.: Discontinuous map of the straight line, Dohk. Ahad. Nauk. SSSR. 143(5), 1038–1041 (1962) 18. Mira C.: Sur les structure des bifurcations des diffeomorphisme du cercle, C.R. Acad. Sci. Paris, Series A, 287, 883–886 (1978) 19. Mira C.: Chaotic Dynamics. World Scientific, Singapore (1987) 20. Mira, C., Gardini, L., Barugola, A., Cathala J.C.: Chaotic Dynamics in TwoDimensional Noninvertible Maps. World Scientific, Singapore (1996) 21. Namatame A.: Adaptation and Evolution in Collective Systems. World Scientific, Singapore (2006) 22. Nusse H.E., Yorke J.A.: Border-collision bifurcations including period two to period three for piecewise smooth systems, Physica D 57, 39–57 (1992) 23. Patton J.H., Stanford M.S., Barratt E.S.: Factor structure of the barratt impulsiveness scale, J. Clin. Psyc., 51, 768–774 (1995) 24. Schelling T.C.: Hockey helmets, concealed weapons, and daylight saving, J. Conf. Res., 17(3), 381–428 (1973) 25. Schelling T.C.: Micromotives and Macrobehavior. W.W. Norton, New York (1978) 26. Wijeratne A.W., Ying Su Y., Wei J.: Hopf bifurcation analysis of diffusive bass model with delay under “negative-word-of-mouth”. Int. J. Bif. Ch. 19, 3, 1059–1067 (2009) 27. Young H.P.: Individual Strategy and Social Structure. Princeton University Press, Princeton (1998)

Modelling opinion formation by means of kinetic equations Laurent Boudin1,2,3 and Francesco Salvarani4 1 2 3 4

UPMC Univ Paris 06, UMR 7598 LJLL, Paris, F-75005, France, [email protected] CNRS, UMR 7598 LJLL, Paris, F-75005, France INRIA Paris-Rocquencourt, REO Project-team, F-78153 Le Chesnay Cedex, France Dipartimento di Matematica F. Casorati, Universit` a degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy, [email protected]

Summary. In this chapter, we review some mechanisms of opinion dynamics that can be modelled by kinetic equations. Beside the sociological phenomenon of compromise, naturally linked to collisional operators of Boltzmann kind, many other aspects, already mentioned in the sociophysical literature or no, can enter in this framework. While describing some contributions appeared in the literature, we enlighten some mathematical tools of kinetic theory that can be useful in the context of sociophysics.

New opinions are always suspected, and usually opposed, without any other reason but because they are not already common. John Locke, An Essay Concerning Human Understanding

1 Sociophysics 1.1 Introduction The success of statistical mechanics as a tool for describing physical systems composed by a great number of interacting elementary entities has induced some researchers to apply the same methodology to study problems of other sciences having the common feature that a global behaviour is obtained as a result of a chain of elementary processes. These new fields of application involve, for example, biology, economy, and sociology. In particular, when describing a social phenomena, the basic entities are individuals, who interact with other members of the population by means of elementary mechanisms. The idea of using methods and concepts from physics to study sociological phenomena appeared in 1971 in a pioneering paper of Weidlich [79]. Some G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 10, c Springer Science+Business Media, LLC 2010

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Laurent Boudin and Francesco Salvarani

years later, Galam, Gefen, and Shapir [38] first used the word sociophysics in order to characterize this kind of approach. They stated a global frame for sociophysics as a new field of research. However, this strategy, even promising, is not exempted of conceptual difficulties. A major problem is that the individuals cannot be thought as simple mindless agents, and that their behaviour does not obey, in general, to some exact formulae. Human beings are indeed the opposite of simple elementary entities. The fine dynamics of their mental schemes is (and may remain) unknown. Moreover, in the real world, it is not guaranteed at all that two individuals, exactly stimulated in the same way, give the same answer. Hence, a program based on the individuation of quantitative mechanisms in social dynamics could seem hopeless: modelling the behaviour of social agents by means of mathematical models implies an extreme simplification of the problem and, as a consequence, the inference of the macroscopic phenomenology starting from these basic models could seem without any predictive value. However, as observed in many applications of statistical physics, many qualitative (and sometimes quantitative) features of the system do not depend on the microscopic details of the processes under examination: only some high-level properties, such as conservation laws and symmetries, are important in order to obtain a coherent and reasonably correct description. Sociophysics is nowadays a recognized field research within statistical physics. Beside the sociophysical approach, a good variety of other view points has been proposed to study the aforementioned questions. For example, whereas many papers adopt the point of view of mathematical statistics (for example, [32, 33, 49, 56]), many other contributions are based on the language of game theory (among others, [44, 64]) or fuzzy systems [24, 66, 67]. In what concern sociophysics, many problems have been studied and various approaches have been proposed. It is worth noting that, in the literature, the quantity of theoretical works is greater than empirical studies. The set of the most explored questions includes, among others, opinion and cultural dynamics, flocking, applause dynamics, hierarchies formation, human dynamics (including social web) and social networks. Because of its interest in the scientific community, a general overview on the research of social dynamics by means of a physical approach is available in some review articles. We quote [20, 37] and their references, that provide an outlook on different problems and methods. 1.2 Towards the kinetic viewpoint of opinion formation Many strategies have been introduced in the literature to study the opinion formation. Some papers describe the phenomenon by using a discrete opinion variable, by working in the context of Ising models (for example, [38, 39, 73, 76]). An alternative strategy consists in describing opinion dynamics by using a continuous variable. Indeed, the opinion of an individual may

Modelling opinion formation by means of kinetic equations

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smoothly evolve in a continuous space of possible opinions. For instance, the position of a social agent with a respect to a “yes/no” binary question can be expressed with nuance from the two extreme positions (“yes” and “no” without any doubt). An individual can also be indifferent regarding the question itself. In [26], the opinions lie in the closed interval [0, 1]. The authors consider a population of N agents, where each individual randomly interacts with one of his neighbours, following a mechanism close to a collisional mechanism in the Boltzmann equation with a cutoff effect. The population eventually reaches a stationary state, which can be a priori determined. This collisional mechanism is not the only one that can be taken into account. As an example, in [3], the binary interaction process of [26] is combined with a diffusion one, which models the spontaneous changes of mind of the individuals. The competition between the two processes generates three possibilities. In absence of diffusion, the results from [26] are of course recovered and the concentration in some predefined opinions happens. When there is a weak diffusion, the favoured opinions again exist, but the concentration is not total. Eventually, when the diffusion is strong, the opinion becomes uniform in the population. Of course, there are many other phenomena which can be considered; some of them are discussed later. We point out also that randomness is a key notion in the opinion formation process. As a matter of fact, the behaviours of the individuals can really randomly change, as well as the dynamics of the interaction between people. Moreover, the effects of external phenomena (such as, for example, mass media) are not necessarily fully predetermined. Hence, any description of opinion formation dynamics must be able to consider random processes. This consideration leads naturally to the kinetic approach, which is essentially the deterministic description of an underlying probabilistic phenomenon. It is based on a partial integrodifferential equation of Boltzmann type which governs the time evolution of an unknown function (normally a probability density) that describes the system. Even if kinetic theory is an active area of research, however, a systematic specialised review describing the state-of-the-art concerning the kinetic approach in opinion dynamics is still missing. Goal of this chapter is to fill in the gaps in the existing literature. This review does not intend to cover all the question and approaches of sociophysics. This work is divided into three parts. In the first one, we briefly present the Boltzmann equation, some of its mathematical properties and some way to discretize it. Then we point out the main phenomenon induced by the interaction mechanism between two social agents, the tendency to consensus. Eventually, we discuss other sociological phenomena which can be taken into account to compete with the natural compromise process.

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2 Kinetic approach in sociophysics: tools and methods The strategies, based on nonequilibrium statistical mechanics, which are used for modelling phenomena of opinion formation are mainly inspired to the classical field of the kinetic theory of rarefied gases. In this section, we briefly explain the basic aspects of the main examples of kinetic equation: the Boltzmann equation and its linear variant. The purpose of both equations is to describe the time evolution of a system – composed by a great number of particles – by means of a distribution function in the phase space of the system, which can depend on a number of variables which is greater than the number of independent variables of the observables at a macroscopic scale. We always assume that such a system is composed by identical particles, obeying the laws of classical mechanics, with only translational degrees of freedom. If the particles are contained in a domain Ω ⊆ R3 , the distribution function f (x, v, t) of such a model should be defined on Ω × R3 × R+ and, for all t, the integral f (x, v, t) dx dv x∈X

v∈V

represents the number of particles contained in the space volume X ⊆ Ω with velocity in V ⊆ R3 . Note that, in order to give a sense to the previous considerations, a reasonable hypothesis on f is: f (x, v, t) ∈ L1loc (Ω; L1 (R3 )),

∀ t ∈ R+ ,

(or, at least, f (·, ·, t) is a positive bounded measure on K × R3v , for every bounded subset K of R3x ), which means that there is a finite number of particles in a bounded domain of the space. We shall here always consider, for the sake of simplicity, that Ω = R3 , and that the system is isolated, to avoid the effects of external forces on the particles. If we suppose moreover that these particles do not mutually interact, then the time evolution of the distribution function f is given by the free transport equation ∂f + v · ∇x f = 0, ∂t which means that the number of particles is conserved along the characteristics dx/dt = v and dv/dt = 0 (that is, f (x, v, t) = f (x − vt, v, 0)). On the contrary, if the effect of the collisions between particles is no longer negligible, the description above does not hold, and one must also take into account these collisions. This leads, as we shall see, to add a non-vanishing second member in the free transport equation. In the next two subsections, we shall consider two situations: the Boltzmann equation and the linear transport equation. Then we provide some insights about the numerical methods which can be used for the Boltzmann equation.


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2.1 The Boltzmann equation If we suppose that the particles of the system interact through elastic and binary collisions, then the time evolution of f is governed by the Boltzmann equation [11, 12] which, in the whole space and without external forces, has the following form: ∂f + v · ∇x f = Q+ (f, f ) − f L(f ). ∂t

(1)

Here Q+ and L are, respectively, a quadratic and a linear operator defined by σ(v − v∗ , ω)f (x, v , t)f (x, v∗ , t) dω dv∗ Q+ (f, f ) = R3

S2

and L(f ) =

R3

S2

σ(v − v∗ , ω)f (x, v∗ , t) dω dv∗ .

The parameter ω is a unit vector of the unit sphere S 2 , so that dω is an element of area on the surface of the sphere, and (v , v∗ ) are the pre-collisional velocities of two incident particles, related to the post-collisional velocities (v, v∗ ) by the following relations: v =

1 (v + v∗ + |v − v∗ |ω), 2

v∗ =

1 (v + v∗ − |v − v∗ |ω). 2

Moreover, the kernel σ is a non-negative function (or, at least, measure) which takes into account the details of the interactions between particles. It only depends on |v − v∗ | and on the scalar product (k · ω), where k = (v − v∗ )/|v − v∗ |. The simplest case is when one deals with Maxwellian molecules. In this situation, the kernel σ is reduced to a function of (k ·ω): σ(v −v∗ , ω) = σ(k ·ω). The spatially homogeneous Boltzmann equation (that is ∇x f = 0) associated to such a kernel has several features: for example, as shown by Wild [81], it is possible to obtain a semi-explicit representation of the solution for the Cauchy problem of (1) under the form of a convergent series. Kinetic equations (in particular, those in which appears a collision kernel) are usually derived (at the formal level) from Hamiltonian systems. While, for linear equations, it is in general possible to give a rigorous proof of those derivations (see for example [41] for the linear Boltzmann equation), nonlinear equations are much more difficult to tackle, and rigorous results only exist for special regimes: in particular, for the Boltzmann equation, we mention the derivation of local (in time) solutions [22,58], or global (in time) small solutions [50, 51, 71]. When singular (nonlinear) kernels are concerned (non cutoff Boltzmann kernel, or Landau kernel), the derivation of the corresponding kinetic equation seems completely open, though the equations themselves have been extensively studied, and the only result obtained up to now, in

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that situation, is the kernel’s characterization performed in [30], which allows to derive, starting from some physical abstract requirements, the collisional kernels of the kinetic equations but not the equations themselves. Texts of reference on the wide subject of the Boltzmann equation, in which the arguments outlined above are more precisely discussed are [21, 22, 78]. Finally, we note that there is a quantity of variants of (1). Among others, we recall Kac’s model [55] and the discrete velocity models (see [42, 57, 65, 70] for additional information and bibliography). 2.2 The linear transport equation for photons The behaviour of the distribution function for a set of photons in a medium can be governed by the linear transport equation. Even if it is simpler, this classical transport equation retains some features of the Boltzmann equation, among which the balance of gain and loss terms. The distribution function for photons n = n(x, ν, ω, t) is defined on R3 × R+ × S 2 × R+ . For all t, its integral n(x, ν, ω, t) dx dν dω x∈D

ν∈[ν0 ,ν1 ]

ω∈Ω

represents the number of photons contained in the space volume D ⊆ R3 , with frequency included in the interval [ν0 , ν1 ] ⊆ R+ and with velocity direction belonging to Ω ⊆ S 2 . The time evolution of the unknown n is governed by a transport equation whose gain and loss terms are linear. In the whole space, the Cauchy problem for this equation has the following form: 1 ∂n + ω · ∇x n = −σ(ν)n + I(n) + S(ν), c ∂t

(2)

with initial condition n(x, ν, ω, 0) = n0 (x, ν, ω)

(x, ν, ω) ∈ R3 × R+ × S 2 ,

where σ(ν) is the total cross-section of absorption and scattering, depending on the frequency, S(ν) is a given source, c is the speed of light and I is a linear scattering operator defined by ν I(n) = σ (ν → ν, ω · ω)n(ν , ω )dν dω . s R+ S 2 ν Here ν and ω are, respectively, the pre-interaction frequency and velocity direction, and σs is the scattering cross-section, which represents the probability of a transition (ν , ω ) → (ν, ω). For a more extensive discussion we refer to the classical texts on the subject (in particular, [19, 23]).


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2.3 Numerical methods The numerical solving of the Boltzmann equation (1) is ensured by a time split algorithm. During a time step one solves, on the one hand, the transport part, and, on the other hand, the collisional part. The latter part is usually the most expansive one, because of the collision operator nonlinearity. There are mainly two ways to discretize the distribution function: the discrete velocity methods and the particle methods. One can find in the literature lots of improvement of the numerical methods briefly presented later, depending on the aims of the user: conserve some physical properties, asymptotical preserving schemes, etc. The reader is invited to refer to [29] for more details. Particle method The distribution function is approximated by a sum of Dirac masses in the phase space [27, 28, 74] f (x, v, t) = fi (t)δXi (t) (x)δVi (t) (v), (3) i

where Xi (t) is the position of particle i at time t, and Vi (t) its velocity. The solving of the transport part is easy, since we just have to follow the particles along their trajectories. The discretization of the collisional operator is more intricate and is more often performed thanks to a Monte Carlo method, which induces a probabilistic treatment [10, 68, 69]. At the end of each collisional step, the locations and velocities of the particles have changed, but the quantity fi (t) has not. The collision process for the numerical particles mimics the behaviour of real physical particles, which ensures the conservation of physical quantities such that the momentum. Nevertheless, this kind of method generates a lot of computational noise. There are mainly two ways to decrease it: using a large number of particles or averaging numerical results. Discrete velocity method Deterministic methods can also be used to discretize the Boltzmann operator: the discrete velocity methods, see [17, 52, 72] for example. The distribution function is given by its value on a uniform and time independent phase space grid. Formula (3) still holds, but this time, the positions and velocities do not change, only fi (t). But a couple of pre-collisional velocities generates a small number of possible post-collisional velocities. The mesh must then be very fine, and the computational cost may be very high [45]. In what concerns the models of opinion formation, the choice of the numerical method widely depends on the phenomena taken into account, and the relevance of each method must therefore be studied. Nevertheless, the time splitting should still be considered since, in most situations, one deals with several different phenomena, with specific modelling features.

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3 The main phenomenon: the compromise As we already pointed out, the feature which appears in almost all models of opinion formation is the tendency to compromise, which mimics the collisions happening in the traditional Boltzmann equation. That means that the binary interactions naturally tend to concentrate the opinions of the population around some values (the average one, or, in other cases, periodic values, for instance). 3.1 Basic models Towards continuous models The introduction of kinetic models in the context of sociophysics goes back to the beginning of the nineties. In [47], Helbing points out that the master equation and Boltzmann-like equations are suitable for the quantitative description of behaviour changes and social processes. In the aforementioned article, the author does not limit himself to consider problems of opinion formation, but introduces a general framework for social situations described by a system of N individuals, whose state y ∈ {1, . . . , S} represents the possible behaviour strategies concerning a certain situation. It is quite clear that the first Helbing model is a discrete velocity model. Note that this kind of model have been used later on to study opinion dynamics phenomena (for example, [6–9]). However, in the absence of spatial phenomena, their structure more remembers a dynamical system than a kinetic model. We therefore invite the interested reader to directly refer to the literature. The dynamics of the system is then described by a master equation, which is a phenomenological set of first-order differential equations describing the time evolution of the probability of a system to occupy each one of a discrete set of states. If we denote by ny the number of subsystems in state y, which must satisfy the constraint ny = N, y

by n = (· · · , ny , · · · ) the configuration vector of the system and by P (n, t) the probability of observing the configuration n at time t, the master equation of the system is dP (n, t) = w(n|n ; t)P (n , t) − w(n |n; t)P (n, t), dt n n

(4)

where w(n |n; t) are the configurational transition rates from configuration n to configuration n .


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There are two types of transitions: spontaneous changes of state of a single subsystem, or a direct pair interaction. The form of the transition rates w(n |n; t) is w(n |n; t) = w1 (y |y; t)ny if n = (· · · , (ny +1), · · · , (ny −1), · · · ), in the case of spontaneous transitions, w(n |n; t) = w2 (y , z |y, z; t)ny nz if n = (· · · , (ny + 1), · · · , (ny − 1), · · · , (nz + 1), · · · , (nz − 1), · · · ), in the case of binary interactions, and w(n |n; t) = 0 otherwise. Under this situation, and in absence of spontaneous transitions, the master equation reduces to be an homogeneous discrete velocity model with quadratic collisional part, whose quadratic structure is a discrete version of the collisional integral of a Boltzmann equation. The states of the system are strategies of individuals playing a game with others, which they randomly meet. As a result of these collisions, they change their strategies by adopting those of their more successful opponents with probabilities proportional to the difference between the expected successes of the latter and their own. The author classifies the pair interactions in four types, denoted as (I), (II), (III), (IV), whose interpretation is the following. 1. Interactions (I) describe imitative processes, that is the tendency to take over the strategy of another individual. 2. Interactions (II) describe avoidance processes, where an individual changes the strategy when meeting another individual using the same strategy (processes of this kind are known as aversive behaviour, defiant behaviour or snob effect). 3. Interactions (III) represent some kind of compromising processes, where an individual changes the strategy to a new one (the “compromise”) when meeting an individual with another strategy (such processes are found, if a certain strategy cannot be maintained when confronted with another strategy). 4. Interactions (IV) describe another kind of imitative processes, different from processes of type (I): an individual changes the strategy despite the fact that he convinces his interaction partner of the strength of his strategy. Social processes of this kind are very improbable and can usually be neglected. Moreover, the author studies the behaviour of the most probable strategy distribution of the previous model and shows that its time evolution is governed by a Boltzmann-like equation.

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In [46], the author deeply explores the properties of the master equation (4), and shows that it is consistent with many models of social theory, e.g., the diffusion models, Lewin’s field theory, the logistic equation, the gravity model, the Weidlich-Haag model or the game dynamical equations. Assuming that the set Ω of possible behaviours x is continuous, the master equation (4) is reformulated as a space-homogeneous Boltzmann equation ∂P (x, t) = [w(x|x ; t)P (x , t) − w(x |x; t)P (x, t)] dx . (5) ∂t Ω A Kramers-Moyal expansion (which is essentially a second-order Taylor approximation) of (5) leads to the Boltzmann-Fokker-Planck equation n n ∂(Ki P ) 1 ∂ 2 (Qij P ) ∂P =− + , ∂t ∂xi 2 i,j=1 ∂xi ∂xj i=1

where

Ki (x, t) =

Ω

(xi − xi )w(x |x; t) dx ,

are the effective drift coefficients and Qij (x, t) = (xi − xi )(xj − xj )w(x |x; t) dx Ω

1 ≤ i ≤ n,

1 ≤ i, j ≤ n,

are the effective diffusion coefficients. Whereas the drift coefficients govern the systematic change of the distribution P (x, t), the diffusion coefficients describe its spreading, due to fluctuations resulting from the individual variation of behaviour changes. This formulation allows to introduce two concepts: the social forces and the social fields. We only remark here that a social field represents, in the model, the influence of the public opinion, social constraints and tendencies. In [48], a method for solving the master equation (4) is presented, by writing the unknown P in an approximate form. The exact solution can then be obtained as an approximate expression of a path integral. Social state for an electoral competition In his works [60–63], Lo Schiavo develops kinetic models of Boltzmann type for social dynamics, to eventually reach a relevant description of an electoral competition. The first model he used [60] is an adapted version of the J¨ ager and Segel population dynamics model [53]. It is fitted to describe interacting agents who are characterized by a continuous variable u ∈ R, called the social state (poor/rich). It opened the road to model an electoral competition in his following articles. Note that the model of [60] is close to Helbing’s, as the variable of the density function is not yet the opinion, but a social state, which is quite similar to the configuration vector n in [46–48].


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In [61], the distribution functions describing the population are modified by two kinds of phenomena: the collisions between individuals, which are an internal process, and external forces. More precisely, the population is divided into groups, e.g., political parties, and the agents can choose their belonging to one group or another, or not belonging to any. The population inside each party is then described by a specific distribution function. The external forces can obviously have an effect each distribution function. They also allow the transition between groups by somehow influencing the choice of each individual with respect to the groups, i.e., the mass exchange between each group related distribution function can happen. Let us give some more details on the model. We denote by p the number of available groups. The variable of the distribution functions this time lies in [−1, 1]. It is not related to a specific group, it only represents the feeling of an agent about his social condition and his opinion with regard to the society where he lives. The positive values of u mean that the associated individual is quite happy, and the negative ones mean he is unsatisfied. If the collision process inside each group is standard, Lo Schiavo also defines another interaction process between individuals with different states and groups. Like in [2], the kinetic system that rises from these assumptions reads, for any 1 ≤ i ≤ p, ∂fi ∂(fi Ki (f )) − + = Q+ i (f ) − Qi (f ), ∂t ∂u

(6)

where each fi is the distribution function for the ith party, f = (fi )1≤i≤p , − Q+ i (f ) and Qi (f ) are, respectively, the gain and loss terms for the ith party due to the collisional processes, and eventually Ki (f ) is a propaganda operator, which draws an individual from a given party to the ith party. Note that this propaganda function is very similar to the social force defined by Helbing in [48]. In the previous operators, some frequency functions are used to drive the probability for which each kind of interaction occurs. Those frequencies allow to describe the asymptotic behaviour of the model, see 3.2. Fully collisional model As the main phenomenon in the opinion formation process is the compromise, it is quite natural to investigate a model where only collisional effects happen. We mainly focus now on systems where the variable of the distribution function is the opinion itself. It is systematically denoted with the letter x in the remaining of the chapter. Ben-Naim et al. [4,5] use a very similar microscopic model to the one from [26], to study the opinion dynamics, in a situation when the individuals reach compromise through binary interactions. They assume that the opinions of the population lie in a closed interval [−Δ, +Δ], Δ > 0. When the opinions (x1 , x2 ) of two individuals are close enough, in that case, |x2 − x1 | < 1, they both acquire the average opinion (x1 + x2 )/2. Otherwise, there is no

256


interaction. The density function of the population f satisfies the following rate equation of Boltzmann type: ∂f (x, t) = f (x∗ , t)f (2x − x∗ , t) dx∗ ∂t |x∗ −x| 0 describes the interaction rate, but is set to a constant value at the beginning. In both papers, some mathematical properties of the model are discussed. The discussion of [77] is focused on the moments of f to prepare for the study of the quasi-invariant limit, see 3.2. In [15], the authors prove the existence of the solution to (13) and (14), with an initial datum in L1 (−1, 1), lying in L∞ (0, T ; L1(−1, 1)), for any T > 0. 3.2 Mean field approximation When one considers a kinetic system with lots of interacting particles, it is most often very difficult to obtain an exact solution, except for very simple cases, such as the one-dimensional Ising model, for instance. The main idea, here, is to replace the microscopic interactions by a unique averaged interaction implying, most of the time, moments of the distribution function. This reduces the kinetic system into a simplified problem. The average behaviour of the kinetic system can then be obtained in an easier way. Consequently, we can also get more easily the time asymptotic behaviour of the distribution function, which is relevant, because it helps to quite accurately describe the stationary solutions of the kinetic equation. In the kinetic framework, the asymptotic models are often quite simple, for example, of Fokker-Planck type. In the final part of [61], Lo Schiavo discusses, with a computational point of view, the asymptotic behaviour of his basic and extended (with external forces) models. He points out that this behaviour mainly depends on the various frequencies of interaction used in the models. The quasi-invariant limit The studies from [77] and [16] tackle the question of the quasi-invariant limit from both mathematical and numerical points of view. They both assume that the collisional and diffusive effects are small, but remain linked. Boudin and Salvarani [16] assume that their attraction function η used in (11) and (12) is a constant close to 1, i.e., η = 1 − ε, ε > 0, and that the diffusion coefficient α in (13) has the form α(x) = εk α0 (x) for any x. They discuss the limit, when ε goes to 0+ , of the distribution function, with respect to the parameter k. They derive three different regimes: the collision-dominated one (k > 1), the diffusion-dominated one (k < 1) and the equilibrated one (k = 1), where the collision kernel is replaced by both linear and nonlinear (because of a moment of f ) terms. In the first case, they obtain an exact solution of the limit equation, which goes to a Dirac mass, when the time grows. In the second one, they provide an estimate of the convergence speed to the equilibrium of the solution to the the limit equation, the existence of which was previously obtained in [18]. Eventually, in the third case, they study the steady states and obtain, under some assumptions on β, an exponential rate of convergence.


259

Toscani [77] shows that the constants γ and σ involved in (8) and (9) are the key quantities to remember the microscopic collision mechanism when passing to the quasi-invariant limit (i.e., when γ and σ both go to 0), through the ratio λ = σ 2 /γ. In fact, he recovers the three same regimes as in [16] with respect to the value of λ (0, in R∗+ or +∞). In the equilibrated regime, i.e. when λ > 0, he obtains the convergence of the time scaled weak solutions f (x, τ /γ) to (10), towards a function g(x, τ ), which is a weak solution to the following Fokker-Planck equation ∂g λ ∂2 ∂ = ((x − m)g), D(|x|)2 g + 2 ∂τ 2 ∂x ∂x where

(15)

1

m=

xf (x, 0) dx −1

is the initial mean opinion. Note that it is really important here to assume that the interaction rate β does not depend on the opinion, and that P = 1. For the other regimes, he also introduces a suitable asymptotic limit of the model yielding a Fokker-Planck equation. An example Aletti, Naldi and Toscani [1] study one of these Fokker-Planck equations: ∂f ∂ =γ (1 − x2 )(x − m(t))f , ∂t ∂x

(16)

where γ is linked to the spreading (γ = −1) or the concentration (γ = 1) of the opinions. For γ = 1, (16) directly comes from the asymptotic limit in the collision-dominated case in [77], choosing P (y) = 1 − y 2 . For γ = −1, it looks like the model presented in [73], obtained by a mean field approximation of the Sznajd model [76], in the case of two opinions. Unlike (15), (16) is really nonlinear, because the time depending mean opinion is involved: 1

m(t) =

xf (x, t) dx. −1

They use some properties of a pseudoinverse of the cumulative distribution function to obtain the well-posedness of their problem, and the existence and uniqueness of solutions. Then they derive results on their large-time behaviour. The two values of the parameter γ call for separate treatments. In the spreading case, they prove that the limit distribution function is given by two masses located in ±1. In the concentration case, the limit has one or three Dirac masses, and in the latter case, two of them are located in ±1 and appear because they already exist at initial time.

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4 Other sociological phenomena Up to now, we pointed out one main phenomenon in the opinion formation process, the compromise. We even add for some basic models the self-thinking process, which appears either in the collision mechanism [1, 77] or in the kinetic equation itself [15, 16]. In fact, there are obviously lots of sociological phenomena which can be a part of the competition with the collisional process. We only present some of them here, because they appeared in the kinetic literature. The reader can refer to the review articles on the topic [20, 37] to find other tracks. 4.1 From the opinion to the choice The question of the choice/vote naturally rises after the opinion formation. Indeed, when someone votes, he does not necessarily only follow his own opinion, he may try to prevent the electoral results to be too far away from his opinion. It is quite clear that a vote model simultaneously depends on the voter opinion and on the electoral system. In the sociophysical literature, voting models are provided, for instance, in [40, 59, 75]. Comincioli, Della Croce and Toscani [25] propose a possible approach to the formation of choice. They mostly follow [77], but they add a fixed distribution of possible choices M (x), which can be seen, in the kinetic theory, as a fixed background of field particles. As they only consider opinions regarding a finite number N + 1 of questions, they can write a typical form of the background: N N M (x) = ωi δx¯i (x), ωi = 1. (17) 0

0

Let us give some explanations about the coefficients in (17). The parameters ωi ∈ (0, 1) are the probability that an agent chooses the ith possibility. The values x ¯i ∈ (−1, 1)\{0} represent the stability of the ith choice. They reflect the fact that extreme opinions are more difficult to change. The population and the background interact, and it is translated in a microscopic collision rule by: x = x − γP (|x − x∗ |)(x − x∗ ) + η D(|x2 |),

(18)

which is very similar to (8). The functions P and D, the random variable η and the constant γ have the same meaning as in [77]. The time evolution of the distribution function f is described by a kinetic equation involving a linear collisional integral of Boltzmann type. It reads ∂f = Q(f, M ), ∂t

(19)

where the collisional operator can be written in its weak form, for any smooth enough test function φ, as:


Q(f, M ), φ =

(−1,1)2

f (x, t)M (x∗ ) (φ(x ) − φ(x)) dx dx∗ .

261

(20)

Then they tackle the quasi-invariant limit of their system (19) and (20). The linear Boltzmann equation is again asymptotically well described by a Fokker-Planck type equation. This Fokker-Planck type equation recovers and generalizes analogous one obtained by mean field approximation of the voter model in [75]. 4.2 Contradictory individuals In [34, 36], Galam introduces and uses the notion of “contrarian” people to explain some major recent electoral phenomena. That kind of individuals cannot be convinced by standard arguments. In fact, they may systematically oppose the majority opinion, whatever it is. In [13], the authors introduce the notion, a little bit different, of contradictory people, opposed to conciliatory ones. When interacting with conciliatory individuals, who tend to compromise, they follow the opposite microscopic opinion, instead of simply going away from the average one. More precisely, if they still consider conciliatory people who tend to compromise, they also take into account two kinds of interactions involving contradictory people. Let x denote the opinion of a conciliatory individual and x∗ the opinion of a contradictory one, before interaction, and x , x∗ the respective post-collisional opinions. The first type of interaction writes: x − x∗ x + x∗ + η(x) , 2 2 x∗ + x x∗ − x + η(x∗ ) . x∗ = −α(x∗ ) 2 2 x =

(21) (22)

If (21) is still the same as in [15], the opinion of the contradictory individual uses the standard post-collisional opinion and shifts it to an opposite value using a reaction function α. In the second kind of interaction, the value of x is the same as in (21), but the new value of x∗ is given by: ⎧ (1 − x )(1 − x∗ ) ⎪ ⎪ 1− if x < x∗ , ⎪ ⎪ (1 − x) ⎪ ⎨ (23) x∗ = x∗ if x = x∗ , ⎪ ⎪ ⎪ ⎪ (1 + x )(1 + x∗ ) ⎪ ⎩ − 1 if x > x∗ . (1 + x) This time, the contradictory opinion tends to go to ±1, just to stay away from the conciliatory one it interacted with. For each kind of interaction, they also define the contradictory– contradictory collision mechanism.

262


In any case, the kinetic system satisfied by f and g is the following: ∂f = Q(f, f ) + R1 (f, g), ∂t ∂g = R2 (f, g) + S(g, g), ∂t

(24) (25)

where Q(f, f ) is the collision kernel associated to the interaction between conciliatory people, R1 (f, g) and R2 (f, g) are the kernels for the mixed conciliatory–contradictory interaction, and S(g, g) is the collision kernel associated to the interaction between contradictory people. The authors obtain an existence result on f and g in L∞ (0, T ; L1(−1, 1)). They next numerically investigate the asymptotic behaviour of the solutions of (24) and (25). If they use (21) and (22) as collision rules, both f and g go to the Dirac mass centred at 0, and if they use (23) instead of (22), they obtain two Dirac masses in ±1 for g, but not necessarily well-balanced, a concentration of f around an opinion which changes periodically in time. This latter behaviour must be underlined, because it means that there is eventually no steady state for f . 4.3 Leadership The concept of leadership is a key point of sociology and many authors have investigated its role in various aspects of the society. For example, many studies recognize that leaders have a crucial role in establishing the organization of pyramidal hierarchies, including large corporations, universities, armies, trade unions or political parties (see, for example, [35]). In [31], the authors explain the formation of opinions in a society by supposing that there exist two categories of people: the opinion leaders (group 2), who are active media users that select, interpret, modify, facilitate and transmit the information, and the less active part of the population (group 1), more passive and ductile. The individuals of the population interact between themselves and modify their opinions by means of a collisional rule which is a variant of Toscani’s [77]. In fact, as in [77], the opinion variable x lies in [−1, 1]. Both groups are described by distribution functions (fi )i=1,2 . If two individuals from the same group i discuss, the post-collisional opinions x , x∗ generated by the interaction of individuals with pre-collisional opinions x, x∗ are obtained through the following formulae: x = x − γi Pi (|x − x∗ |)(x − x∗ ) + ηi1 Di (x),

(26)

x∗ = x∗ − γi Pi (|x∗ − x|)(x∗ − x) + ηi2 Di (x∗ ).

(27)

This collision mechanism (26) and (27) is directly inspired from [77], where there is only one group (of followers). On the other hand, the interaction


263

between a follower with opinion x and an opinion leader with opinion x∗ gives post-collisional opinions x = x − γ3 P3 (|x − x∗ |)(x − x∗ ) + η11 D1 (x),

(28)

x∗

(29)

= x∗ .

The update of a follower’s opinion (28) is exactly the same as in (26). On the contrary, as stated in (29), a leader’s opinion does not evolve at all during an interaction with a follower. We refer to 3.1 for the meaning of γk , Pk , ηij and Dj , for k = 1, 2, 3 and i, j = 1, 2. The distribution functions fi , i = 1, 2, are then governed by the system of two Boltzmann-like equations ∂f1 1 1 = Q11 (f1 , f1 ) + Q12 (f1 , f1 ), ∂t τ11 τ12

(30)

∂f2 1 = Q22 (f2 , f2 ), ∂t τ22

(31)

where τij are the relaxation times, and the collision operators write, under their weak form, 1 Qij (fi , fj )(x, t)φ(x) dx −1 1 [φ(x∗ ) + φ(x ) − φ(x∗ ) − φ(x)] fi (x, t)fj (x∗ , t) dx dx∗ , = 2 (−1,1)2 (32) for all smooth enough test functions φ. Starting from microscopic interactions among individuals, the authors also obtain a quasi-invariant limit, described by a system of Fokker-Planck-type equations, and discuss its steady states. 4.4 Political plurality The political plurality is one of the main characteristics of our societies. Despite the very theoretical viewpoint that the media should be totally indepedent from the political class, it is clear that both are deeply interconnected. Besides, this topic is quite difficult to handle. Indeed, as we can see in the next paragraphs, the models which are used here have, for each party, either several distribution functions or a multidimensional opinion vector. The fact that there are few works (see [43] for example) about that too is quite meaningful. Propaganda and politicians Following [61], in [62], Lo Schiavo designs a specific model to describe the dynamics of a composite, structured society where there are two competing

264


political parties. The model structure once again contains terms with localized interactions and mean field terms. The variable of the standard population distribution function is again the social state u, but, this time, the population division is more intricate. There are now two categories. The first one is constituted of electors, who can be split into three subgroups: party 1, party 2 and the opinionless people regarding the parties. The second one is the political class itself. Its associated variable can be seen as the ideological position ν ∈ [−1, +1], which very much looks alike an opinion from the left (−1) to the right wing (+1). We must emphasize that the ideological position variable of the political class is not at all linked to the social state of the electors. Nevertheless, they may be treated in the same way. Hence, the system which is eventually obtained now is very similar to (6). Indeed, if we denote by f0 the density function for the political class, by f1 and f2 the ones for the electors, respectively, favouring party 1 and 2, by f3 the one for nonvoting electors, and, again, by f = (fi )0≤i≤3 , Lo Schiavo can write, for any t, u and ν, ∂fi ∂(fi Ki (f )) (t, u) + (t, u) = Qi (f )(t, u) + Ri (f )(t, u), 1 ≤ i ≤ 3, ∂t ∂u ∂f0 ∂(f0 K0 (f )) (t, ν) + (t, ν) = S0 (f )(t, ν). ∂t ∂ν

(33) (34)

The functions Ki are again the propaganda functions, whose sociological meaning is discussed in [54]. The operators Qi have the same meaning as in (6), they model the microscopic interactions between electors. The operators Ri model the effect of the political class on the electors, and S0 takes into account the interactions inside the political class. Eventually, in [63], Lo Schiavo presents a reduced version of the previous system (33) and (34). The model structure still contains terms with localized interactions and mean field terms. The main strength of this work is to emphasize the influence of some terms which were only briefly discussed in [62]. Mass media and multipartite situation In [14], the authors propose a kinetic model to describe the evolution of the opinion in a closed group with respect to a choice between multiple options, such as political parties. Two main mechanisms of opinion formation are taken into account: the binary interaction between individuals, as in [15], and the effects of the mass media. In multi-choice situations, a major problem consists in the fact that, in general, it is not possible to rank the options independently on the individual. The authors introduce, therefore, an opinion vector, whose dimension coincides with the number of possible choices. If there exist p ≥ 1 options of choice, xi ∈ [−1, 1] is the agreement variable associated to the choice Pi ,


265

1 ≤ i ≤ p, and the opinion (or agreement) vector x = (x1 , . . . , xp ) ∈ [−1, 1]p gives, for each individual of the population, its opinion about the plurality of options. Nevertheless, the distribution function f , defined on R+ × [−1, 1]p , is still one-dimensional. The first phenomenon taken into account is the binary exchange of opinions inside the population: if x, x∗ are the opinion vectors of two individuals before an interaction, the post-collisional opinions are obtained through a generalization of the collision rule defined in [15]: ⎧ xi + x∗i xi − x∗i ⎪ ⎪ x + η(x = ) ⎪ i i ⎨ 2 2 1 ≤ i ≤ p. (35) ⎪ ∗ ⎪ xi + xi x∗i − xi ⎪ ∗ ∗ ⎩ (xi ) = + η(xi ) 2 2 Once defined the collision rule (35), the interaction between individuals and the corresponding exchange of opinions is described by a collisional integral of Boltzmann type. The weak form of the collision kernel is Q(f, f ), ϕ = β(x, x∗ )f (t, x)f (t, x∗ ) [ϕ(x ) − ϕ(x)] dx∗ dx, (36) (−1,1)2p

where ϕ is a smooth enough test function in the variable x, and β : [−1, 1]2 → R+ is the cross-section, which depends on a suitable pre-collisional opinion distance. The effects of the media on the population are modelled by a background, which can be compared to the background introduced in [25]. This assumption adds a linear kinetic term into the equation. For any media Mj , 1 ≤ j ≤ m, the authors introduce two quantities: its strength αj , which translates the influence of the media on the population and its opinion vector X j ∈ [−1, 1]p , with respect to each option of choice. The effect of each media Mj on the individual is, therefore, described by an interaction rule which reminds the collision rule (35): x˜i = xi + ξj (|Xij − xi |) (Xij − xi ), for all i and j. The functions ξj are the microscopic media attraction functions. The influence of each media is then described by a (possibly time-dependent) linear integral operator, Lj , 1 ≤ j ≤ m, that has the classical structure of the linear Boltzmann kernels (see 2.2). Its weak form is Lj f, ϕ = αj f (t, x) [ϕ(˜ x) − ϕ(x)] dx, (37) (−1,1)p

where ϕ = ϕ(x) is a suitably regular test function.

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By combining the two phenomena, the evolution law of the unknown f is the following integro-differential equation of kinetic type, written in the distributional sense, m ∂f = Lj f + Q(f, f ), (38) ∂t j=1 The authors prove the existence and uniqueness of the solution to (36)–(38) in C 0 ([0, T ]; L1 ((−1, 1)p )) for initial data in L1 ((−1, 1)p ), and provide some numerical tests. In particular, if they use time dependent X j , they also obtain time dependent behaviours for the parts of the population in favour of each party. That means that we may not obtain a relevant asymptotic behaviour of the distribution function.

5 Conclusion In this chapter, we had the opportunity to describe numerous kinetic models fitted to the study of opinion dynamics. We pointed out that the main feature of the opinion formation process is the tendency to compromise, and we recovered it in all the models, obtaining Boltzmann-like equations. We also presented models taking into account the self-thinking, the voting process, the presence of contradictory people or leaders in the society, the propaganda through the media, and a multipartite democracy. We gave some tracks about quasi-invariant limits for those models, leading to Fokker-Planck equations. In many papers we reviewed, some numerical results are given, often regarding simple situations of interest. Those results allow to emphasize the main features of the models under study. We invite the reader to refer to the articles themselves to check that numerical part of the works. The numerical schemes or methods are discussed there, in link with our section dedicated to the numerical methods to solve the Boltzmann equations 2.3. As we already pointed out in Sect. 1.1, there are lots of theoretical contributions, but not so many with sociophysical data. In fact, the auxiliary functions and coefficients are not really investigated from a sociological point of view. Three of the articles we reviewed in this contribution tried to open the road. Helbing [48] worked on the German migration data given in [80]: having the state y ∈ {1, . . . , S} means “living in a region y”, where S was the number of West Germany regions. In [5], the authors discuss their model around the 1993 federal elections in Canada. Eventually, in [31], the authors test the behaviour of their model by confronting them to the results of the state elections in Carinthia (Austria). To enhance the models we investigate, we should try to systematically confront them with real data, such as existing polls, and then obtain more and more realistic models for the opinion formation.


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Part III

Human behavior and swarming

On the modelling of vehicular traffic and crowds by kinetic theory of active particles Nicola Bellomo1 and Abdelghani Bellouquid2 1 2

Department of Mathematics, Politecnico di Torino, Italy, [email protected] University Cadi Ayyad, Ecole Nationale des Sciences Appliquées, Safi, Marocco, [email protected]

Summary. This paper deals with developments and applications of the mathematical kinetic and stochastic games theory to the modelling of the dynamics of vehicular traffic and pedestrian crowds. The mathematical approach is focused on the derivation of the evolution equation for the probability distribution over the state, at the microscopic scale, of vehicles and pedestrians. Models take into account their heterogeneous behaviour.

1 Introduction This paper deals with developments and applications of the mathematical kinetic and stochastic games theory to the modelling of the dynamics of vehicular traffic and pedestrian crowds. The mathematical literature on traffic flow modelling has been developed after the pioneer book by Prigogine and Hermann [55], focused on kinetic type models, as documented in several review papers, among others [10, 11, 34,45]. The literature on crowd dynamics, which is far less developed, has been arguably initiated by Henderson [38, 39], subsequently developed by various authors, as we shall see in the following sections, also under the motivation of modelling panic and evacuation phenomena. Crowds need to be interpreted in a broad sense, namely not only as an assembly of pedestrians but also of individuals who aggregate or disaggregate according to specific strategies, for instance aggregation of criminality [56]. Recent literature is reported in the special issue [7], while an interesting source of information is the material available at the WEB site [14]. Applied mathematicians generally agree that modelling has not yet reached a fully satisfactory unified approach. Therefore, a great deal of additional work


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is still necessary to reach a robust mathematical theory suitable to reproduce, by equations, the non-predictable complexity of vehicular traffic and crowd phenomena. Bearing this in mind, let us anticipate the scaling problem, which identifies different methodological approaches to deal with the mathematical modelling of the systems under consideration. Specifically: The modelling at the microscopic scale consists in deriving, in a framework close to Newtonian mechanics, a differential equation for the dynamics of each single vehicle or pedestrian under the action of the surrounding ones. The solution of a large system of ordinary differential equations can provide the desired description of the overall dynamics. The description at the macroscopic scale is analogous to that of hydrodynamics, which consists in deriving evolution equations for the mass density and linear momentum regarded as macroscopic observable of the flow assumed to be continuous. Mathematical models are stated in terms of nonlinear partial differential equations derived on the basis of conservation equations and phenomenological models used for their closure. The statistical description consists, within a framework close to that of the kinetic theory of gases, in the derivation of a Boltzmann type evolution equation for the statistical distribution function on the position and velocity of the vehicles or pedestrians. Different classes of equations correspond to each type of representation. Moreover, different mathematical structures can be used for each class of equations. The different modelling representations that have been outlined earlier, are characterized by advantages and disadvantages and, in any case, none of them is fully satisfactory. Therefore, the present state of the art does not allow to establish correctly the validity of a class of models with respect to the others, while research activity should look for new approaches suitable to overcome the above outlined technical difficulties. This paper is focused on the use of methods of the kinetic theory that is, in the authors’ opinion, an interesting approach to be properly developed to capture the complexity of the physical reality of the systems under consideration. Specifically, it aims at providing the conceptual basis focused on research developments. The contents are developed through five more sections. In detail, Sect. 2 provides a critical analysis of the common features of traffic and crowds as they appear according to their phenomenological interpretation. Moreover, it focuses on the empirical data obtained by experiments that may be possibly used to validate models. Section 3 introduces the representation of traffic according to the mathematical kinetic theory for active particles. Subsequently, some mathematical structures that may act as paradigms for the derivation of models are proposed and critically analyzed. Section 4 focuses on the approach of the kinetic theory that refers to granular essence of vehicular traffic and reports on mathematical methods based on the discretization of the phase

On the Modelling of Vehicular Traffic and Crowds

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space with the aim of modelling granular flow phenomena of systems that do not satisfy the continuity assumption of the distribution function over the microscopic state of the interacting entities. Section 5 shows how the modelling approach of vehicular traffic can be properly developed to deal with crowd modelling. Finally, Sect. 6 develops a critical analysis mainly focused on open problems and research perspectives concerning both modelling and analytic issues.

2 Common features of vehicular traffic and crowds Classically, the modelling approach needs the interpretation of the complex phenomenology of the systems under consideration, which exhibit common features and remarkable differences. This section provides an insight into these specific characteristics with the aim at investigating how far a modelling approach can effectively deal with all of them. Applied mathematicians involved in traffic modelling often refer to the criticisms from the view point of engineers offered by the sharp paper of Daganzo [29], who observes that the classical continuity assumption of mechanics cannot be applied to traffic flow. Indeed, particle flows in fluid dynamics refer to thousands of particles, while only a few vehicles are involved even in traffic jams. This remark can be extended also to the approach of the kinetic theory and specifically to the assumption of continuity of the distribution function. Further, he observes that a vehicle is not a particle, but a system linking driver and mechanics, so that one has to take into account the reaction of the driver, who may be aggressive, timid, prompt etc. Finally, increasing the complexity of the model increases the number of parameters to be identified that may even be impossible due to the complexity of the setting of experiments. This critical analysis is also reported in various papers, among others [11, 58]. An interesting reading is offered by the book of Kerner [41], who provides various interpretation of physical phenomena which emerge in traffic flow. This book is based on a remarkable experience of the author in the field documented in several papers, among others [42–44]. Bearing all the above in mind, let us select, according to the authors’ bias, the following six, among various ones, characteristics: 2.1. Behaviours of complex systems: Vehicles on roads and individuals in crowds can be regarded as complex living systems who interact in a nonlinear manner. Moreover, interactions follow specific strategies generated by the ability to communicate with the other entities, and to organize the dynamics according both to their own strategy and interpretation of that of the others [5]. Therefore, the knowledge of the interaction of a few entities is not sufficient to describe the collective dynamics of the overall system. A further difficulty is generated by the fact that individual dynamics are not generally observable, while only the overall behaviour can be observed and geometrically interpreted.

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A technical difference, which distinguishes the two systems, is that traffic flow is one directional in one space dimension (or multilane) and over well defined networks, while the dynamics of crowds is in two or three space dimensions, either in bounded domains or in the whole space. Crowds may be constrained by particular geometries that generate different aggregation rules. 2.2. Granular dynamics: The dynamics of vehicles and pedestrians shows behaviour of granular matter with aggregation and vacuum phenomena. Indeed, the continuity assumption of continuum mechanics and of the distribution function in kinetic theory cannot be straightforwardly supposed. 2.3. Influence of the environmental conditions: The dynamics is remarkably affected by the quality of environment including territory and weather. Therefore, the modelling approach needs including this aspect into the mathematical equations. 2.4. Heterogeneous distribution of the individual behaviours: Individual behaviours, and the quality of vehicles, are heterogeneously distributed among interacting entities in traffic and crowds. In the former the microsystem is constituted by the driver-vehicle entity, while in the latter the microsystem is constituted by individuals. The shape of the distribution over the microscopic state has an influence over the strategy developed in the interactions. 2.5. Parameters: The modelling approach needs parameters suitable to model some essential characteristics of the system under consideration. It is important that these parameters are related to specific different phenomena. Moreover, their identification should be technically pursued either by using existing experimental data, or by experiments to be properly designed. 2.6. Influence of panic conditions: Generally, in traffic flow, all drivers have approximately the same strategy, which is not consistently modified by panic conditions. On the other hand, in crowds, the dynamics of the interactions and the overall strategy is modified according to specific situations, for instance the presence of panic can change them consistently. The modelling approach should capture both analogies and differences. The modelling strategy described in the sequel is developed to take into account the above specific characteristics. In general, the approach needs empirical data that can be, namely ought to be, used to validate mathematical models. On the other hand, the difficulty to obtain these data is due to the great variation of the environment and of the individual behaviours. This aspect reduces the amount of available data useful to validate theoretical models. A deep analysis of empirical data is delivered in the book by Kerner [41], which shows a variety of physical phenomena that characterize traffic flow. Some of them are analogous to that of crowds. In particular, the velocity diagram reports, in steady flow conditions, the mean dimensionless velocity ξe ∈ [0, 1] (referred to the maximal value of the mean velocity)


277

vs. dimensionless density ρ ∈ [0, 1] referred to the maximal density ρM corresponding to full packing. The following phases can be observed: ξe ∼ = 1,

I : ρ ≤ ρc

⇒

II : ρ ≤ ρc ≤ ρs

⇒

ξe ↓

with sharp negative derivative ,

III : ρs ≤ ρ ≤ 1

⇒

ξe ↓ 1

with small negative derivative ,

where ρc and ρs are critical densities that identify the above three phases. Moreover, the velocity variance is very small in Phase I, very large in Phase II, while it is significant in Phase III, but smaller than that of Phase II. In general data are very sensitive to the quality of the road or of the environment containing the crowd, as well as to environmental conditions, namely the overall macro-system. Therefore, it is impossible identifying a unique deterministic representation for all roads. This characteristic has suggested [19] to use, in vehicular traffic, ρc as the parameter suitable to describe the quality of the road. It is worth stressing that the above behaviour is common to vehicles and pedestrian as documented in [61] and therein cited bibliography. The report [21] offers a variety of empirical data concerning measurements of the behaviour of pedestrians in a crowd. This data also refer to heterogeneous behaviours including those due to handicaps. Several models use analytic approximation of empirical data by inserting them artificially in the structure of the models instead of modelling a dynamics of interactions that generate a trend to a velocity diagram depending on the parameter modelling the quality of the environment.

3 Mathematical representation and structures The mathematical literature concerning vehicular traffic is widely developed and includes different approaches at all scales as documented in the already cited review papers [10, 11, 34, 45], which report about several papers devoted to this topic. Among others [3, 17, 26, 32, 33, 46–52], and many others. On the other hand, the modelling approaches to crowd dynamics has been developed essentially at the microscopic scale due to Helbing and coworkers [35–37], and to the macroscopic scale, among others [9, 25, 40]. The dynamics of crowds is also related to the dynamics and safety of structures [60, 61]. This section reports about the representation of both systems according to the mathematical kinetic theory for active particles [6] and on the assessment of mathematical structures suitable to act as background paradigm for the derivation of models. The contents are organized through three subsections: the first one deals with the representation of the system, the second one with the assessment of mathematical structures, and the third subsection develops a critical analysis focused on research perspectives.

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3.1 Mathematical representation Let us consider the representation by kinetic theory methods of a large system of interacting entities, namely vehicles or pedestrians regarded as micro-systems, which interact in a suitable environment. Moreover, let us specifically consider [11, 18], the one directional flow of vehicles along a road with length . The time, space and velocity variable define the microscopic state of each vehicle: • t is the dimensionless time variable obtained referring the real time to a suitable critical time Tc to be properly defined by a qualitative analysis of the differential model. Generally, it is convenient identifying the critical time Tc by the ratio between and maximum admissible mean velocity VM in the road that is reached in free flow conditions. • x is the dimensionless space variable obtained dividing the real space by the length of the road. • v is the dimensionless velocity variable referred to the limit velocity V reached by the fastest isolated vehicles. • u ∈ [0, 1] is the dimensionless activity variable, which identifies the quality of the driver-vehicle micro-system. Specifically, u = 0 corresponds to the worst quality, while u = 1 corresponds to the best quality. • ρM is the maximum density of vehicles corresponding to bumper-to-bumper traffic jam. Let us now consider the representation by kinetic theory of active particles, where the state of the whole system is defined by the statistical distribution of position and velocity and activity of the vehicles; f = f (t, x, v, u) :

R+ × [0, 1] × [0, 1] × [0, 1] → R+ ,

(1)

where f (t, x, v, u) is normalized with respect to ρM . Classically, f (t, x, v, u)dxdvdu gives the number of vehicles which, at the time t, are in the elementary domain of the space of the microscopic states: [x, x + dx] × [v, v + dv] × [u, u + du]. Macroscopic observable quantities can be obtained, under suitable integrability assumptions, by moments of the above distribution function. Due to normalization all derived variables are in a dimensionless form. In particular, the dimensionless local density, referred to ρM , is given by:

1

1

f (t, x, v, u) dv du ,

ρ(t, x) = 0

(2)

0

while the total number of vehicles at the time t is computed by integration over space.


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In the same way, the local dimensionless mean velocity can be computed as follows: ξ(t, x) =

1 q(t, x) = ρ(t, x) ρ(t, x)

1

1

v f (t, x, v, u) dv du , 0

(3)

0

while the local speed variance is given by: σ(t, x) =

1 ρ(t, x)

1

0

1

0

2

[v − ξ(t, x)] f (t, x, v, u) dv du .

(4)

Moreover, the speed pressure is defined by the the speed variance multiplied by the local density: p(t, x) = σ(t, x) ρ(t, x). (5) The representation of crowds is analogous in the case of bounded domains. Specifically, let us consider the crowd in a bounded domain Σ, whose largest dimension is again denoted by . The state of the whole system is described by the distribution function: R+ × Σ × Dv × [0, 1] → R+ ,

f = f (t, x, v, u) :

(6)

where Σ ⊂ R2 and v ∈ Dv ⊆ R2 . Reference quantities analogous to that of traffic flow can be used; their definition is essentially the same. For instance the local density is as follows: 1 ρ(t, x) = f (t, x, v, u) dv du , 0

Dv

and analogously for higher order moments. The above representations relay on the assumption that the number of interaction entities, either vehicles or pedestrians, is large enough to justify the continuity assumption of the distribution function. Actually, this is not true in physical reality and an alternative approach needs to be developed as we shall see in the next sections. However, some preliminary reasonings based on the above representation are useful towards a deeper analysis that will be developed in the sequel. 3.2 Mathematical structures In general, the modelling approach should be referred to mathematical structures that can provide the conceptual background for the derivation of kinetic type models of vehicular and crowd dynamics. The existing literature on the kinetic modelling approach has not yet identified a unique structure to be used towards modelling, while a variety of models refer to very different structures, all derived from the classical mathematical kinetic theory [53].

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Only recently, the need of using the kinetic theory for active particles has been postulated to model the heterogeneous behaviour of the microsystems that compose the overall systems. This subsection focus two specific structures that can be possibly developed to deal with the modelling of the above characteristics. Before dealing with the derivation of equations, it is worth fixing some definitions and notations. The interacting entities are called micro-system, in this case vehicle-driver or pedestrians, and are regarded as active particles, namely particles whose state includes the activity variable in addition to mechanical quantities. Interactions involving active particles generate the evolution of the generalized distribution function f . Three types of particles are involved: The test particle, which is representative of the whole system; Field particles that interact with test and candidate particles; Candidate particles that may acquire, in probability, the state of the test particle. The distribution function f is referred to the test particle. Each particle has an interaction domain Ω that corresponds to a visibility zone with Ω ⊆ Σ. Different types of interactions can be considered: Localized binary interactions when the measure of Ω is negligible; Mean field binary interactions when the resultant action on the test particle is the sum of all binary actions included in Ω; Topological interactions with stochastic games when particles modify, in probability, their state according to a strategy based on the analysis of the position and state of all particles in Ω. • Let us first consider models with mean field interactions, which need the definition of the acceleration F applied to the micro-system in x with velocity v and activity u by the one in x∗ with velocity v∗ and activity u∗ . The corresponding structure is as follows: ∂t f (t, x, v, u) + v · ∂x f (t, x, v, u) + ∂v F [f ]f (t, x, v, u) = 0 , (7) where F = F (t, x, v, u) is given by summing all actions applied by particles with state v∗ and u∗ in the visibility zone, x∗ ∈ Ω of the test micro-system. Technical calculations yield: F [f ](t, x, v, u) = F (x, x∗ , v, v∗ , u, u∗ ) f (t, x∗ , v∗ , u∗ ) dx∗ dv∗ du∗ , (8) Γ

where Γ = Ω × Dv × [0, 1], and Ω is the visibility zone of the micro-system in x. Therefore, the derivation of models is based on a detailed modelling of the term F . Of course, technical differences distinguish crowd modelling from vehicular traffic as we shall see in the forthcoming two sections.


281

• Models with interactions modelled by stochastic games Let us now consider a mathematical structure, where the dynamics of the micro-systems is developed by a strategy based on a weighted analysis of the localization and microscopic state of the other micro-systems in its visibility zone. Such a structure can be written, referring to [6] as follows: ∂t f (t, x, v, u) + v · ∂x f (t, x, v, u) = J[f ](t, x, v, u) η[ρ](t, x∗ ) w(x, x∗ ) A(v∗ → v; u∗ → u|v∗ , v∗ , u∗ , u∗ , ρ(t, x∗ )) = Λ

×f (t, x, v∗ , u∗ ) f (t, x∗ , v∗ , u∗ ) dv∗ dx∗ dv∗ du∗ du∗ −f (t, x, v, u) η[ρ](t, x∗ ) w(x, x∗ ) f (t, x∗ , v∗ , u∗ ) dx∗ dv∗ du∗ , (9) Γ

where Λ = Γ × Dv × [0, 1]. This approach considers interactions of a candidate or a test micro-system in x (with velocities v∗ , v and activity u∗ , u, respectively) with the field particles in x∗ , with velocity v∗ and activity u∗ located in its interaction domain, namely x∗ ∈ Ω. Interactions are weighted by the term η[ρ](t, x∗ ) interpreted as an interaction rate, while the distance distribution of the intensity of the interactions is weighted by the function w = w(x, x∗ ). The candidate particle modifies its state according to A, which denotes the probability density that a candidate particles with state (v∗ , u∗ ) reaches the state (v, u) after the interaction with the field particles with state (v∗ , u∗ ), while the test particle looses its state v and u after interactions with field particles with velocity v∗ and activity u∗ . 3.3 Critical analysis As already mentioned, methods of the generalized kinetic theory have been widely applied in vehicular traffic modelling, while only recently have been used in crowd and swarms modelling, see [23]. Models do not take into account the heterogeneous behaviour of the micro-system, while the phenomenology of the systems under consideration, as we have seen in the preceding section, suggests this generalization. It is worth focusing on a critical analysis of a large variety of models available in the literature in view of further research activity in the field: • The continuity assumption of the distribution function does not correspond to the granular structure of the systems under consideration; • A large variety of model plugs into the model a trend to an equilibrium. A simple model is as follows: ∂t f + v ∂x f = Q[f ; ρ] = cr (ρ) (fe − f ) ,

(10)

where f = f (t, x, v), and Q[f ; ρ](t, x, v) describes a trend to equilibrium analogous to the BGK model in kinetic theory, where the rate of convergence cr

282


depends on the local density, and fe = fe (v; ρ) denotes the equilibrium distribution function, that may be parameterized by the local density. On the other hand the dynamics of the interactions microscopic level should naturally generate, as direct predictions of the model, stationary solutions. • Several models are based on localized binary interactions between the test and the field vehicles localized in the point x of the field vehicle. Interactions, similarly to the Enskog equation, can be localized at a fixed distance d from the test vehicles on its front. Moreover, similarly to the Boltzmann equation, factorization of the joint probability related to the two vehicles is assumed. • The dynamics of drivers and pedestrians is not due on pair interactions, but also to the ability of the micro-system to develop a strategy out of the analysis of the state of the surrounding active particles [4]. The contents of the next sections attempts to deal with some of the earlier criticisms.

4 On the modelling of vehicular traffic This section proposes a modelling approach to vehicular traffic based on the preliminary analysis offered by the preceding sections specifically referring to the criticisms due to Daganzo [29] focused in the observation that the assumption of continuity of the distribution function over the microscopic state of vehicles (or pedestrians) can be criticized considering that the number of interacting entities is not large enough to justify this assumption. Therefore, not only the assumption of continuity of the matter has to be put in discussion, but also that concerning the distribution function. Two recent papers Coscia, Delitala and Frasca [26] and Delitala and Tosin [32] have proposed kinetic type models with discrete velocities to take into account the granular nature of traffic. These models are such that the overall state of the system is described by a discrete probability distributions over groups of vehicles with velocity within a certain velocity range. Model [32] has been generalized also to the case of multilane flow [17]. These papers are based on the generalized kinetic theory, namely the heterogeneous behaviour of the micro-system is not considered. This section shows a methodological approach can be developed to design models, where the phase space, namely both space and velocity are discrete variables, while the activity variable is left continuous. Accordingly, the overall state of the system of vehicles, modelled as mass points, is delivered, in each point-cell xi by the following distribution function f (t, x, v, u) =

n m i=1 j=1

fij (t, u)δ(x − xi )δ(v − vj ) ,

(11)


283

which corresponds to the discretizations: Ix = {x1 = 0 , . . . , xm = 1},

and Iv = {v0 = 0 , . . . , vn = 1},

where fij (t, u) = f (t, u; xi , vj ). According to this mathematical representation, the following macroscopic quantities are obtained in each point x = xi : ρi (t) = ρ(t; xi ) = and qi (t) = q(t; xi ) =

n j=1

0

n

1

j=1

0

1

fij (t, u) du ,

(12)

vj fij (t, u) du ,

(13)

where it has been assumed that absence of vehicles with null velocity. Analogous calculations can be developed for higher order moments including velocity variance and entropy functions. The model consists in a set of evolution equations for the densities fij derived according to the following structure: ∂t fij (t, u) + vj Dij [f ](t, u) = Jij [f ](t, u) n 1 η[ρi (t)] Ajhk (u|u∗ , u∗ , ρi )fih (t, u∗ ) fik (t, u∗ ) du∗ du∗ = h,k=1

−

0

fij [ρ(t)]

n k=1

0

1

η[ρi (t)] fik (t, u∗ ) du∗ ,

(14)

where i ∈ Ix , j ∈ Iv and f = {fij }, while the operator Dij [f ] approximates the space derivative of fij and Jij [f ] models interactions among vehicles. In detail, the operator Dij [f ] is computed by conservative schemes for hyperbolic equations, see [16] and therein cited bibliography. Some indications briefly given for the operator Jij [f ] in the case of localized interaction in each cell volume is denoted by xi . Specifically, η[ρi (t)] is the interaction rate, which gives the number of interactions per unit time among the vehicles; and Ajhk (u|u∗ , u∗ , ρi ) = A(vh → vj , u∗ → u|vh , vk , u∗ , u∗ , ρi (t))

(15)

defines the so-called table of games, which models the microscopic interactions among the vehicles by giving the probability density that a vehicle with speed vh and activity u∗ adjusts its velocity and activity to vj and u after an interaction with a vehicle travelling at speed vk with activity u∗ . This term is conditioned by the local density and is, in each point x = xi , a probability density with respect to the outputs. Hence, it satisfies the following condition:

284

Nicola Bellomo and Abdelghani Bellouquid n j=1

0

1

Ajhk (u|u∗ , u∗ , ρi ) du = 1 ,

∀ i, h, k, ρi .

Focusing on the rate η it is assumed to behave as follows: c 1 , exp − c > 0, η[ρi ] 1 − ρi (t) 1 − ρi (t)

(16)

(17)

for ρ ∈ [0, 1]. Namely, this function is first monotonically increasing with respect to ρ and subsequently decays to zero for ρ → 1. Accordingly, the local interaction rate becomes higher and higher as the density increases towards its limit threshold fixed by the road capacity. However, when vehicles approach the packing density the sensitivity of the driver to the other vehicles rapidly decays to zero. The modelling of the terms A, which define the table of games, needs a mathematical interpretation of the microscopic phenomenology of the system. According to [6], the following factorization is suggested: Ajhk (u|u∗ , u∗ , ρi ) = B(vh → vj |ρi (t), vh , vk )C(u∗ → u|ρi (t), u∗ , u∗ ) ,

(18)

which amounts at assuming that the dynamics related to the velocity variable is conditioned by the local density and by the activity variables, while the dynamics related to the activity variable is conditioned only by the local density. A reliable proposal for the table of games B, as it has already given good agreement with experimental data, is offered by that proposed in [32]. Here we replace the parameter α ∈ [0, 1], which models the quality of the macrosystem (road plus environmental conditions) by β = α u. Namely, the quality of the road is measured corresponding to the best quality of the micro-systems, however, it is reduced by u ∈ [0, 1] models the quality of the micro-systems. This table is reported, for sake of completeness in the Appendix. It has been proved by [59] that the mathematical structure of the original model [32] can be related, under suitable assumption to the mathematical structure proposed in [2] of the kinetic theory for active particles that includes, as a particular case, the classical Boltzmann equation. The modelling of the table of games C can be obtained assuming that if the distance between the activities of the interacting pairs is large enough then the microsystems have a trend to mix the quality of their dynamics, while the opposite behaviour is observed in the case of small distances which generate competition. Moreover, the higher is the quality of the road, the more significant is the mixing term. A conceivable model is as follows:

where:

C(u∗ → u|u∗ , u∗ , ρi (t)) = δ(u∗ − u(u∗ , u∗ , ρi (t))) ,

(19)

|u∗ − u∗ | ≥ ε ρ ⇒ u = u∗ + ε α(u∗ − u∗ ) ,

(20)


285

while the opposite behavior is as follows: |u∗ − u∗ | < ε ρ ⇒ u = u∗ (1 − ε α) .

(21)

Finally, this model is characterized by two parameters: • α ∈ [0, 1] that characterizes the quality of the road and environmental conditions; • ε that is a small, with respect to one, parameter characterizing the mixing trend. A particular case which is worth to be considered in the validation of models is the spatially homogeneous case that corresponds to constant density over the whole space. Therefore, the local density is a parameter. Such a model can be formally written as follows: n

∂t f j (t) = Qj [f ](t) = − f j (t)

h,k=1 n 1 k=1

0

0

1

η[ρ0 ]Ajhk (u|u∗ , u∗ , ρ0 )f h (t, u∗ ) f k (t, u∗ ) du∗ du∗

η[ρ0 ]f k (t, u∗ ) du∗ .

(22)

The statement of the initial value problem corresponding to the spatially homogeneous case is obtained by linking (21) to initial conditions: ⎧ ⎨ ∂t f j (t, u) = Qj [f ](t, u) , (23) ⎩ j f (t = 0, u) = ϕj (u) , homogeneous in space. The statement of the initial boundary value problem in the general case is obtained by linking (14) to the initial and boundary conditions. Specifically, using periodic boundary conditions, yields: ⎧ j ⎨ ∂t fi (t, u) + vj Dij [f ](t, u) = Jij [f ](t, u) , (24) ⎩ j j j j fi (t = 0, u) = ϕi (u) , f1 (t, u) = fm (t, u) . It is expected, in particular, that the solution to Problem (23) generates the fundamental velocity and and flux diagrams as they are experimentally observed according to the brief summary of Sect. 2. Preliminary simulations show that this result is achieved not only for the shape of the mean velocity and flux vs. density but also for the velocity fluctuations according to the observations by Kerner [41]. Of course, the model can be further refined according to the critical analysis of Sect. 6.

286


5 On the modelling of crowds This section aims at providing the guidelines to transfer the approach proposed in the preceding section to the modelling of crowds. This is an important topic considering that the literature in the field is almost limited to the approach at the microscopic scale as documented by the activity developed by Helbing and coworkers, among various papers [35, 36]. The approach at the macroscopic scale has been settled by Hughes [40], and subsequently developed by various authors [9, 25], by means of classical methods of continuum mechanics based on the use of mass and momentum conservation equations properly closed by phenomenological models modelling the relation of the acceleration term, or mean velocity, to local flow conditions. Recently, Piccoli and Tosin [54] have considered discrete-time modelling by means of a family of measures, which provide an estimate of the space occupancy by pedestrians in time. The interesting aspect of this approach is the coupling of the microscopic dynamics determined by pedestrians’ strategy and the macroscopic one related to conservation of mass. The interest in this type of modelling is not purely speculative, for instance the papers by Venuti et al. [60, 61], are focused on the modelling of the interactions between crowds and lively footbridges and related structural analysis. Moreover, Helbing studies the effect of onset of panic conditions and evacuation problems [37], while detailed calculations of pilgrim’s Jamarat bridge, where overcrowding is the cause of frequent accidents, have been developed by Coscia and Canavesio in [25].

Q

º0 P Σ

∂Σ

Fig. 1. Geometry of the crowd domain

As already mentioned, the mathematical approach of the kinetic theory to modelling crowds is not yet well settled. Therefore, this section offers some guidelines to be properly developed in a research program. The aim consists in showing how the modelling of interactions can be developed according to the framework of the kinetic theory for active particles. Bearing all the above in mind, let us consider crowd dynamics in bounded domains Σ with boundary ∂Σ, whose geometry is represented in Fig. 1, where


287

ν0 is the straight direction to be followed by a pedestrian in P to the exit Q. Moreover, let us subdivide the domain Σ into finite sub-domains Σi for i = 1, . . . , n. Moreover, let us discretize the velocity of individuals, regarded as active particles, into a finite umber of velocities vj for j = 1, . . . , m. Therefore, the overall state of the system is described by the distribution function: f (t, x, v, u) =

n m i=1 j=1

fij (t, u) δ(x − xi ) δ(v − vj ) ,

which differs from that of (11) simply by the dimension of x and v. The mathematical structure to be used for modelling is a technical generalization of that we have seen in the preceding section: ∂t fij (t, u) + vj Dij [f ](t, u) = Jij [f ](t, u) n 1 = η[ρi (t)]B(vh → vj |ρi (t), vh , vk , u∗ ) h,k=1

0

× C(u∗ → u|ρi (t), u∗ , u∗ )fih (t, u∗ ) fik (t, u∗ ) du∗ du∗ n 1 j − fi (t) η[ρi (t)]fik (t, u∗ ) du∗ , k=1

(25)

0

where the factorization (18) has been used. The above structure is quite general. However, it can be simplified focusing to specific applications. For instance, one can assume that the activity variable is the same for all active particles and that the velocity modulus is the same for all of them, while the direction is identified by the discrete variable θj , for j = 1, . . . , m, corresponding to the discretization: Iθ = {θ1 = 0, . . . , θj , . . . , θm = 2π}. Accordingly, the distribution function, for each θj and in each xi , and the mathematical structure are as follows: ∂t fij (t) + vj Dij [f ](t) = Jij [f ](t) n η[ρi (t)]B(θh → θj |ρi (t), θh , θk )fih (t)fik (t) = h,k=1

− fij (t)

n

η[ρi (t)]fik (t) ,

(26)

k=1

where f (t, x, v, u) =

n m i=1 j=1

fij (t) δ(x − xi ) δ(v − vj ) δ(u − u0 ) ,

(27)

288


with fij (t) = fi (xi , vj , u0 ), and where the term C does not appear in (26) considering that we have assumed, for simplicity, that the activity variable is the same for all individuals. The derivation of specific models needs the modelling of the interaction rates can be developed similarly to the case of vehicular traffic. Moreover, it needs the specific modelling of the table of games, which can be derived by taking into account both the attraction from other individuals of the crowd and the fact that generally a crowd has a target, for instance to reach an outlet zone corresponding to evacuation, for instance a point T of the boundary corresponding to the exit, which is identified by the angle θî . A simple model of the interactions among particles in the crowd is as follows: • Interaction with a upper direction and target: It is actions contribute to an anticlockwise rotation: ⎧ if θk > θh ; ⎨ α (1 − ρi ) (ε1 + ε2 ) j Bhk [ρi ] = α (1 − ρi ) (1 − ε1 − ε2 ) if θk > θh ; ⎩ 0 otherwise.

assumed that both interθî > θh ; θî > θh ;

j = h + 1, j = h,

• Interaction with a upper direction and a lower target: It is assumed that the two interactions contribute, respectively to an anticlockwise and a clockwise rotation: ⎧ α (1 − ρi ) ε1 if θk > θh ; θî < θh ; j = h + 1, ⎪ ⎪ ⎨ α (1 − ρi ) (1 − ε1 − ε2 ) if θk > θh ; θî < θh ; j = h, j Bhk [ρi ] = ⎪ if θk > θh ; θî < θh ; j = h − 1, ⎪ ⎩ 1 − α (1 − ρi ) ε2 0 otherwise. • Interaction with a lower direction and an upper target: It is assumed that the two interactions contribute, respectively, to a clockwise and an anticlockwise rotation: ⎧ α (1 − ρi ) ε2 if θk < θh ; θî > θh ; j = h + 1, ⎪ ⎪ ⎨ ) (1 − ε − ε ) θî > θh ; j = h, α (1 − ρ j i 1 2 if θk < θh ; Bhk [ρi ] = ⎪ if θk < θh ; θî > θh ; j = h − 1, ⎪ ⎩ α (1 − ρi ) ε1 0 otherwise. • Interaction with a upper direction and target: It is actions contribute to a clockwise rotation: ⎧ if θk < θh ; ⎨ α (1 − ρi ) (ε1 + ε2 ) j Bhk [ρi ] = α (1 − ρi ) (1 − ε1 − ε2 ) if θk < θh ; ⎩ 0 otherwise,

assumed that both interθî < θh ; θî < θh ;

j = h − 1, j = h,

j where Bhk [ρi ] stands for B(θh → θj |θh , θk , ρi (t)) and α is a parameter that models the quality of the environment.


289

Of course, the above model is quite simple; however, it can be technically improved by referring to the more general structure (25), where the term C(u∗ → u|ρi (t), u∗ , u∗ ) can be modelled according to the same guidelines offered for the vehicular traffic. Similarly, a finite number of velocities can be introduced with a dynamics of velocity modification by jumps ruled as reported in the Appendix. An important aspect of crowd modelling consists in analyzing how the presence of panic conditions modifies the model. Specifically, we can refer again to the structures of (25) and (26) as well as to the table of games that has been proposed earlier. Basically, we claim that the structure is still valid in panic conditions; On the other hand parameters and table of games are modified according to the following indications: 1. Panic conditions remarkably increase the limit velocity. Therefore, the real values expressed by the dimensionless variable refer to quantitatively higher values. 2. Heterogeneity is reduced, namely all individuals attempt to behave in the same way to escape danger conditions. 3. Parameters ε1 and ε2 that in normal conditions are of the same order attain different values in the case of panic. Specifically, the action of other individuals become more important with respect to the need of reaching the objective. Of course, the evaluation depends on the localization of the source of panic. Further critical analysis and perspectives to refine the above guidelines are given in the last section.

6 Critical analysis and perspectives This chapter has proposed a new approach to modelling vehicular traffic, where both the granular dynamics of the flow and the heterogeneous behaviour of the driver-vehicle micro-system have been taken into account. Subsequently, the approach has been technically generalized to crowd modelling. The content has to be regarded as first step towards the modelling of the complex system under consideration. This final section aims at showing how the common features described in Sect. 2 have been taken into account and subsequently indicate some research perspectives concerning modelling topics. 6.1 Complexity problems Let us consider the sic specific characteristics of traffic and crowd systems, which have been briefly described in Sect. 2, and let us analyze, for each of them, how far the mathematical structures and the modelling approach proposed in the preceding sections has taken them into account.

290


Dynamics of complex systems: The modelling approach is such that the entities, which play the game are regarded as active particle that interacts with all the other particles that are in a suitable domain identified by the visibility zone. Active particles play a game corresponding to the earlier interaction. The output, identified in probability, is technically related to the ability they have to develop a certain strategy. Interactions may not be homogeneous in space, considering that active particles can, in some cases, choose different observation paths. An important consequence is that interactions do not involve pairs of particles, but a number of them, their states, and geometrical distribution. Granular dynamics: The discretization of the phase space overcomes the unrealistic assumption of continuity of the distribution function. The modelling of interactions takes into account groups of vehicles that occupy finite volumes in the phase space. The same reasonings can be referred to the whole space of the microscopic states. Environmental conditions: The parameter α models the quality of the road and environmental conditions, and plays a crucial role in the modelling of interactions. In fact, high values of α contribute to the modifications of velocity, which are prevented by low values of the parameter. This dynamics is technically related to the quality of the micro-system either vehicle-driver of pedestrian. Such a quality identified by the variable u acts in the same way of α. However, interactions modify the probability distribution over the activity variable. Heterogeneity: The heterogeneous distribution of the quality of the drivervehicle micro-systems is taken into account by the activity variable that plays, as already mentioned, a role analogous to α in the interaction dynamics. The modelling takes into account a certain mixing of the behaviors of the microsystems. Parameter identification: Models are characterized by a small number of parameters, each of them with a well defined physical meaning. Their identification can be achieved by comparing the description delivered by the model with experimental data which, however, are obtained by macroscopic measurements. The identification of parameters can take advantage of paper [19], where it has been suggested to use the experimental evaluation of ρc to identify the parameter α. This is possible for positive values of ρc considering that ρc ↑ ⇒ α ↑. Changes in the dynamical behaviour: Modelling panic conditions is an important aspect of the modelling approach, which needs to be developed in the presence of danger in connection with evacuation problems. It has been discussed in Sect. 5 how these conditions do not modify the mathematical structures, but the identification of the parameters of the model leads to substantially different values.


291

6.2 Modelling perspectives The aim of this paper has been focused on the design of guidelines towards the modelling of traffic and crowd dynamics. The validity of the mathematical structures proposed to achieve this objective has been discussed in the preceding subsection. On the other hand, modelling approach leaves various problems still open, which need to be properly treated to refine the simple models delivered in this paper. Of course, the first step consists in improving the predictability of the table of games by observing the ability of the model to reproduce experimental results, mainly the velocity diagram, and emerging collective behaviours. The already cited paper [32] is a good example of model validation. However, further problems can be mentioned. Focusing on vehicular traffic dynamics let us observe that the number of discrete cells and velocities has been presented as a free parameter. On the other hand, the modelling approach should precisely refer the discretization to the local density, for instance by assuming that the size of the cells is adapted to contain a fixed number of vehicles. This problem appears to be technical in the case of models in one space dimension (including crowd models. However, its generalization in more than one space dimension has to overcome remarkable technical difficulties. An additional problem, in the case of crowd modelling, is the presence of obstacles which modify the concept of direction θi , which ends up to depend on the obstacles and shape of the boundary as shown in Fig. 2, which shows how the obstacle modifies the trajectory to be followed by a pedestrian in Q to reach the exit Q.

Q

ν0

P

Σ ν0

∂Σ

Fig. 2. Geometry of the crowd domain

It is worth mentioning that some recent papers propose a macroscopic approach vehicular traffic, where the parameters of the model are generated

292


by a detailed analysis of the interactions at the microscopic level [12, 13, 31]. These models generate the fundamental diagram, similarly to some of the kinetic models, without plugging this experimental datum into the model. Certainly, this approach is interesting and has the advantage of using directly the macroscopic approach that directly provides the observable quantities of interest for the applications. The above cited papers have motivated research activity in the field of engineering sciences to use this type of models at the microscopic scale for a direct derivation of the fundamental diagram [19,20]. Still some mathematical problems remain open such as the modification of the dynamics of interactions in conditions far from the conditions of validity of the velocity diagram, namely steady uniform flow conditions. Possibly, the dynamics at the lower scale can be modelled as suggested in Sects. 4 and 5 and linked to the higher scale as suggested in the two scale approach proposed in [24]. Some perspective ideas proposed in this paper can be extended to the modelling of swarms, which can be observed in the animal world, but also in micro-organisms and cells [8]. The novelty and difficulty of the problem have a consequence of a rather limited literature concerning mathematical approaches. Conceptually, different methods have been used in connection with different objectives. For instance stochastic differential equations [1], macroscopic equations derived from stochastic perturbation of individual dynamics [22,30], modelling swarming patterns [23,57], and flocking phenomena [27,28]. Summarizing, the modelling approach should take into account that fluctuations are an intrinsic feature of the systems under consideration. Finally, let us mention that recent studies [4] conjecture, on the basis of empirical data generated by systematic observations, that some systems of animal world develop a common strategy based on interactions depending on topological rather than metric distances. Namely each individual in the swarm organizes the dynamics looking at a fixed number of neighbours independently on their position. This contributes to avoid fragmentation of the swarm. Technically, this means that interactions involve a number of active particles rather than those in the visibility zone. In general, a swarm has the ability to express a collective intelligence that is generated by a cooperative strategy [15]. Avoiding the fragmentation of the swarm is already an aspect of the strategy that defends the swarm from aggressive attacks. The interested reader can find additional information in paper [6], where it is discussed, among various topics, that the dynamics of the interactions depends both on the localization of individuals (the output of the interactions is different from the border to the center of the swarm) and on the topological distribution of individuals has remarkable influence on the output of the interactions. The above indications need to be regarded as a very preliminary step towards the development of models suitable to describe the complex dynamics of the system under consideration. Possibly, research project can be developed by using the kinetic theory methods reviewed in the preceding sections,


293

where the modelling of the interaction terms should take into account suitable developments of the table of games and of the two scale modelling approach proposed in Sects. 4 and 5. Appendix - A Table of Games for Vehicular Traffic Models • Interaction with a faster vehicle (h < k). This interaction can be modelled according to a follow-the-leader strategy, which implies that the candidate vehicle either maintains its current speed or possibly shift, in probability, to a higher velocity: ⎧ ⎨ 1 − u α (1 − ρ) if j = h, j if j = h + 1 (h, k = 1, . . . , n), Bhk [ρ] = u α (1 − ρ) ⎩ 0 otherwise. • Interaction with a slower vehicle (h > k). It is assumed that the candidate vehicle does not accelerate and, either it is forced to queue, reducing its speed to that of the leading vehicle, or it maintains its current speed, because it has enough free space to overtake: ⎧ ⎨ 1 − u α (1 − ρ) if j = k j Bhk if j = h (h, k = 1, . . . , n), [ρ] = u α (1 − ρ) ⎩ 0 otherwise. • Interaction with an equally fast vehicle (h = k). The interacting vehicles are unlikely to preserve strictly their speed during the motion, for this would imply they do not interact, behaving as if they were alone the road. Therefore, the effect of the interaction is distributed over four possible outcomes: ⎧ uαρ if j = h − 1 ⎪ ⎪ ⎨ 1 − u α if j = h (h = 2, . . . , n − 1). j Bhh [ρ] = u α (1 − ρ) if j = h + 1 ⎪ ⎪ ⎩ 0 otherwise. i • Note that the form of Bhh [ρ] applies only if h = 1, and h = n. A technical modification is needed at the boundary of the velocity grid: when h = 1 or h = n the candidate vehicle cannot brake or accelerate, respectively, because of the lack of further lower or higher velocity classes: ⎧ ⎨ 1 − u α (1 − ρ) if j = 1 j if j = 2 B11 [ρ] = u α (1 − ρ) ⎩ 0 otherwise,

and

⎧ if j = n − 1 ⎨uαρ j Bnn [ρ] = 1 − u α ρ if j = n ⎩ 0 otherwise.

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Particle, kinetic, and hydrodynamic models of swarming José A. Carrillo1 , Massimo Fornasier2 , Giuseppe Toscani3 , and Francesco Vecil4 1 2 3 4

ICREA - Departament de Matem` atiques, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Spain, [email protected] Johann Radon Institute for Computational and Applied Mathematics (RICAM), Altenbergerstrasse 69, A-4040 Linz, Austria, [email protected] Department of Mathematics, University of Pavia, via Ferrata 1, 27100 Pavia, Italy, [email protected] Johann Radon Institute for Computational and Applied Mathematics (RICAM), Altenbergerstrasse 69, A-4040 Linz, Austria, [email protected]

Summary. We review the state-of-the-art in the modelling of the aggregation and collective behavior of interacting agents of similar size and body type, typically called swarming. Starting with individual-based models based on “particle”-like assumptions, we connect to hydrodynamic/macroscopic descriptions of collective motion via kinetic theory. We emphasize the role of the kinetic viewpoint in the modelling, in the derivation of continuum models and in the understanding of the complex behavior of the system.

1 Introduction Everyone at some point in his life has been surprised and astonished by the observation of beautiful swinging movements of certain animals such as birds (starlings, geese, etc.), fishes (tuna, capelin, etc.), insects (locusts, ants, bees, termites, etc.), or certain mammals (wildebeasts, sheep, etc.). These coherent and synchronized structures are apparently produced without the active role of a leader in the grouping, phenomena denominated self-organization [2, 12, 56] and it has been reported even for some microorganisms such as myxobacteria [45]. Most of the basic models proposed in the literature are based on discrete models [23, 36, 68, 72] incorporating certain effects that we might call the “first principles” of swarming. These first principles are based on modelling the “sociological behavior” of animals with very simple rules such as the social tendency to produce grouping (attraction/aggregation), the inherent minimal space they need to move without problems and feel comfortably


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inside the group (repulsion/collisional avoidance) and the mimetic adaptation or synchronization to a group (orientation/alignment). They model animals as simple particles following certain microscopic rules determined by their position and velocity inside the group and by the local density of animals. These rules incorporate the “sociological” or “behavioral” component in the modelling of the animals movement. Even if these minimal models contain very basic rules, the patterns observed in their simulation and their complex asymptotic behavior is already very challenging from the mathematical viewpoint. The source of this tendency to aggregation can also be related to other factors rather than sociological as survival fitness of grouping against predators, collaborative effort in food finding, etc. Moreover, we can incorporate other interaction mechanisms between animals as produced by certain chemicals, pheromone trails for ants, the interest of the group to stay close to their roost, physics of swimming/flying, etc. Although the minimal models based on “first principles” are quite rich in complexity, it is interesting to incorporate more effects to render them more realistic, see [2, 4, 8, 42] for instance. Along this work, we take as reference models the one proposed in [30, 49, 53] for self-propelled particles with attraction and repulsion effects and the simple model of alignment in [25, 26]. On the one hand, the authors in [30] classify the different “zoology” of patterns: translational invariant flocks, rotating single and double mills, rings, and clumps; for different parameter values. On the other hand, in the simpler alignment models [25, 26], we get generically a flocking behavior. We will elaborate starting from these basic bricks to conclude with continuum models capable of simulating the collective behavior of systems with a large number of agents N . Control of large agent systems are important not only for the somehow bucolic example of the animal behavior but also for pure control engineering for robots and devices with the aim of unmanned vehicle operation and their coordination, see [21,25,57] and the references therein. When the number of agents is large as in migration of fish [73] or in myxobacteria [45, 56], the use of continuum models for the evolution of a density of individuals becomes essential. Some continuum models were derived phenomenologically [10, 63–65] including attraction–repulsion mechanisms through a mean force and spatial diffusion to deal with the anticrowding tendency. Other continuum models are based on hydrodynamic descriptions [14, 22] derived by means of studying the fluctuations or the meanfield particle limits. The essence of the kinetic modelling is that it does connect the microscopic world, expressed in terms of particle models, to the macroscopic one, written in terms of continuum mechanics systems. A very recent trend of research has been launched in this direction in the last few years, see for instance [14, 15, 27, 28, 39] for different kinetic models in swarming and [13,14,34,37] for the particle to hydrodynamics passage. The analysis, asymptotic behavior, numerical simulation, pattern formation, and their stability in many of these models still remain an unexplored research territory.

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This book chapter is organized as follows: first we concentrate on particle models, then we consider the next level of modelling going to the kinetic equations of multiagent systems in Sect. 3 and finally we discuss some hydrodynamic descriptions of these systems. The objective of this book chapter is to make a review of the state-of-the-art in this interesting new topic of research with an emphasis in the kinetic model viewpoint.

2 Particle models In the following we detail the description of particle models: first those based on the combination of self-propelling, friction and attraction-repulsion phenomena, and then those based on the Cucker–Smale description. ATTRACTION

ALIGNMENT REPULSION

Fig. 1. Three-zone model

2.1 Regions of influence In collective motion of groups of animals, three fundamental regions of influence are distinguished, see [1, 44]. The first region is characterized by the tendency of moving apart from another individual in near proximity, typically in order to avoid physical collision or being of mutual obstacle. In the immediate further proximity, this region of avoidance or repulsion is then substituted by a region of orientation or alignment, where the individual tries to identify the possible direction of the group and to align with it. When the individual finds itself too far apart from the group, it will try to reach the others which are located in the far distance region of attraction, see Fig. 1 for a scheme

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of this model. The strength of these different effects depends on the distance, and on the number of other individuals located instantaneously in the different zones; hence the direction of the individual will be changed according to the weighted superposition of these effects. In this review, our starting point consists of some particle models, also called Individual-Based models, incorporating attraction, repulsion, and orientation mechanisms. However, these are certainly simplified models which do not take into account several other important aspects: cone of visibility, closed-neighbor interaction, noise, roosting, aerodynamics, etc. It is worth mentioning that these three zones models have been improved by adding many of these different effects and analysed for different types of animals and particular species; see for instance the works of theoretical biologists, applied mathematicians and physicists in [3, 24, 32, 41, 46, 47, 49–51, 53, 70, 71], the works for fish [4, 6, 42, 73] with the aim of studying migration patterns for the capelin around Iceland, and the recent study [2, 43] for birds, focusing on starlings aggregation patterns in Rome.

Fig. 2. Mills in nature and in models!

In the first part of this chapter we address particle, kinetic, and hydrodynamic models for describing mathematically the basic three-zones model. Despite their simplicity, these minimal models show how the combination of these simple rules can produce striking phenomena, such as pattern formations: flocks and single or double mills, as indeed in nature we can observe, see Fig. 2. In the second part of this chapter, we discuss several improvements in the modeling including other important aspects, like the fact that animals are not influenced simultaneously by all the other individuals, either because they have a special cone of visibility or because, in a very short time, they can simply track the motion of a relatively small number of other individuals; for example it is estimated statistically that certain birds, for instance starlings, decide their direction according to mutual topological rather than metrical distance, and in particular to the behavior of the proximal 6-7 birds only [2]. This number is perhaps sufficiently small to allow the individual bird to take quickly a decision, and sufficiently large to avoid that the group easily


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splits. Another effect is due to nondeterministic behaviors, simply related to the impossibility of an individual to apply the “ideal rule” in a very short time, these leads to small random errors in some of the decisions taken. 2.2 Self-propelling, friction, and attraction–repulsion model The first model we review, as proposed in [30], is given as follows: Self-Propelling, Friction, and Attraction–Repulsion Particle Model: ⎧ dxi ⎪ ⎪ ⎪ ⎨ dt = vi , (1) dvi 1 ⎪ ⎪ = (α − β|vi |2 )vi − ∇U (|xi − xj |), ⎪ ⎩ dt N j=i

for i = 1, . . . , N , where α, β are non-negative parameters, U : Rd → R is a given radial potential modelling the short-range repulsion and long-range attraction, and N is the number of individuals. The term associated to α models the self-propulsion of individuals, whereas the term corresponding to β represents a friction following Rayleigh’s law. The combination and the balance of these two terms result in the tendency of the system (if we ignore the effect due to the repulsion and attraction term) to reach the asymptotic speed |v| = α/β, however not influencing the orientation of the velocities. A typical choice for U is the Morse potential which is radial and given by: U (x) = k(|x|),

k(r) = −CA e−r/A + CR e−r/R ,

(2)

where CA , CR and A , R are the strengths and the typical lengths of attraction and repulsion, respectively. The most relevant situations for biological applications are determined for C := CR /CA > 1 and = R /A < 1, which correspond to long-range attraction and short-range repulsion. These quantities rule the combination of both the effects, resulting in different regimes [30]. In particular, there are two fundamental observable situations: the first of stability, for Cd > 1, where individuals form crystalline-like patters. Here, for N sufficiently large, particles find an optimal spacing and maintain a fixed relative distance from each other; in the second regime, for Cd < 1, so-called catastrophic, individuals tend to a rotational motion of constant speed, if initially well separated, |v| = α/β, and single or double mills are observed, see Fig. 2. The area of these rotating patterns stabilizes as N goes to infinity, see [14] and the discussion therein. The regions determining these regimes are readily classified in the graphics of Fig. 3 in two dimensions, in terms of the fundamental parameters C = CR /CA and = R /A . Let us

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Fig. 3. Classification of different regions corresponding to the model in [30], figure reprinted with the permission of authors

remark that the potential is not scaled by 1/N in [30], this fact does not affect the type of patterns obtained in each regime but their behavior for large N is different, more comments in [14]. We have presented the model with the scaled interaction potential to be able to get a meaningful limit as N → ∞ while keeping the total mass of the system finite. 2.3 The Cucker–Smale model The model proposed by Cucker–Smale in [25, 26] takes into account only an alignment mechanism of the individuals by adjusting/averaging their relative velocities with all the others. The strength of this averaging process depends on the mutual distance, and closer individuals have more influence than the far distance ones. For a system of N individuals this model is described by the following dynamical system: The Cucker–Smale Particle Model of Flocking: ⎧ dx i ⎪ = vi , ⎪ ⎪ ⎨ dt N 1 dvi ⎪ ⎪ = ⎪ H(|xi − xj |)(vj − vi ), ⎩ dt N j=1

for i = 1, . . . , N , where the communication rate H is given by H(x) = a(|x|),

a(r) =

for positive parameters K, ς, and γ ≥ 0.

K , (ς 2 + |r|2 )γ

(3)


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For x, v ∈ Rd×N , denote Γ (x(t)) =

1 |xi (t) − xj (t)|2 , 2

(4)

1 |vi (t) − vj (t)|2 . 2

(5)

i=j

and Λ(v(t)) =

i=j

Then, different regimes are achieved in this model too, depending on the parameters K, ς, γ. Theorem 12.1 ([25,26]). Assume that one of the following conditions holds. (i) γ < 1/2; (ii) γ ≥ 1/2 and, 1 2γ 1

2γ−1 2γ−1 1 2γ−1 (K)2 1 − > 2Γ x(0) + ς 2 , 2γ 2γ 8N 2 Λ(0) then there exists a constant B0 such that Γ (x(t)) ≤ B0 for all t ∈ R, while Λ(v(t)) converges towards zero as t → ∞, and the vectors xi − xj tend to a limit vector x ij , for all i, j ≤ N . Let us remark that the theorem was improved in [15,37] in the case γ = 1/2 for any initial data. Let us also note that the regime γ ≤ 1/2 neither depends on N, d nor on the initial configuration of the system; in this case, called unconditional flocking, the behavior of the population is perfectly specified, all the individuals tend to move with the same mean velocity and to form a group with fixed mutual distances, but not necessarily in a crystalline-type pattern and with a spatial profile that depends on the initial condition, see Fig. 4. In the regime γ > 1/2, flocking can be expected under sufficient density conditions, i.e., Γ x(0) is small enough and K is large enough, and the initial relative velocities are not too large, i.e., Λv(0) is small. Despite the theoretical and fundamental nature of this result, surprising and remarkable applications of the Cucker–Smale principle have recently been found in spacecraft flight control [57] in the context of the ESA-mission DARWIN. Darwin will be a flotilla of four or five free-flying spacecrafts that will search for Earth-like planets around other stars and analyze their atmospheres for the chemical signature of life. The fundamental problem is to ensure that, with a minimal amount of fuel expenditure, the spacecraft fleet keep remaining in flight (flock), without loosing mutual radio contact, and possibly scattering.

3 Kinetic models Unlike the control of a finite number of agents, the numerical simulation of a rather large population of interacting agents can constitute a serious difficulty

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12.5 10

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Fig. 4. We illustrate in these figures the dynamics of four particles in 1D, i.e., (d = 1), as modeled by (3) for γ ≤ 1/2, with initial positions and velocities (0, 6), (3, 2), (−3, −2), and (0, −3). Top left: dynamics of positions, Top right: dynamics of velocities, Bottom center: dynamics in phase-space

which stems from the accurate solution of a possibly very large system of ODEs. Borrowing the strategy from the kinetic theory of gases, we may want instead to consider a density distribution of agents, depending on spatial position, velocity, and time evolution, which interact with stochastic influence (corresponding to classical collisional rules in kinetic theory of gases) – in this case the influence is spatially “smeared” since two individuals do interact also when they are far apart. Hence, instead of simulating the behavior of each individual agent, we would like to describe the collective behavior encoded by the density distribution whose evolution is governed by one sole mesoscopic partial differential equation, see [20] for classical references. Let f (x, v, t) denote the density of individuals in the position x ∈ Rd with velocity v ∈ Rd at time t ≥ 0, d ≥ 1. The kinetic model for the evolution of f = f (x, v, t) can be easily derived by standard methods of kinetic theory, considering that the change in time of f (x, v, t) depends both on transport (individuals moving freely if they do not interact with others) and interactions with other individuals. Discarding other effects, this change in density depends on a balance between the gain and loss of individuals with velocity v due to binary interactions.


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3.1 Formal derivation from Boltzmann-type equations Let us assume that two individuals with positions and velocities (x, v) and (y, w) modify their velocities after the interaction, according to: v ∗ = C (x, v; y, w) w∗ = C (y, w; x, v), where C is a suitable interaction rule. This leads to the following integrodifferential equation of Boltzmann type,

∂f + v · ∇x f (x, v, t) = Q(f, f )(x, v, t), (6) ∂t where

Q(f, f )(x, v) = ε R2d

1 f (x, v∗ )f (y, w∗ ) − f (x, v)f (y, w) dw dy. J

(7)

In (7) (v∗ , w∗ ) are the pre-interaction velocities that generate the couple (v, w) after the interaction. In (6) J := J(x, v; y, w) is the Jacobian of the transformation of (v, w) into (v ∗ , w∗ ) via C . The role of the interaction operator Q is to provide a stochastic description of the interactions happening between each pair of individuals. Hence, the mesoscopic behavior is described by means of an averaging process. We are in particular interested in the interaction rules suggested by the particle models (1) and (3), i.e., respectively (C1) C (x, v; y, w) = v + η (α − β|v|2 )v − ∇U (|x − y|) ; (C2) C (x, v; y, w) = [1 − η a(|x − y|)] v + η a(|x − y|)w. Other interaction rules will be considered as well later where, for instance, noise will also be included. Let us remark that we are assuming that the previous change of variables is well-defined, i.e., that C (x, v; y, w) is invertible under suitable assumptions. The presence of the Jacobian in the interaction operator (7) can be avoided by considering the weak formulation. By a weak solution of the initial value problem for (6), corresponding to the initial density f0 (x, v), we shall mean any density satisfying the weak form of (6)–(7) is given by:

∂ ϕ(x, v)f (x, v, t) dv dx + (v · ∇x ϕ(x, v))f (x, v, t) dv dx (8) ∂t R2d R2d

=ε (ϕ(x, v ∗ ) − ϕ(x, v))f (x, v, t)f (y, w, t) dv dx dw dy R4d

for t > 0 and all smooth functions ϕ with compact support, and such that

lim ϕ(x, v)f (x, v, t) dx dv = ϕ(x, v)f0 (x, v) dx dv. (9) t→0+

R2d

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The form (8) is easier to handle, and it is the starting point to explore the evolution of macroscopic quantities (i.e., of moments for ϕ(v) = 1, v, v 2 ). The phenomenon of pattern formation is mainly related to the large-time behavior of the solution of (6). An accurate description can be still furnished as well by resorting to simplified models for large time. This idea has been first used in dissipative kinetic theory by McNamara and Young [52] to recover from the Boltzmann equation in a suitable asymptotic procedure, simplified models of nonlinear frictions for the evolution of the gas density [5,66]. Similar asymptotic procedures have been subsequently used to recover Fokker–Planck type equations for wealth distribution [11], or opinion formation [67]. By expanding ϕ(x, v ∗ ) in Taylor’s series of v ∗ − v up to the second order the weak form of the interaction integral takes the form:

(ϕ(x, v ∗ ) − ϕ(x, v))f (x, v, t)f (y, w, t) dx dv dy dw 4d R

= (∇v ϕ(x, v) · (v ∗ − v)) f (x, v, t) f (y, w, t) dx dv dy dw (10) R4d ⎡ ⎤

d 2 ∂ ϕ(x, v ˜ ) 1 ⎣ (vj∗ − vj )2 ⎦f (x, v)f (y, w) dx dv dy dw, + 2 R4d i,j=1 ∂vi2 with v˜ = θv + (1 − θ)v ∗ , 0 ≤ θ ≤ 1. Now we resume the specific interaction rules (C1) and (C2) and we rewrite the first and second order terms of the Taylor’s series as follows: for v ∗ − v = η (α − β|v|2 )v − ∇U (|x − y|) := η F 1

(ϕ(x, v ∗ ) − ϕ(x, v))f (x, v, t)f (y, w, t) dx dv dy dw (11) R4d

=η (∇v ϕ(x, v) · F 1 ) f (x, v, t) f (y, w, t) dx dv dy dw 4d R ⎡ ⎤

d 2 ∂ ϕ(x, v ˜ ) η2 ⎣ [Fj1 ]2 ⎦ f (x, v)f (y, w) dx dv dy dw, + 2 R4d i,j=1 ∂vi2 for v ∗ − v = η a(|x − y|)(w − v) := η F 2

(ϕ(x, v ∗ ) − ϕ(x, v))f (x, v, t)f (y, w, t) dx dv dy dw (12) R4d

=η (∇v ϕ(x, v) · F 2 ) f (x, v, t) f (y, w, t) dx dv dy dw 4d R ⎡ ⎤

d 2 ∂ ϕ(x, v ˜ ) η2 ⎣ [Fj2 ]2 ⎦ f (x, v)f (y, w) dx dv dy dw. + 2 R4d i,j=1 ∂vi2


307

For η 1, i.e., for the strength of the interaction being very small, such that εη = λ constant and εη 2 1, a regime that is expected at large time, we may approximate the interaction operator with the sole first-order term, i.e.,

∂ ϕ(x, v)f (x, v, t) dv dx + (v · ∇x ϕ(x, v))f (x, v, t) dv dx ∂t R2d R2d ⎧ ⎪ ⎪ (∇v ϕ(x, v) · F 1 ) f (x, v, t) f (y, w, t) dx dv dy dw for (C1) ⎨λ 4d R

≈ ⎪ ⎪ ⎩λ (∇v ϕ(x, v) · F 2 ) f (x, v, t) f (y, w, t) dx dv dy dw for (C2) R4d

or, in the strong form, we derive two corresponding nonlinear friction-type equations Self-Propelling, Friction, and Attraction–Repulsion Kinetic Model: ∂f + v · ∇x f = λ (∇x U ∗ ρ) · ∇v f − ∇v · ((α − β|v|2 )vf ) , (13) ∂t where

ρ(x, t) = f (x, v, t) dv. (14) Rd

and ∗ is the x-convolution, and The Cucker–Smale Kinetic Model of Flocking: ∂f + v · ∇x f = λ∇v · [ξ(f )f ] , ∂t where

K

ξ(f )(x, v, t) = R2d

(ς 2

+ |x − y|2 )

β

(15)

(v − w)f (y, w, t) dy dw,

or equivalently, ξ(f ) = [(H(x)∇v W (v)) ∗ f ] and W (v) = H(x) = a(|x|), and ∗ is the (x, v)-convolution.

(16) 1 2

|v|2 ,

3.2 Formal derivation via mean-field limit The equations derived above via grazing collision limit from the general Boltzmann-type equations (6) can also be computed via mean-field limit. Here we present a simple formal description of this procedure for the Cucker–Smale model, see also [13, 37]. Similar computations can be performed for (1) and more general models [7, 13, 29, 54].

308


Define the empirical distribution density (or the atomic probability measure) associated to a solution (x(t), v(t)) of (3) and given by f N (x, v, t) =

N 1 δ(x − xi (t))δ(v − vi (t)), N i=1

where δ is the Dirac delta and P(Rk ) denotes the space of probability measures on Rk . Let us assume that the particles remain in a fixed compact domain ¯ ⊂ Rd × Rd for all N in the time interval t ∈ [0, T ]. It is (xi (t), vi (t)) ∈ Ω easy to check that this assumption for the Cucker–Smale model (3) is fulfilled if for instance the initial configuration is obtained as an approximation of an initial compactly supported probability measure f0 . Since for each t the measure f N (t) := f N (·, ·, t) is a probability measure in P(R2d ) together with the uniform support in N , then Prohorov’s theorem implies that the sequence is weakly-∗-relatively compact. Hence, there exists a subsequence (f Nk )k and f : R → P(R2d ) such that f Nk → f

(k → ∞)

with w∗ − convergence in P(R2d ),

pointwise in time. We shall give a formal derivation of the evolution equation satisfied by the limit measure. Let us consider a test function ϕ ∈ C01 (R2d ) and we compute N N d N 1 d 1 f (t), ϕ = ϕ(xi (t), vi (t)) = ∇x ϕ(xi (t), vi (t)) · vi (t) dt N i=1 dt N i=1

+

N 1 H(xi − xj ) [∇v ϕ(xi (t), vi (t)) · (vj (t) − vi (t))] N 2 i,j=1

⎛

⎞ N 1 = f (t), ∇x ϕ · v − ⎝ 2 H(xi − xj ) [∇v ϕ(xi (t), vi (t)) · vi (t)]⎠ N i,j=1 ⎛ ⎤ ⎞ ⎡ N N 1 ⎣1 +⎝ H(xi − xj ) vj (t)⎦ · ∇v ϕ(xi (t), vi (t))⎠ N i=1 N j=1 N

⎤ N 1 N f (t), ⎣ H(x − xj )⎦ ∇v ϕ(x, v) · v N j=1 ⎛ ⎞ N 1 N H(x − xj )vj (t)⎠ · ∇v ϕ(x, v) . + f (t), ⎝ N j=1

= f N (t), ∇x ϕ · v −

⎡


309

We can easily compute N N 1 1 H(x − xj ) = H(x − y), δ(y − xj ) x = H ∗ ρf N (x, t), N j=1 N j=1

where

ρf N (x, t) =

N

f (x, v, t) dv = Rd

N 1 1, δ(y − xj )δ(v − vj ) ; N j=1 v

Similarly one deduces N 1 H(x − xj ) vj (t) = H ∗ mf N (t, x), N j=1

where

mf N (x, t) =

N

vf (x, v, t) dv = Rd

N 1 v, δ(y − xj )δ(v − vj ) . N j=1 v

Collecting these formal computations we obtain d N f (t), ϕ = f N (t), ∇x ϕ · v + H ∗ mf N · ∇v ϕ − H ∗ ρf N ∇v ϕ · v . dt After integration by parts, in both x and v, we obtain N ∂f + v · ∇x f N − ∇v · ξ(f N )f N , ϕ = 0, ∂t or, in the strong form, ∂f N + v · ∇x f N = ∇v · ξ(f N )f N , ∂t where ξ is defined in (16). For the limit of k → ∞ of the subsequence f Nk this leads formally to ∂f + v · ∇x f = ∇v · [ξ(f )f ] . ∂t which is exactly (15) for η = 1. The mean-field limit N → ∞ introduced above can be proved rigorously by using the techniques in [13, 37], and also for the model (1) [7, 13, 29, 54].

310


3.3 Rigorous derivation of the mean-field limit Here, we would like to address the main results achieved so far concerning the solutions of (13) and (15). With this goal, we need to introduce further notations and preliminaries. On the probability space P(Rk ) we define the so-called Monge-Kantorovich-Rubinstein distance [18, 69], W1 (μ, ν) = sup ϕ(x)d(μ − ν)(x) , ϕ ∈ Lip(Rk ), Lip(ϕ) ≤ 1 , (17) Rk

where Lip(Rk ) denotes the set of Lipschitz functions on Rk and Lip(ϕ) the Lipschitz constant of ϕ. The main results in [13] show the well-posedness of the kinetic models (13) and (15) in the set of probability measures. Let us emphasize that the main technical issue is to overcome the lack of global Lipschitz character of the fields in phase space not allowing for direct application of known results [7,29,35,54,55,61,62]. In fact, we are able to overcome their lack of Lispchitzianity at infinity but not at the origin for the attraction–repulsion potential U . Therefore, in order to ensure the validity of the following results, assume further that we substitute in (1) and in (13) the Morse potential U with the smoother version U (x) = k2 (|x|),

k2 (r) = −CA e−r

2

/2A

+ CR e−r

2

/2R

.

(18)

This new potential does not change significantly the qualitative behavior of the particle model as described in Fig. 2. The precise theorem in [13] states: Theorem 2. Assume that f0 ∈ P(R2d ) is a compactly supported initial datum. Then there exists a unique measure valued solution f ∈ C([0, +∞); P(R2d )) to (13), respectively to (15), (i.e., it is a solution in the sense of the distributions), and there is a function R = R(T ) such that for all T > 0, supp f (t) ⊂ BR(T ) ⊂ R2d ,

for all t ∈ [0, T ].

Moreover, the solution depends continuously with respect to the initial data in the following sense: Assume that f0 , g0 ∈ P(R2d ) are two compactly supported initial data, and consider the respective measure valued solutions f, g to (13), respectively to (15). Then, there exists a strictly increasing function r : [0, ∞) → R+ 0 with r(0) = 0 depending only on the size of the supports of f0 and g0 such that W1 (f (t), g(t)) ≤ r(t)W1 (f0 , g0 ),

t ≥ 0.

In Theorem 2 the main difference between the self-propulsion model (13) and the kinetic Cucker–Smale model (15) is on the growth on the supports in space and velocity in time, i.e., the function R(T ), as we will see later. The previous theorem does not give any information about the asymptotic behavior of (13) or (15), but it does imply as an important consequence the convergence of the particle method via mean-field limit.


311

Corollary 3. Given f0 ∈ P (R2d ) compactly supported and a sequence of f0N empirical measures associated with the initial data for the particle system (1), respectively to (3), (with xi (0) and vi (0) possibly varying with N ), in such a way that lim W1 (f0N , f0 ) = 0. N →∞

Consider ftN the empirical measure associated with the solutions of the particle system (1), respectively of the particle system (3), with initial conditions xi (0), vi (0). Then, lim W1 (ftN , ft ) = 0, N →∞

for all t ≥ 0, where f = f (x, v, t) is the unique measure solution to (13), respectively to (15), with initial data f0 . Concerning the asymptotic behavior, there are no results so far which establish the convergence to stable patterns for solutions of (13). In particular a classification of the asymptotic behavior of this kinetic equation as for the corresponding particle model (1) is lacking. We propose in Sect. 6 numerical experiments which highlight some of the features of this kinetic model. Moreover, when considering macroscopic quantities (moments) and passing to hydrodynamic equations, it has been possible to show, under special assumptions, that single and double mills (Fig. 2) are stationary solutions for the kinetic model as well [14], although it is widely open, as mentioned earlier, how to establish and predict their formation and to analyze their stability. Differently from the situation encountered in the kinetic model (13), again for the Cucker–Smale model we are able to provide a result of non-universal unconditional flocking which generalizes Theorem 12.1 for measure valued solution of (15), see [15,37,39]. In order to present this result we need a bit of notation: given a measure μ ∈ P(R2d ), we define its translate μh with vector h ∈ Rd by:

ζ(x, v) dμh (x, v) = ζ(x − h, v) dμ(x, v), Rd

R2d

for all ζ ∈ Cb0 (R2d ). We will also denote by μx the marginal in the position variable, that is,

ζ(x) dμ(x, v) = ζ(x) dμx (x), R2d

Rd

Cb0 (Rd ).

With this we can write the main conclusion about the for all ζ ∈ asymptotic behavior, i.e., the convergence relative to the center of mass variables to a fixed density characterized by the initial data and its unique solution. Theorem 4. Assume γ ≤ 1/2. Given f0 ∈ P(R2d ) compactly supported, there exists L∞ (f0 ) ∈ P(Rd ) such that the unique measure-valued solution f to (13) satisfies

312


lim W1 (fxmt (t), L∞ (f0 )) = 0,

t→∞

where m = R2d

v df0 (x, v).

Let us mention that it is also proved in [15] that the support in velocity of the solutions always shrinks in time toward its mean velocity m and it does it exponentially fast. In space, the support of the solutions will not grow in time by fixing our axis in the co-moving frame of the center of mass. In other words, the support in space of the solutions at all times is inside a ball of the form B(xoc + mt, Rx ) with xoc the initial center of mass with Rx fixed. It has not been proved so far what happens in the kinetic model for the regime γ > 1/2, and which density conditions are required in order to obtain again flocking, i.e., convergence to a profile traveling with mean velocity. In such a regime we may expect density phase transitions in order to obtain flocking in the kinetic model as well, i.e., no flocking is achievable for initial low mean-density distributions and flocking is always achievable when the initial distribution has high mean-density.

4 Hydrodynamic models Kinetic models are time-consuming when solving in more than four dimensions (two for position and two for velocity). Therefore, we wish to reduce the dimensionality of the problem by taking a macroscopic limit, in such a way that numerical simulations become affordable. 4.1 Flocking, single-mills and double-mills Let us obtain continuum-like equations by computing the evolution of macroscopic quantities starting from (13) and (15), as usually done in kinetic theory. These macroscopic quantities are the velocity moments of f (x, v, t). The mean velocity field u(x, t) and the temperature T (x, t) are defined by:

ρu = vf (x, v, t) dv and dρT = |v − u|2 f dv, Rd

Rd

respectively. Integrating (13) or (15) in v we obtain the continuity equation ∂ρ + div(ρu) = 0. ∂t Proceeding by integrating (13) or (15) against vdv and using integration by parts, we find the momentum equation which involves the second moment of the distribution function. To close the moment system we assume that fluctuations are negligible, i.e., that the temperature T (x, t) = 0, and that the velocity distribution is monokinetic: f (x, v, t) = ρ(x, t) δ(v − u(x, t)). In this way, we obtain for (13) the hydrodynamic systems


313

Self-Propelling, Friction, and Attraction–Repulsion Hydrodynamic Model: ⎧ ∂ρ ⎪ ⎪ ⎪ ⎨ ∂t + div(ρu) = 0, (19)

⎪ ∂ui ∂U ⎪ 2 ⎪ ρ + div(ρuu ) = ρ (α − β|u| )u − ρ ∗ ρ . ⎩ i i ∂t ∂xi and The Cucker–Smale Hydrodynamic Model of Flocking: ⎧ ∂ρ ⎪ ⎪ ⎨ ∂t + div(ρu) = 0,

⎪ ⎪ ⎩ ρ ∂ui + div(ρuui )= H(x − y)ρ(x, t)ρ(y, t) [ui (y, t) − ui (x, t)] dy. ∂t Rd (20) The system of equations (19) was already proposed in [22] based on computations of the empirical measure associated to N particles. Here, the same description is recovered from the monokinetic ansatz applied to the kinetic equations (13) or (15). In [22] the authors discussed the validity of this approximation based on numerical comparisons of the N -particle system and the hydrodynamic system (19). They concluded that the hydrodynamic system is a good approximation close to the steady state pattern situations and performed a linear stability analysis around the simple infinite-extent flocking solution ρ = ρ0 and |u| = α/β. Also, the system of equations (20) was somehow obtained in [39], although there they include the evolution of the temperature and they did not perform any momentum closure. Let us look for steady single-milling and flocking patterns as particular type of monokinetic solutions or hydrodynamic solutions of (13) or (15). Let us first concentrate on flocking solutions to (13). By imposing that the velocity field is constantly u0 satisfying β|u0 |2 = α in (13), we obtain that the flocking solutions of (19) with density ρ(x, t) = ρ˜(x − tu0 ) are characterized by U ∗ ρ˜ = C,

ρ˜ = 0,

where C is a constant of integration, as verified numerically in [49]. Solutions of this simple looking equation can be very complicated even in one dimension for regular and singular interaction potentials [33, 59]. For the system (20), it is trivially observed that all densities of the form ρ(x, t) = ρ˜(x − tu0 ) with u0 constant are solutions. Now, let us search for single-mill stationary solutions of (19) in two dimensions by setting u in a rotatory state, ! α x⊥ u=± , β |x|

314


where x = (x1 , x2 ), x⊥ = (−x2 , x1 ), and look for ρ = ρ(|x|) radial, then we see that ∇x · u = 0, u · ∇x ρ = 0, which implies the continuity equation, and furthermore, α x (u · ∇x ) u = − . β |x|2 Thus, (19) implies U ∗ρ=D+

α ln |x|, β

whenever ρ = 0,

(21)

where D is a constant of integration, which gives a linear integral equation that can be solved for ρ. Multiple solutions with support filling an interval [R0 , R1 ] with 0 < R0 < R1 were found numerically in [49] and matched to single mill patterns in [22]. Such solutions represent circular swarms in which all particles move with the same linear speed α/β. Given the conditions on the regularity of the potential U allowing for the rigorous existence of annularly supported solutions to (21) and studying their stability seems a challenging problem. Double mills, however, cannot be simply explained with this hydrodynamic approach because of the use of a single macroscopic velocity. They correspond to combination of single-mill solutions seen as distributional solutions of the kinetic model (13), see [14] for more details. 4.2 Fluid dynamic description of flocking via Povzner–Boltzmann equation In [48], Lachowicz and Pulvirenti established an interesting connection between solutions of the Euler equations for compressible fluids, and the solutions of an equation describing the dynamics of a system of particles undergoing elastic collisions at a stochastic distance. More precisely, consider density, velocity and temperature fields ρ(x, t), u(x, t), and T (x, t) which constitute a (smooth) solution of the Euler equations (up to some time t0 before the appearance of the first singularity), and construct a local Maxwellian function M whose mean density, velocity, and temperature are given by ρ, u, and T , respectively [20]

ρ(x, t) (v − u(x, t))2 M (x, v, t) = . exp − 2T (x, t) (2πT (x, t))3/2 Consider also a system of N particles located at the points x1 , x2 , . . . , xN on a domain of R3 , which move freely unless a pair of them undergo an elastic collision, expressed by the formula 1 vi = vi − ((vi − vj ) · nij )nij , 2

1 vj = vj + ((vi − vj ) · nij )nij , 2

(22)


315

where the unit vector nij is given by: nij =

xi − xj . |xi − xj |

(23)

As usual, vi and vj denote the outgoing velocities, where the ingoing velocities are given by vi and vj , provided that (vi − vj ) · nij < 0. Each binary collision takes place according to a stochastic law. The collision times for each pair i and j of particles are independent Poisson processes with intensity given by ϕ(xi , xj , vi , vj )|(vi − vj ) · nij |, and ϕ is given by: 1 1 χ(|xi − xj | ≤ δ) χ(|vi − vj | ≤ θ) N ε δ3 1 Bδ (|xi − xj |) χ(|vi − vj | ≤ θ). = Nε

ϕ(xi , xj , vi , vj ) =

(24)

In (24) χ(I) is the characteristic function of the subset I. The evolution of the system of particles is described by the N -particle distribution function f N (x1 , v1 . . . , xN , vN , t) which gives the probability density for finding the N particles in the points x1 , . . . , xN with velocities v1 , . . . , vN at the time t ≥ 0, see [61, 62]. Let the s-particle distribution functions be defined by the marginals

f N,s (x1 , v1 . . . , xs , vs , t) = f N (x1 , v1 . . . , xN , vN , t) dxs+1 dvs+1 · · · dxN dvN . Then, under some additional hypotheses on the regularity of the solutions to the Euler system in the time interval [0, t0 ], it is proven in [48] that, for all σ > 0 there exist ε0 (σ), δ0 (σ, ε), θ0 (σ, ε, δ), and N0 (σ, ε, δ, θ) such that if ε ≤ ε 0 , δ ≤ δ 0 , θ ≥ θ0 N ≥ N 0 sup M − f N,1 < σ,

t∈[0,t0 ]

where f N,1 is the 1-particle marginal corresponding to the N -particle distribution function f N (x1 , v1 . . . , xN , vN , t) with initial conditions f N,s (x1 , v1 . . . , xs , vs , t = 0) =

s "

M (0; xj , vj ).

j=1

Substituting particles with birds, and changing consequently the interaction intensity ϕ in (24), introduces a reasonable model to study the time-space evolution of a population of birds, and, at the same time it establishes an interesting connection with the fluid dynamic picture, in presence of a large

316


population. On the other hand, since the flocking phenomena is heavily dependent from dissipation, the elastic picture provided in [48] is not suitable to take into account this effect. The idea to generalize the binary interactions of the stochastic system of particles to take into account dissipation phenomena has been recently developed in [34], to end up with a consistent dissipative correction of the Euler system. An intermediate step in the analysis of [48] shows that, as the number of particles tend to infinity, the 1-particle marginal f N,1 = f satisfies the (elastic) Boltzmann–Povzner equation [58]. As already described in Sect. 3.1, Povzner–like equations are based on collision integrals of the form (7). In presence of an intensity function (24) where ϕ(xi , xj , vi , vj ) =

1 Bδ (|xi − xj |), Nε

the elastic Povzner collision operator reads [58]

1 Bδ (|x − y|) (f (x, v )f (y, w )−f (x, v)f (y, w)) dwdy, (25) QP (f, f )(x, v) = ε R6 where the pair v , w is the postinteraction pair obeying to the elastic law of type (22). The variant of the interaction rules (C1) and (C2) introduced in Sect. 3.1 which leads to a dissipation phenomenon consistent with the elastic picture is a binary interaction in which birds dissipate their relative velocity only along their relative direction. This agrees with the realistic assumption that

W Y

V X

Fig. 5. Example of interaction rule as in the Cucker–Smale-Povzner model where the particle with position x and velocity v averages its velocity with a particle in position y and velocity w according to (26)

agents which are approaching tend to diminish their relative velocity along their relative position, and the same happens in the opposite situation where they are going away (Fig. 5).


317

The consequent microscopic dynamics of two birds (x, v) and (y, w) is fully governed by the interaction coefficient 0 < e(|x − y|) < 1 which relates the components of the agents velocities before and after an interaction. The change in velocity now reads 1 v ∗ = v − (1 + e)((v − w) · n)n , 2

1 w∗ = w + (1 + e)((v − w) · n)n, 2

(26)

where the unit vector n is given by (23). This interaction is momentum preserving. Note that, on the contrary to what happens in the Cucker–Smale dynamics, the postinteraction velocities collapse into a standard Povzner–type (conservative) interaction as e = 1 [58]. In this case the pair (v ∗ , w∗ ) becomes the energy preserving pair (v , w ) 1 v = v − ((v − w) · n)n , 2

1 w = w + ((v − w) · n)n. 2

For dissipative interactions e decreases with increasing degree of dissipation. In agreement with Cucker–Smale model, we assume e(|x − y|) = 1 − η a(|x − y|),

(27)

where a(r) is the communication rate given in (3). The constant η has to be chosen so that 0 < e(|x − y|) < 1. If the interactions are such that the dissipation is low, so that η 0 is the noise strength. Here, α ∈ R is the effective friction constant coming from α = α1 − α0 with α0 , α1 > 0, and α0 is the linear Stokes friction component and α1 is the self-propulsion generated by the organisms. Using Ito’s formula to obtain a Fokker–Planck equation for the N -particle distribution and following a BBGKY procedure, it is easy to derive formally the following kinetic Fokker–Planck equation: Self-Propelling, friction, and Attraction–Repulsion with Noise Kinetic Model: ∂f +v ·∇x f +divv (α−β|v|2 )v f ]−divv [(∇x U ∗ ρ)f ] = σΔv f. (41) ∂t Analogously, for the Cucker–Smale model with white noise, one will conclude the equation The Cucker–Smale with Noise Kinetic Model: ∂f + v · ∇x f = λ∇v · [ξ(f )f ] + σΔv f , ∂t

(42)

with ξ(f ) defined in (16). Let us mention that the stochastic particle system related to Cucker–Smale with noise has been recently analysed in [38]. In fact, these Vlasov-Fokker–Planck-like equations can be obtained from the stochastic particle models by coupling methods as it will be proved in [9]. It is an open problem to discuss any property about their asymptotic behavior and existence of any stationary solution. Finally, let us mention that again they serve as bridges between the particle descriptions and the macroscopic descriptions. As shown in [14], one can obtain the macroscopic equations % & ∂t ρ = ∇x · (∇x U ∗ ρ)ρ + Δx ρ (43) and


% & ∂t ρ = ∇x · (∇x U ∗ ρ)ρ ,

323

(44)

as approximations to (41) in different scaling regimes. Both (43) and (44) were proposed in [64, 65] as continuum models for swarming, and thus, recovered through the presented kinetic theory. They are related to models in granular media equations, see [16, 17]. 5.3 Adding nonlinear-dependent noise to Cucker–Smale In a recent paper [72] it is argued that coherence in collective swarm motion is facilitated in presence of a certain degree of randomness (in the sense of imperfect applications of the interaction rules) which has to be weaker at some position around which mean velocity of particles is larger. We would like to describe this special phenomenon, again taking as a reference the model (3). Let us now assume that the dynamics is imperfect in the sense that it is perturbed by random noise: The Cucker–Smale Particle Model with Noise: ⎧ dxi = vi dt, ⎪ ⎪ ⎪ ⎨ ' ( N m ( σ 1 ⎪ ) ⎪ dv = a(|x − x |)(v − v ) dt + 2 a(|xj − xi |) dΓi (t). j i j i ⎪ ⎩ i N N j=1 j=1 (45) Here, a denotes again the communication rate function defined as in (3) and Γi (t) (1 ≤ i ≤ N ) are N independent Wiener processes with values in Rd , and σ ≥ 0 is a constant denoting the coefficient of noise strength. Notice that the strength of noise for ith particle is N σ a(|xj − xi |) N j=1

which is proportional to the summation of distance potentials of ith particle with all particles. In order to address a mesoscopic description, this time we resume the approach via Boltzmann equations. Let us assume that the postinteraction velocities (v ∗ , w∗ ) of two individuals which have positions and velocities (x, v) and (y, w) before interaction are determined by the rule v ∗ = (1 − η a(|x − y|))v + η a(|x − y|)w + 2ση a(|x − y|) θv , w∗ = η a(|x − y|)v + (1 − η a(|x − y|))w + 2ση a(|x − y|) θw , where η > 0 and σ ≥ 0 are constants which will enter into the equation exactly in the same way as in Sect. 3.1, and

324


θv = (θv,1 , θv,2 , . . . , θv,n ) ∈ Rd , θw = (θw,1 , θw,2 , . . . , θw,n ) ∈ Rd . θv,i and θw,i , (1 ≤ i ≤ d) are identically distributed independent random variables of zero mean and unit variance. For now, it is also supposed that supx a(|x|) is finite and 1 (46) η sup a(|x|) < . 2 x Notice that this assumption can be removed in the later grazing limit since a will be scaled up to a small parameter ε > 0. As in Sect. 3.1, the evolution of the density can be described at a kinetic level by the following integrodifferential equation of Boltzmann type: ∂f + v · ∇x f = Q(f, f ), ∂t with

#

Q(f, f ) = δIE Rd ×Rd

(47)

$ 1 f (x, v∗ )f (y, w∗ )−f (x, v)f (y, w) dy dw , J(|x − y|)

where (v∗ , w∗ ) mean the precollisional velocities of particles that generate the pair velocities (v, w) after interaction, and J(|x − y|) = (1 − 2η a(|x − y|))d is the Jacobian of the transformation of (v, w) into (v ∗ , w∗ ). We resume now the weak formulation of the stochastic problem, and we consider any smooth function ϕ(x, v) with compact support; for a weak solution f it holds that

d ϕ(x, v)f (x, v, t) dx dv = v · ∇x ϕ(x, v)f (x, v, t) dx dv (48) dt R2d R2d # $ ∗ + δ IE (ϕ(x, v ) − ϕ(x, v))f (x, v, t)f (y, w, t) dx dy dv dw R4d

for any t > 0 and

lim ϕ(x, v)f (x, v, t)dxdv = t→0+

R2d

R2d

ϕ(x, v)f0 (x, v)dxdv.

To carry out the grazing limit, we scale a as a = εa0 , where ε > 0 is a small parameter. Suppose that f = f (x, v, t) satisfies (47), where f actually takes the form of f δ,ε which depends on parameters δ and ε but the superscripts


325

are omitted for brevity. Let us begin with the weak form (48) and we consider the Taylor’s expansion ϕ(x, v ∗ ) − ϕ(x, v) = ∇v ϕ(x, v) · (v ∗ − v) +

1 β ∂v ϕ(x, v)(v ∗ − v)β 2 |β|=2

1 β + ∂v ϕ(x, v˜)(v ∗ − v)β , 6 |β|=3

where v˜ is a vector between v ∗ and v. Recall also that v ∗ − v = εηa0 (|x − y|)(w − v) + 2σεηa0 (|x − y|) θv . Then, formally one has IE[ϕ(x, v ∗ )−ϕ(x, v)] = ε∇v ϕ(x, v)·ηa0 (|x−y|)(w−v)+εΔv ϕ(x, v)·σηa0 (|x−y|), up to O(ε2 ) terms. Thus, it holds

d ϕ(x, v)f (x, v, t) dx dv = v · ∇x ϕ(x, v)f (x, v, t) dx dv dt R2d R2d

∇v ϕ(x, v) · (w − v)ηa0 (|x − y|)f (x, v, t)f (y, w, t)dx dy dv dw + δε 4t

R + δε σΔv ϕ(x, v)ηa0 (|x − y|)f (x, v, t)f (y, w, t) dx dy dv dw R4d

+ δεO(ε). Taking the so-called grazing limit, so that ε → 0, δε → 1, then the limit function, still denoted by f (x, v, t), satisfies

d ϕ(x, v)f (x, v, t) dx dv = v · ∇x ϕ(x, v)f (x, v, t) dx dv dt R2d R2d

+ ∇v ϕ(x, v) · (w − v)ηa0 (|x − y|)f (x, v, t)f (y, w, t) dx dy dv dw 4d

R + σΔv ϕ(x, v)ηa0 (|x − y|)f (x, v, t)f (y, w, t) dx dy dv dw. R4d

This implies that f satisfies formally the equation

The Cucker–Smale Kinetic Model with Nonlinear Dependent Noise: ∂t f + v · ∇x f = η∇v · (ξ(f )f ) + ησ(a0 ∗ ρ) Δv f.

326


Let us define as usual

m(x, t) =

vf (x, v, t) dv. Rd

Let us now consider the parameter γ > d as introduced in (3). Hence, a0 (|x|) is now a even summable function which describes weak long-range interactions. Moreover, for simplicity, assume η = 1, and recast the equation as follows: ∂t f + v · ∇x f + (a0 ∗ m) · ∇v f = (a0 ∗ ρ)∇v · (∇v f + vf ).

(49)

Note that, except for the presence of the convolution a0 ∗ ρ, the right-handside is given by a Fokker–Planck operator term ∇v · (∇v f + vf ). One simple steady state solution of (49) is provided by the global Maxwellian function M(v) =

% & 1 exp −|v|2 /2 . d/2 (2π)

(50)

In [31, Theorem 1] it is shown that this steady pattern is locally stable. In fact, if we assume an initial datum of the type f0 (x, v) = f (x, v, 0) = M(v) +

M(v)F0 (x, v),

for a smooth and small function F0 , then, under such smoothness and smallness assumptions, it is possible to show that the evolution of f is also of the type f (x, v, t) = M(v) + and F (x, v, t) → 0

M(v)F (x, v, t),

(t → ∞), with a polynomial rate.

6 Numerical experiments In this section we illustrate a few numerical experiments which show the dynamics and asymptotic properties of the particle and kinetic models discussed in the previous sections. Before going into these experiments, we spend a few words in relation to the numerical methods used. The general system of ODEs governing particle models is solved by standard classical third order Runge-Kutta schemes. As for the kinetic models, several strategies are followed for the discretization of the time–space–velocity domain. A robust choice is the use of a homogeneous


327

grid in both space and velocity with suitable finite differences schemes based on nonoscillatory ENO schemes and a Runge-Kutta discretization in order to advance in time. Another possibility is the use of splitting methods, which allow to decompose the problem into transport phases in different variables, solve each part separately and recombine everything to approximate a solution for the whole problem. With splitting methods the phase space (x, v) is decomposed into either dimensions (Dimensional Splitting), therefore the transport equation is decomposed into the transport along the x-dimension and the transport along the v-dimension; at this point, the DS can be coupled to semi-Lagrangian methods for the solution of each transport block. Although direct finite differences solvers for the transport equation are wellestablished and robust, they have the drawback of being constrained by the CFL condition. The splitting methods have the advantage of allowing larger time-steps, but their drawback is that they require the solution of the characteristics, possibly complicated. We refer to [19] for more details on this kind of numerical schemes for kinetic equations. 6.1 Mills, double mills, crystalline structure formation, and flocking The problem (1) depends on six parameters: the self-propulsion coefficient α, the friction coefficient β, the attraction strength CA , the attraction typical length lA , the repulsion strength CR , and the repulsion typical length lR . Different choices for these parameters lead to different configurations of the asymptotics of the problem: when α > 0 and β > 0, the meaningful magnitudes are represented by the ratios C = CR /CA and l = lR /lA . In Fig. 8 we sketch four different situations: the parameters α and β fix the veloci ties of the particles to vi = α/β. If the repulsion strength is larger than the attraction strength and the effects of self-propulsion/friction are weak, then we obtain a single milling, as in the (a) sketch of Fig. 8; if, instead, the self-propulsion/friction effects are stronger, we can obtain a flocking with a crystalline structure, as in the (c) sketch of Fig. 8. Double mills are obtained in a short-range repulsion regime in which the attraction strength is larger than the repulsion strength, as we can observe in the (b) sketch of Fig. 8. Finally, whenever the Cucker–Smale interaction is added, we obtain flocking in which the velocity is fixed by the self-propulsion/friction coefficients, as in the (d) sketch of Fig. 8. 6.2 The Cucker–Smale model both for particle and kinetic regimes For γ < 1/2 the strength of the interactions makes the whole system converge to a uniform velocity. In the case of the discrete problem, this means that all the particles tend to have the same velocity; in the case of the continuous problem, this means that the distribution function tends to concentrate on

328

single mill

b

x1

x1

d

flocking

x2

flocking with crystalline structure

x2

c

double mill

x2

x2

a


x1

x1

Fig. 8. The choice of the parameters in the problem composed by self-propulsion, friction and attraction–repulsion is crucial to obtain completely different asymptotics: (a) single mills, we are in regime α = 0.07, β = 0.05, C = 2.5 > 1, l = 0.02 and Cl2 = 0.01; (b) double mills, we are in regime α = 0.15, β = 0.05, C = 0.5 > 1, l = 0.2 and Cl2 = 0.02; (c) flocking with crystalline structure, we are in regime α = 0.2, β = 0.1, C = 2.5 > 1, l = 0.02 and Cl2 = 0.001; (d) flocking, we are in regime α = 0.07, β = 0.05, C = 2.5 > 1, l = 0.02 and Cl2 = 0.01, plus we added Cucker–Smale interaction with γ = 0.05

a delta function in the velocity space and to be distributed only along the spatial dimension, as we see in Fig. 9. In these simulations we show also that the discrete problem approximates well the continuous problem, as expected theoretically by the mean-field limit. In case we use a parameter γ > 1/2, then the interaction between the particles may not force them to converge to a uniform velocity, and flocking may depend on the density. In Fig. 10 the system is initialized by distributing two groups around velocities +2.5 and −2.5; in the example, we see that there is no convergence to a uniform velocity.

Particle, kinetic, and hydrodynamic models of swarming time = 0

time = 0

4 velocity

2 0 –2 –4 –20 –15 –10 –5 0 5 position

10

15

4 2 velocity

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 –0.05

0 –2 –4 –20 –15 –10

20

–5

0 –2 –4 5

10

15

20

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 –0.05

20

0 5 position

10

15

20

10

15

20

10

15

20

0 –2 –4 –20 –15 –10

–5

time = 4

time = 4

2 0

–2 –4 –20 –15 –10 –5 0 5 position

10

15

4 2 velocity

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 –0.05

4 velocity

15

2

position

0 –2 –4

20

–20 –15 –10

–5

time = 8

0

–2 –4 10

15

20

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 –0.05

4 2 velocity

2

0 5 position

0 5 position

time = 8

4 velocity

10

4

velocity

velocity

2

–20 –15 –10 –5

5

time = 2

4

0

0 position

time = 2

–20 –15 –10 –5

329

0

–2 –4 –20 –15 –10

–5

0 5 position

Fig. 9. Particle vs. Kinetic simulation of the Cucker–Smale model in 1D with γ = 0.05 < 1/2. The kinetic problem is solved through splitting methods


4 velocity

2 0 –2 –4 – 20 – 15 – 10 – 5 0 5 position

10

15

20

time = 1 4 velocity

2 0 –2 –4 –20 –15 –10 –5 0 5 position

10

15

20

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 –0.05

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 –0.05

time = 0 4 2 velocity

time = 0

0

–2 –4 – 20 –15 –10 –5

–2 –4 –20 –15 –10 –5 0 5 position

10

15

20

time = 5 4 velocity

2 0 –2 –4 –20 –15 –10 –5 0 5 position

10

15

20

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 –0.05

10

15

20

10

15

20

10

15

20

0

–4 0 5 position


0

20

–2

0 –2 –4 –20 –15 –10 –5

0 5 position


velocity

2

15

2

–20 –15 –10 –5

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 –0.05

10

time = 1

time = 2 4

0 5 position

4 velocity

330

0 –2 –4 –20 –15 –10 –5

0 5 position

Fig. 10. Particle vs. Kinetic simulation of the Cucker–Smale model in 1D with γ = 0.95 > 1/2. The kinetic problem is solved through splitting methods


331

The convergence rate of the discrete Cucker–Smale system can be influenced by including some nonlinear effects; replace the dynamics in (3) by: N dvi 1 = H(|xi − xj |)(vj − vi )vj − vi p−2 , dt N j=1

as in [40]. Of course, for p = 2 we recover the usual model (3). In Fig. 11 we plot ' (N ( ) |vi (t) − v |2 i=1

against time in logarithmic scale. For p = 2 the convergence toward the asymptotics is exponential, while for p > 2 the convergence has polynomial rate for some of the values studied, and for p < 2 the asymptotics is reached in finite time. A detailed study of this problem will be done elsewhere. l2-convergence towards the average velocity in linear/log scale 100 1 p=2.5 0.01

p=2.25

0.0001 1e-06

p=2

1e-08 1e-10 p=1.95

1e-12 p=1.9

1e-14 1e-16 1e-18

0

2

4

6

8

10 time

12

14

16

18

20

Fig. 11. Flocking convergence rate for the nonlinear Cucker–Smale model in 1D

6.3 Leadership emergence for particle regimes, when visibility cone conditions are considered The Cucker–Smale model with γ < 1/2 forces all the particle to the average velocity given by the initial spatial and velocity configuration. When a vision

332


angle α is introduced, then the conditions for the formation of several groups are set; the particles which cannot see the other ones become leaders of a group. The evolution of the system, when different vision angles α are considered, is sketched in Fig. 12; as the angle decreases, the ability of the particles of feeling the presence of the others decreases, therefore more distinct groups with emerging leaders tend to be formed. angle = 2

x2

x2

angle = 2.5

angle = 1.5

angle = 1

x2

x1

x2

x1

x1

x1

Fig. 12. Cucker–Smale model with γ = 0.05 and different vision angles

Acknowledgment: JAC and FV acknowledge support from the project MTM2008-06349-C03-03 DGI-MCI (Spain) and 2009-SGR-345 from AGAURGeneralitat de Catalunya. MF acknowledges the support of the FWF project Y 432-N15 START-Preis “Sparse Approximation and Optimization in High Dimensions,” and the hospitality of Texas A&M University during the preparation of this work. GT acknowledge support from the Italian MIUR project “Kinetic and hydrodynamic equations of complex collisional systems.” JAC and GT acknowledge partial support of the Acc. Integ. program HI2006-0111.

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Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints Emiliano Cristiani1 , Benedetto Piccoli2 , and Andrea Tosin3 1

2 3

CEMSAC - Universit` a degli Studi di Salerno, Fisciano (SA), Italy and IAC-CNR, Rome, Italy, [email protected] IAC-CNR, Rome, Italy, [email protected] Department of Mathematics, Politecnico di Torino, Turin, Italy, [email protected]

Summary. This paper is concerned with mathematical modeling of intelligent systems, such as human crowds and animal groups. In particular, the focus is on the emergence of different self-organized patterns from nonlocality and anisotropy of the interactions among individuals. A mathematical technique by time-evolving measures is introduced to deal with both macroscopic and microscopic scales within a unified modeling framework. Then self-organization issues are investigated and numerically reproduced at the proper scale, according to the kind of agents under consideration.

1 Self-organization in many-particle systems One of the most outstanding expressions of intelligence in nonclassical physical systems, such as human crowds or animal groups, is their self-organization ability. Self-organization means that the individuals composing the system can give rise to complex patterns without using intercommunication as an essential mechanism. For instance, in normal conditions pedestrians are known to arrange in specific patterns, chiefly lanes (cf. Fig. 1a,b), as demonstrated by many experimental investigations [19, 20, 23, 28, 29]. Lane formation may be fostered by a suitable setup of the space, as reported in [19, 23]: a test performed in a tunnel connecting two subway stations in Budapest showed that a series of columns, placed in the middle of the walkway, induce pedestrians to organize in two oppositely walking lanes, preventing each of them to expand up to the full width of the corridor. More in general, lanes form also spontaneously, i.e., without the need for being triggered by environmental factors, provided the density of pedestrians is sufficiently large [20]. This is particularly evident if


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Emiliano Cristiani et al.

one considers the case of two groups of people, walking in opposite directions, which meet and cross (see also [28]). Grouping and self-organization are well known and largely observed also in animals, see for example [34]. These phenomena are in fact ubiquitous, ranging from bird flocks in the sky to migrating lobsters on the sea floor. Many papers on this subject (see, among others, [7,11,12,14,16,17,32,35,36,48,51]) proved by means of numerical simulations that few simple rules adopted by each animal can give rise to a complex organization of the whole group. Patterns commonly seen in nature are (see [26] for the nomenclature): globular clusters (e.g., starlings, Fig. 1c, surf scoters, Fig. 1d), extended and front clusters (e.g., wildebeests, antelopes, and pigeons), lines (e.g., elephants, lobsters, penguins, Fig. 1e), Vees, Jays, and echelons (e.g., geese). It is commonly agreed that self-organization is the result of elementary actions that each subject performs to fulfill specific wills. Concerning pedestrians, the following basic guidelines can be identified: •

The will to reach specific targets, e.g., an exit or a meeting point, which drives pedestrians along preferential paths, determined mainly by the geometry and the spatial arrangement of the walking area. Unlike animals, pedestrians experience strong interactions with the environment, because they usually move in highly structured spaces scattered with all sorts of obstacles. • The will to not stay too close to one another, with a preference for uncrowded areas (repulsion from other individuals). Pedestrians may agree to deviate from their preferred path, looking for free surrounding room. In addition, it is reasonable to believe that occasionally also a mild form of cohesion occurs, which translates the tendency of pedestrians to not remain isolated. This happens, for instance, in those groups whose individuals share specific relationships, such as groups of tourists in guided tours. For animals, the basic guidelines can instead be outlined as follows: •

The will to not stay too close to one another, in order to avoid collisions (repulsion from other individuals). • The will to not remain isolated (group cohesion). Grouping is in fact advantageous for many reasons, such as predator avoidance or food search, see [34]. Like pedestrians, also animals may want to reach some specific destinations (e.g., when they migrate). However, the direction of motion toward targets is usually almost constant for quite a long time, because animals move in large and basically obstacle-free environments. Therefore, this aspect plays a minor role in the description of their self-organization, corresponding to a simple translation of the center of mass of the whole group. Self-organization can be broken under particular circumstances entailing a dramatic change in the basic interaction rules discussed earlier. For pedestrians, an illuminating example is panic, when individuals tend to cram toward

Modeling self-organization in pedestrians and animal groups

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(c)

(a)

(d)

(b)

(e)

Fig. 1. Self-organizations. (a), (b): oppositely walking lanes of pedestrians. (c): three-dimensional globular cluster of starlings. (d): two-dimensional crystal-like globular cluster of surf scoters. (e): a line-like structure by penguins. Reproduction of these pictures with kind permission of the respective copyright holders. Credits are in the acknowledgments at the end of the paper

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a common target (e.g., an escape) instead of seeking the less congested paths. Lane formation is then ruled out and less organized patterns emerge in the crowd, probably in consequence of a strong simplification of the interaction rules. Nevertheless, the arrangement of the environment may help restore, at least partially, the normal order of the flow. A typical example is an obstacle placed in front of an exit, which in some situations (presumably under panic conditions) may improve the flow of people through the exit itself, provided shape, dimensions, and position of the obstacle are accurately studied. This is a variant of the so-called Braess’ paradox [31], which states that a condition intuitively expected to lead to a worse situation may instead give rise to better outcomes. In animals, the changes in the behavior are even more evident. External conditions (e.g., presence of predators, and weather), group tasks (feeding, exploring) or group speed can modify the interaction rules, leading to great modifications in the resulting patterns of the group. The environment usually does not interfere much, and pattern formation is mainly due to interactions among the individuals. In this paper we are concerned with mathematical modeling of the abovediscussed systems, with the specific aim of describing the spontaneous emergence of self-organization. In particular, we focus on two basic characteristics of the interactions among the individuals, namely nonlocality in space and anisotropy. The inclusion of these factors makes our models able to explain the differences observed in self-organization and pattern formation of various groups of agents in terms of different visual fields and sensing zones. At the same time, we introduce a modeling framework based on the measure theory, that allows for a unified formulation of macroscopic and microscopic models in any space dimension, and for a convenient numerics. This enables us to investigate both macroscopic self-organization, typical of large crowds of pedestrians, and microscopic self-organization, more specific of animal groups, using common modeling principles and tools. In more detail, the paper is organized as follows. After this introduction, Sect. 2 proposes a comparison between classical and intelligent particles, and highlights the main differences of the latter with respect to more standard systems dealt with by classical physics. In view of these considerations, Sect. 3 develops some preliminary modeling strategies to enhance intelligence as a distinctive feature of the systems at hand. These serve as guidelines for Sect. 4, where the modeling technique by time-evolving measures is introduced and specific macroscopic and microscopic models are detailed. Numerical tests on the ability of these models to reproduce spontaneous self-organizing patterns for both human crowds and animal groups are performed and commented in Sect. 5. Finally, Sect. 6 draws some conclusions and briefly sketches research perspectives.


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2 Classical vs. intelligent particles A plethora of physical systems can be basically described as systems of interacting particles, think for instance of fluids, gases, and similar matters. Actually, also human crowds and animal groups are susceptible to this rough characterization, although it should be clear from the previous discussion that the particle analogy is only formal. Indeed, intelligence is what really makes the difference in this context: it gives rise to a decision-based dynamics determined by individual behavioral rules, rather than to a classical dynamics passively ruled by inertia. For this reason, human beings and animals are more properly characterized as Intelligent Particles (denoted IPs in the sequel for brevity). IPs can act directly on the system, rather than being passively subjected to the evolution itself, whence their ability to self-organize and to generate complex patterns. In the following, we summarize the main differences between classical and intelligent particles, in order to outline the main novelties posed by intelligent systems with respect to more standard frameworks. Robustness vs. Fragility. Classical particles are robust, in the sense that they interact almost exclusively through collisions. For instance, gas particles change direction of motion and velocity only when hit by other particles, or possibly when they collide with the walls of the container in which the gas is stored. Conversely, IPs are fragile, they try as much as possible to avoid mutual collisions as well as to steer clear of walls and obstacles scattered along their path. Blindness and Inertia vs. Vision and Decision. Classical particles are blind, indeed they have no information on the environment and on the distribution of the surrounding particles. Therefore, their dynamics is essentially ruled by inertia, i.e., a passive response to the mechanical cues coming from the exterior. On the contrary, human beings and animals feature specific visual fields, hence they can obtain information on the surrounding environment (e.g., presence of obstacles, of walls) and on the current distribution of (a certain number of) other agents. This information is then used to make decisions on the future individual evolution, generally by following some ordinary behavioral rules proper of the kind of agent under consideration. Local vs. Nonlocal Interactions. Collisions among, say, gas particles require the latter to be sufficiently close to hit each other. Because of the typical size of gas molecules, by far much smaller than the environments where they flow, this has been classically understood as if the colliding particles were occupying the same spatial position at the moment of the impact (see, e.g., [49] and the references therein), hence assuming the conceptual approximation of local, i.e., pointwise, interactions. Conversely, IPs do not interact mechanically, rather they are influenced by the presence of other individuals or objects a certain distance away, that they want either to approach or to avoid. The resulting

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interaction is thus nonlocal, because the agents do not need to be in contact to interact. We notice that there exist basically two types of nonlocal interaction: a metric one, such that each agent is influenced by all other agents located at a distance less than a given threshold; and a topological one, such that each agent is influenced by a given number of other agents, no matter how far they are.4 Topological interactions have been used in several models, e.g., [32, 33, 42, 51], this idea being supported by experimental investigations like, for example, [2] on fish schools. Other models [11,12,17,35] do not include this feature in favor of a purely metric approach. However, recent results [3] show that starlings interact topologically with other six/seven group mates. It is reasonable to believe that topological interactions are common in nature, even in pedestrians, since they are mainly due to a limited capacity in processing the information. On the other hand, a purely topological interaction in a wide domain is not realistic, for an individual may not be concerned with very far mates (e.g., because it does not see them at all). Isotropy vs. Anisotropy. Another striking difference concerning the way in which classical and intelligent particles interact is that the first are isotropic, meaning that they are equally affected by mechanical cues coming from all directions. IPs are instead anisotropic, i.e., they are sensitive to stimuli coming from specific directions. This is partly related to the width of their visual field, which in pedestrians coincides with the half-space in front of them, whereas in animals covers often almost all the surrounding space. More precisely, the visual field must be intended as an upper bound for the regions in which cohesion and repulsion are active. Indeed, repulsion can be expected to be mainly felt against the IPs ahead rather than against those on the side or behind, especially in running people or fast-moving animals. Cohesion is instead active toward the group mates in front, if the goal is to follow the head of the group, or in every direction, if the goal is the unity of the group. The ideas of limited visual field, expressed in terms of blind rear zone, and of anisotropic sensing are quite common in the biological literature, see for instance [12, 17, 25, 32, 33, 35, 37]. However, they mainly play the role of passive features of the agents, and are not regarded as active features able to influence the shape of the group. Energy and Entropy vs. Self-organization. Particle collisions allow to describe the mechanics of classical systems by invoking the balance of linear momentum, as well as energy and entropy principles. A straightforward extension of these ideas to intelligent systems is not possible, because the ability of IPs to make subjective decisions continuously puts and removes energy in and from the system, in hardly quantifiable amounts. Entropy criteria may in turn be questioned, because entropy is classically related to the equiprobability of the 4 It is worth noting that in the biological literature authors often regard both metric and topological interactions as local models, in contrast to nonlocal models in which every agent interacts with all other group mates.


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states of a system, whereas self-organization promotes specific configurations of the IPs. The idea that, in living matter, energy and entropy play a somehow atypical role was already suggested by Schr¨ oedinger [46] in the Sixties, who formalized it through the concept of negative entropy. It is useful, at this point, to consider a very borderline system with respect to the “classical vs. intelligent” dichotomy, namely vehicular traffic. Traffic is, in principle, an intelligent mechanical system, because the intrinsic mechanics of the vehicles is tempered by the presence of the drivers, who determine a non-strictly mechanical behavior of the cars. However, many successful models, relying on the fluid dynamical analogy of the flow of vehicles along a road, have been proposed in the literature (see [27,45] for comprehensive reviews on microscopic, kinetic, and macroscopic models). This approach has been possible, despite the nonstandard nature of the system at hand, because traffic is essentially one-dimensional, so that vehicles regulate their speed only. Such a dimensional constraint reduces significantly the possibility of self-organization and pattern formation, therefore fluid dynamical modeling, entropy reasonings, and finally the development of models and theories based on nonlinear conservation laws are feasible.

3 What mechanics for intelligent systems? The discussion of the previous sections has highlighted a crucial feature of intelligent systems: their dynamics is only partially determined by classical (pseudo-) mechanical cues, therefore, in principle, it is not fully describable within the classical Newtonian framework of point mechanics. Concerning this, we note that in the systems we are considering impulsive forces are often present. For example, an animal which starts moving reaches in a very short time its final velocity, which then remains constant for a while. Including in the models such impulsive forces through a Newtonian approach is quite difficult and, after all, needless. The dynamics of intelligent systems calls for a paradigm which enhances intelligence as a distinctive feature, rather than recovering it as a by-product of other principles. In this respect, let us consider that, from the mechanical/ dynamical point of view, intelligence might be regarded as the ability of the agents to control their velocity actively, i.e., without the need for inertial accelerations caused by external actions. In order to take this aspect into account, we suggest to split the velocity v of the agents in two basic contributions: v = w + ν,

(1)

to each of which there correspond specific modeling strategies. In more detail: •

w is the external velocity, i.e., the component of the total velocity depending essentially on actions exerted on the agents by the external environment. For example, in human crowds these may be the repulsion produced

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by obstacles and walls or the attraction produced by targets (e.g., doors, displays). In animal groups this effect is instead greatly reduced, because animals move in much less structured and generally clear environments. For the external velocity, a Newtonian-like modeling is feasible, hence w can be deduced from inertial reasonings up to a careful identification of the external actions. • ν is the intelligent velocity, i.e., the component of the total velocity that the individuals control actively. The intelligent velocity is determined by the behavioral rules that the agents comply to, and by the decisions that they consequently make, therefore it need not be Newtonian. For instance, pedestrians and animals set up the intelligent velocity according to the occupancy of the surrounding space, so as to steer clear of congested areas while possibly preserving the compactness of the group. A convenient manner to model ν is by an equation of state of the form ν(t, x) = f [Q](t, x), where t, x are time and space variables, respectively, Q comprises all state variables which can contribute to the determination of ν (e.g., the distribution of the agents in a suitable neighborhood, the metric and topological structure of the sensing zone), and f is a functional relationship. Many microscopic (i.e., agent-based) models of animal groups available in literature actually fit into such framework, and the same is true for macroscopic models using hyperbolic partial differential equations, see e.g., [15, 47] and references therein. The forces do not appear explicitly, the velocity being expressed directly as a function of the density of the animals, usually in a nonlocal way. On the other hand, different models at both the microscopic and the kinetic scale rely instead, at least formally, on a more Newtonian-like framework, in the sense that they recover the dynamics of the system from generalized forces responsible for the acceleration of the agents [6,18,24]. Similar considerations apply also to models of human crowds, for an overview of which we refer to [21]. In particular, macroscopic models derived by closing the mass conservation equation with suitable relations for the velocity follow the ideas outlined earlier, see, e.g., [8–10, 22, 30, 31]. Other models resort instead more heavily to classical fluid dynamical analogies, indeed they supplement the mass conservation equation by a momentum balance equation invoking concepts like acceleration or generalized forces and pressures [4]. In all of the above-mentioned models, it is in principle possible to introduce topological interactions among IPs, as well as anisotropic sensing zones. Unfortunately, models based on PDEs and nonlinear conservation laws suffer from an important drawback: their mathematical treatment and numerical implementation get rather complicated in dimensions greater than one. Indeed, in realistic cases of interest for applications one must deal with possibly punctured two-dimensional domains (the holes representing the obstacles), which requires to handle boundary conditions in a way that may not be immediately


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suited for hyperbolic equations. Therefore, we prefer to use an alternative modeling strategy, which, at the same time, enables us to develop a modeling framework capable to treat both the macroscopic and the microscopic scale within a unified theoretical formulation.

4 Mathematical modeling by time-evolving measures In this section we set up a modeling framework for the study of the behavior of IPs. In doing this, we describe the occupancy of the space by the agents at different times via a sequence of positive Radon measures. This approach is inspired by [5], where the authors address rendez-vous problems for multiagent systems, and has already been investigated in recent works [43,44]. Here we show furthermore that also microscopic models for animal groups can be recast in the time-evolving measures framework. Therefore, we ultimately provide a unified modeling procedure to treat both macroscopic and microscopic intelligent systems. Let us consider a domain Ω ⊆ Rd , which represents the area where IPs are located and move. In our discussion, d is the dimension of the domain, from the physical point of view it may be d = 1, 2, 3. A great advantage of our approach is that there are basically no differences in the theory for different dimensions. At every time n ≥ 0, we define a Radon positive measure μn over the Borel σ-algebra B(Ω), such that, for each measurable set E ∈ B(Ω), the number μn (E) ≥ 0 measures the occupancy of the area E by the IPs. In other words, the mapping μn provides the localization, i.e., the distribution, of the agents in the domain Ω at time n. The evolution to the next time n + 1 depends on the dynamics of the system, that we describe via a motion mapping γn : Ω → Ω: γn (x) = x + vn (x)Δt,

x ∈ Ω.

(2)

In practice, in the time step n → n + 1 the point x is advected to γn (x) by the velocity field vn : Ω → Rd . The duration of the time step is Δt > 0. Then, the measure μn+1 is constructed by pushing μn forward with γn , that is μn+1 = γn #μn or, more explicitly, μn+1 (E) = μn (γn−1 (E)),

∀ E ∈ B(Ω).

(3)

This corresponds to the simple idea that the number of IPs contained in E at time n + 1 coincides with their number at the previous time n in the preimage γn−1 (E), therefore (3) expresses the conservation of the mass of IPs. By rewriting it in the equivalent form μn+1 (E) − μn (E) = −[μn (γn−1 (E c ) ∩ E) − μn (E c ∩ γn−1 (E))], we recognize the formal statement of a conservation law: the left-hand side represents the variation of the measure of E in a single time step, and the

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right-hand side the difference between the outgoing flux, i.e., the measure of the set of points belonging to E and mapped outside E by γn , and the incoming flux, i.e., the measure of the set of points not belonging to E and mapped inside E by γn . Mathematical models are deduced from the structure (2) and (3) by prescribing the initial distribution of the IPs, that is the measure μ0 , and by specifying the form of the velocity field vn . According to the discussion of Sect. 3, we represent vn as the superposition of external and intelligent contributions: vn (x) = w(x) + νn [μn ](x). The functional dependence of the intelligent velocity on the distribution of the agents, formally expressed by the square brackets in this formula, enables us to account for all of the factors having to do with the perception of the occupancy of the surrounding space from IPs (cf. Sect. 2). In more detail, we distinguish cohesive and repulsive effects in νn , that we model as: νn [μn ](x) = νnc [μn ](x) + νnr [μn ](x) = Fc (y − x) dμn (y) + Bc (x)

Br (x)\{x}

Fr

y−x 2

|y − x|

dμn (y)

(4)

for coefficients Fc ≥ 0, Fr ≤ 0. In this equation, Bc (x), Br (x) ⊂ Ω are the zone of cohesion and the zone of repulsion of the agent in x, respectively, which will be defined in the following. The strength of the cohesive velocity νnc is chosen to be growing linearly with the distance among IPs within the zone of cohesion Bc (x). Cohesion is mainly topological, i.e., it involves a predefined number of IPs. By consequence, for a fixed positive p, the zone of cohesion is chosen to satisfy μn (Bc (x)) ≤ p, but its size may vary from point to point according to the distribution of the agents. However, a maximal size of Bc (x) exists, say s, beyond which interactions among IPs are inhibited due to the distance (cf. Sect. 2), so that also Ld (Bc (x)) ≤ s (Ld being the Lebesgue measure on Rd ). In practice, Bc (x) is adjusted dynamically under the constraints μn (Bc (x)) ≤ p

and Ld (Bc (x)) ≤ s.

(5)

Let us choose, for convenience, Bc (x) to be a circular sector of the ball of radius Rc > 0 (to be detemined), centered in x and with central angle αc ∈ (0, 2π] (Fig. 2). Other choices are possible, for instance in [35] the authors use elliptical regions to take the body form of the agents into account. Because of the symmetry of the body, we assume that the sensing domain of each IP is symmetric with respect to the main direction of motion fixed by the vector w(x). Introducing the family of circular sectors α , S(x, R, α) = y ∈ Ω : |y − x| ≤ R, rˆ(x, y) · w(x) ˆ ≥ cos 2


Bc(x)

Rc x

347

αc 2 αc 2

w(x)

Fig. 2. Definition of the zone of cohesion by means of the velocity w, the topological radius Rc , and the span angle αc ∈ (0, 2π]. Analogous considerations hold also for the zone of repulsion, up to considering a metric radius Rr and an angle αr

where rˆ(x, y) and w(x) ˆ are the unit vectors in the directions of y − x and w(x), respectively, and setting Rc = max{R ≥ 0 : μn (S(x, R, αc )) ≤ p, Ld (S(x, R, αc )) ≤ s}, we can easily define Bc (x) := S(x, Rc , αc ). Notice that the constraint on the Lebesgue measure of each S basically amounts to fixing a metric upper bound Rcmax to the maximum radius allowed for the zone of cohesion Bc (x). This formalizes the topological cohesion complemented with a metric cut-off observed in [3] (see also [16]). Conversely, we take the strength of the repulsive velocity νnr proportional to the inverse of the distance among IPs within the zone of repulsion Br (x). Repulsion is mainly metric, as each IP simply tries to maintain a minimum distance between itself and other IPs. Hence Br (x) has a fixed size Ld (Br (x)), while its measure μn (Br (x)) may vary from point to point according to the crowding of the space. Assuming again for convenience that Br (x) is a circular sector of the ball with center in x and (fixed) radius Rr > 0, we simply have Br (x) := S(x, Rr , αr ), αr ∈ (0, 2π] being the angular span of the zone of repulsion. Finally, the external velocity w depends essentially on the interactions of the IPs with the environment. Instead, it is independent of the distribution of the agents in the domain, because it represents the velocity that an isolated agent would set to reach its targets. Due to the strong differences in the structure of the typical environments where pedestrians and animals move, we refrain from giving here a general structure of w. We will detail it in the next subsections, with reference to specific applications. 4.1 Macroscopic models Macroscopic models are useful to study emergent self-organizing behaviors on large scales. This is the case of human crowds, in which self-organization entails the formation of patterns that are clearly visible only when the density of people is sufficiently high. In addition, these models are particularly

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handy to face the technical complexity of pedestrian flows in structured environments scattered with obstacles, possibly also in connection with safety and optimization issues. Macroscopic models are obtained from the modeling framework outlined earlier via the continuum hypothesis, which amounts to saying that the measure μn is absolutely continuous with respect to the Lebesgue measure Ld (μn Ld ). Therefore, one can introduce a non-negative function ρn ∈ L1 (Ω) such that dμn = ρn dx, and speak of density of IPs. The link with the measure μn is explicitly stated as: ∀ E ∈ B(Ω). μn (E) = ρn (x) dx, E

Starting from an initial measure μ0 Ld , existence of the density for all times n is provided by the following result proved in [43]: Theorem 1. For all n > 0, let a constant Cn > 0 exist such that Ld (γn−1 (E)) ≤ Cn Ld (E),

∀ E ∈ B(Ω).

If μ0 Ld is non-negative, then there is a unique sequence {ρn }n≥1 ⊂ L1 (Ω), ρn ≥ 0 a.e. in Ω, such that μn Ld with dμn = ρn dLd and ρn 1 = ρ0 1 for all n > 0. 1 ∞ 1 ∞ If in addition nρ0 ∈ L (Ω) ∩ L (Ω) then ρn ∈ L (Ω) ∩ L (Ω) as well, with ρn ∞ ≤ ( k=1 Ck ) ρ0 ∞ for all n > 0. Under the continuum hypothesis, the intelligent velocity (4) specializes as: y−x Fc (y − x)ρn (y) dy + Fr (6) νn [ρn ](x) = 2 ρn (y) dy. |y − x| Bc (x)

Br (x)

Concerning the external velocity, to be definite we consider the application to pedestrians. In this case, it is convenient to model w as a normalized potential flow w = ∇u/|∇u|, which does not depend on the distribution ρn of the people but only on the geometry of the domain (cf. Sect. 3), including the possible presence of obstacles. The function u : Ω → R is a scalar potential satisfying, e.g., Laplace’s equation Δu = 0, which returns smooth fields bypassing the holes of Ω. Boundary conditions for this equation may be set to identify attractive (respectively, repulsive) areas, such as doors or displays (respectively, obstacle edges or perimeter walls). For instance, targets along the boundary ∂Ω of the domain may be characterized by the maximum potential, say u = 1, whereas the remaining portion of the perimeter walls by the minimum potential, say u = 0. On the internal boundaries, namely obstacle walls, the Neumann condition ∂u/∂n = 0 may be prescribed instead, which corresponds to zero normal component of the velocity w.


xhM h

h

Ld(E j ∩ γ ˜ hn(E hk))

349

h

Ek

h h

h

Ej

xhj

xh1 h

h Fig. 3. Left: the grid {Ejh }M j=1 in the domain Ω, with the generic cell Ej highlighted. Right: the coefficients of the numerical scheme (7) are determined as the (hyper)volumes of the overlapping parts of adjacent grid cells under the push-forward operated by the approximate motion mapping γñh

Equation (3) is numerically solved on a finite volume-like partition of the domain Ω, using an ad hoc computational scheme deduced from the one suggested in [44] with the inclusion of the new modeling features introduced h h elements of characteristic here. Basically, a grid {Ejh }M j=1 consisting of M size h > 0 is introduced in Ω (see Fig. 3, left), and the density ρn and the velocity vn are discretized as piecewise constant functions: ρn (x) ≈ ρ˜hn (x) ≡ ρhn,j , ∀ x ∈ Ejh , ρhn ](x) ≡ vn [˜ ρhn ](xhj ) vn [ρn ](x) ≈ vñh [˜ xhj being a point of the cell Ejh , for instance its center. This induces naturally a discretization of the motion mappings γn : ρhn ](x)Δt, γn (x) ≈ γñh (x) = x + vñh [˜ where it should be noticed that the γñh ’s act as piecewise translations on the h μhn := ρ˜hn dLd , the scheme grid {Ejh }M j=1 . After defining the new measures d˜ h is obtained by imposing the push-forward μ ñ+1 = γñh #˜ μhn on the grid cells h h h h h −1 h ñ+1 (Ej ) = μ ñ ((˜ γn ) (Ej )) for all j = 1, . . . , M h , which gives Ej only, i.e., μ (cf. Fig. 3, right): h

ρhn+1,j

M 1 = d h ρh Ld (Ejh ∩ γñh (Ekh )). L (Ej ) k=1 n,k

(7)

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If the γn ’s are sufficiently smooth, and if the spatial resolution h and the time step Δt are linked by a CFL-like condition, this scheme turns out to be nicely behaved in terms of stability and localization error: Theorem 2. Let γn be a diffeomorphism and let h, Δt > 0 satisfy Δt

max

j=1, ..., Mh

|vn [˜ ρhn ](xhj )|2 ≤ h,

where | · |2 is the Euclidean norm in Rd . Then: (i) There exists a constant C > 0 independent of h such that h

M

|μn+1 (Ejh ) − μ ˜hn+1 (Ejh )| ≤ C ρn − ρ˜hn 1 + h .

j=1

(ii) For each n > 0, there exists a constant Cn > 0 independent of h such that max

j=1, ..., M h

|μn+1 (Ejh ) − μ ˜hn+1 (Ejh )| ≤

max

j=1, ..., M h

|μ0 (Ejh ) − μ ˜h0 (Ejh )| + Cn hd .

The proof of this result can be recovered again in [43]. The estimates of Theorem 2 rely essentially on the L1 metric, because they assume the existence of the densities {ρn }n≥1 for the measures {μn }n≥1 . However, it can be questioned that such a metric is not the optimal one to evaluate the distance between measures, even when densities are available. This is particularly true in the application to pedestrian flows, where selforganization phenomena may give rise to measures concentrated in thin areas of the domain (though not singular in view of Theorem 1). Small errors in the localization of these measures would be roughly estimated by the L1 distance of the corresponding densities, even when the corresponding distributions of the crowd are intuitively close. To be definite, let us consider two measures μ1 , μ2 with densities ρ1 , ρ2 , respectively, as illustrated in Fig. 4. Assume that ρi ∞ = O(1/), Ld (supp ρi ) = O(), i = 1, 2, for some arbitrarily small > 0, whereas the average pointwise distance between ρ1 and ρ2 is O(δ) for δ > 0. Thus, it is reasonable to expect a localization error of μ2 over μ1 of the order of δ, but if δ is larger than and we use the L1 norm we inevitably get ρ2 − ρ1 1 = O(1), because most mass is concentrated in the nonoverlapping parts of the supports of ρ1 , ρ2 . A more correct way to measure the distance between μ1 , μ2 , which also better matches our intuition on what this distance should be, is offered by the Wasserstein metric, that we can briefly introduce as follows. Let (Ω, d ) be a metric space for which every probability measure on Ω is a Radon measure. Then the Wasserstein distance between two probability measures μ1 , μ2 is defined as: W (μ1 , μ2 ) := inf d (x, T (x)) dμ1 (x), T

Ω


351

O() ρ1(x)

ρ2(x)

O(1/)

O(δ)

x

Fig. 4. Two measures dμi = ρi dLd , i = 1, 2, whose L1 distance is O(1) whereas Wasserstein distance is O(δ). The latter gives then a better estimate of the error produced by μ2 in the localization of μ1

the inf being taken among all transport maps from μ1 to μ2 : μ2 = T #μ1 ,

i.e.,

μ2 (E) = μ1 (T −1 (E)),

∀ E ∈ B(E).

W (μ1 , μ2 ) is the best (i.e., the lowest) transportation cost to move the measure μ1 onto μ2 (and vice-versa). In our previous example, we would actually get W (μ1 , μ2 ) = O(δ). Notice that model (3) can be interpreted as an explicit Euler discretization in time of a gradient flow on the Wasserstein space, at least for sufficiently smooth motion mappings, cf. [1,38,50]. Therefore, the Wasserstein metric may be profitably used to further improve the theory with a more accurate error analysis.

4.2 Microscopic models Microscopic models are useful to study self-organization phenomena at small scale. They are common in biological literature, where a wealth of models were studied in order to understand grouping behavior in fish schools, bird flocks, mammals herds, and bacteria aggregations. Microscopic models can be obtained from our time-evolving measures framework, removing the continuum hypothesis. Denoting by xnj ∈ Ω the position of the jth IP at time n, the measure μn is now chosen to be the counting measure, i.e., μn =

N j=1

δxnj ,

μn (E) = card {xnj ∈ E},

(8)

where N is the total number of agents and δx is the Dirac measure centered in x. Existence issues are now by far much easier than in the macroscopic

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case, because the measure μn+1 can be constructed explicitly. Indeed, after a push-forward by γn it is easily computed that μn+1 =

N j=1

δγn (xnj ) ,

hence at the next time step the new positions of the IPs are xn+1 = γn (xnj ). j Finally, the resulting model turns out to be a classical agent-based model of the form = xnj + vn (xnj )Δt, xn+1 j which corresponds to the discrete-time version of the dynamical system x˙ j (t) = v(t, xj (t)) obtained by an explicit Euler scheme. Inserting the measure (8) in (4) yields the following expression of the intelligent velocity: xnk − xnj Fc (xnk − xnj ) + Fr n (9) νn (xnj ) = 2. |xk − xnj | xn ∈Bc (xn ) xn ∈Br (xn )\{xn } k

j

j

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Besides the formal derivation of (9) from (4), we notice that at the microscopic scale the body size of the IPs matters. Indeed, the latter are not point agents, thus when two of them are closer than a characteristic distance > 0 they touch each other (cf. also [39–41] about microscopic pedestrians). Therefore, we admit that within a small neighborhood of radius repulsion from a certain agent xnj is active against all other agents, regardless of the angular span αr , and that the distance between two IPs is never smaller than , cf. Fig. 5. This also helps avoid singularities, i.e., nonintegrability with respect to the Dirac measure, in the second sum of (9).

αr

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Fig. 5. At the microscopic scale, the body size of the IPs matters. Therefore, repulsion from the jth agent is always active in a complete neighborhood of radius , regardless of the radius Rr and the angle αr . This, however, does not affect the zone of cohesion


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Finally, we assume that the external velocity of the agents is constant (for instance, the vector w = (1, 0) for a rightward motion), as the environment is clear and we are mainly interested in the dynamics of the interactions among the IPs. This way, we can formally drop w from (4) by a simple change of frame of reference.

5 Numerical results In this section we address some relevant case studies, which highlight the ability of our model to reproduce self-organizing patterns at both the macroscopic and the microscopic scale. In particular, we focus on pedestrians for macroscopic self-organization, studying the emergence of oppositely walking lanes in crossing flows, the spontaneous arrangement of people in a group in motion, the effect of cohesion, and finally the dynamics in a crowded environment scattered with obstacles. Conversely, we study microscopic self-organization with specific reference to animals, for which a closer look is necessary in order to catch crystal-like structures and line formations. 5.1 Macroscopic self-organization in pedestrians As recalled in Sect. 1, pedestrians experience, in normal situations, no cohesion among each other. Hence, in most numerical tests later we set Fc = 0 and consider the following overall velocity: ∇u(x) y−x + Fr (10) vn [ρn ](x) = 2 ρn (y) dy, |∇u(x)| |y − x| Br (x)

where Fr < 0 is constant. Due to the ahead-behind asymmetry of pedestrians, their visual field covers a frontal area only. Therefore, we define the zone of repulsion Br (x) to be the half-ball of radius Rr in the direction of the external velocity. Unless otherwise stated, the values of the relevant parameters are, for all tests, αr = π, Rr = 0.1, and Fr = −1. Lane formation in crossing flows (k)

We consider two groups of pedestrians, of density ρn ∈ L1 (Ω), k = 1, 2, respectively, walking in opposite directions. This amounts to defining two (k) (k) measures dμn = ρn dL2 , each of which evolves in time according to the push-forward (3). However, the two evolutions are coupled, since we assume that, within each group, repulsion is oriented against the individuals of the opposite group. This choice is reasonable because interactions among pedestrians

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walking in opposite directions are more dangerous than interactions among pedestrians walking in the same direction, due to higher relative velocities. Specifically, the intelligent velocities are such that: y − x (2) (1) (2) Fr νn [ρn ](x) = 2 ρn (y) dy, |y − x| Br (x)

νn(2) [ρ(1) n ](x) =

Fr Br (x)

y−x

(1) 2 ρn (y) dy,

|y − x|

so that pedestrians try to steer clear of oppositely walking people and to gain instead room in their direction. The external velocities are the vectors w(1) = (1, 0) and w(2) = (−1, 0), thus groups walk rightward and leftward, respectively. Figure 6 shows that the model is able to account for lane formation when the two groups meet at the center of the domain and start to interact. In particular, alternate lanes fully emerge for both uninterrupted and time-periodic flow of people from the boundaries of the domain, and turn out to be a quite stable equilibrium configuration of the system. Such a configuration is however reached in different times, being in particular more delayed in the second case, as a consequence of different intermediate dynamics undertaken by the system. Notice also the spontaneous breaking of symmetry occurring between the two groups, which are instead specular at the beginning.

Fig. 6. Lane formation in crossing flows, with inhomogeneous uninterrupted (first row) and time-periodic (second row) injection of people from the boundaries of the domain. Pedestrians walking rightward are in blue, those walking leftward in red. Negative values of the density of the first ones are for graphical purposes only


(a)

(b)

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Fig. 7. Spontaneous arrangement of a crowd in motion. Starting from a compact cluster, the group expands and the density decreases. Leaders tend however to maintain the initial configuration

Spontaneous arrangement of a crowd in motion Next we investigate the self-organization spontaneously emerging within a group of pedestrians in motion in a clear environment. The total velocity vn is now as in (10), with an external velocity simply given by w = (1, 0) (the group is walking rightward). The group is initially compact and has a homogeneous density (Fig. 7a). As soon as people start to interact, the model predicts an expansion of the crowd in consequence of the repulsion. At the same time, the density decreases and becomes inhomogeneous due to the anisotropy of the visual field (Fig. 7b). In particular, given the orientation of the external velocity, top–bottom symmetry is preserved, because Br is symmetric with respect to w. However, front–rear symmetry is lost, and most people remain initially concentrated in the rear part of the group, where the influence of the mass ahead is stronger. By consequence, in this zone the velocity is lower, hence at successive times the group elongates in the horizontal direction until the distribution of people becomes again substantially homogeneous (Fig. 7c). Only the motion of the leaders seems to be basically unperturbed (Fig. 7b,c), coherently with the fact that they simply follow the external velocity because nobody is in front of them. Effect of cohesion In this test we investigate the effect of cohesion in the macroscopic framework of pedestrians. The intelligent velocity νn is now as in (6), with Fc significantly greater than Fr (|Fc /Fr | = O(102 )) and a sensing domain for cohesion spanning the whole space around the agents (αc = 2π). The test starts with three clusters of pedestrians at the same homogeneous density, located a certain distance away from one another along the left side of the domain (Fig. 8a), which walk rightward (w = (1, 0)). If Rcmax is large enough, pedestrians are allowed to adjust the amplitude of their zone of cohesion Bc so as to interact with a predefined amount of people, which in

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(b)

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Fig. 8. Effect of cohesion in walking pedestrians. (a) Initial condition. (b) Topological cohesion makes the three clusters merge. (c) Metric cohesion with large radius Rcmax gives a result qualitatively similar to the topological one. (d) Metric cohesion with small radius Rcmax produces a merging only of sufficiently close clusters

this simulation is set at 2/3 of the total mass initially present in the domain (i.e., p = 2/3 μ0 (Ω) in (5)). Then the three clusters tend to merge in a unique group, as shown in Fig. 8b (this result can be compared with Fig. 4c in [3], which shows the outcome of a similar experiment performed by a microscopic model). If instead p = +∞, the zone of cohesion is fixed to its maximum size determined by Rcmax (metric cohesion). In particular, for a large radius Rcmax (Fig. 8c) the result is qualitatively similar to that obtained with topological cohesion, while for a small radius Rcmax (Fig. 8d) only the two clusters initially sufficiently close merge, the third one being instead unaffected by the presence of other agents in the domain (see Fig. 4b in [3]). We refer the reader to the next Sect. 5.3 for comments on the importance of topological cohesion in spite of some qualitatively similar metric outcomes. Dynamics in presence of obstacles Finally we study the motion of a crowd in a structured environment, in which some obstacles give rise to bottlenecks and direct pedestrians along preferential paths (the external velocity field is no longer homogeneous in space). In particular, we consider the case of a group of pedestrians wanting to go through a narrow passage, obtained by placing two obstacles in front of each other as in Fig. 9a. The external velocity w is obtained by solving Laplace’s equation for the potential u, along with sliding boundary conditions at the obstacle edges (Neumann conditions). A Dirichlet boundary condition is instead imposed on the right edge of the domain, in order to set the potential at its maximum and to identify pedestrians’ target. In this simulation cohesion is not active (Fc = 0), while repulsion is felt more strongly than in case of motion in clear environments (Fr = −10). Starting from an inhomogeneous density of people, confined in the left area of the domain Ω (Fig. 9a), the self-organization of the crowd predicted by the model is as follows. When approaching the bottleneck (Fig. 9b), people initially pass through at the maximum speed (Fig. 9e), however not all pedestrians can access the bottleneck at the same time and an obstruction


(a)

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(e)

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(f)

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(g)

Fig. 9. Pedestrian dynamics in presence of obstacles. Upper row: a crowd wants to reach the right edge of the domain, going through a bottleneck. An obstruction forms as pedestrians try to access the passage, until all people flow to the opposite side. Lower row: speed map. The speed of the crowd is inhomogeneous, with a sharp transition from low to high values across the bottleneck in correspondence of the opposite transition in the values of the density

forms (Fig. 9c). Speed before the bottleneck is low, some individuals in the middle of the group are even forced to stop, whereas behind the bottleneck it attains again its maximum (Fig. 9f). After a certain time, the whole group flows through the bottleneck and the obstruction is depleted (Fig. 9d,g). This simulation compares qualitatively well with that proposed in [19] by means of a microscopic model. 5.2 Microscopic self-organization in animals For animal groups, we consider the complete structure of the intelligent velocity, in which both cohesion and repulsion are active. We investigate the importance of the topological correction and the effects of the anisotropic interaction, varying p and the angles αr and αc . We choose Rr as few times the body size of the agents, while we allow for a large maximum radius Rcmax for cohesion. Unless otherwise specified, simulations start with a random distribution of the agents in a square. Two-dimensional globular cluster For the first test we set αr = αc = 2π and p = N . The choice p = N , together with a large Rcmax , implies that cohesion is basically metric and all-to-all. The parameters Fc and Fr are of the same order of magnitude. The system reaches a stable equilibrium in few iterations, forming a ball-like group with

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Emiliano Cristiani et al. 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.5

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an irregular internal structure. In Fig. 10 we show the typical outcome. If we introduce the topological correction, choosing p = 7, the system reaches again a stable equilibrium in few iterations, but this time a group forms, in which all IPs are at the same distance from each other. Every internal IP is surrounded by six group mates forming an hexagon, in a crystal-like structure (Fig. 11a) (cf. also [16, 36]). By computing the distribution of the angles between each IP and its neighbors, we investigate the orientation of the hexagons. In Fig. 11b,c we show the angle distribution for 1 run and 100 runs, whence we can deduce that the orientation of the hexagons is mainly random. By switching cohesion off (Fc = 0) and setting up a mild repulsion among the agents (Fr = −0.05) with a frontal visual field only (αr = π), we can also mimic with the microscopic model the spontaneous arrangement of a group of IPs described in Sect. 5.1 by the macroscopic model. Starting from a regular square configuration (Fig. 12a), the IPs move rightward and interact only with the IPs in front of them. In Fig. 12b we show the final stable configuration, directly comparable with that in Fig. 7c.

Modeling self-organization in pedestrians and animal groups 3

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Line formation Here we set αr = π/4, αc = π, and p = N . Cohesion is now greater than repulsion (Fc > |Fr |). The system does not reach an equilibrium (see Fig. 13a), nevertheless the resulting pattern is strongly different from that of the previous test. The new outcome is mainly due to the modification in the relative magnitude of Fc and Fr and to different angles αc , αr , which imply a different anisotropy in the sensing zones. When introducing the topological correction with p = 7, the system reaches an equilibrium after few hundreds of iterations, forming a line oriented in the direction of the motion. Neglecting small contributions of the repulsion component we obtain a rather stable line (but for some little border effect in the head due to the fact that group leaders cannot interact with p group mates ahead), see Fig. 13b. Lines are an example of pattern produced by self-organization of terrestrial animals like migrating penguins or elephants. A statistics on the distribution of the relative angles among the agents for 100 runs (Fig. 13c) shows that a line configuration is always reached.

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In [13] the authors show the effect of a continuous variation of the pair (αr , αc ) from (2π, 2π) to (π/4, π). As we have seen in the previous test, the first choice corresponds to a two-dimensional globular cluster. They find that wide angles induce a stretching of the group along the vertical direction, although in most runs the system does not reach an equilibrium. Conversely, small angles lead to a strong elongation of the group in the horizontal direction, and, in the limit case, to the formation of a line. 5.3 Comments on the effect of topological correction The previous tests clearly show that topological cohesion between IPs greatly changes the resulting pattern. This does not mean that it is impossible to obtain similar structures with a purely metric cohesion, by duly tuning the radius of cohesion (see, e.g., the macroscopic test in Sect. 5.1). However, the topological correction is essential in order to deal with a large value of the maximum radius allowed. Indeed, as we have recalled in Sects. 2 and 4, a metric upper bound Rcmax to the radius of cohesion Rc exists, which translates the fact that IPs are in no case concerned with very far mates, and which should necessarily coincide with the fixed radius of cohesion in a purely metric approach. Now, Rcmax is in general rather large, because IPs are able to see quite far, and can be attracted even by far fellows if necessary. By consequence, once the group is formed, a purely metric cohesion with a large Rcmax would imply attraction with an unreasonable number of other IPs, instead of feeling comfortable with the proper amount of IPs in the surroundings. Thus the topological correction is the only way to stay cohesive with a reasonable number of group mates while keeping a large maximum radius Rcmax . As a further confirmation of this, the test on the effect of cohesion in the macroscopic model (cf. Sect. 5.1) shows that a small value of Rcmax in the purely metric approach distorts the cohesion itself: aggregation is only partial, as little far group mates may not be seen.

6 Conclusions and research perspectives In this paper we have introduced a modeling framework for self-organizing intelligent particles, which takes into account both macroscopic and microscopic points of view, and is suitable to include topological and anisotropic interactions in an easy way and in any dimension. The results we have obtained from our numerical simulations suggest that these two features alone let self-organization emerge spontaneously, without forcing pattern formation via ad hoc agent-specific behavioral rules. In this respect, the most important parameters of the model are the ratio |Fc /Fr |, i.e., the relative strength of cohesive vs. repulsive terms, and the angles αc , αr , in other words the span of the sensing zones for cohesion and repulsion, respectively.


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The modeling technique by time-evolving measures that we have introduced is promising and deserves further investigation, because it enables one to address macroscopic and microscopic modeling by common mathematical structures and tools. Specifically, we have used the macroscopic scale to study self-organization in pedestrian flows, as in this case clearly distinguishable patterns emerge only when the density of people is sufficiently large. In addition, the macroscopic approach may be profitably used in control and optimization problems connected to the improvement of the flow and the safety of crowds. For instance, as recalled in Sect. 1 about Braess’ paradox, this might imply optimization of the locations of some obstacles in crowded environments, like train stations or shopping malls. Conversely, we have studied self-organization in animals at the microscopic scale in order to catch the fine internal structure of the group and to highlight the appearance of regular structures (e.g., crystals) formed by few agents. The microscopic approach may be used to address problems in which the granularity plays an essential role. Furthermore, it may allow to introduce in the behavior of the agents stochastic effects which are not suited to an averaged macroscopic framework.

Acknowledgements Credits for pictures of Fig. 1: (a), (b) Copyright Dirk Helbing. (c) Copyright Bjarne Winkler.5 (d) Copyright Ryan Lukeman.6 (e) Copyright Noah Strycker.7 This research was partially supported by the Network of Excellence project HYCON (2004–2009) and by the FIRB 2005 research project CASHMA. A. Tosin was supported by a postdoctoral research scholarship “Compagnia di San Paolo” awarded by the National Institute for Advanced Mathematics “F. Severi” (INdAM, Italy).

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Statistical physics and modern human warfare Alex Dixon1 , Zhenyuan Zhao2 , Juan Camilo Bohorquez3 , Russell Denney2 , and Neil Johnson2 1 2 3

Cavendish Laboratory, Cambridge University, Cambridge CB3 0HE, U.K. Physics Department, University of Miami, Coral Gables, Florida 33126, USA, [email protected] CEiBA Complex Systems Resarch Center and Industrial Engineering Department, Universidad de los Andes, Bogota, Colombia.

Summary. Modern human conflicts, such as those ongoing in Iraq, Afghanistan and Colombia, typically involve a large conventional force (e.g., a state army) fighting a relatively small insurgency having a loose internal structure. In this chapter, we adopt this qualitative picture in order to study the dynamics – and in particular the duration – of modern wars involving a loose insurgent force. We generalize a coalescence-fragmentation model from the statistical physics community in order to describe the insurgent population, and find that the resulting behavior is qualitatively different from conventional mass-action approaches. One of our main results is a counterintuitive relationship between an insurgent war’s duration and the asymmetry between the two opposing forces, a prediction which is borne out by empirical observation.

1 Introduction It is now more than 50 years since Richardson first uncovered empirical similarities in the total number of casualties for different wars [21]. Building on this, we presented some preliminary empirical analysis several years ago [15] which suggested a common power-law pattern in the frequency distribution of casualties within two individual wars, Iraq and Colombia. Specifically, our preliminary findings suggested that the probability that a violent intra-war event (i.e., a violent incident occurring within a war between the two opposing populations) produced x casualties, is well approximated by the power-law form p(x) = Cx−α with C a positive constant and α ∼ 2.5. Intriguingly, this same approximate value α ∼ 2.5 was also identified by Clauset et al. for the dataset of global terrorist events [3]. Power-law behavior is known to be widespread across many physical and social systems [2, 18], however it was a surprise to find it describing individual events within a single war. Here we explain how coalescence-fragmentation models from statistical physics can be adapted to understand this observation, and hence to develop G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 14, c Springer Science+Business Media, LLC 2010

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promising quantitative descriptions of insurgent warfare. In the rest of Sect. 1, we present a simple model and discuss its attractiveness as a first-order approximation to a modern insurgency [10, 14]. Section 2 and beyond examines multiple population variants including different rules-of-engagement, with a focus on the war’s duration. Comparisons to empirical data for duration of wars and casualties are encouraging, suggesting that this novel application of statistical physics to human conflict could have a very productive future. 1.1 Simple one-population model of insurgent dynamics We refer to our basic one-population insurgent model as the ‘EZ’ model [10, 14, 15]. Consider an insurgent force comprising many ‘agents,’ which are each casualty causing units. In the simplest case each agent is just a single fighter, but the definition also covers equipment such as explosives, or even information. We make the reasonable assumptions that (1) the insurgency does not have any external controller, (2) groups of agents may spontaneously form and/or break up over time. These groups are neither fixed in size nor in number, and we will use the terms ‘group’ and ‘cluster’ interchangeably. A given group can merge with other groups to form larger groups, or it may fragment into single agents. We define the ‘attack strength’ of a given group to be the average number of people who are killed or injured as a result of an attack involving that group. Each agent is taken to have an attack strength of 1, so that a cluster of size s has an attack strength of s. The number of clusters with a given attack strength is denoted by ns , and the total attack strength of the population (which is equal to the total number of agents in it) is taken to be a constant, N = sns . At each timestep in the model, a cluster (including those containing single agents) is selected with probability proportional to its size s. Equivalently, we could choose an agent at random from the population and then select the cluster to which it belongs. The probability of any cluster of size s being selected is therefore P (s) = sns /N . With a probability ν the group selected fragments into single agents. Otherwise, with a probability (1 − ν), a second group is selected with probability proportional to its size and the two clusters coalesce into a single group with size equal to the sum of the two constituent group sizes. The process then repeats at every subsequent timestep. For small fragmentation probability ν 1, this ongoing fragmentation and coalescence process results in a steady-state cluster size distribution [8, 10] featuring a power law with exponential cut-off: s 4(1 − ν) ns ≈ N s−5/2 . (1) (2 − ν)2 For small ν, the dominant s dependence is the power law with slope 5/2 = 2.5 in agreement with the preliminary empirical observation in Ref. [15] for the wars in Iraq and Colombia, and the empirical observation of Clauset et al. for global terrorism events [3].

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Fig. 1. Flowchart of a two-population EZ model (TPEZ)

2 Two-population conflict model We now add a second population comprising type B agents, with similar internal cluster dynamics. The second population comprises ps clusters of size s and a total size sps = P . A cluster is picked from this total population, N + P , with probability proportional to its size. In what follows, we will interchangeably refer to population A as having total size NA or N , and population B as having total size NB or P , respectively – the specific choice will be clear from the context. The cluster then fragments with a fragmentation probability dependent on its population type, νA or νB . If the cluster does not fragment then a second cluster is selected from the total population, with probability proportional to its size. If the two clusters are of the same type (A or B) they coalesce; if they are of different types then they interact. We start by employing very simple rules-of-enagagement for interactions, in order to illustrate the basic results. In an interaction, we assume that the smaller cluster is destroyed and the larger cluster is reduced in size by an amount equal to the smaller cluster’s size. If they are both of the same size (but opposite type) then both clusters are destroyed. In this way, both populations lose the same amount of agents in any given interaction. These lost agents are then removed from the model. A flowchart of this model (referred to as TPEZ) is contained in Fig. 1. The initial A and B populations at timestep t = 0 are N0 and P0 , respectively.

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2.1 Numeric simulation results A typical result from a numerical simulation is shown in Fig. 2. It shows the average cluster distribution (i.e., number of clusters of size s vs. s) for both populations at various timesteps. The A distribution has been rescaled (displaced up the y-axis) to separate it from B for clarity. The larger A population develops into a power-law distribution, and this dependence remains as agents are destroyed. Both distributions continue to move toward the origin at higher timesteps, until no type B agents remain. From this point onward, the total A population stays constant as does its distribution. As both populations have their sizes reduced by the same amount in any interaction, the final A population is equal to the difference between the two initial populations. The power law exponent for both distributions is ≈2.5, which is the same value as for the single population EZ model’s steady-state distribution. The A population also exhibits the finite size effects observed in the EZ model; the power law becomes distorted as the cluster sizes reach the limit imposed by the total population size. This behavior is typical within the model, and is independent of the initial population sizes and fragmentation probabilities (νA , νB ). However, the the amount of time taken for the smaller population to be destroyed by the larger population (i.e., the war’s duration) does depend on all of these variables. It is this duration that now becomes the focus of our discussion.


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2.2 Analytic derivation of a war’s duration The time taken for one population to be destroyed (i.e., the time to extinction, or duration of the war) can be derived as follows. The probability QAB that any cluster of population A is selected and interacts with one of population B is the sum of the probabilities for an A cluster of size s to interact with any B cluster, qAB (s). The first factor in qAB (s) is the probability for a cluster of type A and size s to be selected, the second the probability for this cluster not to fragment, and the third factor is the probability for any cluster of type B to then be selected: sns rpr NP QAB = (1 − νA ) r = qAB (s) = (1 − νA ) (2) N +P N +P (N + P )2 s s

using the fact that sns = N , rpr = P . The probability QBA of selecting a B cluster and it interacting with an A is given by a similar expression, with νA replaced with νB . After an interaction, each population A and B is reduced by an amount equal to the size of the interaction (which is the size of the smallest cluster in the interaction). Introducing an average interaction size c, and starting from timestep t = 0, the populations then become N = N0 − c, P = P0 − c. After i interactions the populations will be N = N0 − ic , P = P0 − ic. Therefore, the probability for an interaction between A and B after i previous interactions is Q(i) = QAB + QBA =

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n n a digamma function, and the fact a+1 = 1 − 1 allows us to express the duration T in this TPEZ model as: T =

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This gives the war’s duration in terms of the initial larger (N0 ) and smaller (P0 ) populations, their fragmentation probabilities (νA and νB ) and the average size of a destructive interaction, c. Note that the interaction does not happen for each timestep, so c is not simply the average cluster/group size – however the variation from 1 is found from numerical simulations to be small and approximately linear, i.e., c ≈ 1 + 0.2(P0 /N0 )(1 − νB )(1 − νA ).

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2.3 Numeric simulation of a war’s duration Figure 3 shows the results from numerical simulations of the TPEZ model, together with the predicted analytic result from from (4). The agreement is good in all cases. T increases with fragmentation probability since interactions between the populations only occur when a cluster is selected and does not fragment, and interactions are required to destroy agents. T also increases with total population N0 + P0 because more agents require more time to be destroyed. The dependence of T on the ratio of N0 to P0 is in stark contrast to the expected behavior from mass-action theory [27], which would have instead suggested that a strong opponent would destroy a weaker one more quickly than if the two sides were of comparable strength. The numerical simulations and the analytic theory show that the opposite is actually true: The larger the relative imbalance in strengths, the longer the fight lasts. A population of 100 and 900 agents takes considerably longer to decay to extinction than two equal populations of 500. This surprising result can be understood by

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looking at the average number of ‘events’ that occur in a given model war as a function of initial A population (keeping total population constant, so B population P0 = 1,000 − N0 ), see Fig. 4. The events are: FRAG – Type A cluster selected and fragments COAL – Type A cluster selected and coalesces KILS – Opposite type clusters selected, A bigger KILD – Opposite type clusters selected, B bigger DRAW – Opposite type clusters selected, same size and END, which is not an event but the final A population. 105

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The number of type A coalescences and fragmentations (interpopulation activity) increases rapidly as A become the majority population. (Note the logarithmic scale). At the same time the number of destructive interactions (KILD, KILS, and DRAW) between the populations decreases. As the probability for any cluster being selected is proportional to its size, the probability for any population being selected is proportional to its total size. For unequal populations there is a greatly increased probability for the larger population to self-interact (i.e., the same population being selected twice) as compared to equal populations where the probability for self and opposite interaction is the same. The larger population effectively gets in the way of its own search for the smaller population. As destructive events only occur with opposite interactions, this results in an increase in the time between agents being destroyed in asymmetric populations. This is not offset by the decrease in time due to fewer agents to destroy, leading to a net increase in the time required for extinction. Numerical simulation shows that the distribution of time intervals between interactions of A and B clusters shifts from exponential to power law, as the portion of N0 : P0 changes (see Fig. 5). While the result is unexpected, remarkably it reflects the empirical observation in Fig. 6 that asymmetric wars take longer to resolve than those in which the sides are of comparable strength [7]. 2.4 Comparison of model and real war durations In Fig. 6, the upper thick curve shows the theoretical T while the lower two curves show the mass-action predictions. The mass-action equations that we employ are those traditionally used for wars of attrition [11,17,19,20,23,25,26]: (1) dN (t)/dt = −aN (t)P (t), and dP (t)/dt = −aN (t)P (t) called Lanchester’s undirected mass-action model, and (2) dN (t)/dt = −bP (t), and dP (t)/dt = −bN (t), called Lanchester’s directed mass-action model, where a and b are constants. We take World War II as the dividing point between ‘old’ wars and ‘new’ wars. ‘Old’ wars are well described by the mass-action models, while ‘new’ wars are closer to our model prediction, implying an absence of grouping dynamics in ‘old wars’ [16, 22]. 2.5 Model variants We consider explicitly two types of variation: (1) a group fragments into two random-sized groups [13]; (2) the group is picked independent of its size. Figure 7 shows the dependences. With the exception of Fig. 7(d), all variants (a), (b), and (c) retain the main features of the basic model, no matter how the group fragments or is picked. However, if we replenish the destroyed population by inactive members in order to keep the population constant, the curve reverts to the classic mass-action shape, i.e., maximum at symmetric point x = 0, as the lower two curves in Fig. 6. For the basic model as well as no-replenishment models, two opposing populations actively seek to fight with


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each other – by contrast, adding an inactive population to keep a constant total population, mimics the situation where the size of the space in which they are fighting is fixed. In short, this inactive sub-population now acts like a solvent that separates or delays the clashes and hence the conflict. Peacekeeper variant One topically pertinent model variant can be generated by adding a third species (population C). Like a peacekeeping force, C can block interactions. For simplicity, we assume the NC members of C are permanently arranged

Fig. 6. Adapted from [27]. Duration T of human conflicts as a function of asymmetry x between the two opposing military populations. x = |N0 − P0 |/(N0 + P0 ). Data are up to the end of 2008; hence, final data points for the three ongoing wars will lie above the positions shown, as indicated by arrows. The lower two lines are the massaction results. The upper thick curve [i.e., (4)] is generated using νA = νB = 0.7 and N0 + P0 = 1, 000 fixed. Changing νA and νB changes the height of the theoretical peak but leaves qualitative features unchanged

into nC groups, each with sC permanent members. A group is selected as in the original model. If it is of type A or B, then it can fragment as usual with a probability νA or νB . If it does not fragment, then a second cluster is selected. If this cluster is A or B then coalescence/fighting proceeds as before. If this second cluster is C, then a third cluster is selected. If this cluster is of the same type as the first then they coalesce. If it is the opposite (A or B) type then

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the C cluster is compared to the size of the selected A and B clusters. If it is greater than or equal in size then there is no clash – otherwise, the A and B clusters fight as before. Figure 8 shows that if C comprises only a few large groups, then T decreases irrespective of the asymmetry. Having a few large C groups means that some sizable battles can be blocked; however, it also allows the buildup of sizeable groups of both A and B, which in turn makes the typical size of interactions bigger. By contrast, if C comprises many small groups, T can be much larger, showing a huge increase around the symmetric populations case (i.e., x = 0). If real-time management of the C population is possible, this duration profile T can be manipulated even further.

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Minority advantage variant We next consider a class of model variant in which we change the behavior in the case of a draw (i.e., when two clusters of opposite species but the same


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size are selected). Previously both clusters were destroyed – but now, only the cluster belonging to the larger total population is destroyed. This could mimic a particular ‘home advantage’ for a cluster from the minority population B, when faced with an equal-sized cluster of the invading army A. The first difference in the results from the original model is that the larger population does not always win any more. It has a larger probability of winning, but it is not guaranteed. The larger population loses members at a faster rate than the smaller, as shown in Fig. 9(a). The total initial population in the graph is constant at 1,000, so the 600 and 400 groups are part of the same simulation as the 700 and 300. If the smaller population has enough members for it to last until the two populations are the same size (e.g., 600 : 400 case) then both

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populations decrease at the same rate until they are both destroyed. If the smaller population is destroyed before it reaches the same size as the larger (e.g., 700 : 300), then the larger population remains at the end on average. It is possible in the 700 : 300 for the two populations to reach the same size and both be destroyed. However, this is very unlikely – in contrast to the original model where it was simply not possible. Despite the difference in the rules when compared to the basic model, the graph of T (Fig. 9(b)) has a similar shape. This gives us further confidence that the results of the basic model are robust. The major difference is the sudden change in gradient which occurs at around N0 = 350 and N0 = 650. Between 0 and 350, A is too small to have any significant chance of becoming the same size as B, so B remains in the final state. Between N0 = 350 and 650, A does have a high probability of catching up with B (or vice versa above 500), with both populations being destroyed. Above 650 then it is the same as below 350 but reversed; A cannot be caught up by B so it remains at the end. These effects determine T . The duration T depends on fragmentation probability in a similar way to before; if either ν is higher then there are less fatal interactions and so the war

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Fig. 9. (a) Minority advantage variant, showing average populations as a function of time. (b) time to extinction (T ) as a function of the initial A population (N0 ) N0 + P0 = 1,000

lasts a longer time, leading to a large peak at νA , νB ≈ 1. The dependence is not quite symmetric though since the time actually depends more strongly on the minority population’s fragmentation probability. This is because if νA is high then A is mostly single and many draws occur, reducing A until it is the same size as B. Both populations then reduce to 0 quite quickly. If νB is high though then B will be mostly single and can be killed off by A. Killing off a smaller population takes longer than two-equally sized populations, and even longer in this model because draws do not reduce the smaller population. The final state population (Fig. 10(b)) varies strongly with νA (majority rate) and slightly with νB . This is for the same reasons as T ’s variation. If νA is high then there are lots of single A agents, so lots of draws and both populations go to 0. If νA is low, then A kills off B effectively so it remains at the end. The νB dependence is similar but less important: If there are lots of single B (high νB ) then there are more draws, and A and B become of the same size and both effectively disappear. The graph of initial population and fragmentation probability vs. T (Fig. 10(c)) is similar to the previous model, with two differences. First, while the center section (N0 ≈ 300 − 700) is the same, the ‘wings’ are much higher. This is because the dominating process for removing agents is draws. If the minority population is too small to be able to reach the majority (which happens at around N = 300) then there are still lots of draws which reduce the majority population but not the minority. This leads to an increase in time to destroy the minority: It can not win but all the draws slow down the process. The other major difference is the huge peak as N0 ≈ 900 and νA ≈ 0.9. The high fragmentation rate means that A is mostly single, while the lack of B agents means they do not get a chance to coalesce. In this case in the basic model, draws would occur and kill off B. In this model however, draws just reduce the much larger A population. In order for B to be destroyed there needs to be a ‘big A meets small B’ event, which is very rare with these distributions, so the time required is very large. The final state population of the A is also lower than for lower νA values, as all the draws reduce A. The final state


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Fig. 10. With minority advantage (a) duration dependence on fragmentation probability, N0 = 750, P0 = 250; (b) final state population dependence on fragmentation probability, N0 = 750, P0 = 250; (c) duration dependence on initial population and fragmentation probability, N0 +P0 = 1, 000, νB = 0.1; and (d) final state population dependence on fragmentation probability, N0 + P0 = 1, 000, νB = 0.1

populations reflect these differences (Fig. 10(d)). When A is the minority the final B population is a straight line, it does not depend on νA . When A is the majority population though, the amount of fragmentation in A has an effect: The final population is slanted, with higher νA leading to lower final populations. A majority population having a low fragmentation probability (forming large clusters) leads to the quickest removal of B, with the most agents left. For the minority population, having a larger ν (forming single agents) is the best plan, as this has the highest chance of destroying the other population along with yours, and also takes the longest for you to be destroyed. There are three possible outcomes to the model; either the majority A population only, the minority B population only or neither population remains at the end. The probability for the majority win outcome is shown in Fig. 11(a) as a function of fragmentation probability, for a fixed initial concentration of N0 = 750, P0 = 250. The probability at the same conditions for the minority win outcome is shown in Fig. 11(b). As the probability for the minority win is very small, the probability for no population to remain is the inverse of Fig. 11(a). The position and shape of the boundary depends on initial population concentrations.

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Fig. 11. Both correspond to the minority advantage variant: (a) Probability for majority population to remain and (b) probability for minority population to remain

Peacekeepers with minority advantage This variation adds a third species (population C) to the minority advantage model as described in flowchart (Fig. 12). The evolution of the system to a steady state with one, or both, populations becoming extinct proceeds as before. The presence of the third population C does increase the time required for a population to be destroyed however. This time increase is most dramatic in the region where N and P are close enough together for both to be destroyed, as shown in Fig. 13(a). This graph is for NC = 100 or 0, N0 + P0 = 1,000. The increase in duration time T with NC is displayed in Fig. 13(b). All C agents were single, and there was no ‘test’ for C, i.e., it always blocked an interaction

Fig. 12. Flowchart for peacekeeper with minority advantage variation. ‘AB m’ means minority advantage


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if it was present. The large increase in time with C when both populations become extinct, can be explained. In these cases both N and P become very small, while NC remains constant. As the probability for a particular population being chosen is proportional to its total size, the most likely event is then two C’s being selected, leading to nothing happening for that time step. The next most likely process involves a C and any other two clusters – because C blocks any fatal interaction with which it is involved, nothing then happens. By contrast when one population remains at the end, this population still has a probability comparable to C of being selected, so the probability of agents being removed is not reduced by the same magnitude.

3 Encounter fragmentation model The EF model also involves two populations of agents, the same as TPEZ, which can group together to form clusters within their populations. On each timestep a cluster is selected from the total population (A+B) with uniform probability, so each cluster has the same chance of being selected. A cluster is then selected out of the total population with probability proportional to its size, resulting in each agent having the same probability of being selected. The two clusters are then compared; if they are of the same type (A or B) they coalesce. If they are of different population

Fig. 13. (a) Time to extinction (i.e., duration T ) and (b) final population for N0 + P0 = 1,000, and νA = νB = 0.1 with peacekeeper and minority advantage. However, there is no ‘test,’ i.e., whenever C is picked, there is no fight

type then the smaller cluster fragments and the larger is unaffected, or if they are the same size both clusters fragment.

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3.1 Numerical simulation results Numerical simulations of the model show that the steady state involves the larger population coalescing into a single massive cluster, with size equal to its total population. The smaller population in contrast is distributed into a range of cluster sizes in the steady state, with the distribution given by a power law with an exponent depending on the relative initial population sizes. We find that the steady state is independent of the initial conditions – for example all agents could start single or both populations could start in a single cluster. Figure 14 shows the cluster distributions obtained for various initial populations, along with power law fits to the data. Note the presence of finite size effects distorting the power law for cluster sizes nearing the population limit, similar to the case in the EZ model. This limit can be raised by using larger populations, albeit at the expense of computation time. Using well established methods [3, 15] the power law coefficients can be robustly determined. As power laws diverge as x → 0 then distributions of the type p(x) = Cx−α only exist above some minimum value xmin [18]. Above xmin , α can be found using a maximum-likelihood function: α=1+n

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where xi is a single measurement (i.e., in our case a single cluster of size s) and n the total number of measurements above xmin (the total number of clusters). The Kolmogorov-Smirnov goodness-of-fit test is used to compare the distribution Cx−α with the simulation data, and by minimizing the D-statistic from this test, xmin (and hence α) is found. As a check α is also estimated using least squares regression. Table 1 summarizes the coefficients determined for different initial population ratios (as power laws are scale independent, α is independent of the total population). The confidence intervals of α determined from the maximum likelihood method were calculated using bootstrap resampling [9] of the distribution with 1,000 replications. The limits stated are at the 0.95 confidence level. The uncertainties given for the leastsquares regression α are the fit standard errors. The estimated and calculated values for α agree in all cases to at least two significant figures (except 5 : 15, where all three values differ by ≈0.1, suggesting that the low upper cut off in cluster size at s ≈ 300 is adversely affecting the statistics). Least squares regression is known to be unreliable in this application [12], and the analytic value is technically only valid for N → ∞, accounting for the minor discrepancies between the results. For comparison, the values obtained for the standard single population EZ model (analytic value −5/2) are α1 = 2.66, α2 = 2.60 [10]. 3.2 Analytic solution We present an outline derivation of the steady state cluster distribution of the smaller population, with full details in the Appendix. Defining ns and ps as the number of clusters with size s from population A and B, respectively, we can construct equations describing their change between timesteps. For s ≥ 2: Table 1. Encounter fragmentation model cluster distribution coefficient, as determined from numerical simulations (α1 , using maximum likelihood (ML) and Kolmogorov-Smirnov test and α2 , using least squares regression) and analytic solution (αe , see Sect. 3.2) N :P α1 α− α+ α2 αe xmin 10:10 2.3261 2.3250 2.3267 2.3498(1) 2.3333 21 9:11 2.3919 2.3907 2.3929 2.4007(1) 2.4337 23 8:12 2.5384 2.5374 2.5398 2.5197(1) 2.5625 24 7:13 2.7463 2.7444 2.7479 2.7020(2) 2.7316 27 6:14 3.0360 3.0328 3.0396 3.0075(7) 2.9608 25 5:15 3.5311 3.5240 3.5393 3.423(3) 3.2857 30 The value in parenthesis is the error in the last digit. Also shown is the minimum x value above which the power law holds (determined by K-S test). α+ and α− are the 95% confidence limits for the ML estimate, determined by bootstrap resampling.

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s ps dns s ns ns s ≥s ps s ≥s = − − dt ns + ps s ns + s ps ns + ps s ns + s ps s ns s ns ns s ≥s ns − − ns + ps s ns + s ps ns + ps s ns + s ps s−1 ns (s − s )ns−s s (5) + ( ns + ps )( s ns + s ps ) and for s = 1:

n1 dn1 n1 s ns ns − = − dt ns + ps s ns + s ps ns + ps s ns + s ps r ≤s 2 r pr r n s ns ps r ≥s r =2 r . + s =2 + s =2 ns + ps s ns + s ps ns + ps s ns + s ps

Similar equations hold for population B. The first two terms on the RHS of (5) are due to a cluster selecting (being selected by) a larger cluster of the opposite population, which causes it to fragment. The next two terms are due to a cluster of size s selecting any other cluster from the same population and coalescing with it. The final term is due to two clusters joining together to form a new cluster of size s. In (6) the first two terms correspond to a single agent being selected to coalesce, and the last two terms with a larger cluster fragmenting into single agents. As we know that the steady state involves the larger population forming a single cluster, then taking B to be the larger population (P > N , where N and P are A and B’s total populations) we can simplifythe above equations by using the fact that in the steady state ps = 1, s ps = P , and s ns = N . We also make the approximation that ns 1, so that the probability of a B cluster being picked first is negligible, which is valid for large N . As derived in detail in the Appendix, the steady-state solution to these equations is: N P N P P P P + N + 3! ( N ni = n1 (6) e P +2 +2) i−( N N +2P +2) (N +P )3 N 2 P +2N P 2 ! which is valid for i 1 and P > N . This gives a power law cluster distribution of the form ns = Cs−α with C an irrelevant constant and α dependent only on the initial population sizes, N and P . 3.3 Variants and modifications We now describe several ways in which the encounter fragmentation (EF) model can be extended. Casualty variation, EFF The EF model is altered to include a fighting element in interactions between opposite populations. This results in agents being destroyed (removed from the


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system) when an A and B cluster are selected. Previously the smaller cluster would have fragmented, but now the smaller cluster loses half its agents before fragmentation occurs. The larger cluster loses an amount of agents equal to the number lost by the smaller cluster, but does not fragment.

1000 900 7000

Initial B population, P 600 500 400 300

800

700

200

300 400 500 600 700 Initial A population, N

200

100

0

800

900 1000

Time to extinction

6000 5000 4000 3000 2000 1000 0

0

100

Fig. 15. War duration dependence T on initial populations in EF model (with casualties). Initial total population N + P = 1,000

Numerical simulations show that the end state of the system involves the initially smaller population being completely destroyed, while the other population remains in a single cluster with size equal to the difference between the initial populations. This is reminiscent of the result of the two population EZ model discussed in Sect. 2, and indeed the time required for one population to be destroyed follows a similar dependence on the initial populations as that model. As shown in Fig. 15, for a constant total population, the duration T is much greater for an asymmetric population (e.g., 900 : 100) than for a more even population (e.g., 600 : 400). Civilian population variation, EFC A third ‘civilian’ population is introduced into the EF model. This population (type C) can only be selected in the second selection step, and is selected in this step with probability proportional to its size, the same as for the A or B population. Note that as the C population can only be selected second it cannot interact with itself (coalesce), only with the A or B population. If an interaction with the C population occurs, C takes damage equal to the size of the A or B cluster selected first, whereas this cluster is unaffected. The size of the damage, and the population which inflicted it, is recorded. The C population has no effect on the behavior of the model. The distribution of

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damage event sizes reflects, in the steady state, the distribution of the population causing the damage. The size of the C population scales this event distribution, but otherwise has no effect. Therefore provided that the C population is large enough so it cannot be wiped out, then it makes no difference if C agents are actually destroyed in interactions. Recruitment variation, EFR In this variant agents are added to one or both populations at the start of each timestep. The mechanism used to do this can be one of several; new agents can be created single, or can be attached to an existing cluster selected at random or with probability proportional to its size. New agents can also either be added at a constant rate or created whenever the population is below its initial level. Additionally, all the above can be done with agents removed instead of added. Without any mechanism for destroying agents then this system cannot achieve a steady state. However, provided the rate of adding new agents is not too fast (less than 1 per timestep) the effect is simply to change the overall population size, regardless of which actual mechanism is used to attach the new agents. The minority population remains distributed in a power law, with an exponent which depends on the ratio of populations. As one (or both) population sizes are changing, then this power law also changes with time. The same is also true if agents are removed, except in this case one population will eventually become extinct. Army variation, EFA As it is observed that the larger initial population invariably forms a single maximum sized cluster, we may take this population (conventionally the A population) as being constantly distributed in one or more large clusters, rather like a conventional army. The model operates as previously, except that now A no longer fragments or coalesces, and as such starts and ends in large groups. The variation produces the same power law distribution of the B population as before, while the A population remains in its initial distribution. The duration T is insensitive to whether the majority population exhibits internal grouping or not [27]. 3.4 Combination of variants Casualties and civilian population This is a combination of the EFF and EFC variants. While the cluster distribution of the two populations does not become stable when both populations are losing agents, the distribution of the size of damage inflicted upon the C population in an interaction does. This event distribution, the number of events which cause a given amount of damage to C, is shown in Fig. 16 for

Statistical physics and modern human warfare 100000

Population Population Population Population Population

Frequency of events

10000 1000 100 10 1

385

5000:5000 5200:4750 5500:4500 6000:4000 6500:3500 s-2.1 s-2.3 s-2.5 s-3.0 s-3.5

0.1 0.01 0.001

1

10

Event size, s

100

1000

Fig. 16. Distribution of B population damage to C in EF model with casualties, for different initial A and B populations. The graph is an average of 104 simulations

several different initial A and B populations. Again, it is the ratio between the population sizes which is important, any change in total population just alters the scale. The graph shows the distribution of events between the minority (B) population and C; the distribution of events between the majority population (A) and C is a power law which varies only from −2.1 (the value for equal populations, shown on the graph) to −2.0 as A becomes larger than B. Interestingly, the distribution of the number of agents killed in an interaction between A and B (the AB event distribution) is identical to the BC event distribution, except for its scale. 3.5 Army, reinforcement, and casualties This is a combination of the EFA, EFR, and EFF variants. The A population is now explicitly a conventional army, as such its distribution does not change. In fighting interactions then the A population does not lose agents, however B (the insurgent population) does as normal. This can be interpreted as the army reinforcing its units to full strength after any clash. The B population in contrast is replenished as in the reinforcement variation; with a probability r each timestep it receives a new agent, which is added to an existing cluster selected with probability proportional to its size. The effect of agents being both introduced and destroyed is that the system achieves a steady state. The distribution of the B population in the steady state is shown in Fig. 17. As can be seen it is a power law with an exponent which depends only on the rate of new agents being recruited. The A population distribution is fixed, and the distribution of fatal interaction events between A and B is identical to the B cluster distributions shown in the figure (except for the scale). In the steady state, the size of the A population has no effect at all on the AB event

Alex Dixon et al.

Average number of clusters, ns

386

B recruitment rate r = 1 B recruitment rate r = 0.33 B recruitment rate r = 0.2 B recruitment rate r = 0.1 s-2.1 s-2.8 s-4.2 s-5.8

1000 100 10 1 0.1 0.01

0.001 1

10

100 Cluster size, s

1000

Fig. 17. Minority (B) cluster distribution in steady state for encounter fragmentation model, with army, casualties and insurgent reinforcements. The AB interaction distribution is identical to the distributions shown. Also shown are approximate power laws. The type A population was 104 , changing this would only affect the scale. The distributions are an average of 104 simulations

distribution, and no effect on the B cluster distribution except for determining the total population size at which B will be stable, so that the ratio of N to P is fixed for a given value of r. 3.6 Comparison with conflict data The final model developed has many features in common with guerrilla warfare. Guerrilla armies are known to organize from the ground up [24], with small groups forming and then joining up with other groups. In encounters with a larger force they also tend to fragment and withdraw [24]. Recruitment can also be erratic, with new members almost always recruited into existing groups (new guerrillas are unlikely to just appear and start attacking). Larger groups are also likely to be more successful at recruiting, or simply be easier for new members to find or hear about. Recalling that agents can represent not only people but also equipment, larger groups are again more likely to be able to acquire more. This is exactly the same in the model, where a new agent is added to a group with probability proportional to its size. A conventional army on the other hand tends to have rigidly organized units. These units operate independently and neither tend to just join together upon meeting, nor do they generally fragment unless they take heavy losses [6]. The conventional army is also assumed to have sufficient resources to be able to reinforce its units immediately in the case of losses: This assumption is generally true as armies engaged in ongoing active war have reserves to draw upon. If the army is starting to lose substantial numbers that cannot be replaced to the insurgents, then the conflict is likely to either develop into conventional warfare or


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end, either of which are not covered by this model. The first cluster selection process also gives each cluster an equal chance of action, which means each group acts as a single unit, and also gives the initiative to the population with more groups; this is almost always the insurgents. The second cluster selection picks a cluster with probability proportional to its size: This means that the first cluster selected is more likely to encounter/interact with larger clusters. This makes sense, as apart from the fact that larger groups occupy more area, they are also easier to detect. In the following, the final EF model variant (EFA+EFR+EFF, with A explicitly an army, B insurgents, and casualties and recruitment occurring) is compared with casualty data for two conflicts using data from CERAC [4]. The model’s casualty event distribution from AB interactions is compared, but the distribution would be exactly the same if we used interactions involving a civilian population. In order to minimize inaccuracies and statistical fluctuations in the data and computer simulation, both are plotted as cumulative distribution functions, P (X ≥ x). This is defined as the probability that an event is greater than or equal to a given size, x. Iraq: March 14, 2003 to October 23, 2005 The data available is divided into two periods, one from the start of the war to 01/05/03, and the second from then up to 23/10/05. Shortly after the war started there were substantial numbers of insurgents (remnants of the army, Ba’athists, etc.) already in place, and subsequently more insurgents have been recruited (foreign fighters, nationalists, etc.) [5]. The model was therefore started with a large initial B (insurgent) population, a relatively large A (coalition army) population, and a constant recruitment rate for insurgents. The numbers used are: N0 = 4,500, P0 = 8,500, r = 0.45. The distribution of events (casualty sizes between A and B) during two different time periods are plotted, along side the corresponding data from Iraq, in Fig. 18. As can be seen the correlation in both cases is good for all values except for events of size 1 and 2. The model has almost reached a steady state by t = 250000, and while we cannot know if the data has reached a steady-state with only two time points, it has been suggested elsewhere [15] that α (the slope) has varied smoothly between these two time points and is tending to a constant value. This is identical to the model, further validating it and the choice of parameters. Colombia: January 01, 1988 to December 31, 2004 For the case of Colombia the guerrilla war has been ongoing for a long period of time, so a steady state model distribution is appropriate. The initial B population in this case does not matter, the recruitment rate was set at a lower rate than Iraq (r = 0.35), as was the army population (N0 = 4,000). The result of this distribution in the steady state is shown in Fig. 19. The match is again good except at very low s values. The power law coefficient (α)

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is known to fluctuate with time in this conflict [15], for this reason the effect of altering the recruitment rate in the steady state to two different values is also shown in the figure. The result of this is to vary α as observed. Iraq data: start up to 01/05/03 Iraq data: 01/05/03 to 23/10/05 Model: start up to t = 17000 Model: t = 17000 to 250000

Cumulative frequency, P(X ≥ x)

1

0.1

0.01

0.001

1

10

100 Event size, x

1000

Fig. 18. Comparison of model with Iraq data at two different time points. The first set of data and model have both been rescaled for clarity

Cumulative frequency , P(X ≥ x)

1

Colombia Data Model: N = 4000, r = 0.35 Model: N = 4000, r = 0.38 Model: N = 4000, r = 0.32

0.1

0.01

0.001

0.0001 1

10

Event size, x

100

1000

Fig. 19. Comparison of model with Colombia data

4 Outlook We hope that the present chapter has given an indication of what statistical physics might have to offer to the daunting challenge of quantifying human conflict. This field is new, in the sense that new insurgent warfare is not


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within the standard military modeling approach using mass-action partial differential equation approaches (e.g., Lanchester). There may also be other areas in which these models could potentially be useful, e.g., the battles of the immune system fighting invading pathogens. Much remains to be explored in these fields, and many exciting new results undoubtedly await discovery. We are extremely grateful to Mike Spagat and Sean Gourley for ongoing collaboration and discussions on modeling conflicts, and to Jorge Restrepo (CERAC) for making CERAC data available to us.

Appendix: derivation of analytic solution for encounter fragmentation (EF) model There are two populations A and B. Population A comprises an average of ns clusters of size s and has a total population N , while population B has ps clusters of size s and total population P . Each timestep, a cluster is selected at random from the total population N + P , so the probability of a cluster of size s being selected is proportional to ns (ps for B). A second cluster is then selected with probability proportional to its size, sns (sps ). The two cluster types (A or B) are compared, if they are the same the two clusters coalesce, and if they are different then the smaller of the two clusters selected fragments (both fragment if they are the same size). The time evolution of the number of clusters of size s, ns , is given for s ≥ 2 by: s ps ns s ≥s (A-1) ns [t+1] −ns [t] = − ns + ps s ns + s ps sns s ns ns s ≥s ps − − ns + ps s ns + s ps ns + ps s ns + s ps s−1 ns (s − s )ns−s sns ns s − + ns + ps s ns + s ps ( ns + ps )( s ns + s ps ) and for s = 1 by:

s ns n1 n1 [t + 1] − n1 [t] = − ns + ps s ns + s ps ns n1 ns r ≥s r pr s =2 s − + ns + ps s ns + s ps ns + ps s ns + s ps r ≤s 2 ps r n r =s r + s =2 (A-2) ns + ps s ns + s ps

Similar equations hold for population B. The first two terms on the RHS are due to a cluster selecting (being selected by) a larger cluster of the opposite population, which causes it to fragment. The next two terms are due to a cluster of size s selecting any other cluster from the same population and

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coalescing with it. The final term is due to to two clusters joining together to form a new cluster of size s. The s = 1 equation’s first two terms correspond to a single agent being selected to coalesce, and the last two terms with a larger cluster fragmenting into single agents. The steady state of this system is found from computer simulation to have the larger of the two initial populations forming a single cluster of maximum size. Taking B to be the larger population, (P > N ) then we can simplify the above equations by using the fact that ps = 1 s ps = P s ns = N giving s−1 −ns P − sns − ns N − sns ns + s (s − s )ns ns−s s>1 ns [t+1]−ns [t] = ( ns + 1)(N + P ) (A-3) −n1 N − n1 ns + P s =2 s ns + s =2 s2 ns s=1 n1 [t + 1] − n1 [t] = ( ns + 1)(N + P ) (A-4) We now make the approximation that ns 1, so that the probability of a B cluster being picked first is negligible. This is applicable for large N . With this approximation, in the steady state these equations become ns ns−s −ns (P + N + s ns ) + s−1 −n1 N − n1 ns + P (N − n1 ) s 0= 0= ( ns )(N + P ) ( ns )(N + P ) 1 s ns ns−s (P + N + s ns ) s−1

ns =

s

n1 =

(N +

NP ns +P )

(A-5)

Introducing the generating function h(ω), and its derivative h(ω) =

∞

nr e−ωr

r=2

∞ dh(ω) rnr eP−ωr =− dω r=2

(A-6)

We note that dh(ω) = e−4ω (2n2 n2 ) + e−5ω (2n2 n3 + 3n3 n2 ) dω + eP6ω (2n2 n4 + 3n3 n3 + 4n4 n2 ) + · · · i−1 i−2 −iω −iω e s ns ni−s = e s ns ni−s − n1 ni−1 − (i − 1)ni−1 n1 = −h(ω)

i=4

s =2

i=4

s =1


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Using (A-5) to replace the inner sum, dh(ω) −iω = ni P + N + i ns − in1 ni−1 e dω i=4 −iω ni P + N + i ns − in1 ni−1 = e −e−2ω n2 P + N + 2 ns − 2n1 n1 −e−3ω n3 P + N + 3 ns − 3n1 n2 −h(ω)

Comparing to the s = 2 and s = 3 terms of (A-5), −h(ω)

dh(ω)

= (P + N )

=

−(i−1)ω e−iω ni P + N + i e ini−1 +n1 n1 e−2ω ns − n−ω e

i=2 −iω

ni e

i=2

= (P + N )h(ω) −

+

n

s

i=2

−ω

ini e

−ω

− n1 e

i=2

e−iω (i + 1)ni + n1 n1 e−2ω

i=1

−iω dh(ω) − n1 e−ω ns e (i + 1)ni − 2n1 n1 e−2ω +n1 n1 e−2ω dω i=2

h(ω) dh(ω) = (P + N − n1 e−ω )h(ω) − − (n1 e−ω )2 ns − n1 e−ω dω dω (A-7) Setting ω = 0, we obtain −h(ω)

h(0) =

∞

nr =

ns − n1

r=2

∞ dh(0) =− rnr = −N + n1 dω r=2

ns − n1 = (P + N + n1 ) ns − n1 (−N + n1 ) − n21 (N + P ) ns 0=P ns − N n1 n1 = ns − n1 + (N − n1 ) N + P + ns

(N − n1 )

Comparing with (A-5) gives

ns =

NP N +P

n1 =

N P

N +P P +N +2

Comparing powers of e−ω in (A-7) e−2ω :

ns − n21 0 = n2 N + P + 2 n2 =

n21 (N + P + 2 ns )

(A-8)

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ns − n2 (n1 + 2n1 ) 0 = n3 N + P + 3

e−3ω :

n3 =

e−4ω :

3n1 n2 (N + P + 3 ns )

ns − n3 (n1 + 3n1 ) 2n2 n2 = n4 N + P + 4 n4 =

e−5ω :

2n2 n2 4n1 n3 + (N + P + 4 ns ) (N + P + 4 ns )

ns − n4 (n1 + 4n1 ) 5n2 n3 = n4 N + P + 5 n5 =

5n2 n5 5n1 n4 + (N + P + 5 ns ) (N + P + 5 ns )

Making the assumption that the number of groups of size 2 is likely to be much smaller than the number of size 1, we can parameterize the n2 and higher terms by a small value, β. From computer simulations the group size distribution is expected to be a power law, and so as 1x (i − 1)x > j x (i − j)x for j < i − 1 and any x, the higher order terms are always smaller than the n1 . With this it can be seen that ni = ni−1 n1

i+β (N + P + i ns )

(A-9)

Iterating this back to n1 ni = ni−2 n1

i−1+β i+β n1 (N + P + (i + 1) ns ) (N + P + i ns )

j+β (N + P + j ns ) N +P + 1 ! ns (i + β)! 1 n1 = ni1 +P (1 + β)! ( ns )i−1 N +i !

j=i n1 = n1 Πj=2 ni

ns

Taking logarithms and applying Stirling’s formula ⎞ ⎛ i−1 N +P + 1 ! n n s 1 ⎠ + ln(i + β)! − ln N+ P + i ! ln ni = ln ⎝n1 (1 + β)! ns ns ⎛ = ln ⎝n1

n 1 ns

i−1

⎞ +1 ! ⎠ + 1 ln(2π) + i + β + 1 ln(i + β) (1 + β)! 2 2

N +P ns


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N +P N +P 1 1 ln − (i + β) − ln(π) − +i+ +i 2 2 ns ns N +P + i + · · · O i−1 + ns ⎞ ⎛ i−1 N +P + 1 ! n n1 s ⎠ + i + β + 1 ln(i + β) = ln ⎝n1 (1 + β)! 2 ns N +P N +P 1 N +P ln − +i+ +i + − β + · · · O i−1 2 ns ns ns P As i becomes large we can neglect the terms of order i−1 . Also as ns = NN+P N +P N P (A-8) then ns = P + N + 2 and therefore for N and P of the same order of magnitude, this is a term of order 1. β is defined as a small quantity, so in the i large limit we can take the two rightmost logarithms as equal to i. ⎛ ⎞ i−1 N +P + 1 ! n N + P n s 1 ln ni ≈ ln ⎝n1 exp −β ⎠ (1 + β)! ns ns 1 1 N +P − lni −i− + i+β+ 2 2 ns ⎛ ⎞ i−1 N +P + 1 ! N +P ns N +P n1 = ln ⎝n1 exp − β i(β− ns ) ⎠ (1 + β)! ns ns i−1 N +P + 1 ! N ns N +P n1 +P exp ni = n1 − β i(β− ns ) . (1 + β)! ns ns Using the value of n1 and ns from (A-8) this becomes ni = n1 1 +

NP (N + P )2

1−i N P

P +N + 3 ! ( N + P +2−β ) (β− N − P −2) P N e P N i . (1 + β)! (A-10)

The term to the power 1 − i can be seen to approach 1 as P becomes larger than N , which was one of our initial assumptions. The dominant i dependence P is then the power law with exponent (β − 2 − N P − N ). Comparing with the power law coefficients determined from numerical simulations it can be seen that β ≈ 1.7, confirming our assumption of it as small compared to i. To remove the β parameter, we can use the form of (A-10) to substitute into (A-7). Comparing powers of e−ω as before, we find for the general case 1 ni = nj (i − j)ni−j N + P + i ns j=1 ni−1 nj ni−j ni = (i − j) N + P + i ns j=1 ni−1

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Substituting in the form of (A-10), nj = AB j j c with A, B, c constants ni =

ni−1 AB j j c AB i−j (i − j)c (i − j) N + P + i ns j=1 AB i−1 (i − 1)c

(i − j)c+1 ni−1 ABj c N + P + i ns j=1 (i − 1)c c+1 i ni−1 i − ni = ABj c (i − 1) N + P + i ns j=1 i−1 j−1 ni =

As i is large we can take i/i − 1 ≈ 1 and expand the bracket using the binomial series, 2 −j ni−1 j (c + 1)c c ni = ABj (i−1) 1 − (c + 1) + ··· + N + P + i ns j=1 i−1 2! i−1

As i becomes large we neglect the terms of order i−1 and lower, ni−1 ni = ABj c (i − 1 − (c + 1)j) N + P + i ns j=1 ⎛ ⎞ ni−1 ⎝i ni = ABJ c − ABj c − c ABjj c − ABjj c ⎠ N + P + i ns j=1 j=1 j=1 j=1 ⎛ ⎞ ni−1 ⎝i nj − nj − c jnj − jnj ⎠ ni = N + P + i ns j=1 j=1 j=1 j=1 which uses the result from (A-10) that B ≈ 1. Then as jnj = N , ni−1 i ns − ns − cN − N ni = N + P + i ns This is similar to (A-9), and it can be iterated in an analogous way to yield cN N N +P i−1− − ns ! ns + 1 ! ns ni = n1 N +P + i ! 1 − 1 − cNns − Nns ! ns Applying Stirlings formula in the i large limit as perviously results in N +P + 1 ! N N ns +P cN N +P cN N e( ns + ns + ns +1) i−(− ns − ns − ns −1) ni = n 1 − cNns − Nns ! (A-11)


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where c was defined as the power law coefficient of ni , so for self-consistency cN N N +P − − −1 c=− ns ns ns N N +P N c 1+ =− − −1 ns ns ns N +P n s − 1. c=− ns N + ns With c determined, we can write (A-11) as N +P + 1 ! N +P N +P ns e( N + ns +1) i−(− N + ns −1) ni = n 1 N +P N ns N + ns !

ns from (A-8) N P +N + 3 ! ( P N +2) −( P P +2) P e N P +2 i N N +2P ni = n1 (N +P )! ! N 2 P +2N P 2

using the value of

P

P

ni = n1 Ci−( N N +2P +2)

(A-12) (A-13)

where C is a constant. This is valid for i 1 and P > N .

References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964) 2. Cederman, L.: Modeling the Size of Wars. Am. Pol. Sci. Rev. 97, 135 (2003) 3. Clauset, A., Young, M.: Scale Invariance in Global Terrorism. http://xxx.lanl. gov/abs/physics/0502014 (2005) 4. Conflict Analysis Resources Center, www.cerac.org.co 5. Cordesman, A.H.: The Developing Iraqi Insurgency. Center for Strategic and International Studies (CSIS.org) (2004) 6. Creveld, V.M.: The Art of War: War and Military Thought. Cassell, Wellington House (2000) 7. Cunningham, D., Gleditsch, K., Salehyan, I.: Dyadic Interactions and Civil War, 46th Annual Convention of the International Studies Association, (2005) 8. D’Hulst, R., Rodgers, G.J.: Transition from coherence to bistability in a model of Fnancial markets. Eur. Phys. J. B 20, 619 (2001) 9. Efron, B.: Computers and the theory of statistics: thinking the unthinkable. SIAM Rev. 21, 460–480 (1979) 10. Eguiluz, V.M., Zimmerman, M.G.: Transmission of information and herd Behavior: an application to financial markets. Phys. Rev. Lett. 85, 5659 (2000) 11. Epstein, J.: Nonlinear Dynamics, Mathematical Biology and Social Sciences. Addison-Wesley, Reading (1997)

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12. Goldstein, M.L., Morris, S.A., Yen, G.G.: Problems with fitting to the power-law distribution. Eur. Phys. J. B 41, 255 (2004) 13. Gueron, S., Levin, S.A.: The dynamics of group formation. Math. Biosci. 128, 243 (1995) 14. Johnson, N.F., Jefferies, P., Hui, P.M.: Financial Market Complexity. Oxford University Press, Oxford (2003) 15. Johnson, N.F. et al., Universal patterns underlying ongoing wars and terrorism. e-print http://arxiv.org/abs/physics/0605035 (2006). See Bohorquez, J., Gourley, S. Dixon, A., Spagat, M. and Johnson, N.F. Nature in press (2009) for a full study across many modern wars. 16. Kaldor, M.: New and Old Wars. Stanford University Press, Stanford, CA, U.S.A. (1999) 17. MacKay, N.: Lanchester combat models. Mathematics. Math. Today, 42, 170 (2006) 18. Newman, M.E.J.: Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46, 323 (2005) 19. Plowes, N., Adams, E.: An empirical test of Lanchester’s square law: mortality during battles of the fire ant Solenopsis invicta. Proc. R. Soc. B 272, 1809 (2005) 20. Radford A., Du Plessis, A.: Territorial vocal rallying in the green woodhoopoe: factors affecting contest length and outcome Animal Behav. 68, 803 (2004) 21. Richardson, L.F.: Variation of the frequency of fatal quarrels with magnitude. J. Am. Statis. Assn. 43, 523 (1948) 22. Robb, J.: Brave New War. Wiley, New York (2007) 23. Tanachaiwiwat, S., Helmy, A.: VACCINE: War of the Worms in Wired and Wireless Networks. Technical Report CS 05-859, University of Southern California, California (2005) 24. U.S. Army Insurgency Doctrine: www.usma.edu/dmi/IWmsgs/doctrinal template 1of3.pdf (2005) 25. Yano, S.: New Lanchester strategy. Lanchester, California, (1996) 26. Wu G., Yan, S.: Virus dynamics in vivo. Am. J. Infect. Dis. 1, 156, (2005); Theoretical Analysis of Drug Treatment in Haematological Disease Using Lanchester (Osipov) Linear Law Comp. Clin. Pathol. 11, 178 (2005) 27. Zhao, Z., Bohorquez, J.C., Dixon, A., Johnson, N.F.: Anomalously Slow Attrition Times for Asymmetric Populations with Internal Group Dynamics. Phys. Rev. Lett. 103, 148701 (2009)

Diffusive and nondiffusive population models Ansgar J¨ ungel Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien, Austria, [email protected]

Summary. This survey is concerned with the modeling and mathematical analysis of continuous population equations. These models describe the change of the number of species due to birth, death, spatial movements, or stage variations. Our main focus is on spatially inhomogeneous models, given by reaction-diffusion equations, but we review also age- and size-structured and time-delayed population models. Results on the existence and stability of solutions as well as their qualitative behavior are given.

1 Introduction The modeling of populations is of great importance in ecology and economy, for instance, to describe predator–prey and competition interactions, to predict the dynamics of cell divisions and infectious diseases, and to manage renewable resources (harvesting). Population models describe the change of the number of species due to birth, death, and movement from position to position (in space) or from stage to stage (age, size, etc.). In this short survey, we review some mathematical results for deterministic and continuous population models. We concentrate on the following model classes: • • • •

Spatially homogeneous population models; Spatially inhomogeneous population models; Age- and size-structured population models; and Time-delayed population models.

The evolution of spatially homogeneous populations may be modeled by ordinary differential equations, for instance, by the logistic-growth model. The interaction of competing populations can be described by a system of coupled equations, one of which is the famous Lotka-Volterra system, introduced in Sect. 2. Important questions, beside the wellposedness of the corresponding problems, include the stability of steady states and the biological consequences. In a spatially heterogeneous setting, the population number varies in space and may diffuse in the environment. This gives the class of reaction–diffusion G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 15, c Springer Science+Business Media, LLC 2010

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equations and their systems. Turing found that the stationary solution to a diffusion system may become unstable even if the steady state of the corresponding system without diffusion is stable. Thus, the stability analysis is much more involved than in the spatially homogeneous case. Roughly speaking, in the long-time limit, one may have extinction or coexistence of the species (see Sect. 3 for details). The solutions of the diffusive Lotka-Volterra competition model do not show pattern formation. Hence, this model is not able to describe segregation phenomena. In Sect. 4, we review several cross-diffusion models which allow for inhomogeneous steady states. Roughly speaking, such stationary solutions exist if cross-diffusion is large compared to diffusion. The existence analysis of cross-diffusion systems is complicated due to the strong nonlinear coupling and because the diffusion matrices may be neither symmetric nor positive definite. Recently, some analytical tools have been developed to prove the existence of global-in-time weak solutions. These tools are explained in detail as they reveal interesting connections between the symmetrization of the diffusion matrix and the existence of an entropy (Lyapunov functional). Section 4 is the key section of this survey. When the individuals of a population are not identical but can be distinguished by their age, size, etc., we need to introduce structured population models. In Sect. 5, we introduce some age-structured and size-structured balance equations of hyperbolic type. Following [119], results on the existence and the long-time behavior of the solutions are presented. The change of a population may be delayed because of maturation or regeneration time, for instance. In Sect. 6 we consider time-delayed population models, following partially [55]. If the time delay is not discrete but distributed, we arrive to nonlocal equations in which the term with the retarded variable is replaced by a convolution in time. The field of population modeling has become so large that this survey can review only a small part of the published modeling and mathematical topics. Many model classes and important issues will not be discussed. For instance, we ignore difference and matrix equations, stochastic approaches, and models including mutations, maturation structures, metapopulations, and demographic or biomedical applications. For details about these topics, we refer to the monographs [15, 37, 72, 99, 111, 112, 115, 119].

2 Initial-value population models First, we consider the population dynamics of a single species without interactions in a homogeneous environment. Let u(t) be the population number at time t ≥ 0. Its rate of change is given by the difference of the birth and death rates. Assuming that these rates are proportional to the population number, we obtain the differential equation du/dt = au, first suggested by Malthus [100] in the 18th century, where a, which is typically a positive number, is

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399

the effective growth rate of the population. Its solution models unlimited exponential growth. The capacity of the environment (limited food supply or other environmental resources) can be taken into account by introducing a self-limiting term: du = u(a − bu), dt

t > 0,

u(0) = u0 ,

(1)

where b > 0 is a measure of the environment capacity. This equation, proposed by Verhulst [141] in the 19th century, is called the logistic-growth model. Its solution u(t) converges to the so-called carrying capacity limit a/b as t → ∞ (if u(0) > 0), which is a stable steady state. Next, let us consider the population of two species u(t) and v(t). When they are not interacting, its evolution is described in the logistic-growth approach by the equations du = u(a1 − b1 u), dt

dv = v(a2 − c2 v), dt

t > 0.

The coefficients b1 ≥ 0 and c2 ≥ 0 are called the intraspecific competition constants. In this situation, the populations are evolving independently from each other. When the species are interacting, we add competition terms proportional to the population numbers: du = u(a1 − b1 u − c1 v), dt

dv = v(a2 − b2 u − c2 v), dt

t > 0,

(2)

where the new coefficients c1 and b2 model inter-specific competition or benefit, depending on their sign. We distinguish three types of interactions: • Predator–prey model: c1 > 0, b2 < 0. This choice decreases the (effective) growth rate of the species u and increases the growth rate for v. As the growth rate for v becomes larger when the population number u is large, v represents a predator species and u the prey. Clearly, taking c1 < 0 and b2 > 0 changes the roles of prey and predator. • Competition model: c1 > 0, b2 > 0. The interaction terms are nonpositive, thus decreasing the growth rates of the species. This means that both species are competing for the (food or environmental) resources. • Mutualistic (or symbiotic) model: c1 < 0, b2 < 0. When the two species benefit from the interactions, the growth rates are enhanced. In this situation, the interaction terms are taken non-negative and the constants c1 and b2 are negative. A particular predator–prey model is obtained when the intraspecific kcompetition vanishes, b1 = c2 = 0: du = u(a1 − c1 v), dt

dv = −v(α − βu), dt

t > 0,

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where we have set α := −a2 > 0 and β := −b2 > 0. The sign assumption on a2 means that the predators will become extinct in the absence of the prey, because in this situation, dv/dt = −αv. This system is usually called the Lotka-Volterra model, proposed by Volterra [142] and independently by Lotka to describe a chemical reaction [93] in the first half of the 20th century. It has the remarkable property of possessing a first integral, βu + c1 v − log(uα v a1 ) = const., showing that the system admits positive oscillating solutions if u(0) > 0, v(0) > 0 [112, Chap. 3.1]. The predator–prey model with limited growth (i.e., (2) with a2 < 0 and b2 > 0) allows for two scenarios. Depending on the choice of the parameters and the initial data, the predator population becomes extinct and the prey population approaches its carrying capacity limit, or there is coexistence of both species, i.e., (u(t), v(t)) converges to the asymptotically stable steady state s∗ =

a c − a c b a − b a 1 2 2 1 1 2 2 1 , b 1 c2 − b 2 c1 b 1 c2 − b 2 c1

(3)

as t → ∞. Notice that our sign assumption on b2 implies that b1 c2 − b2 c1 > 0. Similar results are valid for the competition model [143, Chap. 2]. In the mutualistic model, one distinguishes between the weak mutualistic case b1 c2 > b2 c1 , in which the self-limitation, expressed by b1 and c2 , dominates the mutualistic interaction, expressed by b2 and c1 , and the strong mutualistic case, in which mutualism dominates self-limitation. In the former case, there is generally stable coexistence, whereas in the latter case, there are three scenarios: either at least one of the species become extinct or the solution (u(t), v(t)) converges to the steady state (3) as t → ∞, or (u(t), v(t)) blows up in finite time [95]. The dynamics of Lotka-Volterra systems with more than two species is more involved. For instance, chaos has been observed in models with four competing species (see [140] and the references therein).

3 Reaction–diffusion population models In the previous section, we have considered populations which are spatially homogeneous. However, in a spatially heterogeneous environment, the population density will depend on space. Assuming that populations tend to move to regions with smaller number density, it is reasonable to include diffusive terms in the evolution equations, which may be justified as limiting expressions of a Brownian motion [115]. Then a single-species population density u(x, t) evolves in the bounded domain Ω ⊂ Rn according to ut − dΔu = uf (x, u),

x ∈ Ω, t > 0,

u(·, 0) = u0 ,

(4)


401

supplemented with some boundary conditions, where ut abbreviates the time derivative ∂u/∂t. Often, homogeneous Neumann boundary conditions ∇u · ν = 0 on ∂Ω are taken, where ν is the exterior unit normal on ∂Ω. These conditions signify that the number of individuals is fixed in the domain (no migration occurs). Also homogeneous Dirichlet boundary conditions u = 0 on ∂Ω can be used, expressing a very hostile environment at the boundary. The coefficient d > 0 is the diffusion constant, and f (x, u) is the growth rate per capita, depending on the population and the heterogeneous environment. A typical example is the logistic-growth function f (x, u) = a(x) − b(x)u; then (4) is called the Fisher-Kolmogorov-Petrovsky-Piskunov equation, introduced by Fisher [43] and studied by Kolmogorov et al. [77]. Reaction–diffusion models of type (4) have been also considered in physics, chemistry, ecology, etc.; see the monographs of Okubo and Levin [115] and Murray [111]. The function f (x, u) = a(x) − b(x)u is decreasing in u (if b(x) > 0). Some population ecologists argue that the growth rate f may not be decreasing in u for all u ≥ 0, but it may achieve a maximum at an intermediate density. This so-called Allee effect [3] may be caused by, for instance, shortage of mates at low density, lack of effective pollination, or predator saturation [129]. Whereas in the logistic growth case (with a(x) > 0) there exists a unique non-negative steady state (positive for slow diffusion and zero for fast diffusion; see [18]), there may be two steady states when an Allee effect is present [129]. Matano [101] and Casten and Holland [19] showed that any stable steady state to (4) (with homogeneous Neumann boundary conditions) is constant if the domain Ω is convex. The situation becomes much more complex when we consider systems of equations, ut − Δ(Du) = g(x, u),

x ∈ Ω, t > 0,

u(·, 0) = u0 ,

(5)

where u ∈ Rm is a vector-valued function, D = diag(d1 , . . . , dm ) is a diagonal matrix with constant coefficients di , and g = (gi ) : Ω × Rm → Rm (m > 1). Turing found in his seminal work [139] that different diffusion rates di in a parabolic system, modeling the interaction of two chemical substances, may lead to inhomogeneous distributions of the reactants, which allows one to model a pattern structure. Moreover, even if the steady state of the differential equation without diffusion is stable, the corresponding steady state of the diffusion system may become unstable and bifurcations may occur. This phenomenon is generally called diffusion-driven instability. Similar to the scalar case, Kishimoto and Weinberger [75] proved that, in a convex domain, the system (5) with homogeneous Neumann boundary conditions has no stable nonconstant steady state if ∂gi /∂uj > 0 for all i = j (cooperation-diffusion system); the same conclusion holds for m = 2 if ∂gi /∂uj < 0 for i = j (competition-diffusion system). On the other hand, in nonconvex domains, stable nonconstant steady states may exist, see the works [65, 66, 102] for certain dumbbell-shaped domains. For more references, we refer to [39].

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One may ask if a diffusion-driven instability also occurs in Lotka-Volterra diffusion systems. For this, following Lou and Ni [96], we discuss the competition model ut − d1 Δu = u(a1 − b1 u − c1 v),

vt − d2 Δv = v(a2 − b2 u − c2 v),

(6)

for x ∈ Ω and t > 0, with initial and homogeneous Neumann boundary conditions. The coefficients ai , bi , ci , and di (i = 1, 2) are positive. This model may be supplemented by adding a given common environmental potential φ, modeling territories in which the environmental conditions are more or less favorable. In this situation, the equation for u has to be replaced by: ut − div(d1 ∇u − e1 u∇φ) = u(a1 − b1 u − c1 v), and similar for the equation for v. It is known that the initial-boundary value problem from (6) has a unique non-negative smooth solution, see, e.g., [147] for systems with m equations and general semilinearities. The long-time behavior depends on the values of the reaction coefficients, and there are, in contrast to the Lotka-Volterra differential equations (2), three situations. Set A = a1 /a2 , B = b1 /b2 , C = c1 /c2 . Then [96] • Extinction: A > max{B, C} or A < min{B, C}. The solution (u(t), v(t)) converges to (a1 /b1 , 0) or (0, a2 /c2 ), respectively, uniformly as t → ∞. Thus, one species dominates and the other species becomes extinct. • Weak competition: B > A > C. The solution (u(t), v(t)) converges to the steady state s∗ , defined in (3), uniformly as t → ∞. This means that both species coexist. • Strong competition: B < A < C. The steady states (a1 /b1 , 0) and (0, a2 /c2 ) are locally stable, but s∗ is unstable. If the domain is convex, no stable positive steady state exists. In particular, in the weak competition case, the steady state s∗ is globally asymptotically stable regardless of the values of the diffusion coefficients d1 and d2 . In fact, there exists a Lyapunov functional which allows for a longtime asymptotic analysis [88, 124]. Therefore, no nonconstant steady state exists for any d1 and d2 , and there is no pattern structure. For the Volterra model with diffusion, for any number of interacting populations, Murray [110] has shown that the effect of uniform diffusion is to damp all spatial variations. General reaction rates and the stability of constant steady states have been considered by Conway and Smoller [26]. The situation changes when crossdiffusion terms are present in (6), modeling the population pressures created by the competitors; see Sect. 4 for details. The asymptotic behavior of solutions to reaction–diffusion systems similar to (6) has been studied in several papers. For instance, the existence of traveling wave solutions in one-dimensional diffusive Lotka-Volterra predator–prey


403

models with a logistic growth condition was shown by Dunbar [36]. More recently, traveling waves for a reaction–diffusion system with one diffusion term omitted were analyzed by Ai and Huang [1]. The following diffusive mutualistic model was considered by Lou, Nagilaki, and Ni [95]: ut − d1 Δu = u(a1 − b1 u + c1 v),

vt − d2 Δv = v(a2 + b2 u − c2 v)

in Ω×(0, ∞) with initial and homogeneous Neumann boundary conditions and ai , bi , ci > 0. Compared to (6), the signs of the interaction terms are reversed, expressing mutualistic interactions. Lou, Nagilaki, and Ni prove the interesting result that, in the strong mutualistic case (see Sect. 2), the population of the species may blow up in finite time, although one or both species with exactly the same initial data would die out if no diffusion effects are taken into account. The mathematical reason is that diffusion first averages u and v, possibly increasing the densities, and, after some time, the reaction terms dominate and may force the solution to blow up. As the diffusion initiates the blowup process at the first stage, this phenomenon is called diffusion-induced blowup. A related effect is diffusion-induced extinction. Iida et al. [64] have studied the diffusive Lotka-Volterra model in the strong competition case. In the absence of diffusion, if one species is initially superior to the other one, the superior species wipes out the other species. On the other hand, allowing for diffusion (with the same diffusion rates), Iida et al. proved that the superior species may become extinct. If the diffusion rates are different, the situation is more complicated, and we refer to [114] for details. Finally, we remark that systems with more than two equations have been considered too. For instance, the coexistence of competing species in a reaction–diffusion system with one predator and two competing prey is analyzed in [69], and the existence and stability of stationary and periodic solutions to a reaction–diffusion system consisting of m species is proved in [38].

4 Cross-diffusion population models The diffusive Lotka-Volterra competition model has no nonconstant steady state for all possible diffusion rates, thus excluding biologically reasonable pattern structures. Inhomogeneous steady states may be obtained by taking into account cross-diffusion terms instead of just adding pure diffusion to the population models. In this section, cross-diffusion models are analyzed in detail. First, let us consider cross-diffusion systems with constant diffusion rates, ut − (d1 Δu + d2 Δv) = uf (x, u, v),

vt − (d3 Δu + d4 Δv) = vg(x, u, v)

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in Ω × (0, ∞) with initial and homogeneous Neumann boundary conditions. Clearly, a necessary condition to have – at least local – existence of solutions is that the diffusion matrix d1 d2 d3 d4 is positive definite (see, e.g., [7]). This system shows indeed diffusion-driven instabilities. Farkas [40] has proved that the one-dimensional stationary model undergoes a Turing bifurcation at a certain size of the interval under suitable conditions on the coefficients of the system. This means that a larger domain may lead to a heterogeneous distribution of the steady states even if the conditions are homogeneous everywhere. We also refer to [61,79] for results in this direction. The stability and cross-diffusion-driven instability of constant stationary solutions was studied by Flavin and Rionero [44]. Also wavelike solutions are possible in such cross-diffusion systems. For instance, Kopell and Howard [78] have proved the existence of plane-wave solutions of the type (u, v)(x, t) = (U, V )(k · x − ωt), and they have analyzed the stability and instability of the waves. Summarizing, we may say that diffusion tends to suppress pattern formation, whereas cross-diffusion seems to help creating pattern under suitable conditions. Generally, we expect that the cross-diffusion rate of one species is not constant but depends on the population density of the other species and vice versa. Therefore, we replace the linear term by a nonlinear one involving the product of both populations. This leads to the following system, first suggested by Shigesada, Kawasaki, and Teramoto [130]: ut − Δ (d1 + α11 u + α12 v)u = u(a1 − b1 u − c1 v), (7) vt − Δ (d2 + α21 u + α22 v)v = v(a2 − b2 u − c2 v), (8) ∇u · ν = ∇v · ν = 0 on ∂Ω, t > 0,

u(·, t) = u0 , v(·, t) = v0

in Ω,

(9)

where Ω ⊂ Rn is a bounded domain. The diffusion coefficients di and αij as well as the reaction coefficients ai , bi , and ci are assumed to be constant (and non-negative). The expressions α12 Δ(uv) and α21 Δ(uv) are the nonlinear cross-diffusion terms, and α11 Δ(u2 ) and α22 Δ(v 2 ) describe the self-diffusion of the species. The basic idea is that the primary cause of dispersal is migration to avoid crowding instead of just random motion (modeled by diffusion). In the following, we review some mathematical properties of the above crossdiffusion system. Stability. Mathematicians started to pay attention to the model (7) and (8) from the 1980s on, first examining mainly stability issues. One of the first papers is due to Mimura and Kawasaki [108], who have shown, neglecting selfdiffusion and assuming reaction coefficients such that b1 = b2 , c1 = c2 , and c1 /b1 > a1 /a2 > b1 /c1 , that the stationary one-dimensional system has spatial patterns exhibiting segregation. Matano and Mimura [102] showed that,


405

if the diffusion coefficients d1 and d2 are sufficiently large, bounded stationary solutions must be constant. A segregation result in a triangular diffusion system (i.e., α21 = 0) is shown by Mimura [107]. An important paper on the interplay between diffusion and cross-diffusion was published by Lou and Ni [96], and in the following, we will describe their results (also see the review [113]). We consider the weak competition case B > A > C (see Sect. 3), since in this case, the system (7)–(9) without crossdiffusion has no nonconstant steady states. It holds: • Let B > A > (B + C)/2 and d2 belonging to a proper range. Then, if α21 ≥ 0 is fixed and α12 is sufficiently large, there exists a nonconstant steady state. • Let B > A > C. If one of the diffusion constants d1 or d2 is sufficiently large (compared to the cross-diffusion coefficients α12 and α21 ), the constant vector s∗ , defined in (3), is the only positive steady state. This means that there are nonconstant steady states if cross-diffusion is sufficiently large, and large diffusion coefficients tend to eliminate any pattern. In the strong competition case B < A < C, the situation is more complicate but cross-diffusion has similar effects in helping to create pattern formation; see [96] for details. The existence of positive steady states for coefficients (a1 , a2 ) lying in a certain region was shown by Ruan [125], generalizing results by Mimura [107] and by Li and Logan [89]. See also [24] for the same issue. Hopf bifurcations of coexistence steady states have been analyzed by Kuto [82]. The existence and nonexistence of coexistence steady states of the mutualistic model was analyzed by Pao [116], later generalized by Delgado et al. [31]. In recent years, several works were considered with three-species cross-diffusion systems providing sufficient conditions for the existence of nonconstant positive steady states, see [23, 51, 94, 126, 127]. Partial existence theory. First, we report the mathematical difficulties in the analysis of the time-dependent system (7)–(9). Its diffusion matrix d1 + 2α11 u + α12 v α12 u A= (10) α21 v d2 + 2α22 v + α21 u is neither symmetric nor in general positive definite such that even the local existence of solutions is far from being trival. Moreover, there exists generally no maximum principle for parabolic systems, which would allow one to derive bounds on the solutions. Finally, it is not clear how to prove the non-negativity or positivity of the population densities, which is desirable from a biological point of view. It is, therefore, not surprising that the first existence results in the literature were concerned with special cases, and partial results were obtained only: local-in-time existence, and global-in-time existence for small cross-diffusion constants or for triangular diffusion matrices (α21 = 0). One of the first results on the existence of transient solutions was achieved by Kim [73]. He proved the local existence of non-negative solutions to the

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one-dimensional cross-diffusion system without self-diffusion. If all diffusion coefficients are set equal to one, he obtained the global existence of solutions. The reason for the last result is easy to see: Taking the difference of the equations ut − Δ(u + uv) = u(a1 − b1 u − c1 v), vt − Δ(v + uv) = v(a2 − b2 u − c2 v), the difference w := u − v solves wt − Δw = u(a1 − b1 u − c1 v) − v(a2 − b2 u − c2 v). Thus, for given u and v, the function w is a solution to the linear heat equation and can be easily controlled by the right-hand side. Kim derived H 2 (Ω) estimates for u, v, and w, which enabled him to extend the local solution for any time. Another result is due to Deuring [33]. For sufficiently small crossdiffusion parameters α12 and α21 (or equivalently, sufficiently small initial data) and vanishing self-diffusion coefficients α11 = α22 = 0, he proved the global existence of solutions to (7)–(9). Several papers are concerned with the global existence of solutions in the special case α21 = 0. Then the diffusion matrix is triangular and equation (8) for the second species is only weakly coupled through the reaction terms, which considerably facilitates the analysis. We mention some works in this direction: Pozio and Tesei [121] have assumed rather restrictive conditions on the reaction terms for their global existence results. The conditions have been weakened later by Yamada [151]. Redlinger [123] has neglected self-diffusion but he has chosen general reaction terms of the form uf (u, v) and vg(u, v); Yang [152,153] generalized his results. Lou, Ni, and Wu [97] examined the case of one and two space dimensions and included self-diffusion in the equation for v. The system in any space dimension was treated by Choi, Lui, and Yamada [25] under the hypotheses that the cross-diffusion in the equation for u is sufficiently small and that there is no self-diffusion in the equation for v. This work was generalized by Van Tuoc [138], assuming that the cross-diffusion parameter of one species is smaller than the self-diffusion coefficient of the other species. Considerable progress was made by Amann [6]. He derived sufficient conditions for the solutions to general quasilinear parabolic systems to exist globally in time. The question if a given (local) solution exists globally is reduced to the problem of finding a priori estimates in the space W k,p (Ω). More precisely, if the local solution is bounded in W 1,p (Ω) uniformly in (0, T ), where T > 0 is the maximal time of existence and p > n (n being the space dimension), or if one can control the L∞ and H¨ older norms, then the solution exists globally. These results have been applied to triangular cross-diffusion systems, see the works by Amann [6] and later by Le [83]. Full diffusion matrices were considered in [91, 145, 149]. Li and Zhao [91] proved the global existence of solutions under some restrictions on the (cross)


407

diffusion coefficients, whereas Wen and Fu [145] achieved related results for systems with m species. Yagi [149] studied the two-dimensional problem without self-diffusion and showed a global existence result under the conditions 0 < α12 < 8α11 ,

0 < α21 < 8α22 ,

α12 = α21 .

This hypothesis is easily understood by observing that in this case, the diffusion matrix A in (10) is positive definite, z Az ≥ min{d1 , d2 }|z|2

for all z ∈ R2 .

If the above condition does not hold, there are choices of the parameters such that A is not positive definite. Finally, we mention the work [46] by Fu, Gao, and Cui, who have proved the global existence of classical solutions to a three-species cross-diffusion model with two competitors and one mutualist. Global existence theory. Remarkably, the positive definiteness of A is not necessary to obtain global existence of solutions to (7)–(9). The first global existence result for the one-dimensional cross-diffusion system without any restriction on the diffusion coefficients (except positivity) was achieved by Galiano et al. [48]. Their result is based on two observations which are described in the following. First, there exists a transformation of variables which symmetrizes the problem. This transformation reads as: u = ew1 /α21 ,

v = ew2 /α12 .

Then system (7) and (8) transforms into ∂ ew1 − div(B(w)∇w) = f (w), ∂t ew2 where w=

w1 , w2

f (w) =

w e 1 (a1 − b1 ew1 /α21 − c1 ew2 /α12 ) . ew2 (a2 − b2 ew1 /α21 − c2 ew2 /α12 )

The new diffusion matrix w1 (d1 + 2α11 α−1 + ew2 )ew1 ew1 +w2 21 e B(w) = w2 ew1 +w2 (d2 + 2α22 α−1 + ew1 )ew2 12 e is symmetric and positive definite, det B(w) ≥ d1 ew1 + d2 ew2 , i.e., the operator div(B(w)∇w) is elliptic for all positive di and non-negative αij (but not uniformly in w). Another advantage of the above transformation

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is that if L∞ bounds for wi are available, the functions ui are automatically positive. This idea circumvents the maximum principle which cannot be applied to the present problem. We remark that exponential changes of unknowns have been used in other models to prove the existence of non-negative or positive solutions to elliptic or parabolic systems and to higher-order equations, see [49, 60, 67, 68]. The second idea is that the cross-diffusion system admits a priori estimates via the functional v u E1 (t) = (log u − 1) + (log v − 1) dx. (11) α21 Ω α12 Due to the similarity to the physical entropy, we call this functional an entropy. It satisfies the so-called entropy inequality dE1 2d1 √ 2 α11 2d2 √ 2 α22 +2 |∇ u| + |∇u|2 + |∇ v| + |∇v|2 dt α12 α21 α21 Ω α12 √ + 2|∇ uv|2 dx ≤ C1 , (12) where C1 > 0 is a constant depending only on the reaction parameters. This provides, for positive self-diffusion parameters, H 1 (Ω) estimates for u and v. It is not a coincidence that the symmetrizable system (7) and (8) possesses an entropy functional; see later for a discussion of the relation between symmetry and entropy. Clearly, the above computation can be made rigorous only if u and v are non-negative (or even positive) functions. For this, we need to show L∞ bounds for wi , which cannot be deduced from the above entropy estimate. The idea of [48] was to employ another “entropy” functional, 1 1 (u − log u) + (v − log v) dx, E(t) = E1 (t) + γE2 (t), E2 (t) = α21 Ω α12 where γ = 4 min{d1 /α12 , d2 /α21 }. Indeed, employing Young’s inequality, we arrive to dE2 d1 d2 α11 √ 2 ≤− |∇ log u|2 + |∇ log v|2 + 8 |∇ u| dt α21 α12 Ω α12 √ 2 √ α22 √ 2 |∇ u| + |∇ v|2 dx +8 |∇ v| dx + α21 Ω + (−a1 − a2 + (b1 + b2 )u + (c1 + c2 )v)dx. Ω

The second integral can be estimated by the corresponding terms in E1 , and the last integral is controlled by the reactions terms coming from dE1 /dt. After some manipulations we obtain γd2 dE γd1 + |∇ log u|2 + |∇ log v|2 dx ≤ C2 , dt α21 Ω α12


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where C2 > 0 depends again only on the reaction constants. This estimate gives a bound for log u and log v in L2 (0, T ; H 1(Ω)). Up to now, the arguments are valid in any space dimension n. Now, we need the assumption n = 1. Indeed, in this case, the space H 1 (Ω) embeddes continously into L∞ (Ω), thus showing that u = ew1 and v = ew2 are positive. Unfortunately, there are no L∞ bounds for wi in time. Therefore, Galiano et al. [48] have discretized the cross-diffusion system in time (by the backward Euler scheme), obtaining a sequence of elliptic equations, which are solved recursively in time. As time is discrete, the semidiscrete population densities are strictly positive. The above a priori estimates are sufficient to pass to the limit τ → 0 of the time discretization parameter τ , using Aubin compactness (τ ) (τ ) results for the sequence of semidiscrete solutions exp(w1 ) and exp(w2 ). (τ ) (τ ) Because the compactness holds for exp(wi ) and not for wi , we loose the boundedness of log u and log v and thus the strict positivity of u and v in the limit τ → 0, but still obtaining the non-negativity of u and v as limits of sequences of positive functions. The assumption of one space dimension to define the exponentials ewi is crucial in the above argument. In order to deal with multidimensional problems, Chen et al. [20] have used another idea. They have discretized the crossdiffusion term Δ(uv) = div(uv log(uv)) by the finite differences D−h χh uvDh (log(uv)) , with D−h being an approximation of the divergence, Dh an approximation of the gradient, and χh the characteristic function of {x ∈ Ω : dist(x, ∂Ω) > h}. This discretization is inspired from [76], in which a cross-diffusion problem from semiconductor theory was studied. The approximate √ problem possesses an entropy inequality similar to (12) but with the term |∇ uv|2 replaced by χh uv|Dh log(uv)|2 . In order to avoid problems arising from the logarithm, u and v are replaced by u+ + η and v + + η, respectively, where u+ = max{0, u} is the positive part of u and η > 0 is a parameter. The non-negativity of the aproximate solutions is proved by taking the negative part (u− , v − ) = (min{0, u}, min{0, v}) as a test function in the weak formulation of the system, yielding an estimate of the type u− (·, t)L2 (Ω) + v − (·, t)L2 (Ω) ≤

C | log η|

uniformly in t > 0.

In the limit η → 0 this gives u− = v − = 0 in Ω × (0, ∞) and hence the nonnegativity of the population densities. Further approximations are necessary: The system is discretized in time by the backward Euler scheme, and the diffusion coefficients in B(w) are approximated by bounded functions. Then the discrete entropy estimates allow for the limit of vanishing approximation parameters. In [20], the self-diffusion coefficients need to be positive in order to deduce H 1 (Ω) bounds for u and v. This condition has been weakened later in [21]

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by allowing for vanishing self-diffusion, α11 = α22 = 0. Then there are no H 1 (Ω) bounds for u (and v) which are needed for the Aubin compactness √ argument. This problem is solved by exploiting the bounds on u. Indeed, by the Gagliardo-Nirenberg inequality, u is bounded in L4/3 (0, T ; W 1,4/3 (Ω)). Together with an L1 (0, T ; (H s (Ω))∗ ) bound for ut , where (H s (Ω))∗ denotes the dual space of H s (Ω), one can apply the Aubin lemma [135]. However, there remains a problem: The L1 (0, T ; (H s (Ω))∗ ) bound for ut does not imply weak compactness in the context of Lp spaces since L1 is not reflexive. This problem is overcome by using a weak compactness result in L1 due to Yoshida (see Lemma 6 in [21]). We mention that the approximation procedure in [21], compared with [20], has been simplified. Indeed, instead of discretizing the cross-diffusion terms, a Galerkin approximation, together with a semidiscretization in time, is performed. In order to deal with a possible degeneracy of the diffusion matrix, the elliptic regularizations εΔw1 and εΔw2 are added. Thus, there are three instead of four approximation levels needed in [20]: the dimension of the Galerkin space, the time-discrete parameter, and the regularization parameter ε > 0. Another (simpler) regularization was suggested by Barrett and Blowey [10]. They have derived entropy-type estimates by using an approximate entropy functional Eε , which is quadratic for very small and very large population densities, together with a truncation of the diffusion coefficients to ensure uniform ellipticity. This approximation gives: √ u− (·, t)L2 (Ω) + v − (·, t)L2 (Ω) ≤ C ε, and hence u, v ≥ 0 in the limit ε → 0. These procedures fail in the whole-space case Ω = Rn . Indeed, it is natural to assume that the solutions (u, v) decay to zero as |x| → ∞ which implies that log u(x, t) = ∞ and log v(x, t) = ∞ at infinity. But then the partial integrations needed to derive the entropy estimates have to be justified. This difficulty was overcome by Dreher [35] by introducing a modified entropy which compares the solution (u, v) against an exponentially decaying weight function. Dreher also used a semidiscretization in time but a higher-order elliptic regularization with the operator Δ4 . Relation between symmetry and entropy. We have shown above that the crossdiffusion system (7) and (8) can be “symmetrized,” by a change of variables, and that it possesses an entropy functional. This is not a coincidence. In fact, it is well known from the theory of hyperbolic conservation laws that the existence of a symmetric formulation is equivalent to the existence of an entropy functional [71]. This equivalence was reconsidered for parabolic systems by Degond et al. [29] and exploited for the mathematical analysis of energy-transport systems in nonequilibrium thermodynamics [28]. Consider the system ut − div(A(u)∇u) = f (x, u) in Ω, t > 0,

u(·, 0) = u0 ≥ 0 in Ω,

(13)


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supplemented with homogeneous Neumann boundary conditions. The same results hold for Dirichlet and mixed Dirichlet-Neumann boundary conditions [28]. The vector div(A(u)∇u) is defined by its components j div(Aij (u)∇uj ). The diffusion matrix A(u) ∈ Rm×m may be neither symmetric nor positive definite. Systems (13) have been studied by Alt and Luckhaus [5] but only for positive definite matrices A(u) and for solutions u which may change sign. The case of indefinite diffusion matrices can be mathematically treated if there exists a change of unknowns u = b(w) with b : Rm → Rm such that B(w) := A(b(w))b (w) is symmetric and positive definite. To ensure that (13) is of parabolic type, we assume further that the function b is monotone and a gradient, i.e., (b(w1 ) − b(w2 )) · (w1 − w2 ) ≥ 0 for all w1 , w2 ∈ Rm and there exists a function χ : Rm → R such that χ = b. Then (13) can be reformulated as (b(w))t − div(B(w)∇w) = f (x, b(w)). We claim that this system admits some a priori estimates if the reaction term f can be controlled. Define the entropy E(t) = (b(w) · w − χ(w))dx. Ω

Then, after a formal computation, dE + (∇w) B(w)(∇w)dx = f (x, b(w)) · wdx. dt Ω Ω Because B(w) is positive definite, by assumption, the integral on the left-hand side is non-negative. If the right-hand side can be controlled, this equation provides an a priori estimate for w. When −f is monotone in the sense of f (x, b(w)) · w ≤ 0, E is even a Lyapunov functional. In the population cross-diffusion system (7) and (8), we have b(w) = (ew1 , ew2 ) which is a gradient as χ(w) = ew1 + ew2 satisfies χ = b. (Here, we assumed that α12 = α21 = 1, which can be achieved by a rescaling [48].) The entropy becomes w1 E= e (w1 − 1) + ew2 (w2 − 1) dx Ω u1 (log u1 − 1) + u2 (log u2 − 1) dx, = Ω

which is of the form (11). The advantage of the special transformation u = b(w) = (ew1 , ew2 ) is that it gives automatically non-negative or even positive solutions u. These ideas have been employed in the analysis of systems from various applications, such as thermodynamics [28, 30], semiconductor theory [22], and granular materials [49].

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Regularity theory and long-time behavior of solutions. As Hölder continuity of bounded weak solutions plays an important role in showing the global existence of solutions in the framework of Amann, several authors proved the Hölder regularity of solutions under suitable assumptions. For the triangular system, Le [83] proved that if the L∞ norm of v and the Ln norm of u (n being the space dimension) can be controlled in the sense of [83], then their Hölder norms are also controlled. Furthermore, if the control is possible for every solution, there exists a global attractor with finite Hausdorff dimension. The H¨ older continuity results have been generalized by Le in [84,85] to include more general diffusion coefficients. Shim derived uniform L∞ bounds for the solutions under additional conditions on the diffusion constants in the one-dimensional setting, and he showed the convergence to the steady states as t → ∞ [131–134]. The existence of an exponential attractor (i.e., a compact, finite-dimensional, positively invariant set which attracts any bounded set at an exponential rate) was shown by Yagi [150]. Kuiper and Dung [81] proved the existence of a global attractor for cross-diffusion systems with general diffusion functions. For further results in this direction, we refer to the works [86, 87] of Le and coworkers. The long-time behavior of solutions is connected with the existence of constant or nonconstant steady states, as reviewed above. Lou and Ni [96] discussed the question if nonconstant steady states in the weak competition case still exist if both cross-diffusion constants are strong but qualitatively similar. (The answer is yes if only one of the cross-diffusion parameters is sufficiently large.) A partial answer is given by Chen et al. [21] by studying the long-time behavior of the solutions. More precisely, they showed that for vanishing intraspecific competitions b2 = c1 = 0, which is a special case of weak competition, only constant steady states exist no matter how strong the cross-diffusion coefficients are. The argument is as follows. Define the relative entropy v v u u ER (t) = dx, φ ∗ + φ u α21 v ∗ Ω α12 where φ(s) = s(log s − 1) + 1 for s ≥ 0 and (u∗ , v ∗ ) = (a1 /b1 , a2 /c2 ) are homogeneous steady states. Then a computation shows that for all t ≥ s > 0, since b2 = c1 = 0, √ 2 √ dER +C |∇ u| + |∇ u|2 dx dt Ω ≤− b1 u(u − u∗ )(log u − log u∗ ) + c2 v(v − v ∗ )(log v − log v ∗ ) dx ≤ 0, Ω

where C > 0 depends on the diffusion coefficients. Now, if (u, v) is a stationary solution to the cross-diffusion system, clearly dER /dt = 0 and √ √ ∇ u2L2 (Ω) + ∇ u2L2 (Ω) ≤ 0.


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Thus, u and v are constant in Ω. Since they satisfy the stationary equations, we conclude that u(a1 − b1 u) = v(a2 − c2 v) = 0 in Ω. Hence, either u = 0 or u = a1 /b1 and either v = 0 or v = a2 /c2 . In both cases, (u, v) is a constant stationary solution. Numerical approximation. There are only few papers concerned with the numerical discretization of the cross-diffusion system (7)–(8). The one-dimensional stationary problem was numerically solved in [47] using semi-implicit finite differences. The numerical experiments confirm that segregation of the species occurs for sufficiently large cross-diffusion parameters. As mentioned above, a semidiscretization in time was proposed in [48], and the convergence of the semidiscrete solutions to a continuous one was proved. Barrett and Blowey [10] presented a convergence proof for a fully discrete finite-element approximation. Very recently, Gambino et al. [50] have discretized the onedimensional problem by a particle approximation in space and an operatorsplitting method in time together with an Alternating Direction Implicit (ADI) scheme.

5 Structured population models The population models in the previous sections are based on the assumption that all individuals of a certain species are identical. However, populations typically consist of individuals which can be distinguished by various variables such as age, size, gender, etc. In this section we review some models which include an age or size structure, following [117, 119]. Other structured models can be found in [27, 99, 105]. Age-structured models. A model in which the vital rates depend on the age variable was first given by Sharpe and Lotka [128], known as the LotkaMcKendrick or McKendrick-von Foerster equation [45,104]. Let u(a, t) be the age density of a single-species population, where a ≥ 0 is the age and t ≥ 0 the time. Denote by b(a) and μ(a) the birth and death rate, respectively, of the species of age a. Then the change du = ut dt + ua da of the population of age a in a small increment of time dt equals −μ(a)udt [111, Sect. 1.7]. Here, ut = ∂u/∂t and ua = ∂u/∂a. The birth rate b(a) only contributes to u(0, t) since species are born at age a = 0. Dividing the equation for du by dt and noting that da/dt = 1 since a is the chronological age, u(a, t) satisfies the following hyperbolic equation with a nonlocal boundary condition: ut + ua + μ(a)u = 0, t > 0, u(a, 0) = u0 (a), ∞ u(0, t) = b(a)u(a, t)da, t > 0.

a ≥ 0,

(14) (15)

0

This equation is sometimes referred to as the renewal equation since it describes how a population is renewed [58]. When we assume that the life span

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is finite, a ∈ [0, a+ ] with a+ < ∞, this problem can be formulated as a Volterra equation of second kind (for the variable B(t) := u(0, t)), which is called the renewal or Lotka equation [63]. Using this formulation, it can be shown [63, 144] that the solution of the renewal equation has the asymptotic behavior B(t) = B0 exp(λt)(1 + o(t)), where B0 ≥ 0, λ ∈ R, and o(t) tends to zero as t → ∞. This means that the number of newborns changes exponentially with rate λ, at least for large time. Mischler et al. [109] have proved the existence and long-time behavior of solutions to (14) and (15) without using a maximal life span condition (also see [119]). Their idea is to use a generalized relative entropy method. To illustrate this idea, we first observe that the death rate term can

a be eliminated via the transformation w(a, t) = em(a) u(a, t), where m(a) = 0 μ(a)da, since w solves the equation wt + wa = 0, a ≥ 0, t > 0. Therefore, we may assume, without any loss of generality, that μ(a) = 0. It is convenient to introduce the variable v(a, t) = e−λt u(a, t), where (U, V, λ) with U > 0, V ≥ 0, and λ > 0 are the first eigenelements of ∞ ∞ Ua + λU = 0, a ≥ 0, U (0) = B(a)U (a)da, U (a)da = 1, 0 0 ∞ − Va + λV = V (0)B(a), a ≥ 0, U (a)V (a)da = 1. 0

The function U is the eigenfunction associated with the operator in (15), with the first eigenvalue λ, and V is the eigenvector of the same eigenvalue associated with the adjoint operator. The factor e−λt scales the population density in such a way that v stays bounded for all time. In other words, the population density grows exponentially with rate λ > 0, also called the Malthus parameter. The following entropy inequality holds for all convex functions H satisfying H(0) = 0: ∞ ∞ v(a, t) u (a) 0 V (a)da ≤ V (a)da, t ≥ 0. U (a)H U (a)H U (a) U (a) 0 0 This property allows one to show the long-time limit of v(·, t). The limit is expected to be proportional to the steady state U . In fact, if u0 is bounded by U , up to a factor, it follows that [119, Sect. 3.6] ∞ lim |v(a, t) − v0 U (a)|V (a)da = 0, t→∞

∞

0

where v0 = 0 u0 V da. Exponential decay can be shown under a (restrictive) lower bound on the birth rate B [119, Sect. 3.7], and the decay rate depends on this lower bound.


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To some extend, the Lotka-McKendrick model is an age-structured version of the Malthus model, introduced in Sect. 2. The drawback of both models is the unlimited exponential growth of the population. In the literature, many extensions and variants of the Lotka-McKendrick model have been proposed. Here, we mention some of them. A simple model for a cell division cycle with a single phase is the following variant of the Lotka-McKendrick system [119, Sect. 3.9]: ∞ ut + ua + μ(a)u = 0, t > 0, u(0, t) = 2 μ(a)u(a, t)da, 0

with the initial condition u(·, 0) = u0 , where μ is the mitosis (cell division) rate. In this situation, a cell is withdrawn from the differential equation at age a with the rate μ(a) and it creates two daughter cells at age a = 0 with the same rate. The (mathematical) advantage of this model is that its solutions decay exponentially fast to the steady state under the natural assumption that very young cells do not undergo mitosis (thus avoiding the restrictive assumption on B needed in the model (14) and (15)). When the birth and death rates depend on certain variables (sizes) s1 (t), . . . , sm (t), which represent different ways of weighting the age distribution, we arrive at the system [117] ut + ua + μ(a, s1 , . . . , sm )u = 0, a ≥ 0, ∞ u(0, t) = b(a, s1 , . . . , sm )u(a, t)da, 0 ∞ si (t) = ci (a)u(a, t)da, i = 1, . . . , m, t > 0, 0

together with an initial condition for u(·, 0). For i = 1 and c1 (a) = 1, we obtain the Gurtin-MacCamy model introduced in [58]. The existence and uniqueness of solutions to this model is proved in [144] using a semigroup approach. A numerical analysis was performed in [137]. A special case is given by the logistic-growth model i = 1 with μ(a, s1 ) = μ0 (a). The age-specific fertility b is assumed to be non-negative and decreasing with b(a, ∞) = 0. This means that the birth rate decreases when the weighted population average s1 becomes larger. In order to model the spatial dispersal of population species, one may include diffusive terms leading to equations of the form: ut + ua + μ(a)u = div(d∇u),

x ∈ Ω, a ≥ 0, t > 0,

where d > 0 is a diffusion coefficient and (Dirichlet or Neumann) boundary conditions for x have to be imposed on ∂Ω. The population density u depends on the spatial variable x, the age a, and time t. One of the first works in this direction is due to Gurtin [57]. Gurtin and MacCamy [59] presented age-structured models with random diffusion or directed diffusion to avoid

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crowding. Mathematical results are presented, for instance, by MacCamy [98] who studies nonlinear diffusion processes yielding porous-medium-type diffusion equations. Di Blasio [14] proved the existence and uniqueness of solutions to age-structured diffusion models, and Busenberg and Iannelli [17] analyzed a degenerated diffusion problem. A finite-difference scheme was proposed by Lopez and Trigiante [92]. Kim and Park [74] used finite differences in the characteristic age-time direction and finite elements in the spatial variable. A variable time step method was chosen by Ayati [8], and Pelovska [118] developed an accelerated explicit scheme. Size-structured models. For some organisms, age is not the most relevant parameter, but rather the cell mass or its size. This leads to size-structured population models in the size parameter x. In the following, we review some models from [119]. We distinguish between symmetric mitosis (two daughter cells of size x emerge from a mother cell of size 2x) and asymmetric mitosis (the emerging daughter cells have different size). In the symmetric case, the population number u(x, t) may evolve according to ut + ux + b(x)u(x, t) = 4b(2x)u(2x, t),

u(0, t) = 0,

u(x, 0) = u0 (x), (16)

where x ≥ 0, t > 0, and b is the birth rate. The factor 4 can be understood by computing the change of the population number, ∞ ∞ d ∞ u(x, t)dx = 4 b(2x)u(2x, t)dx − b(x)u(x, t)dx dt 0 0 0 ∞ = b(x)u(x, t)dx, 0

which increases with rate b(x). Similar to the age-structured model (14) and (15), the mathematical analysis relies on a certain eigenvalue problem with the first eigenvalue λ (the Malthus parameter) and the corresponding eigenfunctions U , V of the stationary and dual problem, respectively. Perthame and Ryzhik [120] proved that for constant birth rate b(x) = b0 , it holds λ = b0 , V = 1, and ∞ −b0 t u(·, t) − U u0 dx ≤ Ce−b0 t , t ≥ 0, e 0

L1 (0,∞)

and the constant C > 0 depends on the initial datum u0 . A similar result holds for nonconstant birth rates, see [120]. For a numerical solution, we refer to [34]. When the mitosis is asymmetric, we have to change the term on the righthand side of (16): ∞ ut + ux + b(x)u(x, t) = β(x, y)u(y, t)dy. x

This models the division of a mother cell of size y into two daughter cells of sizes x and x − y with rate β(x, y). Under suitable assumptions on b, β, and u0 , the exponential decay of (a rescaled version of) u(x, t) to the steady state


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is proved by Michel et al. [106]. Furthermore, an equation in which the effects of cell division and aggregation are incorporated by coupling the coagulationfragmentation equation with the Lotka-McKendrick model was analyzed by Banasiak and Lamb [9]. In the literature, also size-structured models with (v(u)u)x instead of ux in (16) and different expressions on the right-hand side have been employed, interpreting v(u) as a growth rate, for instance in [70] for optimal harvest modeling and in [41,42] for linear stability and instability results of stationary solutions. For more models and references, we refer to the monographs of Metz and Dieckmann [105] and Cushing [27].

6 Time-delayed population models The change in the population number of a species may not respond immediately to changes in its population or that of an interacting species, but rather after a certain time lag. Time delay in population dynamics models, for instance, the gestation or maturation time of a species or the time taken for food resources to regenerate. Hutchinson [62] postulated the equation du = u(t) a − bu(t − T ) , t > 0, u(0) = u0 , (17) dt where a, b > 0, which was analyzed by May in [103]. Whithout delay, T = 0, we recover the logistic-growth equation with the stable steady state u = a/b. In case of delay, T > 0, May discovered an interplay between the stabilizing resource limitation and the destabilizing time lag. More precisely, if aT < π/2, u = a/b is still a stable steady state, which becomes unstable if aT > π/2. When the time lag is not constant, one may employ the distributed delay equation t du = u(t) a − b(t − s)u(s)ds , t > 0, u(0) = u0 . dt −∞ The model of May was generalized and applied to the modeling of Australian sheep-blowfly populations by Gurney et al. [56]. Diffusive versions can be found in [136, 146]. Time delay may be used to model immature and mature stages; see [2] for an example. For the analysis of a system of delayed equations, we refer to [4]. More references can be found in the monograph of Kuang [80]. Spatial structures have been also considered in delayed models. A simple diffusive extension of the Hutchinson equation (17) is given by: ut − Δu = u(t) a − bu(t − T ) . More elaborate models have been proposed by Gourley and Kuang [53]. Gourley et al. [55] argue that diffusion and time delays are not independent of each other, since individuals may be at different points in space at past times. Britton [16] has suggested a delay term which involves a weighted spatial averaging over the (infinite) domain in order to account for the drift of

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the individuals from all possible positions at previous times to the present position. The equation becomes ut − Δu = u 1 + αu − (1 + α)g ∗ u , x ∈ Rn , (18) where g is a given function and g ∗ u is a convolution in the spatio-temporal variables. The term αu represents the advantageous local aggregation due to high mating probability, for instance; the convolution −(1 + α)g ∗ u with α > −1 models the intraspecific competition due to resource limitations in a neighborhood of the original position. When g is a delta distribution, we recover the Fisher equation (see Sect. 3). The particular choice g(x) = e−|y| gives the Green’s function for an ordinary differential equation, and (18) can be reduced to a system of two local equations analyzed by Billingham [13]. General kernels were considered by Deng [32], establishing the existence, uniqueness, and long-time behavior of solutions. Furthermore, Gourley and Britton [52] studied the linear stability of a related predator–prey system. Equation (18) is an example of a parabolic equation with a functional term; general nonlocal parabolic problems were treated by Redlinger [122]. Population models in bounded domains have been proposed by Gourley and So [54]. The nonlinear stability of traveling wavefronts in a related singlespecies model was proved by Li, Mei, and Wong [90]. Xu and Zhao [148] showed the existence of a global attractor of a nonlocal reaction–diffusion model with time delay. A survey on nonlocal population models, induced by time delays, and more references can be found in [55]. An equation with a time lag in the spatial variable has been proposed recently by Berestycki et al. [11] in order to study the impact of climate change on the dynamics of an affected species: ut − Δu = uf (x − cte, u),

x ∈ Rn , t > 0,

where c is a constant and e is a unit vector. The space dependence of the growth rate f is affected by the time under the action x − cte, i.e., the zones with favorable climate change shift in the direction e with speed c. Heuristically, we expect that populations manage to persist by migrating in the direction e. Indeed, it is shown in [12] that traveling wave solutions of the type u(x, t) = v(x − cte) exist if the climate shift is not too large (i.e., c > 0 is sufficiently small), otherwise there is extinction.

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Index

A Agent-based models, economic interactions agents and games basic minority game, 18–20 variants, 20–23 book modeling market interactions models, 8–12 order-driven market models, 6–8 complex system, 3 econophysicists, 3–4 statistical physics, 4 wealth distributions models Boltzmann distribution, 15 KWEM, 13–17 Pareto exponent, 12

B Bak, Paczuski and Shubik model, 6–7 Binary opinion dynamics, 208–213 Boltzmann distribution, 15 Boltzmann equation, sociophysics, 249–250 Book modeling market interactions models Cont and Bouchaud model, 8–10 Lux and Marchesi model, 10–11 Preis model, 11–12 order-driven market models Bak, Paczuski and Shubik model, 6–7

Challet and Stinchcombe model, 8 Maslov model, 7–8 Brownian motion expressions, 400 C Challet and Stinchcombe model, 8 Chatterjee-Chakrabarti-Manna (CCM) model, 36 Complexity problems, 289–290 Complex socio-economic systems, kinetic modelling encounter rate, 181–183 explorative models, 184 functional subsystem, 178, 199 mathematical modelling, 176–178 mathematical tools, 179–180 additional reasonings, 183–184 closed systems, 180–182 modelling and problems, 184–185 open systems, 182–183 media influence, 186 model and strategy competition, 188–192 modelling external actions, 186–188 predictive models, 184 simulations strategy 1, 193–194 strategy 2, 194–196 strategy 3, 196–198 transition probability density, 181–183 Cont and Bouchaud model, 8–10

428

Index

Cross-diffusion population models global existence theory, 407–410 Lotka-Volterra competition model, 402 Neumann boundary conditions, 403 numerical approximation, 413 partial existence theory, 405–407 regularity theory and long-time behavior, 412–413 stability, 404–405 symmetry and entropy, 410–411 D Diffusive and nondiffusive population models cross-diffusion population models global existence theory, 407–410 Lotka-Volterra competition model, 402 numerical approximation, 413 partial existence theory, 405–407 regularity theory and long-time behavior, 412–413 stability, 404–405 symmetry and entropy, 410–411 initial-value population models logistic-growth model, 399 Lotka-Volterra model, 400 predator-prey model, 399 reaction–diffusion population models Brownian motion expressions, 400 diffusion-driven instability, 401 diffusion-induced blowup, 403 diffusion-induced extinction, 403 Lotka-Volterra differential equations, 402 structured population models age-structured models, 413–415 size-structured models, 416–417 time-delayed population models, 417–418 Diffusive strategic dynamics, 138–139 algorithm, 159–160 equilibrium behavior, 161–162 Discrete dynamical systems linear models, 214 nonlinear models, 214–215 Discrete-time dynamic model, 224, 242–243

E Economic interactions, agent-based models agents and games basic minority game, 18–20 variants, 20–23 book modelling market interactions models, 8–12 order-driven market models, 6–8 complex system, 3 econophysicists, 3–4 statistical physics, 4 wealth distributions models Boltzmann distribution, 15 KWEM, 13–17 Pareto exponent, 12 Encounter fragmentation model analytic solution, 381–382 army, reinforcement, and casualties, 385–386 combination of variants, 384–385 conflict data comparison, 386–388 derivation of analytic solution, 389–395 numerical simulation results, 380–381 variants and modifications army variation, EFA, 384 casualty variation, EFF, 382–383 civilian population variation, EFC, 383–384 recruitment variation, EFR, 384 Environmental conditions influence, 276 Equilibrium statistical mechanics diffusive strategic dynamics, 138–139 algorithm, 159–160 equilibrium behavior, 161–162 drawbacks, 172 equilibrium behavior, 143–145 Erd¨ os–Renyi random graph, 154 Glauber dynamics, 162 many-body dynamics model of, 139–140 transition rates and Markov process, 140–143 many body interactions statics diffusive dynamics revisited, 171 diluted p-agent imitative behavior model, 163–164

Index free energy properties, 166–168 numerics, 168–171 random diluted p-spin model properties, 165 market trading, application, 171–172 2-body model cavity field interpolation, 146–150 critical behavior and phase transition, 153–156 free energy analysis, 150–153 numerics, 156–159

429

periodicity tongues, 236, 237 Schelling model, 227–234 switching propensity, 239–242 H Herding and diffusion function, 69 Hydrodynamic models flocking, single-and double-mills, 312–314 Povzner–Boltzmann equation, 314–319

F Financial markets kinetic models multiple agents interactions, 67–78 speculative market, 59–67 microscopic models Levy–Levy–Solomon model, 53–55 Lux–Marchesi model, 55–58 Fisher-Kolmogorov-Petrovsky-Piskunov equation, 401 Fokker–Planck asymptotics and wealth distribution, 64–66

I Impulsive agents case global dynamics, adaptive models, 234 piecewise linear maps, one discontinuity, 235–237 piecewise linear maps, two discontinuities, 238–239 Initial-value population models logistic-growth model, 399 Lotka-Volterra model, 400 predator-prey model, 399 Isotropy vs. anisotropy, 342

G Galam’s model, 225–228 Gas-like models analogy to two-particle collision process, 32–33 CCM model, 36 Gamma distributions, 35 Markovian process, 33 mean field approach, 36 models with savings, 34–36 rigorous treatments, 37 Global dynamics, adaptive models border-collision bifurcations, 235, 236 discrete-time dynamic model, 224, 242–243 Galam’s model, 225–228 impulsive agents case, 234 piecewise linear maps, one discontinuity, 235–237 piecewise linear maps, two discontinuities, 238–239

K Kinetic asset exchange models directed networks distributed μ, 44–45 uniform μ, 44 gas-like models analogy to two-particle collision process, 32–33 CCM model, 36 Gamma distributions, 35 Markovian process, 33 mean field approach, 36 models with savings, 34–36 rigorous treatments, 37 income distribution, 32 microeconomic formulation exchange with savings, 39–40 global market, 40–41 random exchange, 39 steady state distribution, 41–42 trading process, 38–39 utility functions, 38

430

Index

Pareto exponent, 31 preferential transactions degree distribution, 47 trade network, 47 weighted network, 47–48 Kinetic equations, modelling opinion formation compromise phenomenon compromise vs. self-thinking, 256–258 continuous models, 252–254 electoral competition, 254–255 fully collisional model, 255–256 mean field approximation, 258–259 quasi-invariant limit, 258–259 sociological phenomena contradictory individuals, 261–262 leadership, 262–263 mass media and multipartite situation, 264–266 opinion to the choice, 260–261 propaganda and politicians, 263–264 sociophysics, 245, 246 kinetic approach, tools and methods, 248–251 kinetic viewpoint, 246–247 Kinetic modeling, complex socio-economic systems encounter rate, 181–183 explorative models, 184 functional subsystem, 178 mathematical modelling, 176–178 mathematical tools, 179–180 additional reasonings, 183–184 closed systems, 180–182 modelling and problems, 184–185 open systems, 182–183 media influence, 186 model and strategy competition, 188–192 modelling external actions, 186–188 predictive models, 184 simulations strategy 1, 193–194 strategy 2, 194–196

strategy 3, 196–198 transition probability density, 181–183 Kinetic models Boltzmann-type equations, formal derivation interaction rule, 305 nonlinear friction-type equations, 307 Taylor’s series, 306–307 mean-field limit, formal derivation grazing collision limit, 307 Prohorov’s theorem, 308–309 mean-field limit, rigorous derivation, 310–312 multiple agents interactions Fokker–Plank limit and kinetic asymptotic behavior, 74–77 kinetic setting, 68–74 numerical tests, 77–78 speculative market Fokker–Planck asymptotics and wealth distribution, 64–66 kinetic equation and property, 62–64 kinetic setting, 59–62 numerical simulations, 66–67 Kinetic setting, multiple agents interactions booms, crashes and macroscopic stationary states, 72–74 distribution function, 68 herding and diffusion function, 69 price evolution, 71–72 strategy exchange chartists-fundamentalists, 70–71 value function, 70 Kinetic wealth distribution models and mathematical tools Fourier based distances, 90–91 one-dimensional Boltzmann models equations, 85 interaction coefficients, 84 saving propensity Chatterjee-Chakrabarti-Manna (CCM) model, 86, 87 extended density function, 86 Fokker-Planck equation, 86–87

Index quenched saving propensity, 85–86 Wasserstein and Fourier based distances densities convergence, 88 distances, 88, 89 Monte Carlo simulations, 87 parseval identity, 89 Kinetic wealth exchange models (KWEM) exponential distribution, 16 reshuffling model, 14 Kolmogorov’s representation theorem, 117 KWEM. See Kinetic wealth exchange models (KWEM) L Levy–Levy–Solomon (LLS) model market clearance and equilibrium price, 54–55 utility function and optimal investments, 53–54 wealth dynamic, 53 LLS model. See Levy–Levy–Solomon (LLS) model LM model. See Lux–Marchesi (LM) model Lotka-Volterra differential equations, 402 Lux–Marchesi (LM) model, 10–11 chartist and fundamentalist strategy, 56–57 optimistic and pessimistic chartists, 55–56 price formation process, 57–58 M Macroscopic models continuum hypothesis, 348 Wasserstein metric, 350 Many-body dynamics interactions statics application, market trading, 171–172 diffusive dynamics revisited, 171 diluted p-agent imitative behavior model, 163–164

431

free energy properties, 166–168 numerics, 168–171 random diluted p-spin model properties, 165 model of, 139–140 transition rates and Markov process, 140–143 Market interactions models Cont and Bouchaud model, 8–10 Lux and Marchesi model, 10–11 Preis model, 11–12 Markovian process, 33 Maslov model, 7–8 Mathematical tools, 179–180 additional reasonings, 183–184 closed systems, 180–182 modelling and problems, 184–185 open systems, 182–183 Microeconomic formulation exchange with savings, 39–40 global market, 40–41 random exchange, 39 steady state distribution, 41–42 trading process, 38–39 utility functions, 38 Microscopic models, 351–353 Levy–Levy–Solomon (LLS) model market clearance and equilibrium price, 54–55 utility function and optimal investments, 53–54 wealth dynamic, 53 Lux–Marchesi (LM) model chartist and fundamentalist strategy, 56–57 optimistic and pessimistic chartists, 55–56 price formation process, 57–58 Modelling opinion formation, kinetic equations compromise phenomenon compromise vs. self-thinking, 256–258 continuous models, 252–254 electoral competition, 254–255 fully collisional model, 255–256 mean field approximation, 258–259 quasi-invariant limit, 258–259

432

Index

sociological phenomena contradictory individuals, 261–262 leadership, 262–263 mass media and multipartite situation, 264–266 opinion to the choice, 260–261 propaganda and politicians, 263–264 sociophysics, 245, 246 kinetic approach, tools and methods, 248–251 kinetic viewpoint, 246–247 Multiple agents interactions, kinetic models Fokker–Plank limit and kinetic asymptotic behavior, 74–77 kinetic setting booms, crashes and macroscopic stationary states, 72–74 distribution function, 68 herding and diffusion function, 69 price evolution, 71–72 strategy exchange chartists-fundamentalists, 70–71 value function, 70 numerical tests, 77–78 N Numerical experiments, particle and kinetic models Cucker–Smale Model, 327–331 leadership emergence, particle regimes, 331–332 mills, double mills, crystalline structure formation, and flocking, 327 Runge-Kutta schemes, 326 O One-dimensional Boltzmann models equations, 85 interaction coefficients, 84 One-dimensional models, wealth distribution conservation, in mean models, 94–95 mathematical details Fourier metrics evolution, 96–97

moments evolution, 97–101 steady state of existence and tails, 101–103 numerical results, 103–104 Pareto tail characteristic function, 91 recursion relation, 92 pointwise conservative models, 93–94 One-population insurgent model, 366–367 Order-driven market models Bak, Paczuski and Shubik model, 6–7 Challet and Stinchcombe model, 8 Maslov model, 7–8 P Panic conditions influence, 276–277 Pareto tail, wealth distribution characteristic function, 91 recursion relation, 92 Particle models Cucker–Smale model dynamical system, 302–303 unconditional flocking, 303 regions of influence ideal rule, 301 mills, 300 three-zone model, 299 self-propelling, friction, and attraction–repulsion model catastrophic, 301 classifications, 302 Rayleigh’s law, 301 Pointwise conservative models, 93–94 P´ olya distribution, Tolstoy’s dream continuous limit, 131–133 exchangeable processes, 127 finite stochastic processes, 126–127 marginal distributions, 128–129 moments, 129–130 P´ olya process, 127–128 thermodynamic limit, 130–131 Povzner–Boltzmann equation, 314–319 Preis model, 11–12 Prohorov’s theorem, 308–309 p-spin model, 165

Index R Rayleigh’s law, 301 Reaction–diffusion population models Brownian motion expressions, 400 diffusion-driven instability, 401 diffusion-induced blowup and extinction, 403 Lotka-Volterra differential equations, 402 Robustness vs. fragility, 341 Runge-Kutta schemes, 326 S Saving propensity Chatterjee-Chakrabarti-Manna (CCM) model, 86, 87 extended density function, 86 Fokker-Planck equation, 86–87 quenched concept, 85–86 Schelling model, 205–208, 227–234 Self-organization model animals, 338 classical vs. intelligent particles blindness & inertia vs. vision & decision, 341 energy & entropy vs. self-organization, 342–343 isotropy vs. anisotropy, 342 local vs. nonlocal interactions, 341–342 robustness vs. fragility, 341 macroscopic self-organization, pedestrians cohesion effect, 355–356 dynamics, 356–357 lane formation, 353–354 spontaneous arrangement, 355 mathematical modeling, time-evolving measures conservation law, 345–346 macroscopic models, 347–351 microscopic models, 351–353 rendez-vous problems, 345 zone of cohesion and repulsion, 346 mechanics, intelligent systems, 343–345 microscopic self-organization, animals

433

line formation, 359–360 two-dimensional globular cluster, 357–358 non-locality and anisotropy, 340 pedestrians, 337, 339 starlings and surf scoters, globular clusters, 339 topological cohesion effect, 360 Sociodynamics simulation binary opinion dynamics, 208–213 developing models, 218 discrete dynamical systems linear models, 214 nonlinear models, 214–215 kinetic approach, 215–217 opinion dynamics, 203 agent-based modelling (ABM), 204 Schelling model, 205–208 Sznajd model, 208–213 Sociophysics, 245, 246 kinetic approach, tools and methods, 248 Boltzmann equation, 249–250 numerical methods, 251 photons, linear transport equation, 250 kinetic viewpoint, 246–247 Statistical physics and modern human warfare encounter fragmentation model analytic solution, 381–382 army, reinforcement, and casualties, 385–386 combination of variants, 384–385 conflict data comparison, 386–388 derivation of analytic solution, 389–395 numerical simulation results, 380–381 variants and modifications, 381–384 one-population insurgent model, 366–367 two-population conflict model analytic derivation of war’s duration, 369–370 minority advantage variant, 374–378

434

Index

numeric simulation results, 368 numeric simulation, war’s duration, 370–372 peacekeepers with minority advantage, 378–379 peacekeeper variant, 372–374 and real war durations, comparison, 372 Structured population models age-structured models, 413–415 size-structured models, 416–417 Sznajd model, 208–213 T Table of games, 283, 293 Taxation-redistribution game basic descriptions, 119–120 block taxation and convergences, 124–126 statistical equilibrium, 121–122 taxation and redistribution, 120–121 wealth distribution, 122–124 Taylor’s series, 306–307 Theme variations leadership, geometrical constraints, and cone of influence, 319–321 nonlinear dependent noise Boltzmann equation, 324 global Maxwellian function, 326 grazing limit, 325 white noise, 321–323 Three-zone model, 299 Tolstoy’s dream P´ olya distribution continuous limit, 131–133 exchangeable processes, 127 finite stochastic processes, 126–127 marginal distributions, 128–129 moments, 129–130 P´ olya process, 127–128 thermodynamic limit, 130–131 statistical equilibrium in economics equilibrium, common notion, 116 importance of, 119 Kolmogorov’s representation theorem, 117 Markov chain, 117

stationary distribution, 118 stochastic matrices, 117 taxation-redistribution game basic descriptions, 119–120 block taxation and convergences, 124–126 statistical equilibrium, 121–122 taxation and redistribution, 120–121 wealth distribution, 122–124 Topological cohesion effect, 360 2-body model, equilibrium statistical mechanics cavity field interpolation, 146–150 critical behavior and phase transition, 153–156 free energy analysis, 150–153 numerics, 156–159 Two-dimensional models, wealth distribution Dirac delta, 107–108 numerical simulations, 108–109 Pareto tail, 104–105 rates of relaxation, 106–107 Two-population conflict model analytic derivation of war’s duration, 369–370 minority advantage variant, 374–378 numeric simulation results, 368 numeric simulation, war’s duration, 370–372 peacekeepers with minority advantage, 378–379 peacekeeper variant, 373–374 and real war durations, comparison, 372 V Vehicular traffic and crowds complexity problems, 289–290 critical analysis, 281–282 features of complex systems behaviours, 275–276 granular dynamics and heterogeneous distribution, 276

Index panic conditions influence, 276–277 parameters, 276 mathematical representation active kinetic theory and micro-systems, 278 speed pressure and variance, 279 mathematical structures, 279–281 modelling of, 282–289 modelling perspectives, 291–293 table of games, 293 W Wasserstein and Fourier based distances densities convergence, 88 distances, 88, 89 Monte Carlo simulations, 87 parseval identity, 89 Wealth distribution kinetic wealth distribution models and mathematical tools Fourier based distances, 90–91 one-dimensional Boltzmann models, 84–85

435

saving propensity, 85–87 Wasserstein and Fourier based distances, 87–89 one-dimensional models conservation, in mean models, 94–95 mathematical details, 96–103 numerical results, 103–104 Pareto tail, 91–92 pointwise conservative models, 93–94 two-dimensional models Dirac delta, 107–108 numerical simulations, 108–109 Pareto tail, 104–105 rates of relaxation, 106–107 Wealth distributions models Boltzmann distribution, 15 kinetic wealth exchange models (KWEM) exponential distribution, 16 reshuffling model, 14 Pareto exponent, 12

Mathematical modeling of collective behavior in socio-economic and life sciences

Mathematical Modeling for the Life Sciences

Mathematical Modeling for the Life Sciences

Mathematical Modeling for the Life Sciences (Universitext)

Theory of Collective Behavior

Collective Animal Behavior

Perspectives in Mathematical Sciences (Interdisciplinary Mathematical Sciences)

Computational and Mathematical Modeling in the Social Sciences

Computational and Mathematical Modeling in the Social Sciences

Computational and Mathematical Modeling in the Social Sciences

Perspectives in mathematical sciences

Principles of mathematical modeling

Energy And Life (Modules in Life Sciences)

Principles of Mathematical Modeling

Principles of mathematical modeling

Principles of Mathematical Modeling