DECISION MAKING
AND PROGRAMMING
V V Kolbin
DECIS ON MAKNG
AND PROGRAMMING
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Author:
V. V. Kolbin

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DECISION MAKING

AND PROGRAMMING

V V Kolbin

DECIS ON MAKNG

AND PROGRAMMING

This page is intentionally left blank

DECIS ON MAKNG

AND PROGRAMMING

V V Kolbin St Petersburg University, Russia

Translated from Russian by KM. Donets

V f e World Scientific « ■

New Jersey • London • Sir Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

DECISION MAKING AND PROGRAMMING Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-379-4

Printed in Singapore by World Scientific Printers (S) Pte Ltd

In memory of V.I. Zubov

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CONTENTS Introduction

1

Chapter 1 SOCIAL CHOICE PROBLEMS §1.1. Individual Preference Aggregation § 1.2. Collective Preference Aggregation § 1.3. Manipulation § 1.4. Examples and Algorithms for Preference Aggregation

11 17 32 51 67

Chapter 2 VECTOR OPTIMIZATION § 2.1. Definition of Unimprovable Points § 2.2. Optimization of the Hierarchical Sequence of Quality Criteria . § 2.3. Tradeoffs § 2.4. The Linear Convolution of Criteria in Multicriteria Optimization Problems § 2.5. Solvability of the Vector Problem by the Linear Criteria Convolution Algorithm § 2.6. The Logical Criterion Vector Convolution in the Pareto Set Approximation Problem § 2.7. Computational Research on Linear Criteria Convolution in Multicriteria Discrete Programming

85 85 97 110 130 139 146 160

Chapter 3 INFINITE-VALUED PROGRAMMING PROBLEMS § 3.1. Basic Notions and Propositions § 3.2. Justification of Numerical Methods for Solving Infinite-Valued Programming Problems § 3.3. Numerical Methods of Solution § 3.4. Separable Infinite-Valued Programming Problems

169 169 181 186 200

Chapter 4 STOCHASTIC PROGRAMMING § 4.1. Stochastic Programming Models § 4.2. Stochastic Programming Methods § 4.3. Solution Algorithms for Stochastic Programming Problem § 4.4. Existence of a Deterministic Analog

vii

. .

211 211 219 245 251

viii

Contents

§ 4.5. Results § 4.6. An Example of Applied Problem

264 285

Chapter 5 DISCRETE PROGRAMMING § 5.1. A Geometric Interpretation of Integer Linear Programming Methods § 5.2. Equivalent Forms and Group-Theoretic Interpretation of Discrete Programming Problems § 5.3. An Algorithm for Solving the Integer Linear Programming Problem § 5.4. The Optimality Condition and the Search Method for Discrete Optimization Problems § 5.5. An Algorithm for Solving Mixed Integer Linear Programming Problems § 5.6. Solving the Large Linear Programming Problem by the Dynamic Programming Method

299 301 307 313 321 329 340

Chapter 6 FUNDAMENTALS OF DECISION MAKING § 6.1. Definition of the Decision Problem § 6.2. Basic Notions of Theory of Choice § 6.3. Fundamentals of Decision Making

345 345 352 395

Chapter 7 MULTICRITERION OPTIMIZATION PROBLEMS §7.1. Multicriterion Problems of Selection § 7.2. Numerical Representation of Preference Relations § 7.3. Preference Representation on Probability Measures

427 427 440 455

Chapter 8 DECISION MAKING UNDER INCOMPLETE INFORMATION . . . § 8.1. Decision Making under Incomplete Information § 8.2. Decision Making under Multiple Criteria § 8.3. The Multilateral Decision Model

469 469 508 516

Contents

ix

Chapter 9 MULTICPJTERION ELEMENTS OF OPTIMIZATION THEORY § 9.1. Lexicographic Optimization § 9.2. The Factor Analysis in Organizational Systems § 9.3. Stability of Principles of Optimality § 9.4. Game-theoretic Decision Models

529 529 538 553 568

Chapter 10 DECISION MODELS § 10.1. Conceptual Setting § 10.2. Generalized Mathematical Programming as a Decision Model § 10.3. Binary Relations in the Space of Binary Relations

583 583 589 608

Chapter 11 DECISION MODELS UNDER FUZZY INFORMATION § 11.1. Extension of the Ordering Aspects of ^Well-Defined Binary Relations to the Fuzzy Case § 11.2. Ordering of Binary Relations, as Based on the Notions of Approximation and Regularization of Principles of Optimality § 11.3. General Methodology for A Priori Investigation of Generalized Mathematical Programming Problems

651 651 662 682

Chapter 12 THE APPLIED MATHEMATICAL MODEL FOR CONFLICT MANAGEMENT § 12.1. Mathematical control Models for Tariff Policy in the Regional Fuel and Energy Complex § 12.2. Computational Experiment and Appraisal of Results . . . .

691 733

Conclusion References

729 733

691

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INTRODUCTION The problem of choosing alternatives or the problem of decision making in modern world becomes the most important class of problems that is a common occurrence in everyday life of businessmen and researchers, doctors and engineers, people in their life. Mathematics, especially when equipped with the computation and information aids, can be crucial to the analysis of the relevant problems. However, it is also important to apply mathematical tools in accordance with their capabilities without overestimating or depreciating the role of mathematics and mathematicians in the decision-making process. It is difficult to imagine a complex system in almost any sphere of activities which is not characterized by a conflict of interests of the parties involved. The description of conflict management using the methods of applied mathematics as well as the development of the optimality principles and algorithms for finding optimal solutions form the basis of methodology. Based on its application, and with the participation of specialists from other fields of science, new approaches are developed to solve a variety of decision problems in various organizational systems. The problems being solved include the establishment of a strategic parity in military affairs, the rating of harmful emissions, the collective environmental safety, the exploitation of biological resources, the development of voting procedures, etc. The content of mathematics can be characterized as a system of formal sign models for the real world. Mathematical applications include the description of various phenomena and processes in the formal language as well as the use of formal and logical tools to develop the best (in one sense or another) interventions in real processes, elaborate reasonable actions, and forecast the development of phenomena of the objective world.

l

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Applied mathematics does not examine the solutions in general, but the optimal solutions, in which case the preimages of optimality can be the objective rationality, expediency, profitability, feasibility, fairness, stability, and other concepts displayed by mathematical aids. In the solution of real-world problems, it is essential that the optimality principle being selected in a model agree with the participants' real notions of optimality. Conflicts may occur between individuals, groups of individuals united by various attributes, classes, fighting parties, movements, blocks, and economic or political partners or opponents. The conflict can be viewed as a struggle against Nature where the latter is a participant. The science has not yet developed the general conceptual theory that is universal, i. e. covers equally a.ll types of conflicts in the above sense. The sciences almost entirely concentrated on conflicts, where applied mathematics is successfully used, are the law, military science, many branches of economics, sociology, political science, and psychology. There are good grounds to believe that medicine and some branches of biology and ethics can also be entered in this list. The science is expected to assist in the solution of two main problems arising in any human community, and the development of the human community is judged by how these problems are solved. The main problems of the community are: creation of something and allocation of the results. Any human society faces the problem of how to produce more goods by utilizing the natural and intellectual potential so that the life of people is made more convenient, comfortable, attractive. Here the problems of allocation of resources are solved for consumption and accumulation. One might say that creation of goods and modification of Nature are largely due to the use of knowledge and experience accumulated by natural and engineering sciences. The solution of allocation problems is primarily related to liberal arts, such as economics, sociology, political science, and social psychology attempting to find a mechanism of reasonable internal organization of the community. Depending on what is the human society, one may judge about the practical achievements of liberal arts and their influence on the consciousness of people and society as whole. The principles of goods allocation are based on some notions of fairness. When the majority of members of the human community reject the fairness of the operating principles of allocation, the society in its existing form ceases to exist or loses very important means to maintain the suppression and penal system. Many principles of fairness have been developed throughout the history of human communities. Any society attempts to prove the fairness of

Introduction

3

the operating system of allocation of goods and seeks to improve this system on the basis of its understanding. Applied mathematics, and primarily decision theory, allows the formulation of ethical categories in the form of relevant statements. Discussion of the fairness principles generally causes a violent and emotional mass reaction. Contemporary life demonstrates numerous examples of how the public opinion can be easily manipulated in the discussion of the category of fairness, and the relevant mechanisms for allocation of benefits and costs. We shall discuss two main principles of various alternatives being collectively compared: equality and efficiency. The principle of equality can be reduced to a paradox: the overall poverty is the peak of social fairness if the main thing is that everything must be equally allocated to everybody. It is reasonable to formulate the principle of equality as follows: the opinion of poor people is the first to take into account. Egalitarianism is the striving for equality by "pulling up" the welfare of poor people, but not through destruction of the welfare of the wealthy. Although the adherent of egalitarianism will support the reallocation from the wealthy to the poor (until the wealthy becomes poorer than the poor), he will not object against social differentiation when the welfare of all members of the human community increases. The principle of egalitarianism is opposed by the principle of utilitarianism which, in comparing the alternatives, rests on the aggregate welfare of society. The utilitarian believes that reallocation of benefits is of secondary importance. In his opinion, the point is that the society has to accumulate more benefits. Which of these two main principles is "fairer" must be examined in each specific case, since each of them can be reasonable or absurd. We may consider instead of the two principles the set of all possible principles of collective comparison and select the principle corresponding to the previously given attributes. Today we often encounter the reports in which the authors use simultaneously several different principles of fairness, thereby misleading the readers, listeners, and audience, since these principles are generally incompatible. One must admit that the skills are often lost or are not used in modern researches in economics, establishments of the factual truth of the propositions, models and theories. We have already got accustomed to the fact that theories exist separately from facts, if any. And since the situation in economics in Russia is considered unsatisfactory, it can be rectified by primarily finding out what we actually know, how this knowledge has been obtained, to what extent the knowledge

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is exact and reliable so that the economic theory takes on practical significance. It is time to abandon the illusion that the economic science is something that requires no special training. The delusive simplicity of fuzzy definitions, the "democratic" accessibility of rhetoric instead of mathematical analysis of clear-cut problems, the ease of manipulation of expert evaluations, all these facts were and still are too costly for Russia. The language of modern economic science is mathematics. It is well known that practice is the criterion of truth. Only in the process of rational activities of human beings one may establish whether the notions of the world in general and the economic processes in particular are true or false. Theories and models are experimentally tested, primarily in economic sciences, by comparing them with the accounting data. Active experimentation, i.e. active intervention in the course of the economic process to obtain information is an exception to the rule rather than the rule. In our opinion, the economic experiments conducted in the recent ten years in Russia feature the following disadvantages: — There is no clear-cut formulation of the question the answer to which must be obtained from experiment. — The meaning of the notion of "successful experiment" is falsified. The experiment is successful if an exact and reliable answer is obtained for the question posed by experimenters. The experiment is unsuccessful if such an answer is not obtained. This is the exact meaning of the notion of "successful experiment". — There is no reference team. Everything is known by comparison. Although the theory of designing social and economic experiments makes its first steps, it cannot be ignored. The need to set up a reference team whose performance can be compared with the performance of an experimental team is now almost an axiom in the theory of designing social and economic experiments. — There is no model-theoretic analysis of experimental results which must include: development of alternative explanations for experimental results to design experimental procedures including the effects of artifacts; development and construction of mathematical models for comparing the performance of the experimental team with that of the reference team so that the designs of experimental procedures are adjusted to raise the accuracy and reliability of experimental results. The economic reforms in Russia aimed at rapid transition to market relations through liberalization of prices, and ignored the problems of

Introduction

5

production development. The current stage of the transitional period is characterized by further reduction of outputs, instability of economic relations, nonpayment crisis, and substitution of monetary relations by other forms of settlements. The modern applied mathematics can produce solutions to many tens of classes of conflicts differing by the composition and structure of participants, specific features of the set of their objectives or interests, various characteristics of the set of their actions, strategies, behaviors, controls, and decisions as applied to various principles of selection or notions of decision optimization. Applications of the relevant models and methods are both the essentially conflict sciences and spheres of activities, such as military arts or jurisprudence, and the prognostics, liberal arts, biology, etc. The content of legal sciences includes formation of the norms regulating the techniques of resolution of conflicts between natural and legal persons. It seems reasonable to analyze these conflicts not only from traditional humanitarian positions, as it has been long done in legal sciences, but also by using the tools of applied mathematics (decision theory). In this case, one may detect the features of the phenomena which, in verbal presentation, seem to be misleading and are hardly liable to clear-cut formulation. For example, the mathematical approach directly indicates that most conflicts regulated by civil laws are antagonistic. Indeed, most property cases (perhaps, with the exception of division of property between more than two claimants), residential cases and labor disputes generally amount to the fact that one party claims some utility (using in this case instead of the legal term the "operational"), and the other party disputes this claim. The law provides each party with a set of actions, strategies whose application leads to one or another outcome of the conflict. Criminal or administrative infringement of the law turns out to be more complicated. Ethical norms are close to legal norms of behavior. As distinct from the law, the ethics is not (or almost is not) codified, therefore mathematical discussions in the ethics must start from general conceptual considerations: revelation of various interacting interests and their carriers, as well as optimality principles as applied to the ethics. Various applications of mathematics also occur in sociology. Differences in some interests of the members of a society and their groups are not only smoothed out and surmounted over time, but even develop and grow as the capabilities of the members of the society are expended. Although the conflicts arising here are far from antagonistic contradictions, they are quite numerous.

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The statement of the question of the necessity to optimally combine personal interests with public interests implies the presence of both interests and their difference. An independent problem is the justification of the voting scheme. The known voting schemes: the majority rule, the relative majority rule, the scoring-based scheme, the method of majority pairwise comparisons, etc. - may bring about quite unexpected results. In the course of formation of democracy, the knowledge of the voting theory may be useful to all conscientious members of society. In the voting, the minority principle is the main concern. The influence of a group of participants on a decision under many voting rules is jump-like dependent on the size of the group: the minority can do nothing, the majority can do everything. Introduction of the right of proportional veto allows this distribution of forces to be smoothed. Even a small group may then reject those candidates who are most unfavorable for it. The influence of a coalition is proportional to its size. The choice of the voting rules is the main ethical problem, since it involves far-reaching consequences for the functioning of the majority of political institutions. The debates on the fairness of various methods of voting were heard even in the times of the Renaissance political philosophy. If there are only two candidates, then the ordinary majority voting rule is the fairest method. This majority principle is the starting point of democratic decision making. The relative majority rule may result in the election of an inadequate candidate who will be a loser for any candidate in the pairwise comparison by the majority rule. When military conflicts are considered, their antagonism must not be confused with their intensity. For example, the conflict in which the parties are eager to annihilate each other is not formally antagonistic: in the antagonistic conflict the eagerness of one party to annihilate the other must oppose the eagerness of the latter not to annihilate the former, but to avoid its own annihilation. Basically, only in antagonistic cases one may introduce and distinguish clearly the categories of the offensive (achievement of one's own goals) and the defensive (prevention of achievement of offensive goals). The categories of offensive and defensive can be thought of as a peculiar interpretation of causal relations. Mathematical models for disarmament and armament restriction are constructed rather violently. An important problem arises in the bargaining theory where one party may not know the true estimates of its own utilities and, moreover, the estimates of utilities for the other party. The problems of disarmament and armament

Introduction

7

reduction are often related to the problems of control over compliance with the agreements on prohibition (e.g., a specific type of nuclear tests) that are a source of various problems, including those of a nonantagonistic type. Any prognostic model can be interpreted in terms of a conflict. In these models, three main types of forecast are possible: the deterministic forecast intended to indicate a specific element in the set of future implementations; the stochastic forecast intended to indicate some probability distribution over the set of future implementations; and the indefinite forecast which states that the implementation of the future consists of the elements of the set of future implementations. The contemporary methods of prediction are based on the use of expert estimates that are essentially the individual estimates of individual experts. All these forecasts, as well as their synthesis in the form of a consolidated forecast, no matter by what rules it is effected, cannot be in themselves good or bad. Their quality shows itself only in the comparison with the actual implementation of the future. The forecasts become very important in the social and political and economic life of the society. The applied mathematics is sometimes defined as the theory of mathematical models and methods of optimal decision making. One of the main organizational principles in the applied mathematics is the integrated "team" approach to the solution of problems. The questions of application of mathematical models must be discussed by the teams consisting of representatives from various professions. The set of professions may be very wide and generally depends on the scope and nature of "departmental membership" of the problem. Although this team must include professional mathematicians, the other members of the team must have an adequate mathematical knowledge. With the decision being taken under uncertainty, e.g., when we do not know exactly our purpose and the result of operation is assessed by many criteria, the exact fixing of the decision itself makes no sense. It would be more reasonable to find a class of "adequate" decisions. This fact has long been used by the analysts of alternative possible decisions. In 1904 the Italian economist Pareto formulated the principle by which possible decisions must be sought among the unimprovable alternatives only, i. e. the alternatives whose improvement does not involve their degradation under other criteria. This principle allows a reduction in the set of alternative decisions and provides an opportunity to assess the losses incurred by the operating party.

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One may recall a variety of approaches that allow the rejection of obviously unacceptable decisions and permit a reduction in the set of alternatives under analysis. Many of these approaches, e. g. the Nash equilibrium principle, are important tools of the analysis and solution of applied problems. Some of the authors emphasize that the decision problems under uncertainty can have only one strict mathematical result, that is, the estimate obtained on the basis of the maximin principle. In their opinion, the guaranteed result is a unique reference point, whereas the other alternatives lie in the sphere of hypotheses and risk. Here it is well to bear in mind that no mathematical tools can provide an exact result of choosing an alternative under uncertainty. It is in this perspective that we must assess the attempt made by one of the recognized specialists in applied mathematics L. Zadeh who suggested the rejection of any clear-cut description in decision problems. According to L. Zadeh, the subjective notions of the goal are always fuzzy and the subject's estimates and constraints he operates with are, in general, also fuzzy and occasionally have no qualitative characteristics. At present, the principles and tools are developed to allow, where possible, a reduction in the set of feasible solutions. Mathematics cannot furnish a final criterion for selection if there are actually several criteria. That is the nature of the conflict. The task of mathematics and mathematicians is to choose and exclude from consideration the noncompetitive solutions and distinguish the most promising sets of alternatives. The idea of the sequential analysis is inherent in the human being. A.Markov, Vald, Isaacs, R.Bellman, and many other scientists took part in formalization of this natural process. The central procedure in this general approach to the problem of choosing decisions is based on various rejection principles. The future will show how this approach can be useful for solution of applied problems. This book reflects my lecture notes on the mathematical theory of decision making and programming I have kept since 1965, when teaching at the Saint-Petersburg (Leningrad) State University. Some of them were executed as a monograph, Stochastic Programming (D. Reidel Publishing Company, 1976). The other materials were published in 1984 in my monograph The Macromodels of the USSR National Economy (D. Reidel Publishing Company). In 1999 and 2000 two volumes of my monograph Systems Optimization Methodology (World Scientific Publishing Company) were published which contain the material on the courses of lectures I sill read at the Chair of Economic Decisions, where I have been the chairman since 1994.

Introduction

9

Six books reflecting the author's results of practical activities in scientific management of national and municipal economy and solution of applied problems were published in the Russian language. This work contains only a part of the material on Decision Theory, the lectures I have read since 1990 at the Saint-Petersburg State University. Some of the statements are deliberately given in this book without proof, because the students at the Chair of Economic Decisions, normally 10 to 15 individuals per year, must prove them independently. The author wishes to express his indebtedness to post-graduates M. A.Suvorova and A.V. Shagov who were very helpful in preparing this book. The author's special thanks go to Mr Y. M. Donets, the translator of my monograph. Vyacheslav V. Kolbin Saint-Petersburg November, 2002

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Chapter 1 SOCIAL CHOICE PROBLEMS In recent years an increasingly large range of problems has been solved collectively. The most popular method of collective decision-making is the voting. The voting has the following features: a) a set of alternatives (e.g., candidates to an elective office) is formed in some way for which a decision has to be taken: the list of alternatives is entered in the voting paper; b) each of the procedure participants chooses his opinion about these alternatives and reflects it in the voting paper in accordance with instructions; c) based on the information received from electors, the collective decision is determined in accordance with one or another formal procedure. Formally, the voting rule solves the collective decision problem in which several individual agents (voters) will jointly choose one of several outcomes on which their opinions differ. The voting rule is the systematic solution completely relying on individual opinions. Let L(X) denote the set of linear orders on X. Then the voting rule is the mapping of L(X)N into X, where N is the finite set of voters. That the voting rule can be defined for any conceivable preference configuration expresses the fundamental principle of freedom of opinion: each voter has the right to rank candidates by any method. However, in some voting models containing economic variables or uncertain outcomes, the voters' preferences can be reasonably thought of as satisfying some general condition. This is especially convenient where preferences are aggregated.

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An interest in preference aggregation as the collective decision method generally arose in ancient times when scientists attempted to study the problems involving the voting procedure at the "philosophical level". Ancient thinkers, such as Plinius Junior, Archimedes, and Aristotle, placed emphasis in their works on the problems of choice which became acute during changes of the "methods of government", transition from ancient forms of dictatorship to those of democracy. For this reason, what we now call the constitutional problems of voting were extensively debated. The debates centered at philosophical and world vision issues such as, e.g., "what must be called the majority?", "why should one reckon with what the majority thinks?". Perhaps that is why the choice by simple majority of votes is considered to be the oldest method. This method and its modifications are discussed below. The first attempt at the comparative and critical analyses of the voting procedures was made at the end of the eighteenth century in France. It was the period when France experienced radical revolutionary and counterrevolutionary transformations: overthrow of monarchy, years of the French Republic, transition to the Napoleon Empire, restoration of the Bourbons, and again their dethronement. The controversial nature of the traditional method of "the majority of votes decides" became clear to a wide circle of scientists when the execution of Louis XVI was decided by the Convent on the one vote majority basis. In this situation, two members of the French Academy were distinguished: Jean-Charles de Borda (1733-1799) and Marie-Jean-Antoine-Nicolas de Caritas, marquis de Condorcet, who were the first to study the voting issues as a scientific problem by applying scientific methods to the analysis of this problem: the analysis of model situations and the use of mathematical tools. And these two scientists are rightly called the founders of the voting theory. The new aggregation procedures proposed by these scientists are presented below. These procedures are still often used and studied. Condorcet stated the problem: synthesize analytically the selection procedure by applying mathematical methods, but not through invention of new voting procedures. This problem is still thought of as being far from complete solution. In his Essay on the Application of Analysis to the Probability of Majority Decisions, Condorcet was the first to introduce into science the notions of paired comparisons as the basis for the theory and methods of voters' preference aggregation. At the same time, the "Founding Fathers of the Constitution" James Madison and Thomas Jefferson were concerned with the collective choice.

Social Choice Problems

13

We may presume that Borda's and Condorcet's assumptions had a profound effect on the "Founding Fathers of the Constitution". The method of paired comparisons, which is still applied by the USA Congress to solve urgent state problems, such as, e.g., adoption of amendments to the legislation in force, was also developed under the influence of Condorcet's works. The merits and demerits of the method and its modifications will be discussed in Chapter II of the book. The circle of voters of new voting procedures was expanded owing to the papers by Charles Lutwidge Dodgson (widely known under the pseudonym of Lewis Carroll, the author of "Alice's Adventures in Wonderland" and "Through the Looking-Glass and What Alice Found There"), James Marshall, Thomas Harry, Carl George Andre, where a new system is proposed for aggregation by using a quota. At the close of the nineteenth century, primarily in connection with economic, social and psychological studies, theory of choice and theory of preference aggregation were based on the extremization-criterial principle, and the natural occurrence of a rnulticriterion situation in the evaluation of the quality of the voting procedure was fully appreciated. The most important achievements in this area are associated with the Italian economist Wilfredo Pareto (1848-1929) who clarified the idea of extremal aggregation in the rnulticriterion setting and introduced the "Pareto set" which is extensively used in individual preference aggregation. This will be demonstrated with examples in Part I of this work. The idea of Pareto set is as follows. If one alternative is superior to the other in accordance with all criteria or at least one criterion (and as good as the other alternative according to the other criteria), then this latter alternative can be eliminated from further consideration and only the former alternative known as nondominated remains in the Pareto set. And this is reasonable. Then who, for example, will produce the products that are as good as the competitive products in all parameters. In order to derive a sufficiently narrow set of nondominated alternatives, we need information about criterion prefer ability. And to reveal this information requires a series of paired comparisons of the alternatives. Pareto is also associated with the formulation of one necessary requirement to the voting procedure — Pareto principle and Pareto optimality. Pareto principle. The alternative, for which the electors have all voted, is included in (accordingly, the alternative, for which no electors have voted, is excluded from) the collective choice.

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Pareto optimality. If candidate a is better for everybody than candidate b, then b cannot be elected. There are two more normative properties for the voting rules. Anonymity. Voters' names do not matter: if two voters change their votes, the voting results do not change. Neutrality. Voters' names do not matter. If candidates a and b are interchanged in every voter's preference, then the voting outcome respectively changes (if a was previously elected, then now b will be elected and vice versa.) Anonymity and neutrality will be implemented if we require that voters and candidates have equal rights. Pareto optimality is similar to the efficiency requirement. The abundance of the accumulated aggregation procedures and the difficulties involved in the recognition of the "best procedure" among them lead to the problem statement for the synthesis of "reasonable" procedures. It was early in the fifties that the American scientist Kenneth Arrow initially synthesized the procedures in which every voter chooses among the alternatives on the voting paper those alternatives that are the best (or correct) in his opinion, and then the individual belief aggregation rule constructs on the basis of this information the collective choice. In contrast to this, Arrow considers the procedures in which the voters do not vote, but order all the alternatives on the voting paper, and the collective ordering is constructed on the basis of this information (so that the collective choice as such is made in terms of this ordering by applying some aggregation rule). Arrow was the first to develop the axiomatic approach to synthesizing the voting procedures. He formulated the set of conditions that became canonical in all subsequent studies and, by proving their incompatibility, determined the central direction in the development of the synthesis problems for the voting systems. The results that have already been published show that none of the known procedures satisfies all the voting criteria being introduced, and the same problem arises in the selection of the aggregation procedure that is initially encountered by researchers in the multicriterion setting. Arrow's axioms allow the preference aggregation problem to be viewed as the fuzzy programming problem. The fuzzy programming problems constitute an important subclass of rational choice problems. These problems, in turn, generally reduce to a simpler problem of choosing numbers among some subset of naturally ordered number axis. This is discussed in detail in part I of this work. Examples are also provided for some types of these problems.

Social Choice Problems

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The voting process often brings up the question of how, when necessary, the aggregating rule has to be replaced. The answer to this question can be found in Part II of this work. However, the aggregating rules that are undoubtedly good under some conditions can be of little use under the other conditions, although the differences in the conditions may appear at first sight to be insignificant. We shall focus only on one aspect of the apparent "insignificance". In the selection of an aggregating rule for voting, we shall take into account the so-called legal restrictions. This term is interpreted to mean the by laws or legislative regulations that initially impose some restrictions on the choice of an aggregation rule. For example, legislative regulations may initially prescribe some conditions to be satisfied by the collectively elected candidate. If the candidate is required to be "approved by majority", then the meaning of these words will be defined more exactly (for example, 75% of votes). Legal restrictions may or may not initially prescribe the form in which the voter will express his opinion. If there are no such legal regulations, the organizer of voting has to decide on the choice of the aggregating rule for voting. Legal restrictions are sometimes associated with the notion of voters' equality or inequality. For example, the law or by-law may prescribe that the voting procedure should equally take into account the opinions of all voters and, in spite of the apparent simplicity of this condition, it claims the voting organizers' attention because some form of voters' inequality may actually result from the procedure effect, although it is not explicitly stipulated in the procedure. There are the classes of voting procedures in which the result of the collective choice always coincides with one specific voter's point of view, although the procedure has no explicit prescription to emphasize the point of view of one specific voter. This gives rise to the phenomenon known as the "hidden dictator" (which is related to the well-known Arrow paradox in theory of voting). There are also the classes of voting procedures in which there is no such "hidden dictator", but there is a group of voters — known as the "hidden oligarchy" — such that collectively can be chosen only the alternative which is agreed upon (or is not protested against) by all the members of this group. Thus, owing to the above factors that are revealed only by the theoretical analysis of the voting procedures by using some aggregating rules, implementation of the legal requirement for voters' equality may actually constitute a problem that is far from being very simple. On the other hand, legal restrictions may, conversely, prescribe voters' inequality. For example, such is the case for the right of veto of the permanent members of the U.N.O Security Council.

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This also applies to legal restrictions on the equality or inequality of the candidates listed in the voting paper (e.g., under age limits). Also, we have to face the fact that, apart from the restrictions that are specified by the by-laws and are unconditionally binding for the organizer who chooses the aggregating rule for voting, he must occasionally take into account the established traditions which, although not fixed by legal acts, are considered to be binding. For example, this applies to the representation of various voters' groups in the elected body. Although this is legally not fixed, the situation is obviously unreasonable where during the election of the council of a labor collective the vacancies are all occupied by the representatives of one, perhaps the most numerous department. Therefore, there are no standard techniques or algorithms that enable one to choose for any situation, while acting formally, the aggregation rule which, under these conditions, could be better than the other rules. And then, depending on a specific situation, we need to choose creatively the voting procedure. The reasonable choice of the aggregating rule constitutes every time the problem which has to be solved by taking into account all the specific features of the situation under consideration. To solve this problem, we often need to process large amounts of information and consider many alternatives where the already available applied programs allowing the analysis and selection by many quantitative and qualitative criteria appear to be very useful. The comparative analysis of objects consists of two stages: construction of the so-called Pareto set and construction of the set of nondominated alternatives. The fuller the information on the criterion preference, the higher the chances to derive the bounded set of the current user-best alternatives. We shall distinguish two types of voting, as based on the number of voters. Constitutional (universal) voting is such that the number of participants (voters) is "unobservablyn large (e.g., millions). The voting is thought of as being the voting in small (medium) groups if there are tens (hundreds) voters. In the constitutional voting, the voters essentially differ in their social status, education, and many other attributes. Here, each voter generally has no reasonable belief concerning the other voter's preferences. Such type voting is conducted at the governmental or regional level. In the selection of procedures, preference is given to world-vision considerations: who can take part in the voting, how the majority has to be interpreted, in what way the minority interests are protected, whether the election is carried by direct or two-stage voting. The comparatively small number of voters makes the

Social Choice Problems

17

collective of electors more homogeneous, permits the use of more complicated voting procedures, the ideas of voters about the candidates listed in the voting paper are more detailed. This enables the voters to order the list of candidates in accordance with their opinions. In the voting in small (medium) groups, along with the world-vision issues, technical issues arise: in what form the information (choice or ordering) has to be received from electors on the alternatives listed in the voting paper. How will the collective decision be taken on the basis of this information? How can the voting organizers' or voters' manipulation be avoided? The answers to these and other questions can be found below. § 1.1. I N D I V I D U A L PREFERENCE AGGREGATION In general, the decision maker has to take into account possible effects of his actions on other individuals. The motives can be both altruistic and egoistic. But since we prefer to consider only positive motives, the decision maker in our interpretation is the person who is benevolent, industrious, and guided by high moral principles. Descriptively, most of the actions taken by decision makers can be explained only as a result of manifestation of various interacting forces and actions taken by other individuals who are guided by various motives. A typical example is the process during which the USA Congress examines legislative proposals. Some of the philosophers hold an opinion that no decision is specifically individual, although in many cases this may appear to be the case (at least as a first approximation). However, when we obtain further insight into the problem, we may see how all the decisions interact and that any decision is the result of collective interactions. On the other hand, since the decision-making process is synthesized in the consciousness of specific individuals, every group's choice of a decision is determined by the personal decision of individuals (perhaps, very many individuals), and hence there are no group decisions as such. But any attempt to solve the philosophers' problem had no effect so that the decision-making process could be improved. This chapter examines preference aggregation from the perspective of an individual decision maker.

18 Decision Making and Programming

1.1.1. Individual Preference Aggregation under Certainty Decision Model under Certainty In order to choose the "best" alternative under certainty, it suffices to construct the decision maker's value function. Let v denote this function for the decision effects x. Since the decision maker primarily seeks to "improve N individuals' weifare", it is desirable to study functional relations between the individuals' value functions and decision maker's value function. That is, if we let Vi, V2,..., VN denote N criteria, the estimates in terms of which the individuals' values v\, 1/2,..., VN are expressed by functions, then we should like to examine the form of the functions v{x) = vD[vi{x),vi(x),...

,vN{x)]>

(1-1.1)

where v and i/p are the decision maker's value functions under certainty. This representation is based on a number of assumptions. First, the decision maker's preference for the effects x is completely characterized by the function Ui under the properly chosen scales. (Since each function vi is determined up to positive monotone transformation, before comparing individual values we need to normalize the scales to be used for their comparison.) Second, for all i the preference structure of individuals i is completely determined by function Vi. Third, v(x) contains the assumption that the decision maker knows the functions Ui, because if he did not know these functions, the problem would be undefined. Individual Preference Aggregation under Certainty Most of the works dealing with the preference aggregation model for several individuals concentrate on decision making under certainty, omitting the case where effects are uncertain. We first briefly state Arrow's impossibility theorem which, perhaps, is the best known result in group preferences. The results obtained by Arrow have made a great impact on almost all the studies of group preferences in the last two decades. Arrow's impossibility theorem. The problem stated by Arrow is generally as follows: if each group member's ranking of the set of alternatives is known, then what is the group ranking of these alternatives? Arrow made assumptions about individual ranking aggregation and then examined some implications of these assumptions [17]. The assumptions are as follows.

Social Choice Problems

19

Assumption A (universality). The group consists of at least two members, there are three or more alternatives, the group ordering (ranking) of alternatives must be determined for all possible orderings to be effected by various members of the group. Assumption B (positive association of group and individual orderings). If the group ordering shows that alternative o is preferred to alternative b under a specified set of individual preferences, and if these individual orderings change so that: 1) for all individuals the results of paired comparison of all alternatives, except a, remain unchanged, 2) the results of paired comparison of a and any other alternative either change in favor of a or remain unchanged, then the group ordering must show that a is still preferred to b. Assumption C (independence of unrelated alternatives). If some alternative is eliminated from consideration, whereas preference relations for the remaining alternatives, from the perspective of all group members, are kept unchanged, then the new group ordering of the remaining alternatives will be identical with the initial group ordering of these same alternatives. Assumption D (sovereignty of group members). For any pair of alternatives a and b there exists a set of individual preferences such that, under the group ordering, alternative a is preferred to alternative b. Assumption E (no dictator). The group must have no member such that, when he prefers alternative a to alternative b, the group also prefers alternative a to alternative b irrespective of the other group members' proposals. Arrow shows in his work (1951) that there is no rule for uniting individual preferences to satisfy assumptions A—E. That is, the following theorem is valid [17]. Theorem 1.1.1 (Arrow's impossibility theorem). Assumptions A, B , C, D, E are inconsistent. It follows that our decision maker cannot rely on the procedure that permits the union of individual rankings into one overall ranking of the decision maker himself and satisfies these five assumptions. One implication of the theorem is that, in general, there is no aggregation procedure for individual rankings that can be used to avoid explicit comparison of preferences of various group members. This result is summarized in Sena, where it is proved that there is no transformation procedure for the structure of group preferences that is consistent with both

20

Decision Making and

Programming

Arrow's assumptions and any formal type representation (1.1) and includes no comparison of individual preferences. Additive group value functions. Suppose we need to derive the value function v for the decision maker having as its arguments the value functions Vi, I = 1,2,3,... ,N where TV ^ 3. We shall make the following assumptions [7]Assumption 1 (preference independence). Criteria {Vi,Vj} are preference-independent from all its complements Vij for alii ^ j , N ^ 3. Assumption 2 (positive association of orderings). Suppose some alternatives a and b are identical in preference for the group. If a decreases and changes to alternative a' so that individual c prefers alternative a' to alternative a, while the other individuals consider them to be equivalent, then the whole group prefers a to b. That is, assumption 1 for two individuals i and j , i ^ j means that if the remaining N — 2 individuals are indifferent to the choice between the pair of implications, then the decision maker's preferences with respect to these implications are determined only by the preferences of individuals i and j and do not depend on the degree of the other individuals' preferences. It follows from assumption 2 that if the value functions vt of individuals i, i = 1,2,..., N increase, while the values of V{, i ^ j , remain unchanged, then the decision v® maker's value function also increases. And this assumption is consistent with Arrow's assumption B about positive association between group orderings and individual orderings. The following theorem is valid [9]. T h e o r e m 1.1.2. For N ^ 3, assumption 1 (preference independence) and assumption 2 (positive association of orderings) hold true if, and only if, N

N

i=l

i=l

where, for all i, 1) Vi is the value function for individual i with the measurement scale from 0 to 1, 2) function v*, which is a positive monotone transformation of argument Vi, is the decision maker's value function Vi and represents the decision maker's results of comparison of individuals' preferences. 3) v~l defined as v^(vi) is another value function for individual i that is scaled so as to capture the decision maker's results of comparison of individuals' preferences.

Social Choice Problems

21

In contrast to Arrow's statement, this statement has additionally the comparison of individuals' preferences that has been introduced by the function v* in (1.1.2). The preference degree is formally expressed in this model as a result of the consistent scaling effected by the decision maker. The decision maker has not only to choose the relevant scales for measuring each individual's preference degree, but also to adjust these scales. This calls for (external) comparisons of various individuals' preferences. Based on the foregoing, we may derive various methods to aggregate individuals' preferences. This issue is discussed in [17]. 1.1.2. Individual Preference Aggregation under Uncertainty Decision Model under Uncertainty In order to choose the best alternative under uncertainty, we need the decision maker's utility function. Let u denote this function for effects x. Also, let Ui,U2,- ■ ■, UN denote the criteria, in terms of which the individuals' utilities u\,U2,... , u/v are expressed by functions. When the "pure" decision model under uncertainty is examined, the problem is posed to find a suitable form of function U£> such that u(x) = uD[ui,v,2, ■ ■ ■ ,uN(x)],

(1.1.3)

where u, UE> are the decision maker's utility functions. This model is based on several assumptions. The most important assumption is that the decision maker's interest in x can be characterized by functions U{. It is also assumed that individual i's preference structure is represented by function Ui for ali. In contrast to model (1.1.1), model (1.1.3) does not necessarily require that the decision maker have a precise knowledge of all Uj. In (1.1.3), ui(x) is individual l's utility function for a specific effect x, and at this stage his preferences with respect to x may include both the benevolent and the malevolent attitude towards other individuals. Expressions (1.1.1) and (1.1.3) in this setting correspond to the setting of the multicriterion decision problem for one person. We agree on the following: let u be the decision maker's utility function, and Ui the utility functions of the persons or groups whose opinions are taken into account by the decision maker during formation of his decision. The decision maker is solely responsible for the choice of alternative and must

22

Decision Making and

Programming

establish various possible substitutions [6] between various values of the utility functions for individuals i = 1,2,.. .,N. These persons take no part on their own in the decision-making process. Only the decision maker will weigh the advantages some individuals receive as a result of his choice of a specific alternative. Individual Preference Aggregation under Uncertainty This section examines structurization of the decision maker's utility function and in terms of its components the utility functions Ui, i = 1,2,... ,N, expressing the preferences of the persons concerned. Additive group utility functions. John C Harsanyi stated the set of necessary and sufficient conditions for the group utility function to be expressed as the weighted sum of utility functions from u\ to UN, that is, N

u(x) = y j XiUi(x) i=l

The following assumption is crucial to Harsanyi's arguments. Assumption H. If two alternatives defined by the implementation probability distributions of effects x are equally preferred by each individual, then they are also equally preferred by the whole group state two more assumptions [18]. Assumption 3 (additive independence). The set of criteria t/i, t/2, - • •, UN is additively independent. Assumption 4 (strategic equivalence). The decision maker's conditional utility function u\ for criterion Ui that is a utility for individual i is strategically equivalent to individual z's utility function [9]. Assumption 4 can be viewed as the "honesty" assumption. Assuming that individual i honestly expressed his preferences, the decision maker further uses his utility function w, as his own utility function for estimation of his decisions from the perspective of their desirability for this individual. The following theorem is valid [9]. Theorem 1.1.3. For N ^ 2, assumption 3 (additive independence) and assumption 4 (strategic equivalence) hold true if, and only if, N

u(x) = Y^Kui(x), t=i

(1.1.4)

Social Choice Problems

23

where Ui, i = 1,2,... ,N is individual i's utility function with the scale from 0 to 1, Xi are positive scalable coefficients and x is implication. This theorem shows that assumptions H, 3 and 4 together are equivalent. The decision maker's comparison of individuals' preferences is necessary for finding the values of scalable coefficients A*. The fact that A, are positive ensures a positive relationship between the decision maker's and individuals' preferences in the sense of assumption 2 [17]. Assumption 2 (positive preference relationship). Suppose some alternatives a and b are equally preferred by the group. If a changes and becomes the alternative a* so that an individual c prefers a* to a while the other individuals consider them to be equivalent, then the whole group prefers a* to b. We shall consider an assumption that is weaker than assumption 3 [22]. Assumption 5 (utility independence). Each of the criteria Uj, j ^ i, j = 1,2,..., N is utility-independent of the other criteria Ui. Under this assumption, the utility function u(x) is [21] u(x) =uD(ui,...,ui:...,uN)

= gi(lii) + fi{ui)u*(ui)

for all i,

(1.1.5)

where u = (u\,... , U J _ I , U J + I , . . . ,UN) are all /i-positive, and u\ is the decision maker's conditional utility function for criterion Ui. These representations of the utility function that is dependent on both two and more variables were obtained for a particular set of assumptions in Mayer R . F . [22], Keeney R.L. [16], [15], [18]. The following theorem is valid [22]. Theorem 1.1.4. For N ^ 2, it follows from assumption 4 (strategic equivalence) and assumption 5 (utility-independence) that N U(x)

- UD(U\,U2,.

.-,UN)

N

= ^AjMi(x) + A i=l H

"^2

Xi,\jUi(x)Uj(x)

+

?=!) 3>i h AW~1A1A2 . • ■ AJVMI(2)112(1)..

.UN(X),

where the scales for measuring u and m are chosen so that the values of these functions are in the interval between 0 and 1, and A are the scaling coefficients and 0 < Xi < 1 for all i. Let us consider another assumption. Assumption 6 (consensus). If all the members of the group have the same utility function, then the group utility function must be the general utility function for all members.

24

Decision Making and

Programming

These instruments enable one to consider individual preference aggregation as some problem of fuzzy programming. Section 1.1.3 examines this issue in more detail. Furthermore, the instruments permit the construction of some group utility function and, through aggregation of every individual's preferences, the construction of group preferences. Pareto optimality. The principle of Pareto optimality is the corner stone of most group decision theories [19]. Definition 1.1.1. Principle of Pareto optimality. An alternative is thought of as being Pareto-optimal if any other alternative that is preferred by some members of the group to a greater extent is preferred at the same time by the other members to a lesser extent. The principle of Pareto optimality states that the alternative that is not Pareto-optimal must never be chosen, otherwise, under this choice, the level of satisfaction of at least some individuals cannot necessarily be raised without prejudice to the interests of the other individuals. As applied to the theory of election, the Pareto optimality is as follows [26]. Pareto optimality. If candidate a is better for all than candidate b, then b cannot be elected. In all cases the choice of the Pareto-optimal alternative (candidate) is said to be rational. 1.1.3. Decision-making under Fuzzy Preference Relation on the Set of Alternatives In many cases the general problem of making a decision can be described mathematically by the set of admissible choices (alternatives) and by the preference relation given on this set, which captures the decision maker's interests. This relation is generally binary, i.e., permits the comparison of two alternatives only, although the settings of ternary relation problems are also possible. Here, the problem of making a decision is to choose an admissible alternative which is superior or not inferior to the other alternatives in the sense of the given preference relation. Binary preference relations on the set of alternatives can be described in two ways: as a subset of the Cartesian product of the set of alternatives onto itself or as the so-called utility function. The utility function is generally the mapping of the set of alternatives onto a number axis [5]. That is, this function assigns a number (an estimate of alternative) to each alternative so that equal numbers (the values of utility function) correspond to equivalent

Social Choice Problems

25

(equally important) alternatives, and a larger number is assigned to the superior alternative between two nonequivalent alternatives. However, not any preference relation and not on any set of alternatives can be described by the utility function. In some cases the relation can be described not by one utility function, but only by a finite set of utility functions, in which case the relevant decision problems are usually called the multicriterion problems. In order to describe these problems and find the rational choice of alternatives, we shall use the results obtained by Berezovsky B. A., Borzenko V. I., Kempner L. M. [24], Kazanskaya G. A. Definition 1.1.2 [13]. The decision problems in which the preference relation is described as the utility function are called the mathematical programming problems. The rational decision in these problems is the choice of an admissible alternative on which the utility function assumes, whenever possible, a larger value. The fuzziness in the setting of the mathematical programming problem can be contained in both the description of the set of alternatives and the description of the utility function. The problems in which the description of the set of alternatives is fuzzy and that of utility function is precise are discussed in Bellman R. and Zadeh L. [3], where the multistage decision problems under fuzzy initial conditions are first analyzed by the method of dynamic programming and the analysis and approach to solving the problems of fuzzy mathematical programming are also discussed. Similar problems are studied in [25]. But here we shall concentrate on some types of problems in which the description of utility function is fuzzy. In general, the problems of mathematical programming constitute an important subclass of rational choice problems. Using the utility function, the original problem reduces to a simpler problem of choosing numbers among some subset of the naturally ordered number axis [10]. The values of the objective (utility) function describe the effect produced by the choice of one or another method of actions (alternatives) [13]. Essentially, the adequacy of this model to reality is largely defined by an extent to which this function captures the relationship among various factors of a real process or system. In the mathematical modeling of a complex system, allowance for a sufficiently large number of real factors is impossible because this would overcomplicate the model. Therefore, we have to introduce only a limited number of factors that are under some considerations deemed most essential. Two approaches are possible here. The factors ignored by the model

26

Decision Making and

Programming

description can be deemed absolutely inessential and completely ignored in the decision-making where this model is used. On the other hand, in the second approach the "inessential factors" may not be explicitly introduced in the model, but their effect can be allowed for under the assumption that the response of the model to one or another effect (the choice of an alternative) can be known only approximately (fuzzily). To describe the fuzzy response, we may resort to the help of experts (advisers) representing the effect on the system performance in the model of factors. The larger is the number of the factors involved in the description of the model, the lower is the degree of fuzziness in the description of system responses to effects. Thus, in the second approach the complex system is described by some fuzzy objective function, which assigns for each alternative (an effect on the system) a fuzzy response of the system to the choice of alternative. It is best to start the study of the process for making a rational decision with the set of admissible decisions or alternatives. Depending on the available information, this set can be successfully described to one or another degree of precision. For example, let X denote a universal set of alternatives and Hc{x) the fuzzy description of its subset of admissible alternatives. The values of function fic describe the admissibility of the relevant alternatives in this problem. If, apart from this function, there is no information about the real situation being studied, then we may take as rational the choice of any alternative out of the set xD \ x\x e X, fj,c(x) = sup nc(y) \ , I j/6X J i.e., any alternative having the maximum admissibility, because there is no reason to prefer a particular alternative among these alternatives to the other alternatives. The information about the decision-making process, on the basis of which some alternatives are preferred to the other alternatives, can be given in various ways [25], [9]. The description of information as a preference relation on the set of alternatives is deemed more universal One way to reveal the preference relation in constructing the mathematical model is to consult with a decision maker or experts [9]. The implication here is that the decision maker or experts has the knowledge or notion of the object being studied which has not been formalized in the model because of excessive complexity of such formalization or for some other reasons.

Social Choice Problems

27

Suppose that the definite preference relation R in the set of admissible alternatives X is revealed with the help of the decision maker or experts. This means that one of the following propositions can be made for any pair of alternatives x,y € X. Proposition 1.1.1. "x is not inferior to y", i.e., x ^ y or (x,y) G R. Proposition 1.1.2. "y is not inferior to x", i.e., y ^ x or (y,x) G R. Proposition 1.1.3. ux and y are incomparable", i.e., (x,y)

^ R and

(y,x) $R. Information in this form enables one to narrow down the class of rational choices by incorporating therein only those alternatives that are not dominated by any alternative of the set X [16]. In order to elucidate what alternatives are deemed nondominated, we distinguish the strict preference relation R and indifference relation R1 corresponding to the preference relation R [24]. Definition 1.1.3. We shall say that alternative x is strictly superior to alternative y if at the same time x ^ y and y ^ x, i.e., (x,y) G R and (y,x) £ R. The set of such pairs (x,y) is called the strict preference relation Rs on the set X. The relevant indifference relation R' in X is defined as follows [15]. Definition 1.1.4. The pair (x,y) G R' if, and only if, neither x ^ y, nor y ^ x, or both preferences are simultaneously satisfied. In other words, (x,y) G R' when the available information in the form of the preference relation is insufficient for making a choice between alternatives x and y. Definition 1.1.5 [15]. If(x,y) G Rs, we say that alternative x dominates alternative y(x >- y). Alternative x G X is called nondominated in the set (X, R) if {y, x) £ Rs for any alternative y G X. In other words, if x is the nondominated alternative, then the set X has no alternative which would dominate x. Nondominated alternatives are in a sense unimprovable on the set (X, R), and their choice in the decision problem is by right deemed rational. Thus, information in the form of the preference relation R enables one to narrow down the class of rational choices in X to the set of nondominated alternatives of the form [18] Xnd- = {x\x G X, (y,x) $ RsVy G X). The larger is the amount of information about the real situation, the

28

Decision Making and

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narrower is the preference relation, the narrower is the set X n d ' , and hence the smaller is the uncertainty in rational choice of alternatives. A more flexible method to formalize the experts' knowledge of a real situation is the method in which they have an opportunity to describe the degree of their confidence in preferences among alternatives, numbers in the interval [0, 1]. As a result, the fuzzy preference relation in the set of alternatives is revealed with the help of experts where each pair (x, y) corresponds to the number describing the degree of preference satisfaction x^y. This method of describing the relation permits to a larger extent the experts' knowledge and notions to be introduced into the model, thereby making the model in a sense more adequate to reality. We shall consider two types of problems. The Rational Choice of Alternatives Subject to Criterion Convolution We will consider the following problem [17]. Given the set of alternatives x, each alternative is characterized by several attributes with numbers j = l , . . . , m . Information about the paired comparison of alternatives per attribute is represented as the preference relation Rj. Thus, there are m preference relations Rj on the set X. Suppose the relations Rj are described by the given utility functions fj : X —r R1, where R1 is the number axis. The value of function fj(x) can be interpreted as the numerical estimate of alternative x by attribute j . The alternative with a higher estimate fj(x) is preferred by attribute j . The problem is to make a rational choice among the alternatives on the set X, Ri,..., Rm, as based on the available information, i.e. choose the alternative having higher estimates in all attributes. The rational choice in this case is the choice of alternative x0 6 X having the following properties fj(y)>

fj(xo),

j = l , - - - , m = $ fj(y)fj{x0),

i-l,...,m.

(1.1.8)

Then each of the utility functions fj describes an ordinary preference relation on X of the following form [25]: Rj = {(x,y)\x,y

6 X, fj(x) > fj(y)},

(1.1.9)

We shall use in the following the results and implications from [10], [6]. From [19] we have: m

Qi = f] Rj

(1.1.10)

Social Choice Problems

29

the set of all nondominated alternatives in the set a;, Q\ that is coincident with the set of nondominated alternatives for the set of functions fj,j = l,...,m. The intersection of relations Rj [10] is now represented somewhat differently. Let

H^y) = [l

for

{x ye

' ^

(i.i.ii)

,yJ

^ [0 for (x,y$Rj) ' be the membership function of relation Rj. Then Q\ corresponds to the membership function MQi(z,2/) =min{ j ui(x,3/),...,// m (a;,2/)},

(1.1.12)

that is similar to the convolution of criteria fj [6] of the form F(x) = min A,/,-,

(1.1.13)

j=l,m

as applied to multicriterion problems. Numbers Xj in this convolution play the role of coefficients in the relative importance of the functions being studied. In the convolution of relations (1.1.13) Xj = 1, j = 1, . . . , m this corresponds to the fact that the given relations are all equally important in the choice of alternatives. If the given relations Rj differ in importance, i.e., the relevant attributes differ in importance by which the alternatives have to be compared, then the convolution (1.1.13) uses the coefficients Xj having different values. The convolution of initial relations Rj with coefficients Xj such that m

^A

j

=

l, Xj^O,

j = l,...,m,

(1.1.14)

i=i

results in the membership function of the form [14] VQi(x>y) =min{Ai/xi(a;,2/),...,A m /x m (a;,y)},

(1.1.15)

i.e., the membership function of the fuzzy preference relation. We now introduce the convolution of another form m

VQ2(x,y) = y}2XjHj(x,y),

(1.1.16)

By the set of utility functions, this convolution provides further information on the relative degree of nondominance of alternatives and thus reduces the class of rational choices to the set

30

Decision Making and Programming

X"d

= \x\xGX, [

u»Qd(x) =

sup n&ix')}, x'€X2"dJ

(1.1.17)

where fiQd(x) = lsup[fiQl (y,x) — fiQ2(x,y)] is the fuzzy subset of nondominated alternatives on the set (X,HQ2). X^'d' is the subset of nondominated alternatives on the set (X, p,Q2). The rational choice is the choice of the alternative having the highest degree of nondominance, i. e., the largest value of function / u nd -. Chapter III discusses two examples and provides an algorithm to construct a rational choice of alternative in terms of a given class of problems. The Rational Choice of Alternatives in Terms of a Set of Attributes Given the set of alternatives (or objects) X and the set of attributes (or experts) P, each of the attributes of the set x € X is to some extent an inherent characteristic of each alternative P. For each fixed attribute p £ P, we know the fuzzy preference relation ((p on the set of alternatives X or, to put it somewhat differently, we know the membership function ip : X x X x P => [0,1] whose value p(xi,X2,p) is interpreted as the degree to which the alternative X\ is preferred to the alternative 22 in terms of attribute p. If P is the set of experts, then cp(xi,X2,p) is the preference relation on the set of alternatives offered by expert p. Thus, the function ip describes the family of fuzzy preference relations on the set X in terms of parameter p. The elements of the set P differ in importance. Let fi : P x P =>■ [0,1] be the given fuzzy relation of importance for attributes (experts); the quantity /i(Pi>P2) is taken to be the degree to which attribute p\ is thought of as being Jess important than attribute P2The problem is to choose rationally among the alternatives in the set X by taking into account the above information. We shall describe one possible way to solve this problem as based on the notion of the fuzzy set of nondominated alternatives. Let (pnd(x,p) be the fuzzy subset of nondominated alternatives corresponding to the fuzzy preference relation

AND PROGRAMMING

V V Kolbin

DECIS ON MAKNG

AND PROGRAMMING

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DECIS ON MAKNG

AND PROGRAMMING

V V Kolbin St Petersburg University, Russia

Translated from Russian by KM. Donets

V f e World Scientific « ■

New Jersey • London • Sir Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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In memory of V.I. Zubov

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CONTENTS Introduction

1

Chapter 1 SOCIAL CHOICE PROBLEMS §1.1. Individual Preference Aggregation § 1.2. Collective Preference Aggregation § 1.3. Manipulation § 1.4. Examples and Algorithms for Preference Aggregation

11 17 32 51 67

Chapter 2 VECTOR OPTIMIZATION § 2.1. Definition of Unimprovable Points § 2.2. Optimization of the Hierarchical Sequence of Quality Criteria . § 2.3. Tradeoffs § 2.4. The Linear Convolution of Criteria in Multicriteria Optimization Problems § 2.5. Solvability of the Vector Problem by the Linear Criteria Convolution Algorithm § 2.6. The Logical Criterion Vector Convolution in the Pareto Set Approximation Problem § 2.7. Computational Research on Linear Criteria Convolution in Multicriteria Discrete Programming

85 85 97 110 130 139 146 160

Chapter 3 INFINITE-VALUED PROGRAMMING PROBLEMS § 3.1. Basic Notions and Propositions § 3.2. Justification of Numerical Methods for Solving Infinite-Valued Programming Problems § 3.3. Numerical Methods of Solution § 3.4. Separable Infinite-Valued Programming Problems

169 169 181 186 200

Chapter 4 STOCHASTIC PROGRAMMING § 4.1. Stochastic Programming Models § 4.2. Stochastic Programming Methods § 4.3. Solution Algorithms for Stochastic Programming Problem § 4.4. Existence of a Deterministic Analog

vii

. .

211 211 219 245 251

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§ 4.5. Results § 4.6. An Example of Applied Problem

264 285

Chapter 5 DISCRETE PROGRAMMING § 5.1. A Geometric Interpretation of Integer Linear Programming Methods § 5.2. Equivalent Forms and Group-Theoretic Interpretation of Discrete Programming Problems § 5.3. An Algorithm for Solving the Integer Linear Programming Problem § 5.4. The Optimality Condition and the Search Method for Discrete Optimization Problems § 5.5. An Algorithm for Solving Mixed Integer Linear Programming Problems § 5.6. Solving the Large Linear Programming Problem by the Dynamic Programming Method

299 301 307 313 321 329 340

Chapter 6 FUNDAMENTALS OF DECISION MAKING § 6.1. Definition of the Decision Problem § 6.2. Basic Notions of Theory of Choice § 6.3. Fundamentals of Decision Making

345 345 352 395

Chapter 7 MULTICRITERION OPTIMIZATION PROBLEMS §7.1. Multicriterion Problems of Selection § 7.2. Numerical Representation of Preference Relations § 7.3. Preference Representation on Probability Measures

427 427 440 455

Chapter 8 DECISION MAKING UNDER INCOMPLETE INFORMATION . . . § 8.1. Decision Making under Incomplete Information § 8.2. Decision Making under Multiple Criteria § 8.3. The Multilateral Decision Model

469 469 508 516

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ix

Chapter 9 MULTICPJTERION ELEMENTS OF OPTIMIZATION THEORY § 9.1. Lexicographic Optimization § 9.2. The Factor Analysis in Organizational Systems § 9.3. Stability of Principles of Optimality § 9.4. Game-theoretic Decision Models

529 529 538 553 568

Chapter 10 DECISION MODELS § 10.1. Conceptual Setting § 10.2. Generalized Mathematical Programming as a Decision Model § 10.3. Binary Relations in the Space of Binary Relations

583 583 589 608

Chapter 11 DECISION MODELS UNDER FUZZY INFORMATION § 11.1. Extension of the Ordering Aspects of ^Well-Defined Binary Relations to the Fuzzy Case § 11.2. Ordering of Binary Relations, as Based on the Notions of Approximation and Regularization of Principles of Optimality § 11.3. General Methodology for A Priori Investigation of Generalized Mathematical Programming Problems

651 651 662 682

Chapter 12 THE APPLIED MATHEMATICAL MODEL FOR CONFLICT MANAGEMENT § 12.1. Mathematical control Models for Tariff Policy in the Regional Fuel and Energy Complex § 12.2. Computational Experiment and Appraisal of Results . . . .

691 733

Conclusion References

729 733

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INTRODUCTION The problem of choosing alternatives or the problem of decision making in modern world becomes the most important class of problems that is a common occurrence in everyday life of businessmen and researchers, doctors and engineers, people in their life. Mathematics, especially when equipped with the computation and information aids, can be crucial to the analysis of the relevant problems. However, it is also important to apply mathematical tools in accordance with their capabilities without overestimating or depreciating the role of mathematics and mathematicians in the decision-making process. It is difficult to imagine a complex system in almost any sphere of activities which is not characterized by a conflict of interests of the parties involved. The description of conflict management using the methods of applied mathematics as well as the development of the optimality principles and algorithms for finding optimal solutions form the basis of methodology. Based on its application, and with the participation of specialists from other fields of science, new approaches are developed to solve a variety of decision problems in various organizational systems. The problems being solved include the establishment of a strategic parity in military affairs, the rating of harmful emissions, the collective environmental safety, the exploitation of biological resources, the development of voting procedures, etc. The content of mathematics can be characterized as a system of formal sign models for the real world. Mathematical applications include the description of various phenomena and processes in the formal language as well as the use of formal and logical tools to develop the best (in one sense or another) interventions in real processes, elaborate reasonable actions, and forecast the development of phenomena of the objective world.

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Applied mathematics does not examine the solutions in general, but the optimal solutions, in which case the preimages of optimality can be the objective rationality, expediency, profitability, feasibility, fairness, stability, and other concepts displayed by mathematical aids. In the solution of real-world problems, it is essential that the optimality principle being selected in a model agree with the participants' real notions of optimality. Conflicts may occur between individuals, groups of individuals united by various attributes, classes, fighting parties, movements, blocks, and economic or political partners or opponents. The conflict can be viewed as a struggle against Nature where the latter is a participant. The science has not yet developed the general conceptual theory that is universal, i. e. covers equally a.ll types of conflicts in the above sense. The sciences almost entirely concentrated on conflicts, where applied mathematics is successfully used, are the law, military science, many branches of economics, sociology, political science, and psychology. There are good grounds to believe that medicine and some branches of biology and ethics can also be entered in this list. The science is expected to assist in the solution of two main problems arising in any human community, and the development of the human community is judged by how these problems are solved. The main problems of the community are: creation of something and allocation of the results. Any human society faces the problem of how to produce more goods by utilizing the natural and intellectual potential so that the life of people is made more convenient, comfortable, attractive. Here the problems of allocation of resources are solved for consumption and accumulation. One might say that creation of goods and modification of Nature are largely due to the use of knowledge and experience accumulated by natural and engineering sciences. The solution of allocation problems is primarily related to liberal arts, such as economics, sociology, political science, and social psychology attempting to find a mechanism of reasonable internal organization of the community. Depending on what is the human society, one may judge about the practical achievements of liberal arts and their influence on the consciousness of people and society as whole. The principles of goods allocation are based on some notions of fairness. When the majority of members of the human community reject the fairness of the operating principles of allocation, the society in its existing form ceases to exist or loses very important means to maintain the suppression and penal system. Many principles of fairness have been developed throughout the history of human communities. Any society attempts to prove the fairness of

Introduction

3

the operating system of allocation of goods and seeks to improve this system on the basis of its understanding. Applied mathematics, and primarily decision theory, allows the formulation of ethical categories in the form of relevant statements. Discussion of the fairness principles generally causes a violent and emotional mass reaction. Contemporary life demonstrates numerous examples of how the public opinion can be easily manipulated in the discussion of the category of fairness, and the relevant mechanisms for allocation of benefits and costs. We shall discuss two main principles of various alternatives being collectively compared: equality and efficiency. The principle of equality can be reduced to a paradox: the overall poverty is the peak of social fairness if the main thing is that everything must be equally allocated to everybody. It is reasonable to formulate the principle of equality as follows: the opinion of poor people is the first to take into account. Egalitarianism is the striving for equality by "pulling up" the welfare of poor people, but not through destruction of the welfare of the wealthy. Although the adherent of egalitarianism will support the reallocation from the wealthy to the poor (until the wealthy becomes poorer than the poor), he will not object against social differentiation when the welfare of all members of the human community increases. The principle of egalitarianism is opposed by the principle of utilitarianism which, in comparing the alternatives, rests on the aggregate welfare of society. The utilitarian believes that reallocation of benefits is of secondary importance. In his opinion, the point is that the society has to accumulate more benefits. Which of these two main principles is "fairer" must be examined in each specific case, since each of them can be reasonable or absurd. We may consider instead of the two principles the set of all possible principles of collective comparison and select the principle corresponding to the previously given attributes. Today we often encounter the reports in which the authors use simultaneously several different principles of fairness, thereby misleading the readers, listeners, and audience, since these principles are generally incompatible. One must admit that the skills are often lost or are not used in modern researches in economics, establishments of the factual truth of the propositions, models and theories. We have already got accustomed to the fact that theories exist separately from facts, if any. And since the situation in economics in Russia is considered unsatisfactory, it can be rectified by primarily finding out what we actually know, how this knowledge has been obtained, to what extent the knowledge

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is exact and reliable so that the economic theory takes on practical significance. It is time to abandon the illusion that the economic science is something that requires no special training. The delusive simplicity of fuzzy definitions, the "democratic" accessibility of rhetoric instead of mathematical analysis of clear-cut problems, the ease of manipulation of expert evaluations, all these facts were and still are too costly for Russia. The language of modern economic science is mathematics. It is well known that practice is the criterion of truth. Only in the process of rational activities of human beings one may establish whether the notions of the world in general and the economic processes in particular are true or false. Theories and models are experimentally tested, primarily in economic sciences, by comparing them with the accounting data. Active experimentation, i.e. active intervention in the course of the economic process to obtain information is an exception to the rule rather than the rule. In our opinion, the economic experiments conducted in the recent ten years in Russia feature the following disadvantages: — There is no clear-cut formulation of the question the answer to which must be obtained from experiment. — The meaning of the notion of "successful experiment" is falsified. The experiment is successful if an exact and reliable answer is obtained for the question posed by experimenters. The experiment is unsuccessful if such an answer is not obtained. This is the exact meaning of the notion of "successful experiment". — There is no reference team. Everything is known by comparison. Although the theory of designing social and economic experiments makes its first steps, it cannot be ignored. The need to set up a reference team whose performance can be compared with the performance of an experimental team is now almost an axiom in the theory of designing social and economic experiments. — There is no model-theoretic analysis of experimental results which must include: development of alternative explanations for experimental results to design experimental procedures including the effects of artifacts; development and construction of mathematical models for comparing the performance of the experimental team with that of the reference team so that the designs of experimental procedures are adjusted to raise the accuracy and reliability of experimental results. The economic reforms in Russia aimed at rapid transition to market relations through liberalization of prices, and ignored the problems of

Introduction

5

production development. The current stage of the transitional period is characterized by further reduction of outputs, instability of economic relations, nonpayment crisis, and substitution of monetary relations by other forms of settlements. The modern applied mathematics can produce solutions to many tens of classes of conflicts differing by the composition and structure of participants, specific features of the set of their objectives or interests, various characteristics of the set of their actions, strategies, behaviors, controls, and decisions as applied to various principles of selection or notions of decision optimization. Applications of the relevant models and methods are both the essentially conflict sciences and spheres of activities, such as military arts or jurisprudence, and the prognostics, liberal arts, biology, etc. The content of legal sciences includes formation of the norms regulating the techniques of resolution of conflicts between natural and legal persons. It seems reasonable to analyze these conflicts not only from traditional humanitarian positions, as it has been long done in legal sciences, but also by using the tools of applied mathematics (decision theory). In this case, one may detect the features of the phenomena which, in verbal presentation, seem to be misleading and are hardly liable to clear-cut formulation. For example, the mathematical approach directly indicates that most conflicts regulated by civil laws are antagonistic. Indeed, most property cases (perhaps, with the exception of division of property between more than two claimants), residential cases and labor disputes generally amount to the fact that one party claims some utility (using in this case instead of the legal term the "operational"), and the other party disputes this claim. The law provides each party with a set of actions, strategies whose application leads to one or another outcome of the conflict. Criminal or administrative infringement of the law turns out to be more complicated. Ethical norms are close to legal norms of behavior. As distinct from the law, the ethics is not (or almost is not) codified, therefore mathematical discussions in the ethics must start from general conceptual considerations: revelation of various interacting interests and their carriers, as well as optimality principles as applied to the ethics. Various applications of mathematics also occur in sociology. Differences in some interests of the members of a society and their groups are not only smoothed out and surmounted over time, but even develop and grow as the capabilities of the members of the society are expended. Although the conflicts arising here are far from antagonistic contradictions, they are quite numerous.

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The statement of the question of the necessity to optimally combine personal interests with public interests implies the presence of both interests and their difference. An independent problem is the justification of the voting scheme. The known voting schemes: the majority rule, the relative majority rule, the scoring-based scheme, the method of majority pairwise comparisons, etc. - may bring about quite unexpected results. In the course of formation of democracy, the knowledge of the voting theory may be useful to all conscientious members of society. In the voting, the minority principle is the main concern. The influence of a group of participants on a decision under many voting rules is jump-like dependent on the size of the group: the minority can do nothing, the majority can do everything. Introduction of the right of proportional veto allows this distribution of forces to be smoothed. Even a small group may then reject those candidates who are most unfavorable for it. The influence of a coalition is proportional to its size. The choice of the voting rules is the main ethical problem, since it involves far-reaching consequences for the functioning of the majority of political institutions. The debates on the fairness of various methods of voting were heard even in the times of the Renaissance political philosophy. If there are only two candidates, then the ordinary majority voting rule is the fairest method. This majority principle is the starting point of democratic decision making. The relative majority rule may result in the election of an inadequate candidate who will be a loser for any candidate in the pairwise comparison by the majority rule. When military conflicts are considered, their antagonism must not be confused with their intensity. For example, the conflict in which the parties are eager to annihilate each other is not formally antagonistic: in the antagonistic conflict the eagerness of one party to annihilate the other must oppose the eagerness of the latter not to annihilate the former, but to avoid its own annihilation. Basically, only in antagonistic cases one may introduce and distinguish clearly the categories of the offensive (achievement of one's own goals) and the defensive (prevention of achievement of offensive goals). The categories of offensive and defensive can be thought of as a peculiar interpretation of causal relations. Mathematical models for disarmament and armament restriction are constructed rather violently. An important problem arises in the bargaining theory where one party may not know the true estimates of its own utilities and, moreover, the estimates of utilities for the other party. The problems of disarmament and armament

Introduction

7

reduction are often related to the problems of control over compliance with the agreements on prohibition (e.g., a specific type of nuclear tests) that are a source of various problems, including those of a nonantagonistic type. Any prognostic model can be interpreted in terms of a conflict. In these models, three main types of forecast are possible: the deterministic forecast intended to indicate a specific element in the set of future implementations; the stochastic forecast intended to indicate some probability distribution over the set of future implementations; and the indefinite forecast which states that the implementation of the future consists of the elements of the set of future implementations. The contemporary methods of prediction are based on the use of expert estimates that are essentially the individual estimates of individual experts. All these forecasts, as well as their synthesis in the form of a consolidated forecast, no matter by what rules it is effected, cannot be in themselves good or bad. Their quality shows itself only in the comparison with the actual implementation of the future. The forecasts become very important in the social and political and economic life of the society. The applied mathematics is sometimes defined as the theory of mathematical models and methods of optimal decision making. One of the main organizational principles in the applied mathematics is the integrated "team" approach to the solution of problems. The questions of application of mathematical models must be discussed by the teams consisting of representatives from various professions. The set of professions may be very wide and generally depends on the scope and nature of "departmental membership" of the problem. Although this team must include professional mathematicians, the other members of the team must have an adequate mathematical knowledge. With the decision being taken under uncertainty, e.g., when we do not know exactly our purpose and the result of operation is assessed by many criteria, the exact fixing of the decision itself makes no sense. It would be more reasonable to find a class of "adequate" decisions. This fact has long been used by the analysts of alternative possible decisions. In 1904 the Italian economist Pareto formulated the principle by which possible decisions must be sought among the unimprovable alternatives only, i. e. the alternatives whose improvement does not involve their degradation under other criteria. This principle allows a reduction in the set of alternative decisions and provides an opportunity to assess the losses incurred by the operating party.

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One may recall a variety of approaches that allow the rejection of obviously unacceptable decisions and permit a reduction in the set of alternatives under analysis. Many of these approaches, e. g. the Nash equilibrium principle, are important tools of the analysis and solution of applied problems. Some of the authors emphasize that the decision problems under uncertainty can have only one strict mathematical result, that is, the estimate obtained on the basis of the maximin principle. In their opinion, the guaranteed result is a unique reference point, whereas the other alternatives lie in the sphere of hypotheses and risk. Here it is well to bear in mind that no mathematical tools can provide an exact result of choosing an alternative under uncertainty. It is in this perspective that we must assess the attempt made by one of the recognized specialists in applied mathematics L. Zadeh who suggested the rejection of any clear-cut description in decision problems. According to L. Zadeh, the subjective notions of the goal are always fuzzy and the subject's estimates and constraints he operates with are, in general, also fuzzy and occasionally have no qualitative characteristics. At present, the principles and tools are developed to allow, where possible, a reduction in the set of feasible solutions. Mathematics cannot furnish a final criterion for selection if there are actually several criteria. That is the nature of the conflict. The task of mathematics and mathematicians is to choose and exclude from consideration the noncompetitive solutions and distinguish the most promising sets of alternatives. The idea of the sequential analysis is inherent in the human being. A.Markov, Vald, Isaacs, R.Bellman, and many other scientists took part in formalization of this natural process. The central procedure in this general approach to the problem of choosing decisions is based on various rejection principles. The future will show how this approach can be useful for solution of applied problems. This book reflects my lecture notes on the mathematical theory of decision making and programming I have kept since 1965, when teaching at the Saint-Petersburg (Leningrad) State University. Some of them were executed as a monograph, Stochastic Programming (D. Reidel Publishing Company, 1976). The other materials were published in 1984 in my monograph The Macromodels of the USSR National Economy (D. Reidel Publishing Company). In 1999 and 2000 two volumes of my monograph Systems Optimization Methodology (World Scientific Publishing Company) were published which contain the material on the courses of lectures I sill read at the Chair of Economic Decisions, where I have been the chairman since 1994.

Introduction

9

Six books reflecting the author's results of practical activities in scientific management of national and municipal economy and solution of applied problems were published in the Russian language. This work contains only a part of the material on Decision Theory, the lectures I have read since 1990 at the Saint-Petersburg State University. Some of the statements are deliberately given in this book without proof, because the students at the Chair of Economic Decisions, normally 10 to 15 individuals per year, must prove them independently. The author wishes to express his indebtedness to post-graduates M. A.Suvorova and A.V. Shagov who were very helpful in preparing this book. The author's special thanks go to Mr Y. M. Donets, the translator of my monograph. Vyacheslav V. Kolbin Saint-Petersburg November, 2002

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Chapter 1 SOCIAL CHOICE PROBLEMS In recent years an increasingly large range of problems has been solved collectively. The most popular method of collective decision-making is the voting. The voting has the following features: a) a set of alternatives (e.g., candidates to an elective office) is formed in some way for which a decision has to be taken: the list of alternatives is entered in the voting paper; b) each of the procedure participants chooses his opinion about these alternatives and reflects it in the voting paper in accordance with instructions; c) based on the information received from electors, the collective decision is determined in accordance with one or another formal procedure. Formally, the voting rule solves the collective decision problem in which several individual agents (voters) will jointly choose one of several outcomes on which their opinions differ. The voting rule is the systematic solution completely relying on individual opinions. Let L(X) denote the set of linear orders on X. Then the voting rule is the mapping of L(X)N into X, where N is the finite set of voters. That the voting rule can be defined for any conceivable preference configuration expresses the fundamental principle of freedom of opinion: each voter has the right to rank candidates by any method. However, in some voting models containing economic variables or uncertain outcomes, the voters' preferences can be reasonably thought of as satisfying some general condition. This is especially convenient where preferences are aggregated.

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An interest in preference aggregation as the collective decision method generally arose in ancient times when scientists attempted to study the problems involving the voting procedure at the "philosophical level". Ancient thinkers, such as Plinius Junior, Archimedes, and Aristotle, placed emphasis in their works on the problems of choice which became acute during changes of the "methods of government", transition from ancient forms of dictatorship to those of democracy. For this reason, what we now call the constitutional problems of voting were extensively debated. The debates centered at philosophical and world vision issues such as, e.g., "what must be called the majority?", "why should one reckon with what the majority thinks?". Perhaps that is why the choice by simple majority of votes is considered to be the oldest method. This method and its modifications are discussed below. The first attempt at the comparative and critical analyses of the voting procedures was made at the end of the eighteenth century in France. It was the period when France experienced radical revolutionary and counterrevolutionary transformations: overthrow of monarchy, years of the French Republic, transition to the Napoleon Empire, restoration of the Bourbons, and again their dethronement. The controversial nature of the traditional method of "the majority of votes decides" became clear to a wide circle of scientists when the execution of Louis XVI was decided by the Convent on the one vote majority basis. In this situation, two members of the French Academy were distinguished: Jean-Charles de Borda (1733-1799) and Marie-Jean-Antoine-Nicolas de Caritas, marquis de Condorcet, who were the first to study the voting issues as a scientific problem by applying scientific methods to the analysis of this problem: the analysis of model situations and the use of mathematical tools. And these two scientists are rightly called the founders of the voting theory. The new aggregation procedures proposed by these scientists are presented below. These procedures are still often used and studied. Condorcet stated the problem: synthesize analytically the selection procedure by applying mathematical methods, but not through invention of new voting procedures. This problem is still thought of as being far from complete solution. In his Essay on the Application of Analysis to the Probability of Majority Decisions, Condorcet was the first to introduce into science the notions of paired comparisons as the basis for the theory and methods of voters' preference aggregation. At the same time, the "Founding Fathers of the Constitution" James Madison and Thomas Jefferson were concerned with the collective choice.

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We may presume that Borda's and Condorcet's assumptions had a profound effect on the "Founding Fathers of the Constitution". The method of paired comparisons, which is still applied by the USA Congress to solve urgent state problems, such as, e.g., adoption of amendments to the legislation in force, was also developed under the influence of Condorcet's works. The merits and demerits of the method and its modifications will be discussed in Chapter II of the book. The circle of voters of new voting procedures was expanded owing to the papers by Charles Lutwidge Dodgson (widely known under the pseudonym of Lewis Carroll, the author of "Alice's Adventures in Wonderland" and "Through the Looking-Glass and What Alice Found There"), James Marshall, Thomas Harry, Carl George Andre, where a new system is proposed for aggregation by using a quota. At the close of the nineteenth century, primarily in connection with economic, social and psychological studies, theory of choice and theory of preference aggregation were based on the extremization-criterial principle, and the natural occurrence of a rnulticriterion situation in the evaluation of the quality of the voting procedure was fully appreciated. The most important achievements in this area are associated with the Italian economist Wilfredo Pareto (1848-1929) who clarified the idea of extremal aggregation in the rnulticriterion setting and introduced the "Pareto set" which is extensively used in individual preference aggregation. This will be demonstrated with examples in Part I of this work. The idea of Pareto set is as follows. If one alternative is superior to the other in accordance with all criteria or at least one criterion (and as good as the other alternative according to the other criteria), then this latter alternative can be eliminated from further consideration and only the former alternative known as nondominated remains in the Pareto set. And this is reasonable. Then who, for example, will produce the products that are as good as the competitive products in all parameters. In order to derive a sufficiently narrow set of nondominated alternatives, we need information about criterion prefer ability. And to reveal this information requires a series of paired comparisons of the alternatives. Pareto is also associated with the formulation of one necessary requirement to the voting procedure — Pareto principle and Pareto optimality. Pareto principle. The alternative, for which the electors have all voted, is included in (accordingly, the alternative, for which no electors have voted, is excluded from) the collective choice.

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Pareto optimality. If candidate a is better for everybody than candidate b, then b cannot be elected. There are two more normative properties for the voting rules. Anonymity. Voters' names do not matter: if two voters change their votes, the voting results do not change. Neutrality. Voters' names do not matter. If candidates a and b are interchanged in every voter's preference, then the voting outcome respectively changes (if a was previously elected, then now b will be elected and vice versa.) Anonymity and neutrality will be implemented if we require that voters and candidates have equal rights. Pareto optimality is similar to the efficiency requirement. The abundance of the accumulated aggregation procedures and the difficulties involved in the recognition of the "best procedure" among them lead to the problem statement for the synthesis of "reasonable" procedures. It was early in the fifties that the American scientist Kenneth Arrow initially synthesized the procedures in which every voter chooses among the alternatives on the voting paper those alternatives that are the best (or correct) in his opinion, and then the individual belief aggregation rule constructs on the basis of this information the collective choice. In contrast to this, Arrow considers the procedures in which the voters do not vote, but order all the alternatives on the voting paper, and the collective ordering is constructed on the basis of this information (so that the collective choice as such is made in terms of this ordering by applying some aggregation rule). Arrow was the first to develop the axiomatic approach to synthesizing the voting procedures. He formulated the set of conditions that became canonical in all subsequent studies and, by proving their incompatibility, determined the central direction in the development of the synthesis problems for the voting systems. The results that have already been published show that none of the known procedures satisfies all the voting criteria being introduced, and the same problem arises in the selection of the aggregation procedure that is initially encountered by researchers in the multicriterion setting. Arrow's axioms allow the preference aggregation problem to be viewed as the fuzzy programming problem. The fuzzy programming problems constitute an important subclass of rational choice problems. These problems, in turn, generally reduce to a simpler problem of choosing numbers among some subset of naturally ordered number axis. This is discussed in detail in part I of this work. Examples are also provided for some types of these problems.

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The voting process often brings up the question of how, when necessary, the aggregating rule has to be replaced. The answer to this question can be found in Part II of this work. However, the aggregating rules that are undoubtedly good under some conditions can be of little use under the other conditions, although the differences in the conditions may appear at first sight to be insignificant. We shall focus only on one aspect of the apparent "insignificance". In the selection of an aggregating rule for voting, we shall take into account the so-called legal restrictions. This term is interpreted to mean the by laws or legislative regulations that initially impose some restrictions on the choice of an aggregation rule. For example, legislative regulations may initially prescribe some conditions to be satisfied by the collectively elected candidate. If the candidate is required to be "approved by majority", then the meaning of these words will be defined more exactly (for example, 75% of votes). Legal restrictions may or may not initially prescribe the form in which the voter will express his opinion. If there are no such legal regulations, the organizer of voting has to decide on the choice of the aggregating rule for voting. Legal restrictions are sometimes associated with the notion of voters' equality or inequality. For example, the law or by-law may prescribe that the voting procedure should equally take into account the opinions of all voters and, in spite of the apparent simplicity of this condition, it claims the voting organizers' attention because some form of voters' inequality may actually result from the procedure effect, although it is not explicitly stipulated in the procedure. There are the classes of voting procedures in which the result of the collective choice always coincides with one specific voter's point of view, although the procedure has no explicit prescription to emphasize the point of view of one specific voter. This gives rise to the phenomenon known as the "hidden dictator" (which is related to the well-known Arrow paradox in theory of voting). There are also the classes of voting procedures in which there is no such "hidden dictator", but there is a group of voters — known as the "hidden oligarchy" — such that collectively can be chosen only the alternative which is agreed upon (or is not protested against) by all the members of this group. Thus, owing to the above factors that are revealed only by the theoretical analysis of the voting procedures by using some aggregating rules, implementation of the legal requirement for voters' equality may actually constitute a problem that is far from being very simple. On the other hand, legal restrictions may, conversely, prescribe voters' inequality. For example, such is the case for the right of veto of the permanent members of the U.N.O Security Council.

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This also applies to legal restrictions on the equality or inequality of the candidates listed in the voting paper (e.g., under age limits). Also, we have to face the fact that, apart from the restrictions that are specified by the by-laws and are unconditionally binding for the organizer who chooses the aggregating rule for voting, he must occasionally take into account the established traditions which, although not fixed by legal acts, are considered to be binding. For example, this applies to the representation of various voters' groups in the elected body. Although this is legally not fixed, the situation is obviously unreasonable where during the election of the council of a labor collective the vacancies are all occupied by the representatives of one, perhaps the most numerous department. Therefore, there are no standard techniques or algorithms that enable one to choose for any situation, while acting formally, the aggregation rule which, under these conditions, could be better than the other rules. And then, depending on a specific situation, we need to choose creatively the voting procedure. The reasonable choice of the aggregating rule constitutes every time the problem which has to be solved by taking into account all the specific features of the situation under consideration. To solve this problem, we often need to process large amounts of information and consider many alternatives where the already available applied programs allowing the analysis and selection by many quantitative and qualitative criteria appear to be very useful. The comparative analysis of objects consists of two stages: construction of the so-called Pareto set and construction of the set of nondominated alternatives. The fuller the information on the criterion preference, the higher the chances to derive the bounded set of the current user-best alternatives. We shall distinguish two types of voting, as based on the number of voters. Constitutional (universal) voting is such that the number of participants (voters) is "unobservablyn large (e.g., millions). The voting is thought of as being the voting in small (medium) groups if there are tens (hundreds) voters. In the constitutional voting, the voters essentially differ in their social status, education, and many other attributes. Here, each voter generally has no reasonable belief concerning the other voter's preferences. Such type voting is conducted at the governmental or regional level. In the selection of procedures, preference is given to world-vision considerations: who can take part in the voting, how the majority has to be interpreted, in what way the minority interests are protected, whether the election is carried by direct or two-stage voting. The comparatively small number of voters makes the

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collective of electors more homogeneous, permits the use of more complicated voting procedures, the ideas of voters about the candidates listed in the voting paper are more detailed. This enables the voters to order the list of candidates in accordance with their opinions. In the voting in small (medium) groups, along with the world-vision issues, technical issues arise: in what form the information (choice or ordering) has to be received from electors on the alternatives listed in the voting paper. How will the collective decision be taken on the basis of this information? How can the voting organizers' or voters' manipulation be avoided? The answers to these and other questions can be found below. § 1.1. I N D I V I D U A L PREFERENCE AGGREGATION In general, the decision maker has to take into account possible effects of his actions on other individuals. The motives can be both altruistic and egoistic. But since we prefer to consider only positive motives, the decision maker in our interpretation is the person who is benevolent, industrious, and guided by high moral principles. Descriptively, most of the actions taken by decision makers can be explained only as a result of manifestation of various interacting forces and actions taken by other individuals who are guided by various motives. A typical example is the process during which the USA Congress examines legislative proposals. Some of the philosophers hold an opinion that no decision is specifically individual, although in many cases this may appear to be the case (at least as a first approximation). However, when we obtain further insight into the problem, we may see how all the decisions interact and that any decision is the result of collective interactions. On the other hand, since the decision-making process is synthesized in the consciousness of specific individuals, every group's choice of a decision is determined by the personal decision of individuals (perhaps, very many individuals), and hence there are no group decisions as such. But any attempt to solve the philosophers' problem had no effect so that the decision-making process could be improved. This chapter examines preference aggregation from the perspective of an individual decision maker.

18 Decision Making and Programming

1.1.1. Individual Preference Aggregation under Certainty Decision Model under Certainty In order to choose the "best" alternative under certainty, it suffices to construct the decision maker's value function. Let v denote this function for the decision effects x. Since the decision maker primarily seeks to "improve N individuals' weifare", it is desirable to study functional relations between the individuals' value functions and decision maker's value function. That is, if we let Vi, V2,..., VN denote N criteria, the estimates in terms of which the individuals' values v\, 1/2,..., VN are expressed by functions, then we should like to examine the form of the functions v{x) = vD[vi{x),vi(x),...

,vN{x)]>

(1-1.1)

where v and i/p are the decision maker's value functions under certainty. This representation is based on a number of assumptions. First, the decision maker's preference for the effects x is completely characterized by the function Ui under the properly chosen scales. (Since each function vi is determined up to positive monotone transformation, before comparing individual values we need to normalize the scales to be used for their comparison.) Second, for all i the preference structure of individuals i is completely determined by function Vi. Third, v(x) contains the assumption that the decision maker knows the functions Ui, because if he did not know these functions, the problem would be undefined. Individual Preference Aggregation under Certainty Most of the works dealing with the preference aggregation model for several individuals concentrate on decision making under certainty, omitting the case where effects are uncertain. We first briefly state Arrow's impossibility theorem which, perhaps, is the best known result in group preferences. The results obtained by Arrow have made a great impact on almost all the studies of group preferences in the last two decades. Arrow's impossibility theorem. The problem stated by Arrow is generally as follows: if each group member's ranking of the set of alternatives is known, then what is the group ranking of these alternatives? Arrow made assumptions about individual ranking aggregation and then examined some implications of these assumptions [17]. The assumptions are as follows.

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Assumption A (universality). The group consists of at least two members, there are three or more alternatives, the group ordering (ranking) of alternatives must be determined for all possible orderings to be effected by various members of the group. Assumption B (positive association of group and individual orderings). If the group ordering shows that alternative o is preferred to alternative b under a specified set of individual preferences, and if these individual orderings change so that: 1) for all individuals the results of paired comparison of all alternatives, except a, remain unchanged, 2) the results of paired comparison of a and any other alternative either change in favor of a or remain unchanged, then the group ordering must show that a is still preferred to b. Assumption C (independence of unrelated alternatives). If some alternative is eliminated from consideration, whereas preference relations for the remaining alternatives, from the perspective of all group members, are kept unchanged, then the new group ordering of the remaining alternatives will be identical with the initial group ordering of these same alternatives. Assumption D (sovereignty of group members). For any pair of alternatives a and b there exists a set of individual preferences such that, under the group ordering, alternative a is preferred to alternative b. Assumption E (no dictator). The group must have no member such that, when he prefers alternative a to alternative b, the group also prefers alternative a to alternative b irrespective of the other group members' proposals. Arrow shows in his work (1951) that there is no rule for uniting individual preferences to satisfy assumptions A—E. That is, the following theorem is valid [17]. Theorem 1.1.1 (Arrow's impossibility theorem). Assumptions A, B , C, D, E are inconsistent. It follows that our decision maker cannot rely on the procedure that permits the union of individual rankings into one overall ranking of the decision maker himself and satisfies these five assumptions. One implication of the theorem is that, in general, there is no aggregation procedure for individual rankings that can be used to avoid explicit comparison of preferences of various group members. This result is summarized in Sena, where it is proved that there is no transformation procedure for the structure of group preferences that is consistent with both

20

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Arrow's assumptions and any formal type representation (1.1) and includes no comparison of individual preferences. Additive group value functions. Suppose we need to derive the value function v for the decision maker having as its arguments the value functions Vi, I = 1,2,3,... ,N where TV ^ 3. We shall make the following assumptions [7]Assumption 1 (preference independence). Criteria {Vi,Vj} are preference-independent from all its complements Vij for alii ^ j , N ^ 3. Assumption 2 (positive association of orderings). Suppose some alternatives a and b are identical in preference for the group. If a decreases and changes to alternative a' so that individual c prefers alternative a' to alternative a, while the other individuals consider them to be equivalent, then the whole group prefers a to b. That is, assumption 1 for two individuals i and j , i ^ j means that if the remaining N — 2 individuals are indifferent to the choice between the pair of implications, then the decision maker's preferences with respect to these implications are determined only by the preferences of individuals i and j and do not depend on the degree of the other individuals' preferences. It follows from assumption 2 that if the value functions vt of individuals i, i = 1,2,..., N increase, while the values of V{, i ^ j , remain unchanged, then the decision v® maker's value function also increases. And this assumption is consistent with Arrow's assumption B about positive association between group orderings and individual orderings. The following theorem is valid [9]. T h e o r e m 1.1.2. For N ^ 3, assumption 1 (preference independence) and assumption 2 (positive association of orderings) hold true if, and only if, N

N

i=l

i=l

where, for all i, 1) Vi is the value function for individual i with the measurement scale from 0 to 1, 2) function v*, which is a positive monotone transformation of argument Vi, is the decision maker's value function Vi and represents the decision maker's results of comparison of individuals' preferences. 3) v~l defined as v^(vi) is another value function for individual i that is scaled so as to capture the decision maker's results of comparison of individuals' preferences.

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In contrast to Arrow's statement, this statement has additionally the comparison of individuals' preferences that has been introduced by the function v* in (1.1.2). The preference degree is formally expressed in this model as a result of the consistent scaling effected by the decision maker. The decision maker has not only to choose the relevant scales for measuring each individual's preference degree, but also to adjust these scales. This calls for (external) comparisons of various individuals' preferences. Based on the foregoing, we may derive various methods to aggregate individuals' preferences. This issue is discussed in [17]. 1.1.2. Individual Preference Aggregation under Uncertainty Decision Model under Uncertainty In order to choose the best alternative under uncertainty, we need the decision maker's utility function. Let u denote this function for effects x. Also, let Ui,U2,- ■ ■, UN denote the criteria, in terms of which the individuals' utilities u\,U2,... , u/v are expressed by functions. When the "pure" decision model under uncertainty is examined, the problem is posed to find a suitable form of function U£> such that u(x) = uD[ui,v,2, ■ ■ ■ ,uN(x)],

(1.1.3)

where u, UE> are the decision maker's utility functions. This model is based on several assumptions. The most important assumption is that the decision maker's interest in x can be characterized by functions U{. It is also assumed that individual i's preference structure is represented by function Ui for ali. In contrast to model (1.1.1), model (1.1.3) does not necessarily require that the decision maker have a precise knowledge of all Uj. In (1.1.3), ui(x) is individual l's utility function for a specific effect x, and at this stage his preferences with respect to x may include both the benevolent and the malevolent attitude towards other individuals. Expressions (1.1.1) and (1.1.3) in this setting correspond to the setting of the multicriterion decision problem for one person. We agree on the following: let u be the decision maker's utility function, and Ui the utility functions of the persons or groups whose opinions are taken into account by the decision maker during formation of his decision. The decision maker is solely responsible for the choice of alternative and must

22

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establish various possible substitutions [6] between various values of the utility functions for individuals i = 1,2,.. .,N. These persons take no part on their own in the decision-making process. Only the decision maker will weigh the advantages some individuals receive as a result of his choice of a specific alternative. Individual Preference Aggregation under Uncertainty This section examines structurization of the decision maker's utility function and in terms of its components the utility functions Ui, i = 1,2,... ,N, expressing the preferences of the persons concerned. Additive group utility functions. John C Harsanyi stated the set of necessary and sufficient conditions for the group utility function to be expressed as the weighted sum of utility functions from u\ to UN, that is, N

u(x) = y j XiUi(x) i=l

The following assumption is crucial to Harsanyi's arguments. Assumption H. If two alternatives defined by the implementation probability distributions of effects x are equally preferred by each individual, then they are also equally preferred by the whole group state two more assumptions [18]. Assumption 3 (additive independence). The set of criteria t/i, t/2, - • •, UN is additively independent. Assumption 4 (strategic equivalence). The decision maker's conditional utility function u\ for criterion Ui that is a utility for individual i is strategically equivalent to individual z's utility function [9]. Assumption 4 can be viewed as the "honesty" assumption. Assuming that individual i honestly expressed his preferences, the decision maker further uses his utility function w, as his own utility function for estimation of his decisions from the perspective of their desirability for this individual. The following theorem is valid [9]. Theorem 1.1.3. For N ^ 2, assumption 3 (additive independence) and assumption 4 (strategic equivalence) hold true if, and only if, N

u(x) = Y^Kui(x), t=i

(1.1.4)

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where Ui, i = 1,2,... ,N is individual i's utility function with the scale from 0 to 1, Xi are positive scalable coefficients and x is implication. This theorem shows that assumptions H, 3 and 4 together are equivalent. The decision maker's comparison of individuals' preferences is necessary for finding the values of scalable coefficients A*. The fact that A, are positive ensures a positive relationship between the decision maker's and individuals' preferences in the sense of assumption 2 [17]. Assumption 2 (positive preference relationship). Suppose some alternatives a and b are equally preferred by the group. If a changes and becomes the alternative a* so that an individual c prefers a* to a while the other individuals consider them to be equivalent, then the whole group prefers a* to b. We shall consider an assumption that is weaker than assumption 3 [22]. Assumption 5 (utility independence). Each of the criteria Uj, j ^ i, j = 1,2,..., N is utility-independent of the other criteria Ui. Under this assumption, the utility function u(x) is [21] u(x) =uD(ui,...,ui:...,uN)

= gi(lii) + fi{ui)u*(ui)

for all i,

(1.1.5)

where u = (u\,... , U J _ I , U J + I , . . . ,UN) are all /i-positive, and u\ is the decision maker's conditional utility function for criterion Ui. These representations of the utility function that is dependent on both two and more variables were obtained for a particular set of assumptions in Mayer R . F . [22], Keeney R.L. [16], [15], [18]. The following theorem is valid [22]. Theorem 1.1.4. For N ^ 2, it follows from assumption 4 (strategic equivalence) and assumption 5 (utility-independence) that N U(x)

- UD(U\,U2,.

.-,UN)

N

= ^AjMi(x) + A i=l H

"^2

Xi,\jUi(x)Uj(x)

+

?=!) 3>i h AW~1A1A2 . • ■ AJVMI(2)112(1)..

.UN(X),

where the scales for measuring u and m are chosen so that the values of these functions are in the interval between 0 and 1, and A are the scaling coefficients and 0 < Xi < 1 for all i. Let us consider another assumption. Assumption 6 (consensus). If all the members of the group have the same utility function, then the group utility function must be the general utility function for all members.

24

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These instruments enable one to consider individual preference aggregation as some problem of fuzzy programming. Section 1.1.3 examines this issue in more detail. Furthermore, the instruments permit the construction of some group utility function and, through aggregation of every individual's preferences, the construction of group preferences. Pareto optimality. The principle of Pareto optimality is the corner stone of most group decision theories [19]. Definition 1.1.1. Principle of Pareto optimality. An alternative is thought of as being Pareto-optimal if any other alternative that is preferred by some members of the group to a greater extent is preferred at the same time by the other members to a lesser extent. The principle of Pareto optimality states that the alternative that is not Pareto-optimal must never be chosen, otherwise, under this choice, the level of satisfaction of at least some individuals cannot necessarily be raised without prejudice to the interests of the other individuals. As applied to the theory of election, the Pareto optimality is as follows [26]. Pareto optimality. If candidate a is better for all than candidate b, then b cannot be elected. In all cases the choice of the Pareto-optimal alternative (candidate) is said to be rational. 1.1.3. Decision-making under Fuzzy Preference Relation on the Set of Alternatives In many cases the general problem of making a decision can be described mathematically by the set of admissible choices (alternatives) and by the preference relation given on this set, which captures the decision maker's interests. This relation is generally binary, i.e., permits the comparison of two alternatives only, although the settings of ternary relation problems are also possible. Here, the problem of making a decision is to choose an admissible alternative which is superior or not inferior to the other alternatives in the sense of the given preference relation. Binary preference relations on the set of alternatives can be described in two ways: as a subset of the Cartesian product of the set of alternatives onto itself or as the so-called utility function. The utility function is generally the mapping of the set of alternatives onto a number axis [5]. That is, this function assigns a number (an estimate of alternative) to each alternative so that equal numbers (the values of utility function) correspond to equivalent

Social Choice Problems

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(equally important) alternatives, and a larger number is assigned to the superior alternative between two nonequivalent alternatives. However, not any preference relation and not on any set of alternatives can be described by the utility function. In some cases the relation can be described not by one utility function, but only by a finite set of utility functions, in which case the relevant decision problems are usually called the multicriterion problems. In order to describe these problems and find the rational choice of alternatives, we shall use the results obtained by Berezovsky B. A., Borzenko V. I., Kempner L. M. [24], Kazanskaya G. A. Definition 1.1.2 [13]. The decision problems in which the preference relation is described as the utility function are called the mathematical programming problems. The rational decision in these problems is the choice of an admissible alternative on which the utility function assumes, whenever possible, a larger value. The fuzziness in the setting of the mathematical programming problem can be contained in both the description of the set of alternatives and the description of the utility function. The problems in which the description of the set of alternatives is fuzzy and that of utility function is precise are discussed in Bellman R. and Zadeh L. [3], where the multistage decision problems under fuzzy initial conditions are first analyzed by the method of dynamic programming and the analysis and approach to solving the problems of fuzzy mathematical programming are also discussed. Similar problems are studied in [25]. But here we shall concentrate on some types of problems in which the description of utility function is fuzzy. In general, the problems of mathematical programming constitute an important subclass of rational choice problems. Using the utility function, the original problem reduces to a simpler problem of choosing numbers among some subset of the naturally ordered number axis [10]. The values of the objective (utility) function describe the effect produced by the choice of one or another method of actions (alternatives) [13]. Essentially, the adequacy of this model to reality is largely defined by an extent to which this function captures the relationship among various factors of a real process or system. In the mathematical modeling of a complex system, allowance for a sufficiently large number of real factors is impossible because this would overcomplicate the model. Therefore, we have to introduce only a limited number of factors that are under some considerations deemed most essential. Two approaches are possible here. The factors ignored by the model

26

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description can be deemed absolutely inessential and completely ignored in the decision-making where this model is used. On the other hand, in the second approach the "inessential factors" may not be explicitly introduced in the model, but their effect can be allowed for under the assumption that the response of the model to one or another effect (the choice of an alternative) can be known only approximately (fuzzily). To describe the fuzzy response, we may resort to the help of experts (advisers) representing the effect on the system performance in the model of factors. The larger is the number of the factors involved in the description of the model, the lower is the degree of fuzziness in the description of system responses to effects. Thus, in the second approach the complex system is described by some fuzzy objective function, which assigns for each alternative (an effect on the system) a fuzzy response of the system to the choice of alternative. It is best to start the study of the process for making a rational decision with the set of admissible decisions or alternatives. Depending on the available information, this set can be successfully described to one or another degree of precision. For example, let X denote a universal set of alternatives and Hc{x) the fuzzy description of its subset of admissible alternatives. The values of function fic describe the admissibility of the relevant alternatives in this problem. If, apart from this function, there is no information about the real situation being studied, then we may take as rational the choice of any alternative out of the set xD \ x\x e X, fj,c(x) = sup nc(y) \ , I j/6X J i.e., any alternative having the maximum admissibility, because there is no reason to prefer a particular alternative among these alternatives to the other alternatives. The information about the decision-making process, on the basis of which some alternatives are preferred to the other alternatives, can be given in various ways [25], [9]. The description of information as a preference relation on the set of alternatives is deemed more universal One way to reveal the preference relation in constructing the mathematical model is to consult with a decision maker or experts [9]. The implication here is that the decision maker or experts has the knowledge or notion of the object being studied which has not been formalized in the model because of excessive complexity of such formalization or for some other reasons.

Social Choice Problems

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Suppose that the definite preference relation R in the set of admissible alternatives X is revealed with the help of the decision maker or experts. This means that one of the following propositions can be made for any pair of alternatives x,y € X. Proposition 1.1.1. "x is not inferior to y", i.e., x ^ y or (x,y) G R. Proposition 1.1.2. "y is not inferior to x", i.e., y ^ x or (y,x) G R. Proposition 1.1.3. ux and y are incomparable", i.e., (x,y)

^ R and

(y,x) $R. Information in this form enables one to narrow down the class of rational choices by incorporating therein only those alternatives that are not dominated by any alternative of the set X [16]. In order to elucidate what alternatives are deemed nondominated, we distinguish the strict preference relation R and indifference relation R1 corresponding to the preference relation R [24]. Definition 1.1.3. We shall say that alternative x is strictly superior to alternative y if at the same time x ^ y and y ^ x, i.e., (x,y) G R and (y,x) £ R. The set of such pairs (x,y) is called the strict preference relation Rs on the set X. The relevant indifference relation R' in X is defined as follows [15]. Definition 1.1.4. The pair (x,y) G R' if, and only if, neither x ^ y, nor y ^ x, or both preferences are simultaneously satisfied. In other words, (x,y) G R' when the available information in the form of the preference relation is insufficient for making a choice between alternatives x and y. Definition 1.1.5 [15]. If(x,y) G Rs, we say that alternative x dominates alternative y(x >- y). Alternative x G X is called nondominated in the set (X, R) if {y, x) £ Rs for any alternative y G X. In other words, if x is the nondominated alternative, then the set X has no alternative which would dominate x. Nondominated alternatives are in a sense unimprovable on the set (X, R), and their choice in the decision problem is by right deemed rational. Thus, information in the form of the preference relation R enables one to narrow down the class of rational choices in X to the set of nondominated alternatives of the form [18] Xnd- = {x\x G X, (y,x) $ RsVy G X). The larger is the amount of information about the real situation, the

28

Decision Making and

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narrower is the preference relation, the narrower is the set X n d ' , and hence the smaller is the uncertainty in rational choice of alternatives. A more flexible method to formalize the experts' knowledge of a real situation is the method in which they have an opportunity to describe the degree of their confidence in preferences among alternatives, numbers in the interval [0, 1]. As a result, the fuzzy preference relation in the set of alternatives is revealed with the help of experts where each pair (x, y) corresponds to the number describing the degree of preference satisfaction x^y. This method of describing the relation permits to a larger extent the experts' knowledge and notions to be introduced into the model, thereby making the model in a sense more adequate to reality. We shall consider two types of problems. The Rational Choice of Alternatives Subject to Criterion Convolution We will consider the following problem [17]. Given the set of alternatives x, each alternative is characterized by several attributes with numbers j = l , . . . , m . Information about the paired comparison of alternatives per attribute is represented as the preference relation Rj. Thus, there are m preference relations Rj on the set X. Suppose the relations Rj are described by the given utility functions fj : X —r R1, where R1 is the number axis. The value of function fj(x) can be interpreted as the numerical estimate of alternative x by attribute j . The alternative with a higher estimate fj(x) is preferred by attribute j . The problem is to make a rational choice among the alternatives on the set X, Ri,..., Rm, as based on the available information, i.e. choose the alternative having higher estimates in all attributes. The rational choice in this case is the choice of alternative x0 6 X having the following properties fj(y)>

fj(xo),

j = l , - - - , m = $ fj(y)fj{x0),

i-l,...,m.

(1.1.8)

Then each of the utility functions fj describes an ordinary preference relation on X of the following form [25]: Rj = {(x,y)\x,y

6 X, fj(x) > fj(y)},

(1.1.9)

We shall use in the following the results and implications from [10], [6]. From [19] we have: m

Qi = f] Rj

(1.1.10)

Social Choice Problems

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the set of all nondominated alternatives in the set a;, Q\ that is coincident with the set of nondominated alternatives for the set of functions fj,j = l,...,m. The intersection of relations Rj [10] is now represented somewhat differently. Let

H^y) = [l

for

{x ye

' ^

(i.i.ii)

,yJ

^ [0 for (x,y$Rj) ' be the membership function of relation Rj. Then Q\ corresponds to the membership function MQi(z,2/) =min{ j ui(x,3/),...,// m (a;,2/)},

(1.1.12)

that is similar to the convolution of criteria fj [6] of the form F(x) = min A,/,-,

(1.1.13)

j=l,m

as applied to multicriterion problems. Numbers Xj in this convolution play the role of coefficients in the relative importance of the functions being studied. In the convolution of relations (1.1.13) Xj = 1, j = 1, . . . , m this corresponds to the fact that the given relations are all equally important in the choice of alternatives. If the given relations Rj differ in importance, i.e., the relevant attributes differ in importance by which the alternatives have to be compared, then the convolution (1.1.13) uses the coefficients Xj having different values. The convolution of initial relations Rj with coefficients Xj such that m

^A

j

=

l, Xj^O,

j = l,...,m,

(1.1.14)

i=i

results in the membership function of the form [14] VQi(x>y) =min{Ai/xi(a;,2/),...,A m /x m (a;,y)},

(1.1.15)

i.e., the membership function of the fuzzy preference relation. We now introduce the convolution of another form m

VQ2(x,y) = y}2XjHj(x,y),

(1.1.16)

By the set of utility functions, this convolution provides further information on the relative degree of nondominance of alternatives and thus reduces the class of rational choices to the set

30

Decision Making and Programming

X"d

= \x\xGX, [

u»Qd(x) =

sup n&ix')}, x'€X2"dJ

(1.1.17)

where fiQd(x) = lsup[fiQl (y,x) — fiQ2(x,y)] is the fuzzy subset of nondominated alternatives on the set (X,HQ2). X^'d' is the subset of nondominated alternatives on the set (X, p,Q2). The rational choice is the choice of the alternative having the highest degree of nondominance, i. e., the largest value of function / u nd -. Chapter III discusses two examples and provides an algorithm to construct a rational choice of alternative in terms of a given class of problems. The Rational Choice of Alternatives in Terms of a Set of Attributes Given the set of alternatives (or objects) X and the set of attributes (or experts) P, each of the attributes of the set x € X is to some extent an inherent characteristic of each alternative P. For each fixed attribute p £ P, we know the fuzzy preference relation ((p on the set of alternatives X or, to put it somewhat differently, we know the membership function ip : X x X x P => [0,1] whose value p(xi,X2,p) is interpreted as the degree to which the alternative X\ is preferred to the alternative 22 in terms of attribute p. If P is the set of experts, then cp(xi,X2,p) is the preference relation on the set of alternatives offered by expert p. Thus, the function ip describes the family of fuzzy preference relations on the set X in terms of parameter p. The elements of the set P differ in importance. Let fi : P x P =>■ [0,1] be the given fuzzy relation of importance for attributes (experts); the quantity /i(Pi>P2) is taken to be the degree to which attribute p\ is thought of as being Jess important than attribute P2The problem is to choose rationally among the alternatives in the set X by taking into account the above information. We shall describe one possible way to solve this problem as based on the notion of the fuzzy set of nondominated alternatives. Let (pnd(x,p) be the fuzzy subset of nondominated alternatives corresponding to the fuzzy preference relation

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