Information Gap Decision Theory: Decisions under Severe Uncertainty

Preface to the 1st edition Information-Gap Decision Theory

Everyone makes decisions, but not everyone is a decision analyst. A decision analyst uses quantitative models and computational methods to formulate decision algorithms, assess decision performance, identify and evaluate options, determine trade-offs and risks, evaluate strategies for investigation, and so on. This book is written for decision analysts. The term "decision analyst" covers an extremely broad range of practitioners. Virtually all engineers involved in design (of buildings, machines, processes, etc.) or analysis (of safety, reliability, feasibility, etc.) are decision analysts, usually without calling themselves by this name. In addition to engineers, decision analysts work in planning offices for public agencies, in project management consultancies, they are engaged in manufacturing process planning and control, in financial planning and economic analysis, in decision support for medical or technological diagnosis, and so on and on. Decision analysts provide quantitative support for the decision-making process in all areas where systematic decisions are made. The power of analysis in any field derives from generality and abstraction. This is true of decision analysis, and it gives rise to generic theories such as info-gap decision theory. The decision analyst thus faces a double challenge: to master the abstractions as well as to bring them down to earth. This book assumes familiarity with university mathematics. However, the emphasis is not on theorems and proofs but rather on how to formulate and analyze decisions. The book is practical, but from a theoretical perspective. As Boltzmann put it, nothing is more practical than theory. This book is therefore full of equations as well as examples from a plethora of fields. Each reader will find the most suitable path among the diversity of material presented. The decision analyst is different from other decision makers such as executives, administrators, and similar individuals responsible for public, corporate or professional decisions of many sorts. The decision analyst often fills a supporting role to the executive decision maker, and as such the analyst must be able to translate quantitative results into terms which XI

are meaningful and relevant in the wider decision context. This is another aspect of the decision analyst's double responsibility as both keeper and expounder of the equations. (The fact that the decision analyst and the decision executive may be one and the same individual in no way relieves the decision analyst's responsibility to explain what the equations mean in qualitative and personally or socially meaningful terms.) The executive decision maker also has something to learn from this book. The person responsible for coordinating and finalizing a complex decision must understand what can be provided by quantitative decision analysis, and what cannot. Furthermore, the types of decisions we consider here are often open-ended, unstructured and accompanied by severe lack of information. This implies that the analysis and finalization of a decision is not a single-loop input/output process. It cannot be done by sending specifications to the analyst and waiting for the answer. Difficult decision-making, especially under severe uncertainty, is a process of evaluating and revising assumptions, goals, methods, information, preferences, etc. The executive decision maker must know what sort of guidance the decision analyst needs, and what sort of interaction is not only possible, but often essential, in the development of good decision strategies. Chapters 1 and 12 will be helpful in understanding what info-gap decision theory can do, without the details of how. Info-gap decision theory is radically different from all current theories of decision under uncertainty. The difference originates in the modelling of uncertainty as an information gap rather than as a probability. The need for info-gap modelling and management of uncertainty arises in dealing with severe lack of information and highly unstructured uncertainty. What is an information gap? How is it quantified? How does one use info-gap ideas to analyze such central (and traditionally probabilistic) concepts as risk, gambling, reliability and so on? This book addresses these and many other questions. New theories are like virgin orchards, and much fruit is ripe and ready to eat. But the quest for fuller answers, and even for new questions, is still at full steam. The reader is invited to join the search. Yakov Ben-Haim The Technion Haifa, Israel June 2001

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Preface to the 2nd edition This second edition entails changes of several sorts. First, info-gap theory has found application in several new areas — especially biological conservation, economic policy formulation, preparedness against terrorism, and medical decision-making. This calls for the inclusion of pertinent examples. Second, the combination of info-gap analysis with probabilistic decision algorithms has found wide application. Consequently "hybrid" models of uncertainty, which were treated exclusively in a separate chapter in the previous edition, now appear throughout the book as well as in a separate chapter. Finally, info-gap explanations of robust-satisficing behavior, and especially the Ellsberg and Allais "paradoxes'', are discussed in a new chapter together with a theorem indicating when robust-satisficing will have greater probability of success than direct optimizing with uncertain models. Numerous people have contributed to the development of info-gap theory in recent years. I especially wish to acknowledge useful comments and criticisms by Farooq Akram, Yael Ben-Haim, Zvika Ben-Haim, Mark Burgman, Yohay Carmel, Cliff Dacso, Susan Haack, Jim Hall, Wendell Iverson, Joe Moffitt, Izuru Takewaki and Miriam Zacksenhouse. Yakov Ben-Haim The Technion Haifa, Israel February 2006

Chapter 1

Overview Fm asking that possibility be placed above scripture, that we be unhampered by the conceit of existing knowledge. The only danger from the unknown is in our turning our backs to it. Elliot Baker [6, p.64]

The freedom to decide is an opportunity to err; but every opportunity is also a potential for success. Decision-making is an utterly human activity. A decision theory must reflect the range of human propensities and characteristics, even if the decision agent is an ensemble of individuals, or an institution or an automated algorithm serving a human need. As the decisions become more difficult, dangerous or promising, the homunculus embedded in the theory must become more realistic, subtle and multi-dimensional. This book lays out the concepts and methods of info-gap decision theory. These tools can be used to formulate and evaluate decision algorithms. But decision-making is rarely a closed one-directional procedure, but rather an iterative process of exploration, learning and evaluation. Info-gap theory assists a decision maker to develop preferences, assess risks and opportunities, or choose sources of information and lines of exploration in light of the analysis of severe lack of information.^ Information and uncertainty. Foremost among the properties of a good decision methodology is that it must be driven by available information, as distinguished from suppositions which are reasonable but unverified. The decision-making process must adapt naturally to the type of information accessible to the decision maker. The central emphasis of info-gap decision theory is that decisions under severe uncertainty must not demand more information, or at least not much more, than the decision maker can ^We use the term 'information' in its broadest lexical meaning, referring not only to facts or data, but also to knowledge and understanding.

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reliably supply, and that is not much in the conditions under which the decision must be made. As such, info-gap decision theory is not a universal theory suitable to all situations. Classical statistical decision theory, or von Neumann-Morgenstern expected utility theory, remain bulwarks for a plethora of decision problems, especially those endowed with fairly well understood and highly structured uncertainties. Info-gap theory comes into its own when significant consequences hinge upon decisions made under severe lack of information. The emphasis which info-gap decision theory places on informationdriven decisions has a profound eff'ect on the manner in which the theory is formulated. Where classical decision theories typically begin by stressing the axiomatic basis of rational decision-making, info-gap theory begins by asking: what does the decision maker know, and where are the gaps in the extant knowledge? What is the extent of our fundamental understanding, how much historical evidence do we have, and is the process prone to surprises? Uncertainty is the complement of knowledge, and info-gap models of uncertainty are formulated minimalistically with respect to the agent's information. Of course, info-gap decision theory has a logic of its own which is rational in several senses of that term. If it lacked an intuitively satisfying axiomatic basis it would fail to reflect the human side of decision-making. But the starting point of our development, in chapter 2, is the heuristic study of the formulation of information-gap models of uncertainty. An information-gap model of uncertainty is a non-probabilistic quantification of uncertainty. Info-gap models entail no measure functions: neither probability densities nor fuzzy membership functions.^ Info-gap models concentrate on the disparity between what is known and what could be known, while making very little commitment about the structure of the uncertainty. In formulating an info-gap model, prior information about the uncertain phenomena is invested in determining the structure of a family of nested sets of events. Clustering of events is the central organizing concept of info-gap models of uncertainty, rather than frequency of recurrence, likelihood, plausibility or possibility of events. This book develops a theory of decision, including ramifications of risk, gambling, learning and other subtleties, in the context of this very sparse theory of uncertainty. Our focus throughout this book will be on how to make good decisions with very sparse information. We are primarily interested in prescriptive support for decision makers. The information-driven nature of info-gap decision theory is particularly important for this goal. What the decision maker knows, or could learn, is critical in formulating a decision model, in evaluating or estimating its performance, and in devising wise resource investment and development strategies for improving the decision model. •^ We will encounter info-gap models which represent lack of knowledge about probability functions. These info-gap models will contain probability functions as set-elements, but the quantification of the info-gaps surrounding these probability functions will be non-probabilistic.

C H A P T E R 1.

OVERVIEW

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Furthermore, fragmentary information about complex phenomena, or even extensive data about processes which are prone to structural change, is often quite incompatible with representation as a lottery. While the lottery is the paradigm of uncertainty for most classical decision theories, info-gap models of uncertainty, whose formulation is driven by the decision maker's highly deficient information, are different from lotteries in many ways. Info-gap decision theory is based on quantitative models and provides numerical decision-support assessments. However info-gap theory is not a closed computational methodology. Rather, the quantitative assessments assist the decision maker to evaluate options, to develop strategies, and to evoke and evolve preferences in light of the analysis of uncertainties, expectations and demands. Info-gap models of uncertainty originated in the technological sciences, and the early work on decision-making with info-gap uncertainty concentrated on engineering analysis and design. This is in rather marked contrast to the development of most current decision theories, which have been intensively pursued by economists and other social scientists, psychologists, management and operational researchers and related scholars in the supporting disciphnes of mathematics and statistics. Info-gap decision theory has been heavily influenced by the classical theories, primarily in the identification of the roles and goals of a decision theory, and much less in the formulation of questions or methods of solution. Many concepts from classical theories such as risk aversion, value of information, and learning, have identifiable but different manifestations in info-gap theory. Not surprisingly, the ease with which classical topics find expression in the new theory has led to the reciprocal application of info-gap ideas outside of the technological domain in which they originated, as evidenced in this book by the range of examples. Info-gap decision theory is not a technological theory per se, but rather a methodology with a much wider domain of relevance. Competition as well as mutual stimulation of theories is a major catalyst for development. Robustness and opportuneness. The next stage in our formulation of info-gap decision theory is to address two contrasting consequences of uncertainty: catastrophic failure and windfall success. Uncertainty may be either pernicious, entailing the threat of failure, or propitious, entailing the possibility of unimagined success. Info-gap theory addresses these two confiicting potentials with two immunity functions. The robustness function assesses the immunity to failure, while the opportuneness function assesses the immunity to windfall. These are the basic decision functions of the theory. The immunity functions are formulated in chapter 3, along with the paradigm of the info-gap decision model and a series of examples from engineering design, project management, manufacturing, economics, biological conservation and other areas. The immunity functions of robustness and opportuneness are the heart of info-gap decision models. The immunity functions express the basic knowl-

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edge and uncertainties, as well as the needs and expectations, which the agent brings to the decision problem. The immunity functions facilitate the decision maker's study of various trade-offs inherent in the decision. In particular, the robustness and opportuneness functions enable the decision maker to formulate preferences on the options in light of the uncertainties. A central conclusion is that the analysis of uncertainties can alter the decision maker's preferences. Info-gap decision theory is not normative or nomological in the rigid sense of dictating decisions based on formalized input from the decision maker. The immunity functions are the basis for a process of exploration, by the decision analyst, of options and implications, leading to an evolution of preferences. In info-gap decision theory this evolution of preferences is catalyzed by the immunity functions, and the examination of this evolution recurs frequently throughout the book. In many situations the decision maker will choose to concentrate only on survival: the avoidance of failure as expressed by the robustness of the decision. However, excessive concentration on failure prevention can lead to lethargic behavior and to the loss of promising opportunities. An important question is whether or not robustness to the pernicious side of uncertainty can be enhanced while also promoting the propitious potential for windfall. Interestingly, the answer is that sometimes robustness and opportuneness are antagonistic — one at the expense of the other — and sometimes they are sympathetic and mutually supporting. The antagonism or symbiosis between thwarting failure and currying success is explored in chapter 5. Value judgments. The immunity functions — robustness and opportuneness — are quantitative, but numbers are not enough. The decision maker must make value judgments: how much robustness to the perniciously uncertain environment is needed, and how much should the opportunities from ambient uncertainty be facilitated and at what sacrifice of robustness? The answers to these questions cannot be unique or algorithmic. They can at best be qualitative and imprecise. But the responsible decision maker must forge a connection between quantitative decision analysis and qualitative, linguistic and even subjective values comprising the context of the decision. Chapter 4 explores various methods for calibrating the immunity functions. The elementary methods of dimensional analysis and of calibrating from past experience are discussed first. Then the more ambitious task is begun, of evaluating quantitative increments of robustness in terms of natural language whose meaning is anchored in the values of everyday life. This is based on reasoning by analogy which, though outwardly similar to deduction, is in fact not a method of rigorous proof but rather of plausible explanation aimed at obtaining qualitative assessments based on personal or collective values. Gambling and risk-taking. The exercise of the human will lies at the heart of all challenging and difficult decisions. Nothing evokes the humanity of a decision maker more than risk, and nothing distinguishes different de-

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5

cision makers more sharply than their responses to a gamble. In chapter 6 we explore the info-gap conceptions of risk and gambling. Current technical usage of the term 'gambling' is overwhelmingly probabilistic, and quite understandably so. The early mathematical theory of probability sprouted from the study of games of chance, and probability has flourished ever since. But probability is not synonymous with uncertainty, and probabilistic models do not capture all facets of the phenomena associated with risk-taking. No single theory can reflect the variegated texture of human reactions to and management of risk. In chapter 6, perhaps more than in any other chapter, we take care not to allow existing answers and methods to overwhelm the formulation of our questions and solutions. The info-gap intuitions and quantifications of risk and gambling are numerous, and depend intimately upon the robustness and opportuneness functions and their properties which are studied in preceding chapters. Different formulations emerge in response to different problems. Sometimes the parallel to probabilistic ideas is self-evident, and sometimes the distinctively info-gap flavor is dominant. In any case, the need to quantify and characterize the element of gambling in decisions with severe uncertainty motivates the development of intrinsically info-gap tools. The info-gap analysis of gambling behavior and of perceptions of risk hinges on establishing how the decision maker uses the immunity functions to make decisions. Where, on the various trade-ofl"s, does the agent choose to operate, and how are different choices perceived? When is the decision maker displaying an aversion to or proclivity for risk, and what is riskneutrality? Value of information. When information is scarce its value can be quite great. A small hint may lead to a different course of action which may tip the scale from failure to success. In chapter 7 we quantify the value of information based on the info-gap robustness function. Again we are treading on terrain already covered by powerful probabilistic theories, as in the study of gambling. Again we must follow the logic of info-gap uncertainty to its natural conclusions, allowing the exigencies of severe uncertainty to claim their due. We begin by discussing the informativeness of an info-gap model of uncertainty. An uncertainty model expresses what we do not know or cannot assert about the phenomena in question. An uncertainty model is informative to the degree that it constrains the ambient possibilities, to the degree that it imposes flne-grained distinctions between what can and cannot occur at any given horizon of uncertainty. Then, building on the properties of the robustness function, a demand value of information is formulated: the amount of reward which a decision maker would be willing to relinquish in exchange for new information. The demand value need not be monetary but rather can express other values which are intrinsic to the system under study. While these ideas are classically economic in flavor, the discussion is entirely info-gap rather than probabilistic.

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INFO-GAP DECISION THEORY

Learning. Learning is the process of acquiring valuable information. Learning can take different forms: either the assimilation of data, or the fundamental revision of structures for information-processing. The latter is our primary concern in chapter 8. This is a broad category, and we consider a number of aspects and applications. The process of learning is impelled by error, and the fundamental questions around which our analysis crystallizes are: when is a given decision algorithm inadequate, and what constitutes evidence against the uncertainty model upon which the decision algorithm is based? The underlying theme is that strategic learning involves the revision of info-gap models of uncertainty; the frontier of our understanding moves as we learn, causing modification of our models of uncertainty. The updating of info-gap models is itself an info-gap decision, thus embedding one decision algorithm in another. Coherent uncertainties and consensus. A central idea in the theory of info-gap decisions is that the perception and modelling of uncertainty plays a critically formative role in the establishment of preferences. An immediate consequence is that conflicting preferences among decision makers can arise from disparate understandings of ambient uncertainty. Conversely, agreement can be reached if perceptions of uncertainty can be brought closer together. In chapter 9 we explore the question: by how much and in what ways can assessments of info-gap uncertainty differ and still allow agreement among decision makers? The main idea is to evaluate the degree of coherence between the info-gap models of the competing parties. We explore what one can learn from this coherence about the possibilities for agreement. Hybrid uncertainties. This book is a study of decision-making with info-gap uncertainty. It is recognized, however, that no single theory can supply all the needs of decision analysts, and that some information is well modelled probabilistically. If one has probabilistic information it should be exploited; if it is fragmentary, it can be combined with info-gap models. We illustrate this sort of hybrid-uncertainty analysis in examples throughout the book. Chapter 10 explores in greater depth three distinct approaches to the construction of hybrid probabilistic/info-gap decision algorithms. Robust-Satisficing behavior. In this book we are concerned almost exclusively with prescriptive decision theory: helping decision makers formulate and evaluate decisions. In chapter 11 we deviate from this pattern and study the ability of info-gap theory to explain two famous anomalies in human behavior under uncertainty: the seemingly paradoxical observations of Ellsberg and of Allais. We show that info-gap theory provides simple explanations of these results. We then develop an info-gap model of expected-utility risk aversion, and conclude the chapter with a theorem showing that robust-satisficing behavior has a better chance of survival than direct optimization in a wide class of situations. This helps to explain the wide occurrence of robust-satisficing strategies in competitive environments.

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R e t r o s p e c t i v e essay. We have now completed our quantitative study of the basic elements of info-gap decision theory, and chapter 12 reviews the concepts in an entirely qualitative discussion. The context of this retrospective examination is the assessment of risk in managing a multi-task project. However, this specific format will not obscure the generic nature of the analyses involved in making decisions under severe uncertainty. Chapter 13 is devoted to a discussion of imphcations of uncertainty in general, and of info-gap uncertainty in particular. Various philosophical as well as practical issues are explored.

The challenge of decision-making under severe uncertainty is as old as thinking and as current as each moment's passing thought. The large issues are clear and constant, while the solutions are incomplete and transient. No book, no theory, can or should hope to close the dialog on decision-making, and this book is no different. We will forever strive to enhance our ability to understand and manage the gap between what we do know and what we could know.

INFO-GAP DECISION THEORY

The following table shows the 13 chapters in road-map form. Chapters 3 through 6 are the core of the book, and section 3.1 is the kernel of this core. Chapter 2 provides a conceptual and quantitative foundation in info-gap uncertainty. The reader may choose to consult the quantitative parts of this chapter as the need arises later in the book. Chapters 7 through 11 develop some aspects of more advanced topics. Chapter 12 is a qualitative discussion of many of the concepts developed earlier. Chapter 13 concludes the book with discussion of various foundational implications of info-gap uncertainty. Many examples from a very broad range of topics are discussed throughout the book. The reader is encouraged to focus on those examples whose topical domain is of special interest. Problems are provided to help deepen the reader's understanding and to explore additional directions.

Chap. 1 Overview

Chap. 2 Uncertainty ^

Chap. 3 Robustness and Opportuneness

Chap. 5 Antagonistic and Sympathetic Immunities

Chap. 4 Value Judgments

Chap. 6 Gambling and Risk Sensitivity

^ Chap. 7 Value of Information

Chap. 8 Learning

Chap. 9 Coherent Uncertainties and Consensus

Chap. 10 Hybrid Uncertainty

^ Chap. 12 Retrospective Essay: Risk Assessment in Project Management ^ Chap. 13 Implications of Info-Gap

Uncertainty

Chap. 11 RobustSatisficing Behavior

Chapter 2

Uncertainty It is evident ... that we can have no reason to think that every phenomenon in all its minutest details is precisely determined by law. That there is an arbitrary element in the universe we see, — namely, its variety This variety must be attributed to spontaneity in some form. Charles Sanders Peirce [128] Maybe the task ahead of us is to live rigorously with uncertainty, and bridges need not fall down in the attempt. Gian-Carlo Rota, [148, p.3]

2.1

Historical Perspective

In many ancient societies it was believed that names control their subjects. Curses as well as blessings were directed at an object through its name. The act of naming is close to the act of creation itself, for giving a name gives shape and quality to the subject, forming it. filling it with real living dimension. We tend to take a more skeptical view of names today, but the modern counterpart is: give it a number. Passports, identification cards, social security accounts, credit cards, driver's licences, auto and health insurance, the list of items by which we are numbered is. it would seem, almost numberless. We are an age of number-givers, and the first advice to a novice in the modern world would be: if it stands still, measure it; if it moves, clock its speed. But the modern world didn't start yesterday. 400 years ago Galileo wrote that "this grand book the universe . . . is written in the language of mathematics . . . without [which] one wanders about in a dark labyrinth." 9

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[62, p.70]. 200 years ago Pierre Simon, Marquis de Laplace, completed his monumental three-volume Celestial Mechanics in which he develops the equations of motion of the planets, extending with powerful new mathematical tools the earlier work of Newton and others. Laplace presented a copy of his book to Napoleon, who remarked that in all these hundreds of pages about the heavenly bodies there is no mention of the Creator of the universe. Laplace haughtily replied that he had no need for such an hypothesis. Throughout the first half of the 19th century social scientists counted and measured everything that came to mind: crime rates, suicides, anatomical dimensions, population size. Even the number of undelivered letters in the Paris central post got attention because it was nearly constant from year to year [138, p.51]. People rejoiced in the belief that society could be mastered with numbers just as Laplace's equations ruled the skies. The postal discovery was exciting because it indicated the power of a new idea: averages. One can never imagine all the factors which cause a letter to be undeliverable, but somehow a simple orderliness emerges out of a chaos of complexity and uncertainty. The implications were tremendous. Discoveries such as this led to the rise of insurance programs, the use of census data for social planning, predictions of crime, disease and natural disasters, and countless other concerns growing from the bubbling confusion and variability of human society. The 19th century discovered the usefulness of statistics for managing uncertainty. At first sight it seemed impossible. The border between what is known and what is unknown seemed impenetrable. What can one 'know' about the 'unknown' ? But experience showed that simple universal patterns exist, even for phenomena whose details are beyond human understanding. Laws can be found which govern even what is hidden from our knowledge. This was the first step across the frontier of firm knowledge into the misty wilderness of uncertainty. We are flooded with statistics today, from the average of just about anything to its cross-correlation with just about anything else. We have national agencies for compiling and processing statistics. We have statistical consulting firms which sample, predict or explain everything under the sun. We have professors of statistics who expound their discipline in universities throughout the world. What began as 'statistics' (meaning a number) has become 'Statistics' (a profession). We don't imagine the statistics professor standing at the lectern reciting tables of data. The theory of statistics deals with numerical data no more than the bank director concerns himself with bank notes. What started with Parisian letter-counting and simple averaging has developed into a science for modelling and managing uncertainty. Statistics has become a profession, quite apart from the specific problems to which it is applied. Uncertainty had "arrived" by the end of the 19th century and it has continued to blossom ever since. New patterns of uncertainty emerge due to shifts in the frontier between the hinterland of solid understanding and the uncharted regions of the un-

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known. Since the end of the 19th century the physical sciences have led the way to new understandings of uncertainty. The subject of statistical mechanics relates bulk properties of matter, like the boiling point of a liquid, to the frothy microscopic confusion of countless molecular trajectories. Like the Parisian dead-letter problem, the statistical theory of gases and liquids relates macroscopic observable occurrences to microscopic phenomena of boundless complexity. Statistical mechanics reveals order emerging out of chaos. The newer science of quantum mechanics shows us uncertainties even in the position and momentum of individual molecular motion. Unlike earlier theories, quantum mechanics suggests a basic physical indeterminacy of natural phenomena, perhaps even a violation of traditional views of causality. Finally, nonlinear dynamics reveals chaotic uncertainties and indeterminate bifurcations arising out of what seem to be simple and perfectly deterministic equations of motion. New perceptions of uncertainty have also arisen in response to technological challenges. Engineers use information of many sorts to design and analyze technical systems. Sometimes this information is qualitative, nonnumerical, linguistic. A simple example is reasoning based on rules of thumb like Tf it rattles, tighten it'. The uncertainty associated with information of this sort is different from classical statistical uncertainty and has to do with the vagueness of human language: what is a rattling sound, and how tight is tight? In other situations the information is exceedingly sparse. Many high-tech devices are one-of-a-kind, based on accumulated experience from similar but nonetheless distinct devices. In these situations it is necessary to draw conclusions on the basis of evidence and understanding which is dramatically deficient. This is again another form of uncertainty. Everywhere we turn we find, on the one hand complexity, uncertainty and indeterminacy, and on the other hand patterns and rules which dictate the course of these phenomena. Furthermore, what we mean by 'uncertainty' has changed enormously since 'statistics' first became a household word. In this book we concentrate on the fairly new concept of information-gap uncertainty, whose differences from more classical approaches to uncertainty are real and deep. Despite the power of classical decision theories, in many areas such as engineering, economics, management, medicine and public policy, a need has arisen for a different format for decisions based on severely uncertain evidence. It is interesting to note a parallel between the recent development of non-probabilistic models of uncertainty and the emergence of probabilistic thinking in the 17th and 18th centuries. Hacking writes that new modes of dealing with evidence arose in the low sciences, alchemy, geology, astrology, and in particular medicine. By default these could deal only in opinio. They could achieve no demonstrations and so had to resort to some other mode of proof. The high sciences, such as optics, astron-

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omy, and mechanics still lusted after demonstration and could, in many cases, seem to achieve it. They could scorn opinio and any new mode of argument. New modes of argument arose, perforce, among the students of opinion. [85, p.35] There is an interesting parallel here between the emergence of probability in the mid-17th century, and the divergence of modern conceptions of uncertainty from classical probabilistic thinking. Quite naturally, modern non-probabilistic mathematical models of uncertainty have grown out of the exigencies of new problems. But the parallel is closer than that. Just as the low sciences in the 17th century could not aspire to anything like scholastic demonstrations and had to develop new modes of reasoning, so modern engineering, management and social decision-making face degrees and qualities of uncertainty which are unparalleled in the physical sciences. Model-building in these areas is different from in the physical sciences, and the modes and meanings of proof are different as well. Probability and info-gap modelling each emerged as a struggle between rival intellectual schools. Some philosophers of science have tended to evaluate the info-gap approach in terms of how it would serve physical science in place of probability. This is like asking how probability would have served scholastic demonstrative reasoning in the place of Aristotelian logic; the answer: not at all. But then, probability arose from challenges different from those which faced the scholastics, just as the info-gap decision theory which we will develop in this book aims to meet new challenges.

2.2

Is Ignorance Probabilistic?

The place to start our investigation of the difference between probability and info-gap uncertainty is with the question: can ignorance be modelled probabilistically? The answer is 'no'. The ignorance which is important to the decision maker is a disparity between is known and what needs to be known in order to make a responsible decision; ignorance is an info-gap. Our discussion in this section will focus on the principle of indifference [24]. The 'Principle of Indifference' [123] (or 'Insufficient Reason' [85] or 'Maximum Entropy' [164]) all provide solutions to the same problem: select a probability distribution when the prior information is insufficient to do so. That is, the prior information is augmented by appeal to a 'Principle' whose validity (it is claimed) is prior to the information in hand. It is patent that a distribution obtained in this manner is not a 'maximum ignorance' distribution since it depends on knowledge of the truth entailed in the Principle which is employed. For instance, knowing that an uncertain phenomenon is described by some particular distribution, e.g. a uniform or a triangular distribution, is clearly more knowledge than not knowing at all what distribution describes the phenomenon, as can be illus-

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trated with a simple probability riddle [14, chap.7]. We use two riddles to illustrate the perils of indiscriminately adopting the Principle of Indifference and assuming that ignorance can, always, be modelled probabilistically. Keynes' uncertain density. The merits of the principle in question have been debated for decades, and there is no intention here to review the arguments (see for instance [123, p.75-79]). It is illuminating however to recall one of Keynes' examples of the contradictions inherent in indiscriminate use of the Principle of Indifference [99, p.45]. I choose Keynes because of his staunch adherence to probability as the best, indeed in his day the only, means of arguing from uncertain evidence to reasonable conclusions. Consider a group of engineers using a new and incompletely studied material whose specific volume v takes values between 1 and 3cm'^/g, but about which no other information is available. The Principle of Indifference allows us to adopt the uniform distribution for v on the interval [1, 3]. From this we conclude that the specific volume of an arbitrarily selected sample of the material has a probability of 1/2 to lie between 1 and 2: Prob(l 0

(2.3)

U{a^ r) is the set of all functions whose deviation from F(x, t, w) is nowhere greater than a. For a fixed value of a, this set represents uncertainty in the rate function by specifying a range of variation oi r{x^t^w) around the nominal rate function r{x,t,w). The larger the value of a, the greater the range of unknown variation, so a is called the uncertainty parameter or horizon of uncertainty. However, quite often the value of a itself is not known, so in fact (2.3) is not a single set but rather an unbounded family of nested sets of functions. The degree of nesting, as well as the level of uncertainty, is expressed by the uncertainty parameter a. The family of nested sets, U{a^7)^ a > 0, is one specific type of information-gap model of uncertainty. Section 2.5 summarizes a wide selection of types of info-gap models. The info-gap model in eq.(2.3) represents uncertainty at two levels: unknown variation of the r-function at fixed horizon of uncertainty a, and unknown horizon of uncertainty. Since the sets W(a,r) in this family are in fact convex sets,^ this is a convex info-gap model of uncertainty, sometimes called a convex model. The info-gap model in (2.3) is only one of a wide range of commonly used infogap models. The formulation and choice of an info-gap model depends on the type of initial information. Info-gap models with applications to nuclear assay [11], mechanical analysis [32], engineering design [25], reliability theory [14] and many other areas have been studied. When we organize our information (and our ignorance) in terms of families of sets or clusters like this, the decision maker faces sets of rate functions (or whatever is uncertain) which may be confronted. Which function will actually be confronted is unknown. Probability and possibility theories also have us think about sets of events. In probability theory, we might evaluate the frequency of recurrence of rate functions whose value is less than p, for instance. In possibility theory we might ask whether the set of functions with low rates is highly possible. The diff'erence is that with info-gap models of uncertainty we organize the events into clusters but we do not employ ^ A set is convex if the line segment joining any two elements in the set is itself entirely in the set.

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I N F O - G A P DECISION THEORY

distribution functions to measure them; we simply don't have sufficient information to formulate a probability density or a membership function. The distribution functions in probability or fuzzy theory are designed to measure uncertainty and are related in particular ways to the size of sets. Large sets will tend to have a large probability of including events which frequently recur or which quite possibly will happen. But here again is the crux of the difference between distribution-based uncertainty models (probability and fuzzy logic) and info-gap set-models. In info-gap models of uncertainty we can rank degrees of information gap in terms of the size of the uncertainty parameter a, but this is much weaker information than probability or possibility where the distribution functions indicate recurrence-frequency or plausibility. In info-gap set models of uncertainty we concentrate on cluster-thinking rather than on recurrence or likelihood. Given a particular quantum of information, we ask: what is the cloud of possibilities consistent with this information? How does this cloud shrink, expand and shift as our information changes? What is the gap between what is known and what could be known? We have no recurrence information, and we can make no heuristic or lexical judgments of likelihood. The distinction between probabilistic and info-gap modelling of uncertainty is particularly pronounced in the treatment of rare and unusual events [12; 24; 32, section 1.2]. The procedure by which one formulates an info-gap model is basically different from the usual method for specifying a probabilistic model. In probabilistic formulations one often chooses the form of the model, e.g. Gaussian, and then determines the coefficients of that model (mean and covariance in the Gaussian case). This procedure can work quite well when the form of the model is correct, because then the model parameters can usually be estimated accurately without the need to sample too extensively. This is because, as in the Gaussian model, the parameters can be related to the bulk of events which hover around the mean. On the other hand, if the form of the probabilistic model is only approximately correct, then the tails of the calibrated probabilistic model may differ substantially from the tails of the actual distribution. This is because the model parameters, related to low-order moments, are determined from typical rather than extraordinary events. In this case, decisions will be satisfactory for the bulk of occurrences, but may be less than optimal for rare events. It is the rare events — catastrophes, for example — which are often of greatest concern to the decision maker. The sub-optimality may be manifested as either an over-conservative or an unsafe decision.

2.4

Uncertainty and Convexity

An info-gap model of uncertainty is a family of nested sets. The family of sets is usually unbounded. Nearly all of the info-gap models described in section 2.5 are made up of convex sets. One is led to ask why this occurs? Why do uncertain variations tend to cluster in convex aggregates? What is

C H A P T E R 2.

UNCERTAINTY

19

the relation between uncertainty and convexity? In discussing 'The Doctrine of Chances' the American philosopher Charles Sanders Peirce makes the following observation [127, p.62]: When a naturalist wishes to study a species, he collects a considerable number of specimens more or less similar. In contemplating them, he observes certain ones which are more or less alike in some particular respect. They all have, for instance, a certain S-shaped marking. He observes that they are not precisely alike, in this respect; the S has not precisely the same shape, but the differences are such as to lead him to believe that forms could be found intermediate between any two of those he possesses. He, now, finds other forms apparently quite dissimilar — say a marking in the form of a C — and the question is, whether he can find intermediate ones which will connect these latter with the others. This he often succeeds in doing in cases where it would at first be thought impossible; whereas, he sometimes finds those which differ, at first glance, much less, to be separated in Nature by the non-occurrence of intermediaries. Peirce employs the concept of 'occurrence of intermediaries' to describe variation around a norm. He is employing a spatial analogy to uncertainty, as we will discuss on p.22, and as suggested by his example of the shapes of specimens. He generalizes the gradation of one form into another by "the idea of continuity" [127, p.63]. By continuity Peirce seems to mean a generic 'connectedness' or 'betweenness' which characterizes a genus of similar but varying entities. But in this sense the vertex at the bottom of a "V" is 'between' the tips of the "horns", since it is 'continuously connected' to them via the sides of the "V". This is evidently not what Peirce intends, so neither connectedness nor continuity is quite the right idea. We can recognize that convexity, rather than continuity, is actually a much better mathematical metaphor for similar but varying entities. A set is convex if it entirely contains the line segments joining any two points in the set. That is, a set is convex if it contains all linear intermediaries between every two points in the set. It is convexity, and not simply continuity, which characterizes the clustering tendency of diverse but related phenomena. The intuition which Peirce reveals in the above quotation is suggestive of the connection between uncertainty and convexity. However, one cannot attribute to Peirce an acceptance or even an awareness of the info-gap concept of uncertainty. His 'Doctrine of Chances' is a thoroughly frequentist analysis of probability. But why do uncertain entities tend to cluster in convex aggregates? One partial and somewhat mathematical answer comes from the complexity of the entities themselves: convexity arises asymptotically from the superposition of myriad microscopic factors. Without getting into the mathematics (for which one may consult [20] and [32, chapter 2]), the incoherent com-

20

I N F O - G A P DECISION THEORY

bination of a vast array of elements tends, in particular circumstances, to produce a coherence of intermediate forms. This, we have seen, is the defining property of convexity. Another approach to understanding the prevalence of convexity in uncertain aggregates is through the "stone grinder's theorem" [14]. This theorem states that all convex shapes, and only convex shapes, can, when disjoint, always be separated from one another by flat surfaces.^ In other words, convex shapes are those which can be formed by grinding them on a flat surface. For instance, a standard diamond (which is a convex body) is formed on a flat grinding wheel, while the grinding of a heart-shaped stone (which is not convex) requires access to the sharp edge of the wheel. (A^dimensional convex shapes would require an A^ — 1-dimensional flat surface, but that is a technicality which three-dimensional stone grinders happily avoid.) In the spirit of the stone grinder's theorem, we can speculate that convex aggregates of uncertain variates arise from the elimination of involuted or protruding anomalies. When we formulate an info-gap model of uncertainty by asking, as on p. 18, 'What is the cloud of possibilities consistent with a particular quantum of information?', we flnd that this cloud tends to be convex, as in most of the info-gap models of section 2.5.

2.5

Some Info-Gap Models

In this section we define a fairly wide selection of info-gap models of uncertainty, and discuss typical considerations for choosing a model. The reader may wish to skip this section on a first reading, and refer to it as the need arises. On the other hand, some generic issues will be discussed, such as the spatial analogy to info-gap uncertainty on p.22. An info-gap model is an unbounded family of nested sets all sharing a common structure. A frequently encountered example is the family of nested ellipsoids all having the same shape. The structure of the sets in an info-gap model derives from the information about the uncertainty. In general terms, the structure of an info-gap model of uncertainty is chosen to define the smallest or strictest family of sets whose elements are consistent with the prior information. Ingenuity is sometimes needed to formulate an info-gap model capturing all the prior information, which is often in part linguistic, without introducing unwarranted assumptions. After presenting the first class of info-gap models we pause to discuss a spatial analogy to info-gap uncertainty. This will demonstrate that one's conception of uncertainty can be divorced entirely from the concepts which underlie the mathematical theory of probability. Energy-bound models. Among uncertain dynamic phenomena, transients are perhaps the most difficult to pin down. Large deviations which ^Exceptions to the hyperplane separation of convex sets can arise if the sets are unbounded, but this need not concern us.

C H A P T E R 2.

UNCERTAINTY

21

slip away, great but temporary excursions from nominal values, fleeting anomalies, quirks: the prior information about such occurrences is usually quite sparse and imprecise. Furthermore, the "event" is not a single number but rather a function, usually of time or space or both. Sometimes the function is vector-valued rather than having a one-dimensional scalar representation. Energy-bound uncertainty models arise in many situations, where 'energy' is loosely defined as a quadratic function, in analogy to the energy per unit time of an electric current which is quadratic in the current, or in analogy to the energy of deflection in a rigid beam which is quadratic in the curvature. One type of energy-bound info-gap model for a scalar function u{t) which deviates in an unknown transient manner from the nominal function u{t) is: U{a,u) = lu{t) : f

[u{t)-u{t)f

dt0

(2.4)

This is a family of nested sets of uncertain functions. The uncertainty parameter a determines the level of nesting, so a < a' implies that U{a,u) C U{a'^u). The deviation u{t) — u{t) is transient, that is, it must vanish at least asymptotically, because the integral of the squared deviation is bounded by the square of the finite horizon of uncertainty a. Uncertainty in the function u(t) is represented at two levels. For fixed a, the set ZY(a, u) specifies the unknown deviation of u(t) from u{t). Furthermore, the value of the uncertainty parameter a is unknown, which represents the unknown horizon of uncertainty. Energy-bound models can be defined for vector functions as well: U{a, u) = I u{t) : /

[u{t) - u{t)]^ V [u{t) - u{t)] dt < a^\ ,

a > 0

(2.5) where, throughout this section, F is a known, positive definite, real, symmetric matrix. The info-gap models of eqs.(2.4) and (2.5) are often called "cumulative energy-bound models", to distinguish them from the "instantaneous energy-bound model" which is the set of functions whose "energy" is constrained at each instant: U{a,u) =: ^u{t) : [u{t)-u{t)]^V[u{t)-u{t)]0

(2.19)

U2{oi^u) is precisely the instantaneous energy-bound info-gap model which we encountered in eq.(2.7). A remarkable thing happens to the Minkowski norm \\u\\r as r tends to infinity [88, p. 15]: lim ||ix||^ = max |i^^| (2.20) r—^oo

This property of the Minkowski norm allows us to formulate an envelopebound info-gap model as follows.

C H A P T E R 2. UNCERTAINTY

25

Assuming the envelope functions ipn{t) in eq.(2.10) are positive everywhere, define the diagonal matrix ^ = diag[l/'0i(t), . . . , l/'0iv(t)]. From eq.(2.20) we see that: \^[u{t)-u{t)]\\^=

m^x l - ^ v2A a>l

(2.52) (2.53)

Note that -4= ^ 1. What does this imply about the coherence or compatibility of this function {u{t) in eq.(2.51)) with each of the infogap models? 4. Fourier expansion. Consider the function:

(a) Express f{x) as a Fourier sine series: oo

f{x) = y

CnSinnTTX

(2.55)

n=l

That is, find the Fourier coefficients Cn- (Hint: exploit the orthogonality of the sine functions.) (b) Draw the approximation to f{x): K

fKJx) — y ^Cn sin riTTX

(2.56)

n= l

for K = 5, 10, 20, 100. (c) Can f{x) be represented as a Fourier cosine series? Explain. 5. Properties of ellipsoids. (a) Draw the ellipsoid: ax^ -^by^ = 1

(2.57)

What are the directions and lengths of the principal axes? (b) Draw the ellipsoid: 2x^ + 2xy -\-2y^ = 1

(2.58)

What are the directions and lengths of the principal axes? (c) Given an A/'-dimensional ellipsoid: x^Wx

= 1

(2.59)

where VK is a real, symmetric, positive definite matrix. What are the directions and lengths of the principal axes? (Hint: start with part (c).)

34

I N F O - G A P DECISION

THEORY

6. i^ Uniform-bound and Fourier-ellipsoid info-gap models. Consider the following info-gap models for uncertain scalar functions u{t) defined on the domain 0 0

(2.62)

(a) At any fixed positive value of the uncertainty parameter a, is one of the sets, Ui{a^u) or U2{o:^u). contained in the other? (b) Show that, at all values of the uncertainty parameter a, the following inclusion holds: U2{a,u) C Ui{ay/N,u) (2.63) where A^ is the number of modes in the Fourier expansion in eq.(2.61). What is the interpretation of this inclusion? What is the significance of this constant, \/]V, which scales the uncertainty parameter? That is, why does the scale parameter depend on the order of the Fourier expansion? 7. t Convex models. Show that the following are convex info-gap models. Energy bound: U{a, u) = I u{t) :

/

[u{t) - Z{t)f

dt < a H ,

a > 0

(2.64)

Uniform bound: U{a, u) = {u{t) : \u{t) - u{t)\ < a } ,

a > 0

(2.65)

Slope bound: U{a, u) = I u{t) : u{0) = 0, ^ ^

< a I ,

a > 0

(2.66)

a>0

(2.67)

Ellipsoid bound: U{a,u) = {u:

{u-ZfW{u-u)0

(2.70)

36

I N F O - G A P DECISION THEORY

where V is the set of all mathematically legitimate probability distributions p. U{a,p) is a family of sets of prior distributions. Each set contains the nominal distribution p. The sets become more inclusive as a increases. Given completely certain knowledge of the probabilities p we would choose the box whose probability is highest. (a) Evaluate the robustness of this decision with respect to the uncertainty in p. That is, what is the greatest value of a such that the decision is the same for any distribution p G U{a^p)l In other words, how much uncertainty can the decision algorithm tolerate without altering the decision? (b) Show that the robustness is zero when p^ — P2 —P3, — 1/3. Discuss the implications of this for the original 3-box problem.

Chapter 3

Robustness and Opportuneness We love to toyl for uncertainties, and in this are worse than children. J. Hall [86] Whether the world be the better or the worse for having either chances or gifts in it will depend altogether on what these uncertain and unclaimable things turn out to be. William James [95] What is a King? ... To blind Events and fickle Chance a Slave: Seeking to Settle what for ever flies; Sure of the Toil, uncertain of the Prize. Matthew Prior [141]

Info-gap modelling is a stark theory of uncertainty, motivated by severe lack of information. It does, however, have its own particular subtlety. It is facile enough to express the idea that uncertainty may be either pernicious or propitious. That is, uncertain variations may be either adverse or favorable, and we can assess the intensity of these characteristics. Adversity entails the possibility of failure, while favorability is the opportunity for sweeping success. Info-gap decision theory is based on quantifying these two aspects of uncertainty, and choosing an action which addresses one or the other or both of them simultaneously. The pernicious and propitious aspects of uncertainty are quantified by two "immunity functions": the robustness function expresses the immunity to failure, while the opportuneness function expresses the immunity to windfall gain. We formulate these functions in 37

38

INFO-GAP DECISION THEORY

section 3.1. In section 3.2 we run through a series of simple examples drawn from a wide range of disciplines, and then consider an extended example in section 3.3. The definitions of robustness and opportuneness developed in section 3.1 are sufficient for most practical purposes. Nonetheless, one sometimes needs more general definitions of these functions, which we present in section 3.4.

3.1

Robustness and Opportuneness

3.1.1

A First Look

The robustness function expresses the greatest level of uncertainty at which failure cannot occur; the opportuneness function is the least level of uncertainty which entails the possibility of sweeping success. The robustness and opportuneness functions address, respectively, the pernicious and propitious facets of uncertainty. Let g be a decision vector of parameters such as design variables, time of initiation, model parameters or operational options. We can verbally express the robustness and opportuneness functions as the maximum or minimum of a set of values of the uncertainty parameter a of an info-gap model: a{q)

=

max{a : minimal requirements are always satisfied} (robustness)

P{q)

(3.1)

— min{a : sweeping success is possible} (opportuneness) (3.2)

We can "read" eq. (3.1) as follows. The robustness a{q) of decision vector q is the greatest value of the horizon of uncertainty a for which specified minimal requirements are always satisfied. a{q) expresses robustness — the degree of resistance to uncertainty and immunity against failure — so a large value of a{q) is desirable. Eq. (3.2) states that the opportuneness f3{q) is the least level of uncertainty a which must be tolerated in order to enable the possibility of sweeping success as a result of decisions q. (5{q) is the immunity against windfall reward, so a small value of (3{q) is desirable. A small value of (3{q) reflects the opportune situation that great reward is possible even in the gresence of little ambient uncertainty. The immunity functions a{q) and I5{q) are complementary and are defined in an antisymmetric sense. Thus "bigger is better" for a{q) while "big is bad" for I3{q). The immunity functions — robustness and opportuneness — are the basic decision functions in info-gap decision theory. The robustness function in eq.(3.1) involves a maximization, but not of the performance or outcome of the decision. The greatest tolerable uncertainty is found at which decision q "satisfices" the performance at a critical

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

39

survival-level.^ One may select an action q according to its robustness a{q), whereby the robustness function underlies a satisficing decision algorithm which maximizes the immunity to pernicious uncertainty. The opportuneness function in eq.(3.2) involves a minimization, however not, as might be expected, of the damage which can accrue from unknown adverse events. The least horizon of uncertainty is sought at which decision q enables (but does not necessarily guarantee) large windfall gain. Unlike the robustness function, the opportuneness function does not satisfice, it "windfalls" .^ When P(q) is used to choose an action q, one is "windfalling" by optimizing the opportunity from propitious uncertainty in an attempt to enable highly ambitious goals or rewards.

3.1.2

I m m u n i t y Functions

Quite often the degree of success is assessed by a scalar reward function R{q, u). The reward may be in monetary units, or it may have other dimensions expressing the performance demanded of the system. R{q^ u) depends on the vector q of actions or decisions as well as on an uncertain vector u whose variations are described by an info-gap model U{a,u), a > 0. We will refer rather vaguely to u as an 'ambient uncertainty'. It may be an outcome which depends in some way upon the decision vector q, ov u may be entirely indifferent to how the decision maker acts. The uncertain u may be the essence of the outcome which the decision maker seeks (dollars of profit, or millimeters of displacement, etc.) or u may simply be an auxiliary variable of no inherent significance which nonetheless influences the overall reward. Given a scalar reward function R{q^u), the minimal requirement in eq.(3.1) is that the reward R{q,u) be no less than a critical value r^- Likewise, the sweeping success in eq.(3.2) is attainment of a "wildest dream" level of reward r^ which is much greater than r^. Usually neither of these threshold values, r^, and r^, is chosen irrevocably before performing the decision analysis. Rather, these parameters enable the decision maker to explore a range of options. In any case the windfall reward r^ is greater, usually much greater, than the critical reward TC: r-w > re

(3.3)

The robustness and opportuneness functions of eqs.(3.1) and (3.2) can ^Etymologically, 'satisfice' is an alteration of 'satisfy'. The word was introduced to the psychological and economic literature by Herbert Simon with the meaning: "To decide on and pursue a course of action that will satisfy the minimum requirements necessary to achieve a particular goal." [126] ^While a windfall is, in its original meaning, simply something blown down by the wind, it has come to mean such a thing of value. The Oxford English Dictionary [126] gives the following quaint usage from 1705: The grizly Boar is hunting round. To see what Windfals may be found.

40

INFO-GAP DECISION THEORY

now be expressed more explicitly: S(9, Tc)

=

max la:

niin_ R{q,u)

[

/^(^^'^w)

=

\ueU{a,u)

niin TC > J

(3.4)

J

niax_ R{q,u) \ > r^ > \ueu{a,u) J J

(3.5)

a{q^ Tc) is the greatest level of uncertainty consistent with guaranteed reward no less than the critical reward re, while /3(g,rw) is the least level of uncertainty which must be accepted in order to facilitate (but not guarantee) windfall as great as r^. The complementary or anti-symmetric structure of the immunity functions is evident from eqs.(3.4) and (3.5). The definitions of robustness and opportunity functions in eqs.(3.4) and (3.5) assume that the sets of a-values are not empty. We denote these sets as: -4(^,rc)

B{q,r^)

=

=

\(^'

\

[

\ueU{a,u)

mm_ R{q,u)\

>rc\

{o^' [

{ mdix_ R{q,u)] >r^> \ueu{a,u) J J

J

(3.6) J

(3.7)

A{q^ Vc) is the set of a-values whose least upper bound is the robustness a{q,rc). If A{q,rc) is empty then the decision is completely vulnerable — no realization of the uncertain u can lead to obtaining the demanded reward — and we define the robustness function as zero. Likewise, B{q^r^) is the set of a-values whose greatest lower bound is the opportuneness /5(g,rw). If B{q^r^) is empty then the uncertain variation entails no opportunity for windfall, and we ascribe to the opportuneness function the value of infinity. That is: a{q,r,)

=

0

if

A{q,r,)^^

(3.8)

^(9,rw)

=

oo

if

i3(g,rw) = 0

(3.9)

In some situations the "natural" reward requirement is that the performance function R{q^u) must not exceed a specified value r^ rather than being required to be no less than r^ as in eq.(3.4). For instance, if R{q,u) represents a measure of instability of the system then a small value may be preferred rather than a large value. In this case, eq.(3.4) is modified so that the robustness is the greatest value of the uncertainty parameter such that the maximum reward is no greater than r^2(^5'^c) = max la: 1

I

max_ R{q,u) I < TC >

\ueU{a,u)

J

J

(3.10)

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

41

This is still consistent with the verbal formulation of the robustness in eq.(3.1). In like manner, the opportuneness function is the least value of a so that the reward can possibly be as small as TW, SO that eq.(3.5) is modified to: P{Q^ ^W) = niin \ueU{a,u)

J

(3.11)

J

where r^ is less, usually much less, than TC- The anti-symmetric relation between robustness and opportuneness is retained and, as before, "bigger is better" for a{q,rc) while "big is bad" for (3{q,r^).

3.1.3

Generic Decision Algorithms

The immunity functions of eqs.(3.4) and (3.5), or eqs.(3.10) and (3.11), can be modified to apply to generic decision algorithms, as we now explain. A decision algorithm supplies a response or answer to an input of some kind. A decision algorithm D{q,u) may depend on many quantities, including a design vector q which specifies the structure and properties of the algorithm, as well as an uncertain vector u which may include measurements and other data as well as auxiliary uncertainties. The uncertainty of the variables u is represented by an info-gap model of uncertainty, U{a, u). The decision algorithm returns a value which may be a number or a linguistic variable. Decision algorithms come in myriad forms, and may be either intermediate decisions in a long sequence of analyses, or the algorithm may be a 'bottom line' decision. The decision will take different forms in different applications, but typically the algorithm infers something about the system in question for a given value of the uncertain u. For instance, the algorithm may determine if the maximum response is less than a critical value, in which case D{q,u) takes the value 'y^s', and otherwise 'no'. Or, a decision algorithm may select one from a collection of alternative hypotheses about the system. Or, the algorithm may choose one from among a number of alternative courses of action. The robustness of a decision algorithm D{q^u) can be formulated in various ways. For example, the robustness can be defined as the greatest value of the uncertainty parameter a for which the decision is the same for all events in the info-gap model U{a, u). Formally, the robustness will then be expressed as follows: a{q) — max {a : D{q, u) — D{q, u)

for all

u G U{a, u)}

(3.12)

In other words, the robustness, 3(g), is the supremum of the set of a-values for which the decision is constant for all values of u in ZY(a, u). In some situations the robustness can be evaluated as the greatest a for which the decision varies by no more than a value r^ representing a degree

42

INFO-GAP DECISION THEORY

of error or variability of the decision algorithm. In this case: a{q,rc) =max

r^?, then the overall robustness is: 2(9, Vc) — max < a : niin_ Ri{q, u) [ \ueuia,u) J

> rc,i, i = 1, . . . , / > ' J

(3.15)

where TC represents the vector of critical rewards rc,i. This overall robustness is the robustness with respect to the most vulnerable performance

44

INFO-GAP DECISION THEORY

requirement: S(g,rc) = mm ai{q,rc,i)

(3.16)

i

Likewise, the overall opportuneness function is the lowest horizon of uncertainty at which all the windfalls are possible (though not guaranteed):

MQ^'^W)

= niin< a : [

max_ Ri{q,u) \ueU{oc,u)

> r^^i, z = 1, . . . , / > J

(3.17)

J

where r^ represents the vector of windfall rewards rw,2. This is evaluated as the immunity to windfall of the least opportune performance requirement: ^(^,^w) = max^^((7,rw,i)

(3.18)

i

We will consider examples in sections 3.2.4 and 3.2.6.

3.1.5

T h r e e Components of Info-Gap Decision Models

We summarize the discussion of sections 3.1.1-3.1.4 by noting that the immunity functions are obtained by combining three distinct elements: a system model, a performance requirement, and an uncertainty model. An info-gap analysis begins with these three components. System model. The reward function R{q,u) (or the decision algorithm D{q, u)) expresses or entails the input/output structure of the system to which the decision is applied. R{q, u) assesses the response of the system, in terms of reward or quality of performance, to the decision maker's choices q and to the ambient uncertainty u. We will consider system models covering a very wide range of applications, including engineering design, software reliability, project management, manufacturing, biological conservation, terrorism preparedness, evolutionary success, and more. Performance requirements. The inequalities in eqs.(3.4) and (3.5) (or eqs.(3.10) and (3.11)) express rewards which are demanded or desired from the system. As explained in section 3.1.2, the term 'reward' must be construed broadly. Some requirements are indeed expressed as monetary demands, but sometimes other performance requirements arise such as duration of task implementation, or magnitude of vibrational motion, or level of pollutant concentration, or probability of extinction. In all these cases the metaphor of 'reward' is used to describe the requirements or anticipations from the system, even though sometimes, as in eqs.(3.10) and (3.11), a small value of 'reward' is more desirable than a large value. Uncertainty model. We will focus on info-gap uncertainty, represented by an info-gap model. In some situations the info-gap model may be supplemented with probability models, or the info-gap model may in fact quantify uncertainty about a probability model such as uncertain tails of a distribution. An info-gap model U{a,u) embodies the prior information

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

45

about the uncertain vector u. In many applications a single info-gap model is sufficient, though sometimes several models are used. The decision vector q is chosen to effect some change, hopefully causing a beneficial outcome. The actions which q represents may be design choices such as the selection of a construction material or of a pesticide, or organizational choices such as the logistical flow chart of a multi-task project, or an operational decision to begin or delay some action, or the formulation of a fault-detection algorithm, and so on. The range of activities encompassed by q is as great as human activity itself. But within this universe of possibilities, we must distinguish between two categories of actions: those which alter the uncertainty models and those which do not. This will make a fundamental difference in the calibration of the immunity functions, as we will see in various examples (pp.196, 218).

3.1.6

Preferences

The immunity functions, a{q, TC) and f3{q, TW), are the basic tools in info-gap decision theory. For given values of critical or windfall reward, TC or TW, each immunity function induces a preference ranking on the set of available decisions. More importantly, the immunity functions enable the decision maker to explore the desirability of different options q and different requirements, Vc and Tw, and thus to alter earlier preferences. The robustness S(g, TC) is the greatest level of uncertainty at which action q guarantees reward no less than TC- AS we have noted before, this means that "bigger is better" for the robustness function.^ Consequently, a decision maker will usually prefer a decision option q over an alternative decision q^ if the robustness of q is greater than the robustness of q' at the same value of critical reward TC. We can express this preference more succinctly as: q yr q' if 3(g, re) > a{q', r^) (3.19) Let Q be the set of all available or feasible decision vectors q. A robustsatisficing decision is one which maximizes the robustness on the set Q of available g-vectors and satisfices the performance at the critical level TC'. qdrc) = argmaxa(g',rc)

(3.20)

qeQ

We note that usually, though not invariably, the robust-satisficing action qdrc) depends on the critical reward TC. It must be stressed that the robustness function does not necessarily determine the decision maker's behavior, since both a{q,rc) and qdrc) depend on the critical reward TC, which is a free parameter. That is, S(g, TC) does not necessarily establish a unique preference ordering on the set Q of available actions. It often happens that the decision maker chooses both Tc and the optimal action §c(^c) in an iterative (and introspective) fashion ^This is true also when small "reward" is sought, as in eq.(3.10).

46

INFO-GAP DECISION THEORY

from consideration of the robustness function. In section 3.1.7, throughout chapter 6, and elsewhere, we examine methods by which the decision maker uses the robustness function, sometimes together with the opportuneness function, to explore the implications of alternative scenarios. The opportuneness function (3{q,r^) generates a preference ranking on the available actions in a similar way, though the resulting ranking may be different. /3(^, r^) is the lowest level of uncertainty which must be accepted in order to facilitate windfall reward as great as r^. Thus, unlike the robustness function, "big is bad" for f3{q^r^). Consequently, a decision maker who chooses to focus on windfall opportunity will prefer a decision q over an alternative q' if the opportuneness with q exceeds the opportuneness with q' at the same level of reward r^. Formally: q^oq'

if

^(g,rw) r'^ implies

3(g,rc) < a{q,r'^)

(3.23)

Recalling that bigger values of 3 are preferred over smaller ones, this relation expresses the trade-off between demanded-reward and robustness-touncertainty: if large reward is required for survival then only low immunity to uncertainty is possible [15]. In addition, as also illustrated in fig. 3.1, the opportuneness function of eq.(3.5), /5(g, r^), increases monotonically with increasing wildest-dream reward r^\ rw > r'^ implies ^{q, r^) > P{q, r'J (3.24)

C H A P T E R 3.

Robustness

ROBUSTNESS AND OPPORTUNENESS

47

Uncertainty

high

high

S(9,rc) (Opportuneness)

T

i

(Robustness)

low

low

low (modest)

-• high

Reward, TC or r^

(demanding)

Figure 3.1: Two robustness curves (3) and one opportuneness curve {(3).

Sweeping success cannot be attained at low levels of ambient uncertainty. This is also a trade-off, since "big is bad" for /3. (The immunity functions in eqs.(3.10) and (3.11) are monotonic in reversed directions, though they represent the same trade-offs.) The location of the robustness and opportuneness curves on the plane of uncertainty-vs.-reward, as in fig. 3.1, reveals one type of gambling which is expressed by these trade-offs. Consider the uppermost of the two robustness curves, 3(g, TC), which falls to low and vulnerable levels of immunity only at relatively high demanded reward. Different prior information leads to the lower robustness curve which, though still decreasing monotonically with Tc, runs more closely to the origin. The upper robustness curve represents bolder behavior than the lower curve: at any given level of demanded reward (re) a greater level of ambient uncertainty (S) is tolerable according to the upper curve. Conversely, at fixed ambient uncertainty the upper robustness curve allows greater demanded reward than the lower curve. Ascribing these two robustness curves to two different decision makers operating with different information, we can say that the lower decision maker is more vulnerable to uncertainty than the upper decision maker. The upper curve will lead the decision maker to behavior which would look risky or rash when viewed through the strategy of the lower robustness curve. The trade-offs portrayed in fig. 3.1 demonstrate gambling-like behavior in the sense that the decision maker must choose a position on the immunity curves: this requires deciding how much security can be exchanged for reward, or how much reward can be relinquished in return for security. The choice reflects the extent of the decision maker's propensity to gamble, even though no concepts of chance are involved in the evaluation of the immunity

48

INFO-GAP DECISION THEORY

functions. We will discuss info-gap gambling further in chapter 6. T h e monotonic increase of the opportuneness function which is portrayed in fig. 3.1 also shows a gamble-like trade-off: t h e decision maker's anticipation of greater windfall reward r^ must be accompanied by acceptance of greater ambient uncertainty. T h e trade-offs illustrated in fig. 3.1 show a particular type of coherence between t h e robustness and opportuneness functions. As the decision maker's expectations are reduced, whether they be for windfall reward TW or for critical survival-level r e t u r n re, b o t h 3(g,TC) and P{q^r^) indicate a rosier picture of the effect of uncertainty. T h e robustness function gets larger and indicates greater immunity to failure as TC is reduced, and t h e opportuneness function gets smaller and shows less immunity t o windfall as Tw gets smaller. 3 ( ^ , T C ) and I3{q^r^) are 'cooperative' or 'sympathetic' in t h e sense t h a t they share the same trends with varying expectations r^ and However, we will see in chapter 5 t h a t the variation of robustness and opportuneness with varying decision q need not be sympathetic at all. A change in t h e choice of q which enlarges a ( g , TC) need not simultaneously decrease /?(g, TW). These immunities may be either sympathetic or antagonistic as a function of t h e actions available to t h e decision maker.

3.1.8

Zero Robustness and Preference Reversal Robustness high

low Reward, Vc low (modest)

high (demanding)

Figure 3.2: Two robustness curves. Robustness decreases as aspirations get higher, as we have explained in connection with eq.(3.23). In fact the robustness equals zero for aspirations above some level, as illustrated in fig. 3.2.

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

49

The robustness reaches zero at a very special level of aspiration: the value of reward anticipated by the best-estimate of the uncertain quantities. The robustness, 3(g, TC), depends on an info-gap model U{a^u) whose centerpoint u is the anticipated or estimated value of the uncertain vector u. The anticipated reward from action q is R{q, u). Under very weak assumptions one can show that aspiration as large as R{q, u) has zero robustness [25, section 11.6]: r — R{q^u)

implies

3(^, r) = 0

(3.25)

This is illustrated in fig. 3.2 where the robustness curve for decision q reaches zero when the aspiration equals r. Eq.(3.25) holds for any decision q. In particular, it holds for the decision which optimizes the estimated reward: g* = arg max R{q, u)

(3.26)

g* is the decision whose reward is greatest, based on the nominal or bestestimated value u of the uncertain vector u. The associated reward, r* = R{q^^u)^ has zero robustness as implied by eq.(3.25) and illustrated in fig. 3.2 with the curve labeled q^. Eq.(3.25) says that the anticipated or estimated reward R{q^u)^ of any action q, cannot be relied upon to occur because its robustness to uncertainty is zero. A decision maker who values the reliability with which aspirations are achieved, may tend to "migrate up" the robustness curve to lower levels of aspired reward r^ with larger robustness a(g,re). For instance, consider action q^ in eq.(3.26) which, based on the estimate u, optimizes the reward, q^ is, nominally, expected to result in reward as large as r*, but the robustness of this outcome is zero. If r is adequate reward, where r < r^^ and if a{q^^r) is sufficient immunity against uncertainty, then the decision maker may adopt action q^ with aspiration r. But now we note that the robustness curves of different actions may cross, as illustrated in fig. 3.2. If aspiration r is less than r^ then action q will be preferred over action q'^, at aspiration r, according to the robustness ranking in eq.(3.19), even though nominally q^ would be preferred over q. On the other hand, if r > r^ then q^ is preferred over q^ but not because g* is the best-estimate optimum, but because q^ adequately satisfices the reward and is more robust than q. In other words, the crossing of robustness curves entails the reversal of preferences as the decision maker's aspirations change.

3.2

Simple Examples

We now proceed to examine a sequence of simple examples of info-gap analysis of design and decision problems, chosen from a broad range of disciplines. The reader is invited to select whatever examples are of particular interest. The examples in this section can all be read independently.

50

3.2.1

INFO-GAP DECISION THEORY

Engineering Design: Cantilever

Fig. 17 Figure 3.3: Galileo's portrayal of a cantilever. Fig. 17 in [77], p.116. Reproduced with the kind permission of Dover Publications. The cantilever — a projecting beam supported only at one end — is one of the most widely occurring engineering structures. Galileo initiated the modern study of the mechanics of cantilevers, and his depiction of an end-loaded cantilever is shown in fig. 3.3. The cantilever is the structural paradigm for a plethora of artifacts ranging in size and function from giant buildings, radio towers, cranes, airplane wings, turbine blades, diving boards, canon barrels, car antennas, all the way down to microscopic electromechanical devices. A central consideration in the design of a cantilever is the control of extraneous deflections resulting from unplanned ambient forces acting on the beam. In the broadest terms, there are two conceptually and practically distinct approaches: prevent vibration by stiffening the beam, or absorb vibration by dissipating energy. These concepts are not mutually exclusive and can in fact be implemented together, resulting in a spectrum of design options. We will employ the robustness function in a simplified illustration

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

51

of the choice of a design concept for vibration control of a cantilever. In the simplest of all dynamic models of the cantilever, vibration is considered to occur as completely rigid rotation around the fixed base of the beam. Stiffness of the beam is represented by a linear rotational spring and vibrational energy is dissipated viscously: in proportion to the angular velocity. The angle of deflection 0{t) is related to the moment of force u{t) acting on the beam by the differential equation:

J ^ + c « + « ( „ ^ . ( 0

,3.27)

where J is the moment of inertia of the beam with respect to the base, and c and k are the damping and stiffness coefficients. The solution of this equation for zero initial displacement and velocity, or for times long after onset of motion, is, assuming sub-critical damping: Ou{t)=^ I u{T)h{t-T)dT Jo

(3.28)

where the impulse response function is: /i(t) = - ^ e - ^ ^ ' s i n c J d t

(3.29)

JcJd

The undamped natural frequency \s u = yjk/J, the damping ratio is C = c/{2Juj) and the damped natural frequency is u;d = ^ A / T ^ . The assumption of sub-critical damping is that C^ < 1All that is known about the moment of force which will act on the beam is the typical or nominal transient waveform u{t). and that the actual load events, u{t)^ will be transient in nature and may deviate greatly from u(t). An info-gap model for highly uncertain transient functions is the cumulative energy-bound model of eq.(2.4): U{a,u) ^ lu{t) : I

[u{t)-u{t)f

dt0

(3.30)

In many applications the performance requirement is that the absolute amplitude of vibration not exceed a specified value: \0{t)\ c

(3.31)

We now have the three components of the decision analysis: the system model of eq.(3.28), the uncertainty model in eq.(3.30), and the performance requirement in eq.(3.31). Let q represent the design-decision vector containing the stiffness and damping coefficients k and c. To evaluate the robustness as in eq.(3.10) we must find the greatest absolute defiection as a function of the horizon of uncertainty a.

52

INFO-GAP DECISION THEORY

The angular deflection, eq.(3.28), can be re-written as: Ou{t)= f [u{T)-u{r)]h{t-T)dT-^ Jo

[ u{T)h{t-T)dr Jo ^

-V

(3.32) '

where 9{t) is the nominal response of the cantilever. Using the Schwarz inequahty, the first integral is found to be bounded as: ( / [u{r) - U{T)] h{t - r) dr")

< / [u{r) - u{r)]^ dr / h^r) dr

(3.33) with equality if and only if U{T)—U{T) is proportional to h{t—r). Comparing the first integral on the right of (3.33) with the info-gap model of (3.30), and then using eq.(3.32), one sees that the greatest absolute angular deflection, up to uncertainty a, is: max^ \eu{t)\ = a J [ h^{r)dr + \e{t)\ ueU{a,u)

(3.34)

V Jo

Equating this relation to the critical angular deflection 9c and solving for the uncertainty parameter a yields the robustness: a{q^Oc) = ^ ^ z M L ^Jlo^'ir)dr

(3.35)

provided that the numerator is non-negative, which means that the nominal response does not violate the performance requirement. More demanding performance (smaller ^c) entails lower robustness (smaller a) as expected from the discussion in section 3.1.7. The robustness vanishes, a{q,9c) = 0, if the £equired deflection 9c equals or is less than the anticipated deflection, 9c ^ 1^1, as predicted by eq.(3.25). The denominator in eq.(3.35) can be expressed as: h^T)dT= Jo '' ' ' ' " '

V^ £4>{t) 2Juj^J^

(3.36)

where (f>{t) is a function whose square is: A2M ^l"^^ 4> {t} = —;;:

-2C.t ( e ^

2C

.2 ,^ . 0 . , ^l-C^^ sin wai + sm zwa* + —;: (3.37) To study an exphcit reahzation let us consider a nominal input u{t) which is a step function of known amplitude So and known duration T:

=(.)={«»'

"^{11

(3.38)

C H A P T E R 3.

53

R O B U S T N E S S AND O P P O R T U N E N E S S

The nominal response now becomes:

m = ^^4^7(0 l-^(t),

0 a^in + A 3

(3.80)

If this relation does not hold, then the sample should be extended by measuring more features. In chapter 4 we will pursue a more thorough study of reasoning by analogy and qualitative calibration of increments of robustness.

3.2.6

Project Scheduling with Uncertain Task Durations

The management of a complex multi-task project requires decision-making in many different areas. One of the more quantitative managerial decisions, for which info-gap decision-support tools can be useful, is the design of the flow chart of task execution. A simple flow chart is shown in fig. 3.8. Each box represents a task and the arrows indicate the order of implementation. While all the tasks must be completed, and while many constraints exist on the task-sequencing, alternative flow charts are usually possible for the same project, sometimes based on different technologies for task implementation and incurring different costs. The problem facing the manager is to choose the best among the possible alternatives. The difficulty in making such a choice is that the factors which may retard the implementation of the tasks are often highly uncertain. Some tasks may involve engineering R & D whose resource requirements are hard to estimate. Some tasks may be vulnerable to unknown external factors — economic anomalies, natural catastrophes, political developments — which are impossible even to name or enumerate. In order to choose from among alternative flow charts, the manager needs a concise assessment of the project reliability against highly uncertain adversity. The robustness function provides precisely this. We will consider a simple example in this section, and return to study various other aspects of the problem later. In section 5.4 we will study both robustness and opportuneness functions in attempting to choose between alternative task flow charts. In section 7.5 we will examine the enhanced robustness of an entire project which results from gathering information about a highly uncertain task.

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

65

43J—4i ,/26=i

S-UIA]—^[el-r^^

J16

lOLJllJ

K

12 |__^ 13 |__J 14 i__J 15 Figure 3.8: A 16-task project schedule. Fig. 3.8 shows the plan of a hypothetical 16-task project: a flow chart of the sequence of execution of the tasks. The project is organized into five task paths: Path 1: 1 2 -^ 3 ^ 4 -^ 16. Path 2: 1 5-^6-^3^4->16. Path 3: 1 5^6^7->8^16. Path 4: 1 5 -^ 6 ^ 9 -^ 10 ^ 11 -^ 16. Path 5: 1 12 ^ 13 -^ 14 ^ 15 ^ 16. We note that several tasks appear in more than one path because the paths interlace. The project is completed successfully if all the tasks are completed within the allowed duration, tc- The basic question which the manager asks is: how robust is the successful completion of this project to the uncertain duration of the tasks? We will address that question in this section. Once we are able to answer this with the aid of the robustness function, other questions arise: Can the robustness be increased by organizing the tasks into different task paths? And if so, by how much? This is considered in section 5.4 and in [33]. How can the robustness be improved by gathering information? What information would be most valuable? We defer the treatment of these questions to section 7.5. We now formulate this generically, and present a solution based on the robustness function [33]. The analysis involves three components, as in section 3.1.5. The system model expresses the relation between the task flow-chart and the project duration. The failure criterion (which is the converse of the performance requirement) states the duration within which the project must be completed. Finally, the uncertainty model quantifies the uncertainties which accompany the project; in this case the uncertain durations of the tasks. We first formulate the system model, which tells us how long the project takes. The actual (unknown) duration of the nth task is denoted t^, for

66

INFO-GAP DECISION THEORY

n = 1, . . . , A^ where A^ is the number of tasks. For the example in fig. 3.8 N — 16. The vector of task times \s t — {ti, ... ^t^)^. The tasks are organized into M task paths. In fig. 3.8, M = 5 as explained earlier, fmn is the fractional participation of task n in path m. This means that, in path m, the task following the nth starts when task n is a fraction fmn complete. F is the task-path participation matrix. This is the M X N matrix of numbers fmn between zero and one. For instance, the participation matrix for the project plan in fig. 3.8 is: 1 1 1 F = 1 V l

1 0 0 0 O

1 1 0 0 O

1 0 0 0 0 1 1 ^ 0 0 0 1 1 1 1 0 1 1 0 0 O O O O

0 0 0 1 O

0 0 0 1 O

0 0 0 1 O

0 0 0 0 O

0 0 0 0 l

0 0 0 0 l

0 0 0 0

1 \ 1 1 1 l l l /

(3.81)

The mth row represents the mth task path of the flow chart of fig. 3.8. For example, the value 1/2 in the 6th column of the second row represents the fact that, in the second task path, the task following task 6 (which is task 3 in this path) begins when task 6 is one-half done. The duration of the mth task path is the sum of durations of the tasks in that path, weighted by their fractional participation times: N

Cm = ^

fmntn^

m = 1, . . . , M

(3.82)

n=l

For instance, the duration of the second path is C2 = 1 • ti + 1 • ^3 + 1 • ^4 + 1 • ^5 + 2 * ^6 + 1 ' ^16The system model is the duration of the longest path: N

Tit)

max \Cm\ l<m<M'

'

max l<m<M

/ ^ Jmn^n

(3.83)

n=l

The failure criterion states that the project fails if the duration of the longest path exceeds a critical value. Failure is: T>t,

(3.84)

We may know very little about the variation in the task durations. That is, the gap between what we do know about the durations, and what we need to know in order to make a perfect plan, may be quite substantial. This info-gap results from our incomplete familiarity with the conditions under which the project will be executed, and from the fact that surprises — unexpected occurrences — are bound to arise. This information gap is a form of uncertainty which is usefully represented by an info-gap model. Let us suppose that we have some prior knowledge about the typical or nominal duration of each task, but that we know very little about how much

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

67

the actual duration will deviate from the nominal value. Also, we may have some rough information about the relative variability of the different tasks. Let tn denote the nominal duration of the nth task, for n = 1, . . . , A/^. The coefficients wi, ...,WN are positive numbers expressing the relative variability of the tasks. If we have no prior information about the relative variability of the tasks, then all the Wn will equal unity. If the nth task tends to vary more than the others, then its uncertainty coefficient, Wn^ will exceed 1. Conversely, tasks which tend to vary less than most will have Wn less than 1. A simple uncertainty model based on this information states that each task duration may deviate by an unknown fraction of its nominal value. Consider the following "uniform-bound" info-gap model, which is a discrete ^6. version of eq.(2.9)^ U{a,t) = lt:

^^""^ ^""^ <Wna,

n = l, . . . , A ^ l ,

a>0

(3.85)

U{a, t) is an infinite set of values of the vector t of task durations, and the info-gap model is the unbounded family of nested sets U{a, t), a >0, Each element tn of the vector t, representing the duration of the nth task, varies within the interval: In - Wntna

< tn \teu{oc,t) / J

(3.87)

^This info-gap model allows negative task durations which, though unfounded, is of no consequence for evaluating the robustness. However, when we come to consider the opportuneness function with this info-gap model, as we will do in section 5.4, we will have to exclude negative task durations.

68

INFO-GAP DECISION THEORY

as in eq.(3.10) with R{q,t) = T{t) and TC = tc. The robustness, a{q,tc), is the maximum of the set of a-values for which the greatest path duration is acceptable (no failure) for all task times in the uncertainty set U{a^t). We see in this expression for the robustness that all three components are combined: the system model, the failure criterion and the uncertainty model. We now develop an explicit algebraic expression for a(g, tc). The greatest duration of the mth task path, at horizon of uncertain a, occurs when each task runs maximally overtime. From eq.(3.86) we see that the greatest duration of task n is tn-^WntnOt where, of course, the value of a is unknown. So, the maximum duration of the mth path, up to uncertainty a, is: N

max_ Cyn =

niax_ V ] fmntn = Yl f^^ (^^ + ^^ntnOt) (3.88) N

=

22

N

f^^^^

r

-

N

+ ^ Zl

frnn'^ntn

(3.89)

JT^

Cm^afm

(3.90)

where c^ and fm are defined in eq.(3.89). We have dropped the absolute value signs appearing in eq.(3.83) because all the terms are non-negative. The robustness is found by equating the maximum path duration to the allowed project duration and solving for a: max \cm + afm] = U

(3.91)

l<m<M

In order to do this, define am as the greatest acceptable uncertainty which the mth task path can tolerate without reaching the failure criterion in eq.(3.84). In other words, 3 ^ is the robustness of path m. Equating eq.(3.90) to the critical time, t^ and solving for a yields: a^ = h ^ ^

m = l , ...,M

(3.92)

We require all paths not to fail, so the logical structure of this problem is similar to a serial network of subunits, where each subunit is essential for operation of the network.^ The robustness of the entire project is the robustness of the most vulnerable path. That is, the robustness is the least of these Srn's: S(9,tc) = min dim (3.93) l<m<M

Consider the 16-task project plan whose structure is shown in fig. 3.8 on p.65 and whose participation matrix is presented in eq.(3.81). The nominal times tji and uncertainty weights Wn are recorded in table 3.1. We note ^This is a multi-criterion decision, as discussed in section 3.1.4.

C H A P T E R 3.

n t-n

Wn

1 2 3 4 5 6 7 8

[T 1

69

ROBUSTNESS AND OPPORTUNENESS

9

4 2 2 3 6 3 5 4 2 1 1 2 2 2 1 0.5

10 11 12 13 14 15 16

1 1

4 1

2 1

3 1

1 1

1 1

2 1

Table 3.1: Nominal durations and uncertainty weights.

that the nominal duration of task 6 is 6 units, the longest of the nominal task times, which is significant since this a pivotal task included in three task paths. The uncertainty weights Wn for tasks 2, 5, 6 and 7 all equal 2 since prior information indicates that the durations of these tasks tend to deviate much more than the other tasks. The uncertainty weight for task 9 is wg = 0.5 since this task tends to deviate less than the others.

tc 1 [ _Si 21 0.67 26 1.00 28 1.13 30 1.27

S2

Ss

34

Ss

0.42 0.68 0.79 0.90

0.031 0.19 0.25 0.31

0.00 0.18 0.25 0.32

1.10 1.60 1.80 2.00

Table 3.2: Path robustnesses with various allotted task durations. The reliability of the network will change as a function of the total time allotted for project completion. Table 3.2 shows the robustnesses of the five paths, S ^ calculated according to eq.(3.92), for several values of the allotted total task duration, tc. The boldfaced number in each row is the lowest path robustness which, according to eq.(3.93), is the robustness of the entire project schedule for that value of fc- When the allotted duration is tc = 21 time units the fourth path has zero robustness (S4 = 0). This occurs because the nominal duration of task path 4 precisely equals 21 time units. Consequently, even the slightest time over-run results in a project over-run, according to condition (3.84), so this path has zero immunity to task-duration uncertainty. We will refer to path 4 as the 'nominal critical path'. All the path robustnesses increase as the allotted project duration tc increases, as expected from eq.(3.92): a plot of Sm vs. tc is a linearly increasing line whose slope expresses the trade-off between robustness and performance. Good robustness (large Sm) comes at the expense of poor performance (large tc). At tc = 26 the project robustness is a = 34 = 0.18 and path 4 is still critical. At tc — 28 the project robustness is 3 = S3 = ^4 = 0.25, indicating that paths 3 and 4 are equally vulnerable to uncertainty, and are more vulnerable than all other paths. For project durations greater than 28 time

70

INFO-GAP DECISION THEORY

units the criticality is transferred from path 4, the nominal critical path, to path 3, which is the 'uncertainty critical path'. We see here an essential and unique element of the uncertainty analysis. The path which is critical with respect to the nominal conditions is not necessarily the critical path when sensitivity to uncertainty is considered in the presence of some extra time. When tc = 30 path 3 becomes maximally sensitive, and the project robustness is entirely determined by this path: S = Ss = 0.31. The transfer of path-criticality from path 4 to path 3 is an example of 'reversal of preferences' resulting from intersection of robustness curves, as discussed in section 3.1.8. We note that in all four cases considered in table 3.2, the range of pathrobustness values is quite large. For instance, at tc = 28, the ratio of the most to the least robust path is S5/S3 = 7.2. It is worth noting that the computations for this example take a fraction of a second on a standard personal computer. Even much larger networks are readily analyzed. Further discussion of this example is found in [33].

3.2.7

Portfolio Investment

A typical simplified portfolio investment problem requires the decision maker to choose the dollar amount to buy or sell for each of a number of securities, where the future values of these securities are uncertain. If the (unknown) future unit value of the zth security is Ui and the dollar amount purchased or sold is qi (positive for purchase, negative for sale), then the net change in the future worth of the portfolio after the transaction is: N

1=1

The question is how to choose the investment vector q given uncertainty in the future security-value vector u, as well as constraints such as budget limitations. Furthermore, one may be able to consider alternative investment portfolios: different sets of securities with different uncertainties. How does one assess the relative riskiness of such investment alternatives? Uncertainty model. For a given investment scenario we know the anticipated future values of the securities, 5i, . . . , UN, which we combine in a nominal vector u. Furthermore, we typically have information indicating the relative degree of variability of the securities and the propensity for correlated or anti-correlated variation. Specifically, we will often know an historical covariance matrix for the values of the securities. Let W denote the inverse of the covariance matrix, which is real, symmetric and positive definite. While historical values are often a poor indication of future behavior, we can use this information to formulate an ellipsoid-bound info-gap model for the uncertain future variation of the security values.

C H A P T E R 3.

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The ellipsoid-bound info-gap model for uncertain variation of the actual security-value vector u around the nominal value vector u is: U{a,u) = {u = u + v: v^Wv '^c,iHowever, each robustness function equals zero when the critical reward, re, equals the anticipated return. Fig. 3.9 assists the decision maker to assess the relative riskiness of the two portfolios. Both robustness functions vanish for critical rewards in excess of rc,2 5 so neither portfolio is acceptable if rewards this large are needed. For critical rewards between r^ and rc,2 7 and especially above rc,i, the second portfolio is the clear favorite over the first, since the second portfolio has greater robustness. T h e riskiness of the two portfolios becomes equal when the robustness curves cross, and if values of Vc less t h a n r^ are acceptable then the first alternative becomes increasingly preferable because it aff'ords greater immunity at the same level of guaranteed return. In this example we see how crossing of robustness curves entails reversal of preference, as discussed in section 3.1.8. O p p o r t u n e n e s s f u n c t i o n . We now consider the opportuneness function /?(g, Tw), which is the least level of uncertainty needed to sustain the possibility of reward as large as Vsfj, as expressed in eq.(3.5) on p.40. The opportuneness function assesses the immunity to windfall gain TW , so a small value oi (5 — low immunity to windfall — is desirable, unlike the robustness function for which a large value is needed to assure large immunity to failure. Windfalling, upon which the opportuneness function is based, is diff'erent from satisficing which underlies the robustness function, though on the surface the mathematics looks quite similar. To evaluate the opportuneness function we need the greatest possible

74

INFO-GAP DECISION THEORY

reward up to uncertainty a, which is found to be: max_ q^u = q^u + ayq^W~^q

(3.105)

uE.U{cx,u)

whose similarity to the minimum reward in eq.(3.96) is evident. The opportuneness function is obtained by equating this maximum to the windfall reward r^ and solving for the uncertainty parameter a, leading to: ^(,,r.) = - ^ ; ^ = £ i

(3.106)

(or zero if this expression is negative.) This relation displays the usual trade-off between opportuneness (small P) and windfall reward (large r^)\ large windfall is obtained only at the expense of accepting large ambient uncertainty. If we impose the budget constraint of eq.(3.99) and if, as in eq.(3.100), we assume that the nominal security-values are all equal, then the opportuneness function becomes: ^Jq^W-^q which is similar to the robustness function of eq.(3.101). rr^\r^ qT\^i-i^^y^

^^^/j^TA/-1/J = constaiit

9^1 = Q

Figure 3.10: Schematic illustration of constrained optimization o{ q W

q.

Because windfalling is different from satisficing, and because opportuneness is different from robustness, we can now see that optimizing (3{q^r^) is very different from optimizing a{q,rc). The opportuneness function f3{q^r^) is optimized (minimized ) by maximizing q^W~^q^ while the robustness is optimized (maximized) by minimizing this same quadratic term. First of all, we obviously cannot do both optimizations simultaneously. Furthermore, the first — optimizing f3{q.r^) — cannot be done at all if only the budget constraint of eq.(3.99) is imposed. There simply is no maximum of q^W~^q subject to g'-^l = Q- This is illustrated in fig. 3.10. No matter how large we make the quadratic term (which defines an ellipsoid) it still intersects the plane defined by the budget constraint. This is unlike the minimization of q^W~^q, which occurs when any further constriction of the ellipsoid would cause it to disconnect from the budget-constraint plane.

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In practice of course the budget limitation of eq.(3.99) is not the only constraint. Additional constraints become active as the investment vector q ranges further from the origin: limitations in the supply of securities which can be purchased or constraints on the quantity of holdings which can be sold. Nonetheless, this example demonstrates some of the fundamental differences between windfalling with the opportuneness function and satisficing with the robustness function. Let us leave the attempt to optimize robustness and opportuneness and note that any improvement in one function is obtained at the expense of deterioration in the other. Comparing the robustness and opportuneness functions in eqs.(3.101) and (3.107) we note that any change in the investment vector q which increases one will increase the other, and likewise any decrease in one function will be accompanied by a decrease in the other. However, "big is better" for a while "big is bad" for p. These immunity functions are antagonistic in this example: either immunity can be improved only at the expense of the other.

3.2.8

M o n e t a r y Policy

Economies experience adverse shocks and surprises which are unpredictable from historical data. Info-gap theory is well suited for the selection of monetary policy to counter these surprises, as we illustrate in this adaptation of Brainard's example [39] which is discussed by Blinder [36, pp. 11-12]. We will show that policies which, based on best-estimated models would seem to optimize the outcome, should sometimes be avoided in favor of less aggressive policies, as was suggested by Brainard. Consider the macroeconomic model: y = Gx-^z

(3.108)

where G and z are both highly uncertain, with best-estimates G and ?, respectively. The central bank wishes to choose x so as to pilot y towards a target value, y'^. _ We have no probabilistic model for the error in the estimates G and z, and what we can say is that the fractional error in these estimates is unknown. That is, true (or truer) values G and z deviate from the estimated values G and z by no more than a fraction a. However, the horizon of uncertainty a is unknown. An info-gap model for this uncertainty is the following unbounded family of nested sets of G and z values: ZY(a,G,ri = J G , Z :

G-G ^ ^ G

z—z < < a, a, | ^ ^ | < a L z

a>0

(3.109)

At any horizon of uncertainty, a, the estimates G and ? may err fractionally by as much as a. However, the value of a is not known. Thus an infogap model does not allow a 'worst case' analysis: these is no known worst

76

INFO-GAP DECISION THEORY

case since the horizon of error is unknown. We are deep in the domain of Knightian uncertainty. The performance function is the squared difference between the desired value y^ and the reahzed value y: f{x,G,z) = [y{x,G,z)-y*f

(3.110)

In the spirit of Simon's bounded rationality and the concept of satisficing, we desire the output error, / ( x , G, z)^ to be no greater than the critical value El'. f{x,G,z)<El (3.111) E^ can be chosen to be small or large to express demanding or modest performance aspirations. The robustness of policy choice x is the greatest fractional error in the estimates G and 2, up to which every realization G and z results in acceptable squared error. Formally, the robustness of decision x with aspiration Ec is: a{x,Ec)=msixla:

max

f{x,G,z)]<El\

(3.112)

Large robustness a{x,Ec) implies that policy choice x is immune to error in the estimated model while satisficing the outcome-error at Ec. Low robustness implies that outcome-error as small as Ec cannot be confidently expected with choice x. Let y{x) — Gx -h ? denote the best estimate of the outcome, given choice X, and let us consider values of x for which y{x) < y*. We will assume that G > 0 and J > 0. The robustness of choice x is found to be: a{x^Ec)={

y{x) 0

'^^^-y else

^(^^

(3.113)

Outcome-error no greater than Ec is guaranteed with policy choice x if the horizon of uncertainty is no larger than a{x^ Ec). As illustrated in fig. 3.11, the robustness gets worse (S decreases) as the aspired output error improves (^c gets smaller). That is, robustness trades-off against performance. Furthermore we see in eq.(3.113) and fig. 3.11 that the robustness vanishes when the aspiration Ec equals or is less than the best-estimate of the output error, y'*' — y[x). This is true for any choice of x. We can have little confidence in attaining fidelity as good as the best-estimated fidelity; only poorer fidelity has positive robustness. Since this is true for any x, it is also true for the choice of x which minimizes the estimated error, / ( x , G, J). This is beginning to sound like Brainard's conclusion that policies which optimize the outcome should sometimes be avoided, but there is more.

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a{x,Ec)

a{x,Ec)

y*-y{x)

Figure 3.11: Schematic illustration of the robustness function a{x,Ec) in eq.(3.113).

Figure 3.12: Comparison of two policy choices: reversal of preferences.

Let us now consider two policy alternatives, x and x', where: y{x) > y{x^) > 0

(3.114)

Note that, if G > 0, then eq.(3.114) implies that x' < x. That is, x' is a less aggressive intervention that x. In particular, let x be the policy choice which, based on the bestestimated model, causes the outcome to precisely match the required value: y{x) — y^. This choice of x is what would normally be called the optimal policy. Eq.(3.114) means that the estimated fidelity is worse with the less aggressive policy x' than with x\ 0 = 2/*- y[x) A

(3.124)

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INFO-GAP DECISION THEORY

Eq.(3.123) means that the robustness decreases with chase-time when hunting nearby prey, while eq.(3.124) shows that the robustness increases with time when chasing distant prey. This very simpUstic example is suggestive of the different effort and technology requirements for "ambushes", u{0) < A, as opposed to "search and destroy missions", u{0) > A. Robustness S(g,A)

Duration, T Figure 3.13: Robustness versus time, eq.(3.122). The value of q increases from the bottom to the top curve. u{0) < A. The robustness function of eq.(3.122) is plotted in fig. 3.13 versus the duration of the chase, T, for various values of the hunter's design variable q and for u{0) < A. At fixed effort g, the monotonic decrease of the curves shows that the robustness to uncertain evasive behavior decreases as the duration of the hunt increases. With search-effort q, longer chases which began as a close encounter {u{0) < A) enable the prey to evade the hunter more easily. The figure provides the decision maker with quantitative assessment and insight into the significance of different choices oi q. In a long chase the hunter must exert greater effort (larger q) in order to prevent the prey from escaping.

3.2.10

Assay Design: Environmental Monitoring

The assay of material which is distributed spatially in an unknown manner is a common problem. The design of the assay system must confront the spatial uncertainty of the analyte, but little may be known about the spatial distribution before measurements are made. Consider the following example. The local municipality is planning to release treated sewage into a nearby river. A monitoring system will be installed to detect adverse effects by measuring the density of a species which is an indicator of contamination. The aim of the assay is to determine if the total biomass of the indicator species, over a distance L downstream of the release point, exceeds a critical value. Be. The assay system will measure the local biomass density at A^ equally spaced points. Remedial action will be taken if one or more measurements yield a local density in excess of a specified value, po- The problem is that the spatial variation of the density is uncertain, and the test-point measurements may not exceed po and yet the total biomass may exceed the critical limit, Br.

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It is anticipated that the biomass density, p(x), wih vary gradually along the length of the river, with density rising and falling with a maximum slope which is no more extreme than ±'s. However, the estimate of the maximal rate of change of the density, ?, is highly uncertain. Consider the following slope-bound info-gap model for the biomass density profile, p(x), 0 < x < L, including the no-alarm assay result that density is no greater than po at all of the N test points Xi: W(a,po,S) = | p ( x ) : p(x,) < po,i = 1, • • •, ^ ; I'^'^"^1'~ ^ I < c^j ,

a>0

(3.125) where p'(x) is the slope of the biomass density function. The inequality on the slope, p\ means that, at horizon of uncertainty a, the slope lies either in the interval [(1 — a)?, (1 + a)S] or in the interval [-(1 -ha)?, ( - 1 + a)S]. However, the horizon of uncertainty, a, is unknown. The assay-system designer must choose the number of test points, iV, and the trigger density po^ so as to be confident that, when all N measurements indicate local density less than po^ the total biomass does not exceed Be- The robustness of N measurements with trigger density po is the greatest horizon of uncertainty in the density profile such that the total biomass will not exceed Be when none of the test-point densities exceed po' S(iV,po,^c) = niaxi Of : [

max _ /

p{x) dx] < B^ >

(3.126)

M{a)

We now indicate how the robustness function can be conveniently evaluated. Denote the inner maximum in eq.(3.126) by M{a). The robustness is the greatest value of a such that M{a) < Be- The uncertainty sets U{a^po^^ become more inclusive as a increases, so M{a) also increases monotonically with a. Thus the robustness is the greatest value of a such that M{a) = Be. In other words, M{a) is the inverse of a(A^, po^ ^c)M{a) = Be

implies

3(7V, po, B^) = a

(3.127)

A plot of M ( a ) versus a is the same as a plot of B^ versus a{N^ po.Bc). The evaluation of M{a) is illustrated in fig. 3.14, based on the info-gap model of eq.(3.125) which assumes that the slope of the biomass density is bounded, but that the value of this bound is unknown. Given measured densities of po at two adjacent test points, the greatest possible biomass between these points, at horizon of uncertainty a, occurs when the slope is extremal as indicated in fig. 3.14. The biomass in this interval is readily calculated and the maximum possible biomass, at uncertainty a, in the N — I equal intervals between 0 and L is then found to be: M(a) = Lpo + ^^^^^{l

+ a)

(3.128)

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INFO-GAP DECISION THEORY

Robustness p{x)

S(A^,Po,^c) ^ J p p e r envelope / \ a t uncertainty a

(l+a)s//

Po

/A-(l+a)7

f

^

Xi

X^^l

Bo{N*)

Bo{N)

Position, X

Critical biomass, Be

Figure 3.14: Evalution of M ( a ) , eq.(3.126), showing an upper envelope and three possible density curves.

Figure 3.15: Robustness curves for N and N* test points with trigger densities po and pj, eq.(3.129). A^* > N, p*o> po-

Equating eq.(3.128) to the critical biomass Be and solving for a yields the greatest tolerable error in the estimated slope, which is the robustness:

4(iV-l) S(7V,Po,^c) -
Lpo

+

L's

4(A^-1)

Bo{N)

0

else

(3.129) We will discuss the term Bo{N) shortly. Eq.(3.129) reveals the usual trade-off: the robustness increases ( 5 gets larger) as the performance gets worse {Be gets larger), as illustrated by the positive slopes in fig. 3.15. Eq.(3.129) also demonstrates the value of increased assay-effort: the robustness increases with increase in the number of test points in the length L along the river. Eq.(3.129) shows the unreliability of attaining the estimated performance. Bo{N) is the biomass of a distribution whose measurements all equal po and whose slope between test points precisely equals the anticipated values of ±'s. However, as illustrated in fig. 3.15, a{N,po,Bc) — 0 if Be = BQ{N). T h a t is, arbitrarily small deviation of the slope from the estimated value can cause the actual biomass to exceed the anticipated value. This has strong implications for selecting the number of measurements: in order to reliably detect the critical biomass, more measurements will be needed t h a n is indicated by the anticipated slope variation, as we now explain. Suppose Be is t h e value of biomass which experts deem to be the largest

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acceptable value. Choosing the number of test points, N, from the relation Bo{N) = Be is unreliable because the robustness for detection of B^ is zero. Let 3d denote a demanded robustness to slope-uncertainty. The number of test points should be chosen from: a(7V,Po,5c) = Sd

(3.130)

Choosing N from Bo{N) = Be is equivalent to choosing A^ from eq.(3.130) with 3d = 0. Finally, the intersecting robustness curves in fig. 3.15 demonstrate the trade-off between the number of test points, A^, and the density po at which an alarm is triggered. The reduction in robustness resulting from decreasing the number of measurements, N < N'^, may be partially compensated for by decreasing the trigger density, po < Po- More precisely, the robustness curves for {N,po) and (N'^.PQ) cross at ( 5 x , 3 x ) . This means that design {N,po) is preferred over design (A^*, p^), in terms of robustness, for biomass detection-requirement stricter than 5 x , Be < Bx, while the preference is reversed for laxer biomass requirement Be > B^-

3.2.11

Bio-Terror Preparedness with Epidemiological Models

The world today faces serious and credible threat of the use of biological disease agents against civilian populations. Governments must formulate policy for both strategic preparedness and tactical response to such attacks. Available policy tools include immunization of first-responders and the public, surveillance, control of population movement, assumption of extraordinary police powers and other suspension of civil liberties. The policy maker must choose a combination of tools which is effective, reliable and socially acceptable. Planning and response to bio-terror-induced disease can be assisted by both prior and real-time forecasts of the spread of the disease. Real-time forecasting can assist the authorities in tailoring tactical response to the specific incident. Preparedness and response to bio-terrorism is based in part on computational models of the spread of disease [40, 53, 60]. The epidemiological phenomenon is extremely complex and incompletely understood, especially in the aftermath of a terror attack, and event-specific data are lacking, so these models are accompanied by tremendous info-gaps. We will develop a simple example of info-gap policy analysis. The number of deaths resulting from a bio-terror attack, A^, depends on the number of initial infections /, the number of subsequent secondary infections 5, and the fraction / of infected individuals who die. That is, the mortality following the terror incident is: N = {I^S)f

(3.131)

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INFO-GAP DECISION THEORY

/ can be roughly estimated from intelligence data. S can be estimated from epidemiological models, but considerable uncertainty surrounds the disease agents involve, the population-mixing behavior during and after the incident, and other factors. / depends on the diseases involved, the extent of prophylaxis, and the timeliness of treatment. The estimates and the realized values of / , S and / all depend on the policy adopted by^he^govermnent. The best available estimates of these quantities are / , S and / , with estimated relative errors cr/, as and cr/. An info-gap model for these highly uncertain quantities is: U{a,I,Sj) = \l,S,f: ^•^^ 0 [ CT/ as cFf J (3.132) The robustness of a policy with estimated values / , S and / , and with the requirement that the mortality not exceed A^c^ is: S(7, 5 , / , A^c) = max i a : (

max

A^(/, 5, / ) ) < A^c >

(3.133)

One finds the robustness to be: -c, + 3(7, 5, / , N,) =

^cl+4c2{N,-N) '-

(3.134) 2C2

if A^c > A' and zero otherwise. A = (/ + 5 ) / is the estimated number of deaths, ci = (/ + S)af + {aj + as)f and C2 = {aj + crs)af. The robustness curve of eq.(3.134) is shown schematically infig.3.16. Two points should be noted. First, the robustness to uncertainty in the epidemiological dynamics is zero when the required mortality equals the anticipated value: S(7,5,/,A^c)=0

if Nc = N

(3.135)

That is, the outcome which is anticipated based on the best available estimates has no immunity to error in these estimates. Since anticipated outcomes cannot be relied upon to materialize, due to the info-gaps in the data and models upon which these anticipations are based, we must reduce our aspirations. Only larger values of mortality can be reliably anticipated. This means that we must "migrate up" the robustness curve infig.3.16 to higher (that is, worse) A'c in order to obtain higher (that is, better) a. Performance, Nc, trades-off against robustness, a. The slope of the curve represents the rate at which robustness can be purchased in exchange for reduced performance. The second point to note infig.3.16 is that the slope of the robustness curve is likely to be very small:

85

CHAPTER 3. ROBUSTNESS AND OPPORTUNENESS

Robustness

Robustness

ails, IN,)

a(7,5,/,iVc)

N

N' A^x

Critical mortality, A^c

Critical mortality, A^c

Figure 3.16: Schematic robustness curve, eq.(3.134).

Figure 3.17: Schematic robustness curves for two policies, P and P\ illustrating reversal of preference.

Typical values of the slope might be on the order of 1/100. This means that increasing A^c by 10 (accepting 10 more deaths) increases the robustness by only 0.1. The robustness is the greatest fractional error in the estimated quantities, each as a fraction of its a-value, up to which A^c is not exceeded. A robustness-increment of S = 0.1 means tolerance to an additional 10% error in / , 5 and / . Now consider ^wo different policies, P and P\ corresponding to estimated values (7,5',/) and ( / ' , 5 ' , / ' ) . Fig. 3.17 shows an important situation which can arise in practice: intersection of robustness curves. The nominal or anticipated mortality with policy P is better (smaller) than the nominal mortality with policy P':

N{P) = {I + S)f < {r + S')f = N{P')

(3.137)

Based on the best estimates, policy P is preferred over policy P ' because P has lower anticipated mortality. However, as explained in connection with eq.(3.135), anticipated mortality has zero robustness. Since the robustness curves cross, P' is preferred over P for any value of A^c greater than the value A^x at which the curves cross. For instance, if robustness no less than Sx is required in order to have adequate confidence, then design P ' is preferred over P if Nx is acceptable mortality. The robustness curve for policy P' can cross the robustness curve for policy P , despite the nominal preference for P as expressed in eq.(3.137), if 3 ( P ' , A^c) is steeper than a(P, A^c) as shown in fig. 3.17. Examination of eq.(3.136) shows that steeper slope results from better information about the epidemiological dynamics: cr-, a-, cr-, are smaller than a^, cr^, ar-

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INFO-GAP DECISION THEORY

The nominal preference for P , eq.(3.137), results from more intensive policy measures such asjmmunization or population control causing / , S and / to be smaller than /', S' and / ' . The crossing of the robustness curves shows that severity of policy measures can be replaced by modelling accuracy, and vice versa. The less aggressive policy, P ' , can be preferable when it is accompanied by better information.^ The robustness function provides the policy maker with a quantitative tool for deciding how to allocate policy resources between intrusive governmental activity (which may entail public resentment or non-compliance) and scientific modelling (which may not).

3.2.12

D r u g Selection

Control of blood cholesterol is generally accepted as central in prevention of heart disease. Common measures for cholesterol control include diet and an array of pharmaceuticals. Clinical outcomes are roughly proportional to lowering of LDL-C cholesterol. Cholesterol lowering is essential for longevity in people having high risk of heart disease such as diabetics. Data indicate that cholesterol can be greatly reduced, but the marginal benefit of large reduction is highly uncertain. Furthermore, toxicity is increased by higher doses of drugs or by adding more drugs [170]. We will consider a simplified info-gap robustness analysis for selecting a drug dosage.^^ We have data for the probability of ischemic heart disease, p^j^ at four levels of cholesterol reduction, 0, 20, 40 and 62 mg/dL. The estimated standard errors of these probabilities are SQJ. Likewise, we have much weaker and more uncertain evidence of the increase in the probability of serious complications, p^^, resulting from the drugs used in reducing the cholesterol concentration by the j t h amount. The standard errors of these probabilities are Sij. These data are shown in table 3.3. LDL-C reduction [mg/dL] 0 20 40 62

Poj

SOj

Pij

SOj

.05 .04 .03 .02

.05 .04 .03 .02

0 .001 .003 .01

0 .01 .03 .1

DU^P)

.10 .081 .063 .050

T a b l e 3 . 3 : P r o b a b i l i t y of h e a r t disease, pg^, ^ ^ d complications, p^^, w i t h s t a n d a r d errors Sij, as a function of r e d u c t i o n in blood cholesterol.

The actual values of the probabilities in table 3.3 are highly uncertain due to individual and sub-group variability, imperfect knowledge of the ^This is a variation on Brainard's dictum discussed in section 3.2.8. ^^The author is indebted to Prof. Cliff Dasco for suggesting this example.

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drugs, and variability of the pathologies involved. We use the data in the table to formulate the following interval-bound info-gap model: U{a,p) = {p: pij e [0,1], \pij -Pij\ 0 (3.138) Heart disease and adverse side effects of drugs are both undesirable. Considerable uncertainty surrounds the quantitative evaluation of the degree of dis-utility of these outcomes. In this example we will ignore this uncertainty and assume that we know the dis-utility of the outcomes, do and di for heart disease and complications.^^ The average dis-utility, with cholesterol-reduction by the j t h amount, is: D{j,p) = Pojdo+pijdi

(3.139)

We require this expected dis-utility to be small, and in any case no larger than the critical value Dc. However, we are quite uncertain about the probabilities pij. The robustness of cholesterol reduction by the j t h amount is the greatest horizon of uncertainty in the estimated probabilities p^j up to which all realizations pij result in dis-utility no greater than Dc'. a{j, Dc) = max \oc: [

[ max^ D{j,p) \ < DA \p^U{cy,p) J J

(3.140)

A large value of 3(j, Dc) implies that dis-utility no greater than Dc can be confidently anticipated if cholesterol-reduction by the j t h amount is attempted. A low value of S(j, Dc) implies that the required outcome cannot be confidently expected. The robustness function, S(j, Dc), thus generates preferences on the drug-selections which are available. The robustness function is displayed in fig. 3.18 for the four levels of LDL-C cholesterol reduction. The positive slopes of these curves indicate the usual trade-off: large robustness, 3(j, Dc), is obtained only by accepting large dis-utility, Dc. The robustness equals zero at the estimated value of dis-utility: a{j,D,) = 0 if D, = D{j,p) (3.141) Based on the estimated probabilities, cholesterol reduction of 62 mg/dL ( in the figure) has the lowest dis-utility as seen in the final column of table 3.3. One might therefore be inclined to aim at this large cholesterol reduction. However, this outcome has no robustness to uncertainty in the probabilities as indicated in eq.(3.141). Since the probabilities are highly uncertain, only greater dis-utility can be reliably anticipated with a 62 mg/dL reduction. Furthermore, the robustness curves cross one another. In particular, the curves for 40 and 62 mg/dL cholesterol reduction intersect at a = 0.25. ^In section 10.3 we consider a similar situation in which this simphfication is not made.

INFO-GAP DECISION THEORY

oc{j,Dc)

Figure 3.18: Robustness vs. critical dis-utility for four levels of cholesterol reduction: 0 ( ), 20 ( ), 40 (— •), 62 ( ) mg/dL. do = 2di = 2.

The standard errors in table 3.3 are far greater than 25% of the mean probabilities, so one needs robustness far in excess of 0.25. This indicates that a 40 mg/dL reduction will be more reliable than all other options for any realistic level of dis-utility. The 40 mg/dL choice dominates all other options and is thus the clear preference. We are able to choose a cholesterolreduction goal, and thus a drug type and dosage, without having to choose a desired level of dis-utility, which may be a difficult choice to make.

3.2.13

Estimating an Uncertain Probability Density

In many situations one wishes to estimate the parameters of a probability density function (pdf) based on observations. A common approach is to select those parameter values which maximize the likelihood function for the class of pdfs in question. In this section we develop a simple example to show how to deal with the situation in which the form of the pdf is uncertain. This is a special case of system identification when the structure of the system model is uncertain. A simple example is found in [25, section 11.4]. Consider a random variable x for which a random sample has been obtained, X = (a^i, . . . , XN)- Let p(x|A) be a pdf for x, whose parameters are denoted by A. The likelihood function is the product of the pdf values at the observations because the observations are statistically independent of one another: N

L{X,V) = \{v{x,\\)

(3.142)

z= l

The maximum likelihood estimate of the parameters is the value of A which maximizes L(X,p): A* = argmaxL(X,p) (3.143)

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But now suppose that the form of the pdf is not certain. Let p{x\X) be the most reasonable choice of the form of the pdf, for instance p might be the normal or exponential distribution, but the actual form of the pdf is unknown. We will still estimate the parameters A of the nominal pdf p'(x|A), but we wish to choose those parameters to satisfice the likelihood and to be robust to the info-gaps in the shape of the actual pdf which generated the data, or which might generate data in the future. Let V be the set of all normalized and non-negative pdfs on the domain of X. Thus the actual pdf must belong to V. Let U{a,p) denote an infogap model for uncertainty in the actual form of the pdf. For instance the envelope-bound info-gap model is: U{a,p) = {p{x) : p{x) e V, \p{x) - p{x\\)\ < a^{x)} ,

a>0

(3.144)

where ip{x) is the known envelope function and the horizon of uncertainty, a, is unknown. Now the question is, given the random sample X, and the info-gap model for uncertainty in the form of the pdf, how should we choose the parameters of the nominal pdf ^(xIA)? We would like to choose parameter values for which the likelihood is high. However, since the form (not only the parameters) of the pdf is uncertain, we wish to choose A so that the likelihood is robust to the info-gaps in the shape of the pdf. The robustness of parameter values A is the greatest horizon of uncertainty a up to which all pdfs in U{a,p) have at least a critical likelihood Lc'. 3(A,Lc)=max Lc > \peU{cy,p) J J

(3.145)

To develop an expression for the robustness, define fi{a) as the inner minimum in eq.(3.145). For the info-gap model in eq.(3.144) we see that ji{a) is obtained with the following choices of the pdf at the data points X:

Define:

_ c.max = m i n ^

(3.147)

Since fi{a) is the product of the densities in eq.(3.146) we find: N

/j.{a)

Y[[p{xi) - aipixi)]

if a < a^ax

/3 ^^gx

z=l

0

else

According to the definition of the robustness in eq.(3.145), the robustness of likelihood-aspiration Lc is the greatest value of a at which fi{a) > Lc-

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INFO-GAP DECISION THEORY

Since /j.{a) strictly decreases as a increases, we see that the robustness is the solution of /i(a) = Lc. In other words, fi{a) is the inverse of 3(A, Lc): /i(a) = Lc

implies

(3.149)

a(A,Lc) = a

Consequently a plot of /i(a) vs. a is the same as a plot of Lc vs. a(A,Z/c). Thus, eq.(3.148) provides a convenient means of calculating robustness curves. Robustness S(A,Lc)

Sx/O^n

0.03h 0.02

0.2

0.01

O.lh 1

1.5

2

0

'*\

0.2

0.6

1.0

Critical likelihood, log^g ^c

Lx/L[X,p(x|A*)]

Figure 3.19: Robustness curves. A* = 3.4065.

Figure 3.20: Loci of intersection of robustness curves a(A*,Lc) and S(l.lA*,Lc).

Robustness curves are shown in fig. 3.19 based on eqs.(3.148) and (3.149). The nominal pdf is exponential, p{x\X) — Aexp(—Ax), and the envelope function is constant, '\\){x) — 1. An exponentially distributed random sample containing A^ = 20 data points is generated with A = 3. Thus the nominal distribution is in fact correct, and the uncertainty pertains to the future data generating process, which may not be exponential. The maximum-likelihood estimate (MLE) of A, based on eq.(3.143), is A* = 1/x where x — (l/N) X]2=i ^i is the sample mean. Robustness curves in fig. 3.19 are shown for three values of A, namely, 0.9A'*', A*, and I.IA"^. Given a sample, X, the likelihood function for exponential coefficient A is L[X,p{x\X)]. Each robustness curve in fig. 3.19, 3(A,Lc) vs. Lc, reaches the horizontal axis when Lc equals the likelihood: S(A, Lc) = 0 if Lc = L[X,p{x\X)]. In other words, the robustness of the estimated likelihood is zero for any value of A, as expected from eq.(3.25) on p.49. A* is the MLE of the exponential coefficient. Consequently, for any A, L[X,p{x\X^)] > L[X,p{x\X)]. Thus 3(A*,Lc) reaches the horizontal axis to the right of 3(A, Lc). For the specific random sample whose robustness curves are shown in fig. 3.19, the robustness of the MLE, A*, is greater than the robustness of

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the lower value, 0.9A'*', at all likelihood aspirations Lc- In other words, A* is a better robust-satisficing choice than 0.9A* at any Lc- However, the robustness curves for A* and 1.1 A* cross at (Lx, Sx), indicating that the MLE is preferable at large Lc and low robustness, while LIA"*" is preferred elsewhere. The crossing of robustness curves entails the reversal of preference between A* and I.IA"*", as discussed in section 3.1.8. This pattern is repeated for all of 500 random samples: S(A*, Lc) dominates 3(0.9A*,Lc), while a(A*,Lc) and S(l.lA*,Lc) cross. The coordinates of the intersection of S(A'*', Lc) and 3(1.lA*, Lc) are plotted in fig. 3.20 for 500 robustness curves, each generated from a different 20-element random sample. In each case. A* is the MLE of that sample. The vertical axis is the robustness at the intersection, Sx, divided by the maximum robustness for that sample, Qmax, defined in eq.(3.147). The horizontal axis is the likelihood-aspiration at the intersection, Lx, divided by the maximum likelihood for the sample, L[X,p(x|A*)]. The center of the cloud of points in fig. 3.20 is about (0.5, 0.2). What we learn from this is that the robustness curves for A"^ and 1.1 A* typically cross at a likelihood aspiration of about half the best-estimated value, and at a robustness of about 20% of the maximum robustness. We also see that curve-crossing can occur at much higher values of Lc, and that this tends to be at very low robustness. This happens when L[X,p{x\l.lX'^)] is only slightly less than L[X,p{x\X*)]. Curve-crossing can also occur at much lower Lc and higher robustness, typically because L[X,p{x\l.lX*)] is substantially less than L[X,p{x\X^)]. Note that the data in this example are generated from an exponential distribution, so there is nothing in the data to suggest that the exponential distribution is wrong. But things change, so a model which fits historical data may "become mis-specified . . . [which is] the main problem in economic forecasting" [9, p.246]. The motivation for the info-gap model of eq.(3.144) is that, while the past has been exponential, the future may not be. The robust-satisficing estimate of A accounts not only for the historical evidence (the sample X) but also for the future uncertainty about the relevant family of distributions. In short, the curve-crossing shown in fig. 3.19 is typical, and robustsatisficing provides a technique for estimating the parameters of a pdf when the form of the pdf is uncertain.

3.3

Production Volume With Uncertain Costs

Many decisions under uncertainty involve one or another form of cost-benefit analysis. The prototype of such deliberations is the direct calculation of profit: the difference between monetary earnings and costs. We will illustrate some aspects of the robustness and opportuneness functions with a

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simple economic example of a small firm which needs to choose its production volume despite severely uncertain production costs. Consider a static production situation in which the manufacturer must choose the quantity q which will be produced, and let us assume this to equal the quantity which will be sold. p{q) is the known price per item, which may depend on the quantity produced, and c{q) is the total cost of producing q items. We will suppose that the production cost c{q) is an uncertain function of the production volume q. The profit R{q^ c) depends on both the manufacturer's decision q and the uncertain costs incurred, c: R{q,c) = qp{q)-c{q)

(3.150)

Classical theory: profit maximization. If p{q) and c(g) were both completely known, then the manufacturer could maximize the profit by choosing q to satisfy: dR

dp

dc

/^ . ^.x

However in our case, while the price function p{q) is known, the cost function c{q) is highly uncertain, so eq.(3.151) cannot be directly solved to find the optimum production volume. Satisfice profit, maximize robustness. We now consider a robustsatisficing strategy which both guarantees no less than a specified minimum profit (if possible) and which maximizes the robustness or immunity to uncertainty. Suppose we have chosen an info-gap model U{a,c), a > 0, to match the limited information which we possess about the uncertainty of the production-cost function c{q). Let TC be the lowest profit which the producer tentatively considers to be acceptable. A robust-satisficing strategy for choosing the volume of production is to seek the value of q which guarantees a profit of at least TC and which maximizes the producer's immunity to the unknown variation of the cost function. This "robust rationality" satisfies the producer's need for a minimum profit while also maximizing the avoidance of uncertainty. TC is a parameter which need not be chosen unalterably, but which folds into the firm's decision process. After evaluating the robustness function, a{q,rc), the producer can explore the desirability of different values of critical reward TC by examining the robustness to uncertainty as a function of reward. As in fig. 3.1, a{q^rc) will decrease monotonically as TC increases. The implementation of this strategy proceeds in two stages. First we find the robustness as a function of q. The robustness is the greatest horizon of uncertainty, a, which is consistent with obtaining a profit no less than re, which is expressed by eq.(3.4) on p.40. The second stage is to choose the volume of production to maximize the robustness. This optimal volume, $c(^c), is defined in eq.(3.20) on p.45.

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This approach is quite different from a max-min strategy wherein one maximizes a minimum profit. What one maximizes here is the robustness to uncertainty, making this strategy attractive for avoidance of uncertainty. Of course, one can still gamble since setting a high value for the least acceptable profit Tc will reduce the available robustness a. In subsequent examples (and extensively in chapter 4) we will discuss "calibration" of the robustness: while a may be either dimensionless or have problem-specific units, its value can be scaled to give a feeling of whether the robustness is great or small. Example 1 Uniform-bound info-gap model. Let us suppose that p{q) is known exactly and that we know a nominal or typical value of the production-cost function, c{q), but we also know that the actual cost function can deviate from c{q) in some unknown way. This very limited amount of information about the variability of the cost function can be represented by a uniform-bound info-gap model: ZYu(a,2)-{c(9): \c{q) - c{q)\ < a} ,

a>0

(3.152)

Uu{cy,c) is the set of cost functions c{q) whose deviation from c{q) is no greater than a. What we know about the cost function c{q) is the nominal function c{q) and that the deviation of c{q) from c{q) is bounded by a. What we do not know is the value of the horizon of uncertainty a, so the info-gap model is an unbounded family of nested convex sets Uu{(^,^, for a > 0. The minimum profit, up to uncertainty a, in the evaluation of the robustness in eq.(3.4) is readily seen to occur for the greatest cost function allowed by the info-gap model at horizon of uncertainty a, which is simply c{q) -t- a: min _ R{q, c) = qp{q) - [c{q) + a] (3.153) c{q)eUu{a,c)

For fixed production volume g, the robustness is the greatest value of the uncertainty parameter a for which this minimum is at least TC- Equating the minimum profit to the critical value Vc and solving for a results in the robustness function: qp{q) - [c{q) + a] = r^

=>

a^iq, r^) = qp{q) - c{q) - r^

(3.154)

unless this expression for a^iq.rc) is negative, in which case a^{q,rc) — 0. The robust-satisficing production volume, defined in eq.(3.20), is found from: _

0 = ^^4^

= p{q) + 5 ^

- ^

(3.155)

dq dq dq which is precisely the classical full-information solution, eq.(3.151), if the nominal cost function, c(g), is employed in the classical theory. In other words, the classical profit maximization with precise information leads to the same production decision as the robust-satisficing strategy with this

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particular representation of uncertainty in the cost function. Furthermore, the robust choice of q is in fact independent of the producer's lowest acceptable profit, Tc- However, the info-gap robustness will equal zero for the classical best estimate of the profit. Hence the firm will evaluate its decision differently in the classical and in the robust-satisficing analyses. Example 2 Envelope-bound info-gap model. The equivalence of the classical and the robust-satisficing decision strategies in example 1 results from the structure of the uniform-bound info-gap model of cost-function uncertainty. Suppose, as before, that the demand function p{q) is known exactly, and that the nominal production-cost function ?(g) is known, and that the shape of the envelope of uncertain variation of the actual cost function changes in a known way with the quantity produced. For instance, we may have information about the relative accuracy of the estimated cost, c(g), at different values of q. The envelope-bound info-gap model is suitable to this somewhat different prior information. Instead of eq.(3.152) we use: U,{a,^

= {c{q) : \c{q)-Z{q)\

< ai;{q)} ,

a >0

(3.156)

where c{q) and 'ip{q) are known, and again we have an unbounded family of nested convex sets of possible production-cost functions. V^(g) determines the shape of the envelope within which c{q) varies in an unknown way, while the horizon of uncertainty a (whose value is unknown) determines the size of the envelope. Now, instead of eq.(3.153), the least possible profit up to uncertainty a becomes: min _ R{q, c) = qp{q) - \c{q) -f aV^(g)]

(3.157)

c(qf)G^^e(a,c)

And, instead of eq. (3.154), the robustness of production volume q becomes: Se(g,rc)= r^

^3^^^^^

where R[q,Z{q)] is the estimated profit function, eq.(3.150). Seeking the most robust production volume, as in eq.(3.155), we will (usually) obtain the classical profit-maximization result only if the uncertainty-envelope function ?/;(Q') is constant (so that Ue reverts to the uniformbound info-gap model U^). The robustness of the decision can be calibrated by noting from eq.(3.156) that c(g'), c(g) and a all have monetary units (^(g') is a dimensionless envelope function). 3e(^,rc) is the greatest uncertainty in the cost function Z{q) which is consistent with minimum profit TC. The ratio Se(g, rc)/c(g) is a dimensionless expression of the greatest acceptable fractional uncertainty in c(g). If Se(^, rc)/c{q) ^ 1 then large fractional variation of c(q) with respect to the nominal function does not jeopardize attainment of the critical

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profit Tc, and t h e production volume q is very robust. On t h e other hand, if ae{q,rc)/c{q) q, are shown in fig. 3.21, where \f^^ yjq > co/po implying t h a t R{q,c) < R{q',c). The robustness decreases as t h e critical profit, r^ increases, illustrating the trade-off asserted in eq.(3.23). Furthermore, t h e robustness vanishes precisely when t h e aspired profit Vc equals t h e anticipated profit as stated in eq.(3.25). Finally, these robustness curves cross, indicating t h e potential for preference reversal. T h e nominal profit at production volume q^ exceeds the nominal profit at volume q, suggesting t h a t q' might be preferred over q. However, t h e slope of t h e robustness curve is more negative for q t h a n for g', so t h e curves cross at r x - This means t h a t q will be preferred over q' at profit aspiration less t h a n TX •

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Information and robustness. One would generally expect that more information would enable the choice of a more robust strategy. We can give this intuition a precise statement, and also understand when to expect surprises. The extent of our information about the uncertain elements of the analysis is represented by an info-gap model. The relation "more information" in one formulation than another is expressed by set-inclusion of the corresponding info-gap models: ZYI(Q;,C) C ZY2(a,c). Model T is more informative than model '2' because the former constrains the uncertain quantity more tightly than the latter. In WI(Q;,C) the information gap is smaller than in U2{(^^c). It then results from eq.(3.4) that the corresponding robustnesses are ranked in reverse order: Si(^,rc) > S2(^, ^c). (We will explore this ranking further in chapter 7.) For instance, suppose that the envelope function ip{q) in eq.(3.156) satisfies: 0 < ip{q) < 1. Then the envelope-bound info-gap model is more informative than the uniform-bound model, Ue{a,c) C Z//u( Su(95'^c)- No robust strategy based on the uniform-bound model will yield a robustness greater than that obtainable from the envelope-bound model (with the same values of q and TC). Surprises can occur in the relative robustnesses of different formulations as a result of the incautious assessment of how informative an info-gap model really is. A Fourier ellipsoid-bound info-gap model, such as eq.(2.26) on p.26, is certainly more complicated and intricate than the uniform-bound model, eq.(3.152). However, it is not so clear that the Fourier-bound model is more informative than the uniform-bound model in the strict sense of more tightly constraining the uncertain cost function for all values of ce > 0. In fact, these info-gap models can be chosen so that neither includes the other, though they do intersect. While one model may lead to a more robust decision than the other at some aspiration levels, we are unclear about which is which until we perform the analysis. Information surrogates. Given two info-gap models, where Wi (a, c) C U2{(^^c), we can now simply say that Z//i(a,c) is more informative than 2^2(0^,2). However, it is nonetheless still possible that the maximum robustnesses of these two models are the same: 32(^25'^c) = 3i(g^^,rc). The decision maker will have the same vulnerabihty to info-gaps with U2{o6,c) as with ^/i(a,c), so we can use the terminology of Simon [160, p.239] and refer to ZY2(a,c) as an information surrogate for Z//i(a,c). In fact, given any two info-gap models, regardless of whether one is included in the other, if they have the same maximum robustness we can say that either model is an information surrogate for the other, or that the two models have the relation of information equivalence. The decision maker's robustness will be identical, whichever information set is used, though different robust-satisficing strategies may be adopted. That is, g^ may differ from q^2- (We will explore a related issue in chapter 9 when we ask: by how much can the info-gap models of two decision makers differ and yet they

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still agree on a course of action.) We might also say that the uniform-bound info-gap model is an information surrogate for precise knowledge of the cost function in example 1 of this section, since the robust-satisficing strategy based on ZYu leads to the same choice of q as the classical profit-maximization strategy (compare eqs.(3.151) and (3.155)). While the decision maker makes the same choice with a robust-satisficing strategy based on Uu as with complete knowledge, this is serendipitous, since the optimization goals are different in these two situations. In one case profit is optimized while in the other case robustness is optimized and profit is satisficed. Such serendipitous information surrogates are not uninteresting or unimportant, but we need not be disappointed when they are lacking, since the decision maker may in fact deliberately prefer the robust-satisficing strategy even though it fits a less informative model. A note of caution. We have said that Ui{a,c) is more informative than U2{