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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INFO-GAP DECISION THEORY

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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INFO-GAP DECISION THEORY

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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OVERVIEW

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.

C H A P T E R 1.

OVERVIEW

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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
= -

(2.1)

A different group of engineers is using the same new material and has the same information which, however, they express by saying that the specific gravity p, which is the inverse of the specific volume, lies between 1/3 and Ig/cm'^. Again, nothing else is known so (incautious) application of the Principle of Indifference leads them to conclude that p is uniformly distributed on the interval [|, 1]. Hence the probability is 3/4 that the specific gravity of a sample lies between 1/2 and 1.

Prob Q < p < l) = ^

(2.2)

But these two events, \ < v < 2 and ^ < p ^ 1, are the same event and therefore must have the same probability. We have proven, then, that i — 3 2

4-

Examples such as this abound, and the precise pitfall in applying the Principle of Indifference is not immediately evident. Keynes' ultimate limited warranty on the Principle of Indifference is an insightful defense of subjective probability based on 'Intuition or direct judgment" [99, p.52]. He justifies the Principle when it is used in conjunction with "our knowledge of the form and meaning of the alternatives" [99, p.61] (itahcs in the original). Keynes provides a rule — indivisibility of the alternatives into "sub-alternatives of the same form.'' [99. p.60] (italics in the original) — for testing the Principle of Indifference. Keynes' defense of the Principle of Indifference, and of the tenets of subjective probability in general [153], has been the subject of wide debate [123, p.75]. Wherever one stands in that polemic, Keynes' argument is a fog-horn warning against the indiscriminate

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use of the Principle of Indifference to select a probability distribution based on complete ignorance of how the random variables are distributed. Two envelopes. Here's a simple way to earn some extra cash, compliments of the Principle of Indifference. Two envelopes are placed on the table, about which you know only that each contains a positive amount of money and that one envelope contains twice as much money as the other. You are invited to choose one envelope and then to open and examine its contents. You do so, and find $100 inside. Now you are given the option of exchanging this envelope for the other one, which contains either $50 or $200. Employing the principle of indifference, you might be tempted to assign equal probabilities to these two contingencies and to evaluate the expected reward if you make the exchange as 0.5 x $50 -f 0.5 x $200 = $125. Worth doing! Notice that this argument is valid regardless of the amount you found in the initial envelope (provided it is positive, as stipulated). In other words, once you had chosen an envelope, you should be willing to exchange it for the other envelope even without opening the first envelope: the expected reward is thereby augmented by 25%. Now if this 25% expected gain works once without opening the envelope, as the principle of indifference assures that it does, then it will work again. And again. And again. Until, I suppose, the envelopes split at the seams. Leveraging ignorance. The Principle of Indifference states that if you do not know anything about the events in question, in particular, how they are distributed, then you do know something about these events, namely, that they are uniformly distributed. The Principle of Indifference allows one to leverage one's ignorance, to transform an epistemic limitation (lack of knowledge) into an ontological assertion (statement about the world). While Keynes is right that there are classes of problems where this is logically legitimate or at least intuitively plausible, it is clearly a methodological error in many situations. Not all ignorance can be modelled probabilistically.

2.3

Info-Gap Uncertainty, Probability and Fuzziness

Classically, uncertainty is represented by probability theory, which quantifies lack of information either in terms of the frequency of recurrence of events, or in terms of an observer's subjective degrees of belief [123]. In recent decades the theory of fuzzy logic has been used to represent various types of uncertainties with fuzzy membership functions: the linguistic ambiguity of a proposition, or the possibility or the necessity (rather than the probability) of occurrence of an event [8, 65, 102, 179, 180]. In the historical evolution of uncertainty-thinking, fuzzy logic is a major break with traditional probability theory. These theories reflect different

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aspects of imperfect or fragmentary information, and each does so in a different way. Furthermore, each theory has developed from different classes of applications. But despite all their differences, they share a similarity of form; both fuzzy logic and probability quantify uncertainties with normalized mathematical functions: the probability density or the membership function. Can we quantify uncertainty without using distribution functions at all? The answer, of course, is yes, and the need to do so arises when either information is really scarce or, even if plentiful, the system is prone to fundamental change so the past is a weak indication of the future. Specifically, the need for a sparse model of uncertainty occurs when we find ourselves far over the frontier of our firm knowledge and deep in the realm of the unknown. In such situations we must deal with information which is much more deficient than is customarily handled either by probability or fuzzy logic. Even though info-gap models of uncertainty emerged in engineering — control theory [156], seismic design of structures [63, 64], nuclear measurements [11], mechanical analysis [32] and mechanical reliability [14] — they match an intuition of uncertainty which is prevalent among economists and management theorists. We will mention just a few examples. Knight distinguishes between risk, which can be modelled probabilistically and then insured against, and what he called "true uncertainty" which characterizes and motivates many central business decisions. We will discuss Knight again in section 13.5. Shackle likewise distinguishes probability, which is based on knowledge and experience, from uncertainty which is "un-knowledge" or "novelty". Like Knight, Shackle recognizes the economic "truth that a world which looks to tomorrow is a world bent upon exploiting uncertainty, un-knowledge. It is a speculative world." [157, pp.20, 151, 164]. Bonen studies the planning of innovative technological development projects and remarks that the "difficulty and uncertainty of development . . . increase with increasing know-how gap." [37]. March analyzes individual and organizational decision-making and writes that "Uncertainty is a limitation on understanding and intelligence." [118, p. 178]. Mack writes that "Uncertainty is the complement of knowledge. It is the gap between what is known and what needs to be known to make correct decisions." [116, p.l]. Galbraith, in discussing the design of complex industrial organizations, defines uncertainty as an information gap, as "the difference between the amount of information required to perform the task and the amount of information already possessed by the organization." [76, p.5]. Laufer identifies nine principles for managing projects in an era of uncertainty. He writes that one central principle in project management is that, "in dynamic conditions, influencing the future is not just making decisions early. It is more about the ability to reduce uncertainty and to minimize the impact of surprises." (emphasis in the original) [110, p.74]. Y.B. Choi writes that "Uncertainty is a state of doubt. It describes an indeterminate relationship between us and

16

I N F O - G A P DECISION THEORY

the environment we face. . . . The very existence of uncertainty presupposes the need for judgement and decision making that will resolve this state." [45, p.27]. Consider the following situation [92]. Toxic waste is released from an industrial plant into a river and flows downstream causing damage to plant and animal life, as well as reducing the quality of the water for human use. In assessing the environmental impact it is necessary to account for the local indigenous rate of absorption, consisting of adsorption and chemical reaction of the toxic materials with the river bed and with other materials dissolved in the river water. These processes are complex and depend on many factors, such as flow rate, degree of turbulence, temperature, concentration, and so on, which are poorly known, and which vary in time and place along the river. Long-range factors such as climate change and economic development, which are far from predictable, can impact these processes. In short, the rate of toxic waste removal from the river depends on a function of time, space, and other factors. Hydrological and water quality models can be developed to describe these removal processes, but enormous resources are required to verify the reliability of a comprehensive model. And even a model which is a verified against past behavior may perform much worse in predicting future events. In lieu of extensive resources of money and time needed to develop a well-grounded reahstic model, and in light of unpredictable future surprises, an approximate model can be adopted with the realization that it will deviate by an unknown but possibly substantial margin from the actual behavior. In employing an approximate model we are acknowledging a large information gap between what is known and what needs to be known in order to make fully competent decisions in the management of the toxic eflluent. This info-gap is a severe form of uncertainty and occurs often in practice. In many innovative technological projects very little is known which really should be known in order to make wise and reliable decisions. In environmental planning, in industrial management, in social and economic decision-making, in medical diagnosis and in other areas, quite often the gap between what we know and what we need to know is substantial. How can we organize both our information and our lack thereof? How can we represent the size of this gap to get a meaningful and robust quantification of the uncertainty confronting the decision makers? Conversely, how can we recognize and exploit the favorable surprises which may arise from ambient uncertainty? Returning to the toxic-effluent example, we will demonstrate how an info-gap model of uncertainty can be formulated. Let r(x, t^w) denote the removal rate as a function of position x. time t and other quantities w, based on the best available (but possibly quite simple) model. The actual removal rate function r{x,t.w) deviates in an unknown manner from the nominal model r(x, t, w). We have no information with which to express the likelihood of various alternative rate functions, so we are unable to specify

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a probabilistic model for the uncertainty in the function r{x,t,w). In fact, since the phenomenon in question is quite complex, and since r(x, t, w) itself is a multi-argument function and not simply a fixed number or vector, we are very hard-pressed to make any useful assertions about plausibility among the infinite continuum of alternative r-functions, so a fuzzy model of the uncertainty in the removal rate function is inaccessible without introducing far reaching and unverified assumptions. In light of this severe lack of information about the possible variations of the rate function, a simple info-gap model of uncertainty could be formulated as an unbounded family of sets of all functions consistent with the nominal function, r{x,t,w): U{a,r) = {r{x,t,w)

: \r{x,t,w)-r{x,t,w)\

< a] ,

a>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

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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.

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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

dt
a>0

(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)]
^^^

(^.6)

The nominal and actual vectors, u{t) and u{t) in eq.(2.6), need not be functions of an independent variable t, but simply constant vectors. In this case, eq.(2.6) becomes the elhpsoid-bound info-gap model for vectors, studied extensively by Schweppe [156]: U{a,u) = [u:

[u-u\^V[u-^
^^^

^^'^)

The cumulative and instantaneous energy-bound info-gap models are distinguished from each other primarily by the fact that the former contains

22

I N F O - G A P DECISION THEORY

functions of unbounded magnitude while the latter does not. Furthermore, the instantaneous-bound info-gap models contain persistently deviant functions, while the transient component of elements of the cumulative-bound models must vanish asymptotically. Spatial analogy to info-gap uncertainty. We notice that in all these info-gap models, eqs.(2.4)-(2.7), it is the deviation between a nominal and an actual quantity which is uncertain. The nominal function u{t) is known and any actual realization u{t) is a concrete entity; it is the potential deviation of u{t) from u{t) which is uncertain. Uncertainty is a relation between anticipation and realization; uncertainty is the potential for deviation from the norm. Anticipated or normative behavior will be completely realized in the absence of uncertainty. Unanticipated deviations — surprises — are the essence of info-gap uncertainty. In placing the focus of info-gap uncertainty upon the potential for deviation from the norm we find a striking analogy to Leibniz' conception of space. Leibniz wrote that "I hold space to be something merely relative, as time is. . . . For space denotes, in terms of possibility, an order of things which exist at the same time, considered as existing together, without inquiring into their particular manner of existing." [151, p.249]. For Leibniz, space has no independent existence of its own. One cannot talk about space in and of itself. Space, for Leibniz, is the potential for relation between objects. It is the order of things, the spatial relation among them, which is space. In the same way, info-gap uncertainty is purely relative, a relation between normative and actual behavior. Uncertainty is the potential for deviation of an actual realization from its normative form. Neither the norm nor any specific potential realization is uncertain; it is the potential for deviation of one from the other which is info-gap uncertainty. This spatial analogy for info-gap uncertainty demonstrates that we need no concept of chance, frequency of recurrence, likelihood, plausibility or belief in order to speak of uncertainty. Envelope-bound models. Closely related to the energy-bound infogap models, the envelope-bound models constrain the uncertain deviations to an expandable envelope. For scalar functions we have: U{a,u) = {u{t) : \u{t)-u{t)\
a>0

(2.8)

where '0(t) is a known function determining the shape of the envelope, and the uncertainty parameter a determines the size. U{a^u) is the set of functions u{t) whose deviation from the nominal function u{t) is no greater than aip{t). The envelope function jp{t) is chosen from prior information. For instance, the uncertain deviations can be constrained to particular regions of t-values by choosing '0(t) = 1 in those regions and ip{t) = 0 outside. In other situations ip{t) is used to represent the relative magnitude of variation. For instance, sometimes the maximal variation may be expected to be proportional to the nominal function, so V^(t) oc u{t). In this case, the

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info-gap model in eq.(2.8) becomes the set of functions u{t) whose fractional deviation from the nominal function u{t) is no greater than a: U{a, u) = L{t)

I

: I ""^^^-^^^^ \
I

^(0 I

a > 0

(2.9)

J

In other cases the envelope may be constant everywhere, ^/^(t) = 1, in which case the envelope-bound model of eq.(2.8) reduces to a scalar version of the instantaneous energy-bound model, eq.(2.6). Envelope-bound info-gap models for A/'-vectors can be formulated by constraining each element of the uncertain vector to lie within an expanding envelope: U{a,u) ^ {u{t) : Kit)

- Un{t)\
n = l, ...,7V},

a>0 (2.10) These envelope-bound models are applicable to constants u as well as to functions u{t). In the former case we obtain expanding interval-uncertainty models. For scalars, eq.(2.8) becomes the set of li-values within the expanding interval [5 — a, S + a], where this interval itself is of uncertain size zba. In the vector case, eq.(2.10), such intervals apply to each element of the vector u, producing a family of A^-dimensional expanding boxes. The scalar envelope-bound info-gap models can be generalized for vector functions in another way as well. To understand how this is done, we return to the info-gap model of eq.(2.8) and formulate it differently. Define the following set of functions: S = {u{t) : -ilj{t) < u{t) < ij{t)}

(2.11)

S' is a set of functions in a fixed envelope defined by ±V^(t), centered at zero. For any set, such as 5, and any real number, such as a, we define aS as the set obtained by multiplying each element of S by a. From eq.(2.11) we obtain: aS = {u{t) : -aipit) < u{t) < aij{t)} (2.12) aS is an expanding envelope of functions bounded by ±aip{t) and centered at zero. Likewise, for any set, such as aS, and any function, such as u{t)^ we define aS + u{t) as the set obtained by adding u{t) to each element of aS. Thus (2.12) becomes: aS + u{t) = {u{t) : u{t) - a^{t) < u{t) < u{t) + a'0(t)}

(2.13)

This is a shifted, expanding envelope of functions. Comparing eqs.(2.8) and (2.13), we see that the envelope-bound info-gap model of eq.(2.8) can be expressed as: U{a, u) = {u(t) : u{t) G a 5 + u{t)} ,

a > 0

The set S in this example has two essential properties:

(2.14)

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I N F O - G A P DECISION THEORY

1. S contains the zero-function: OeS

(2.15)

2. 5 is contained in its own expansions. That is: 0 < a < a'

imphes that

aS C a'S

(2.16)

We can now define a very general class of envelope-bound info-gap models for vector-valued functions. Eq.(2.14) is an envelope-bound info-gap model for any set S of functions which obeys eqs.(2.15) and (2.16). This info-gap model is a shifted expanding envelope of functions whose "shape" is determined by the set S. The expanding boxes of eq.(2.10) are a special case. Perhaps the most distinctive feature of the envelope-bound info-gap models in eq.(2.14) is that the sets U{a,u) need not be convex, unlike all the info-gap models discussed up to now. We will present an example of non-convex info-gap models later in this section. Minkowski-norm models. We have already noted that envelope- and energy-bound info-gap models are closely related. The Minkowski norm enables us to develop this further. The quadratic expression v^Vv is the square of a vector norm. This weighted Euclidean norm, \Jv^Vv, is a generalization of the Euclidean length of the vector f, which results when V is the identity matrix. But the weighted Euclidean norm is itself a special case of the Minkowski norm, which is useful in defining a wide class of info-gap models. For r > 1, the Minkowski norm of a real A^-vector u is: (2.17) When r = 2 and u = V^^'^v, we obtain the weighted Euclidean norm:

Ml = \\V'/Ml = v'^Vv

(2.18)

A Minkowski-norm info-gap model is defined as: Uria,u)

= lu:

\\v^^^{u-u)\\

< a\ ,

a>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.

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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
Un{t) -Un{t)

Mt)

(2.21)

Consequently, the envelope-bound info-gap model for vectors, eq.(2.10), becomes: U{a,u) = {u{t) : \\^[u{t)-u{t)]\\^ 0 (2.22) Slope-bound models. The rate of variation has not been constrained in any of the info-gap models considered up to now. For instance, all the models we have considered allow the inclusion of step-like functions with sharp corners. In some applications, prior information precludes such functions. The envelope-bound concept can be applied to the slope rather than to the magnitude of the uncertain function, thereby constraining the rate of variation of the function. The Fourier-bound models to be considered shortly provide an alternative method for constraining rates of variation. For example, if u{t) is the spatial variation of absorptiveness of chemicals along a river bed, or the temporal variation of the effluent flow from a smoke stack, the function may be expected to display some lethargy in its rate of change. A simple slope-bound info-gap model for uncertain variations is: U{a,u)^L{t):

\^M^l^^^\
a>0

(2.23)

where u{t) is the nominal form of the function and ip{t) defines the envelope of uncertain variation of the slope. U{a,u) is the set of functions u{t) whose uncertain deviations from u{t) vary no more rapidly than aip{t). Fourier-bound models. In many situations the decision maker has partial spectral information for characterizing an uncertain phenomenon. For example, geometric shape-imperfections of thin walled shells are constrained to fairly well defined spatial frequency ranges [2, 100]. Likewise, temporal variations in the response of a dynamic system such as an economy or an eco-system are often frequency-band limited. It is usually possible to express an uncertain function as a truncated Fourier series: u{t) = Y^ [an COS nirt ^bn sin riTTt]

(2.24)

n=ni

where the frequency range, represented by the values rii and n2, is known but the contributions an and bn of the diff'erent modes are uncertain. The expansion in eq.(2.24) can be expressed succinctly by defining (/)(t) as a vector of the cosine and sine functions appearing in the relation, and c as the vector of the corresponding Fourier coefficients. Then eq.(2.24) becomes: u{t) = J (Pit)

(2.25)

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I N F O - G A P DECISION THEORY

Uncertainty in the Fourier coefficient vector c can be expressed using an ellipsoid-bound model such as eq.(2.7), or an adaptation of an envelope model such as eq.(2.10). The Fourier ellipsoid-bound model can be written as: U{a,u) = {u{t) = u{t) + c^(j){t) : JWc

< a^} ,

a > 0

(2.26)

where VF is a known, real, symmetric, positive definite matrix. Alternatively, the Fourier envelope-bound model can be represented as: U{a,u) = {u{t) = u{t) -h c^(t>{t) : |cn - Cn| < Q^, ri = ni, ...,722} ,

a > 0 (2.27) where Z is the vector of Fourier coefficients of the nominal function u{t). In some situations one has partial information about the Fourier transform u{uj) of the uncertain function u{t). Again, all the info-gap models discussed earlier can be employed to quantify the uncertainty in U{IJJ). Hybrid info-gap models: Info-gaps in probability functions. Consider a random variable x whose probability density function (pdf) is p{x). The best known estimate of this pdf is p(x), but it is recognized that this estimate may err substantially, especially in the far tails which represent rare events never yet observed. The uncertainty in the estimated pdf can be quantified by an info-gap model in various ways. We will examine one possibility here, and encounter others later on. Let V denote the set of all non-negative and normalized pdfs on the domain in which x is defined. An envelope-bound info-gap model for uncertainty in the estimated pdf, hke eq.(2.8), is: U{a,p) = {p{x) : v{x) G P , \p{x) - p{t)\ < « ^ ( 0 } ,

^ > 0

(2.28)

V({a^p) is an unbounded family of sets of pdfs. The estimated pdf, p(x), belongs to all sets in this family, but the sets become more and more inclusive as the horizon of uncertainty, a, increases. An info-gap model for a discrete random variable can be similarly devised. Let n be a discrete random variable whose value can be any of the integers in the set {1, ...,A^}. The probability distribution of n is P = (Pi, . . . , PN)' The best known estimate of the probability distribution of n is P . Let V denote the set of all normalized and non-negative probability distributions for n. An ellipsoid-bound info-gap model for uncertainty in P is, like eq.(2.7), is:

U{a,P) = lP:

PeV,

(P-py

V(P-P)


a>0

(2.29)

Combined info-gap models. The structure of each info-gap model discussed above derives from one specific type of information: an energybound, an envelope of variation, a frequency window, etc. One may some-

C H A P T E R 2.

UNCERTAINTY

27

times have several such constraints. For instance, the envelope- and energybound models may be combined as: U{a,u)

=


/

[u{t) - u{t)]^ dt < a^ \ , a > 0

(2.30)

This info-gap model is the set-intersection of the models in eqs.(2.4) and (2.8). A non-convex info-gap model: Pendulum-like systems. On p.23 we formulated a class of envelope-bound info-gap models whose sets need not be convex. In particular, if the set S in the info-gap model of eq.(2.14) is not convex, then neither is the set U[a,u). We illustrate this now by considering pendulum-like systems. Consider a simple pendulum: a heavy weight suspended from a point on a long thin string and swinging frictionlessly back and forth. The angle between the string and the vertical direction, 0{i), oscillates in time. If the greatest angle is small, less than about 20 degrees, then 0{t) is very nearly a cosine function, while for larger angles the oscillation is periodic but not trigonometric. In any case, the angular acceleration is related to the angular deflection by: 6>'(t) = -u;^sin(9(t) (2.31) where a; is a constant. The dynamic behavior of the pendulum is completely characterized by the values of deflection and acceleration at any instant, so the ordered pair (^, 6) is a "state vector" for this system. We notice two characteristics of 6 and 6. First, they are always of opposite sign: when 0 < ^ < TT eq.(2.31) implies that —u;^ < ^ < 0, while when —TT < ^ < 0 we see that 0 < ^ < u;^. In short: 00 < 0

(2.32)

That is, if we plot the acceleration versus the deflection, the trace of points will lie in the 2nd and 4th quadrants (upper left and lower right quadrants), as well as passing periodically through the origin. The second feature to notice is that the difl'erence between 0 and 0 is bounded: "' <^{0,0) (2.33) where ip{0^0) is an envelope function delimiting the domain of 0 and 0 values. Taken together, relations (2.32) and (2.33) assert that a plot of 0 versus 0 lies in a bounded region in the 2nd and 4th quadrants. The frictionless nonlinear pendulum described by eq.(2.31) is a simple prototype of a wide range of physical and engineering phenomena. These include the Duffing and van der Pol equations of nonlinear oscillation, heat conduction, off-shore ship-mooring systems, dynamics of nuclear particle accelerators, aerodynamic flutter and instability as demonstrated by the

28

I N F O - G A P DECISION THEORY

dramatic failure of the Tacoma Narrows bridge, orbital and attitude instabilities encountered in the control of spacecraft, as well as many other examples [119, 168, 169]. We now move beyond the simple pendulum and formulate an envelopebound info-gap model for "pendulum-like" systems. The info-gap model will be in the form eq.(2.14) where the set S satisfies relations (2.15) and (2.16). This can be done in many ways. As an illustration we will use a set S of two-dimensional vector functions u{t), where the elements of u{t) are denoted as ui{t) and U2{t). The set S is: S = {u{t) : ui{t)u2{t) < 0, \ui{t) - U2{t)\ < 1}

(2.34)

Figure 2.1: Schematic representation of the set S in eq.(2.34). 5 is a set of two-dimensional functions u{t), but it can be schematically represented as two triangles portraying the domain of instantaneous values of (i^i, 1^2), as in fig. 2.1. S indeed satisfies relations (2.15) and (2.16). S contains the zero-function, which is represented by the origin in fig. 2.1. Also, S is contained in its own expansions: the set aS is simply the expansion (or contraction) of S represented by the two triangles reaching out to the lines passing through ±a on the ui and U2 axes. Furthermore, it is evident that S is not a convex set. A convex set is one which entirely contains the line segments joining any two points in the set. The hourglass shape in fig. 2.1 clearly is not convex. Consequently, the envelope-bound model defined by eq.(2.14), with S taken from eq.(2.34), is a non-convex info-gap model. A non-convex info-gap model: Linear system with uncertain coefficients. Many dynamical systems, arising in fields from the social, natural and physical sciences as well as engineering, are described by a set of linear differential equations with constant coefficients. In vector form:

x{t) = Ax{t) + fit)

(2.35)

C H A P T E R 2.

UNCERTAINTY

29

whose solution is: XA{t) = e^'x(O) + / f{T)e^^'-^^ dr Jo

(2.36)

It not infrequently happens that the coefficient matrix A is incompletely known, though the function f{t) is known. We will consider the situation where A belongs to an expanding family of sets: AeA

+ aSA.

a>0

(2.37)

where A is the known nominal coefficient matrix and the set SA satisfies relations (2.15) and (2.16). SA need not be a convex set of ^-matrices. The response of the system, x^(t), likewise belongs to an expanding family of sets, and can be put in a form analogous but not identical to eq.(2.14). We first define the set: Sa:{a,A) = lx{t) : x{t) =XA{t)-x-{t),

forall

A e A-\-aSA]

^a >0

(2.38) Sx{cx, A) is the set of functions x{t) which represent the difference XA{t) — x-{t) between the actual and nominal response of the system, for any coefficient matrix A in the set A + aSA- Slightly less precisely (but more intuitively) we say that Sx{(^, A) is the set of deviations of the response for all coefficient matrices "up to uncertainty a". Sx{ot,A) obeys conditions similar to relations (2.15) and (2.16): 1. Sx{oi^A) contains the zero-function: OeSx{a,A) 2. Sx{a,A)

(2.39)

expands with a, and is contained in its own expansions: 0 < a < a'

implies that

Sx{a, A) C Sx{a', A)

(2.40)

The set of system responses, for all coefficient matrices m A-\- aSA, can be expressed: A'(a,x~) = { x ( t ) : x{t) e Sx{a, A) + x-it)^

c^ > 0

(2.41)

The structural similarity of this info-gap model to the one in eq.(2.14) is clear. The difference between these two models is that aS in eq.(2.14) expands linearly in the uncertainty parameter a, while the functional form of the expansion of Sx{(y^A) in eq.(2.41) is more general. In both cases, convexity of either U{a^u) or A'(a, x j ) depends on convexity of S or SX{OL^ A), respectively.

30

I N F O - G A P DECISION THEORY

Discrete info-gap models. All the info-gap models we have considered up to now entail continuous variation of the horizon of uncertainty a. However, in some situations the uncertainty is discrete. For instance, ordinal preferences among several options can be specified by positive integers, where a low integer represents low desirability of the corresponding option. The preferences may be uncertain, but the uncertainty is discrete: errors are in integer units. We will illustrate a simple example here; further discussion is found in [30]. Let J be the number of options. A preference vector TT is a J-vector whose mth element is the integer preference-rank for the mth option. That is, T^m — 'T means that option m has rank r. A large value of r implies high preference. If states n and m have the same rank then TT^ = TT^ and the value of this integer is their ordinal rank. Thus a preference vector TT is a J-vector whose elements are chosen from among the first J positive integers. We will iteratively define a sequence of sets of preference vectors which are successively further from the known nominal preference vector, which we denote as TI^^\ First define a column J-vector e* which has 1 in the ith position and 0 elsewhere. Thus, for any preference vector TT, the vector TT + e* raises the rank of the zth state by one. Likewise, TT — e* lowers the rank of the ith state by one. We must add a technical proviso: these ' + ' and ' —' operations do not raise a preference rank above J or lower it below L Let n^^^ be the set containing the nominal preference vector: H^^^ = {TT^^^}. We recursively define the following sets of preference vectors: n(^) = {7rd=e\ for all TT G n ^ ^ - i \

z = l, . . . , j } ,

A: = 1, 2, . . . (2.42)

Thus n^^^ is the set of preference vectors which diff'er from the nominal by no more than k single preference changes. Now we define our info-gap uncertainty model as:

U{a, ^(°)) = y n ( ^ \

a = 0, 1, 2, . . .

(2.43)

ZY(a,7r^^^) is the set of all preference vectors which diff'er from the known nominal preference vector by no more than a changes in rank. U{a^'K^^^) is an info-gap model of uncertainty whose horizon of uncertainty, a, is a non-negative integer. The meaning of a is the 'distance' from the known nominal preference vector. In the absence of uncertainty, when a — 0, only the nominal preference vector is possible, so ^/(0,7r^^^) = {TT^^^}. Like all info-gap models, ZY(a,7r^^^), o; = 0, 1, 2, ..., is a family of nested sets: a
implies that

U{a.

TT^^^) C

U{a',

TT^^^)

(2.44)

C H A P T E R 2.

2.6

UNCERTAINTY

31

X Axioms of Info-Gap U n c e r t a i n t y

We now have a pretty clear idea of what we mean by an info-gap model of uncertainty, both on the intuitive level and quantitatively through examples. We conclude this discussion of info-gap uncertainty with a brief discussion of an axiomatization which is suitable for most of the info-gap models we have encountered. However, we will not employ an axiomatic approach in our development of info-gap decision theory. R is the set of the non-negative real numbers and D is a subset of a Banach space S. Info-gap models of uncertainty are formally represented as maps, U{a,u), from R x D into the power set of 5. Thus U{a,u) is a setvalued function, ascribing a subset of S to each point (a, u) m Rx D. For reasons which are evident from the examples discussed earlier, we refer to a as the uncertainty parameter and to u as the center point of the info-gap model. We say that U{a,u) is "centered" at u and has "size" or "horizon of uncertainty" a. Most of the info-gap models of uncertainty employed in practice have the first two of the following four properties; many have all four properties [20]: Axiom 1 that

Nesting. The info-gap model U{a,u) is nested: a < a^ implies U{a,u)

C U{a',u)

(2.45)

Axiom 2 Contraction. The info-gap model U{0,u) is a singleton set containing its center point: U{0,u) = {u} Axiom 3

(2.46)

Translation. Info-gap models translate linearly: U{a,u)=U{a,0)+u

(2.47)

where U -\- u means that u is added to each element of hi. Axiom 4 Linear expansion. Info-gap models centered at the origin expand linearly from the origin: a' U{a', 0) = —U{a, 0), for all a, a' > 0 (2.48) a where ^U means that ^ multiplies each element of hi. The first axiom — nesting — imposes the property of 'clustering' which is characteristic of information-gap uncertainty. Furthermore, the nesting axiom implies that the uncertainty sets hi{a,u) become more inclusive as a grows, thus endowing a with its meaning as an horizon of uncertainty. Axioms 2-4 specify info-gap models with particular structural properties. Many features of info-gap analysis depend only upon axiom 1. In fact it is sometimes useful to modify the structural axioms 2-4, as would be needed to represent the info-gap model in eq.(2.41). More examples are discussed in [22].

32

I N F O - G A P DECISION

2.7

THEORY

Problems

1. Formulating info-gap models. Formulate an info-gap model for uncertainty in the cost of raw materials in each of the following scenarios. Explain your choice and specify any additional information needed to verify your selection. (a) The cost of raw material varies erratically throughout the year. (b) The cost of raw material varies in a reasonably predictable manner throughout the year, except during the summer months when large erratic fluctuations are observed. (c) The cost of raw material varies in an unknown but gradual manner throughout the year. (d) The cost of raw material increases in an unknown but gradual manner throughout the year. (e) The cost of raw material varies in a reasonably predictable manner throughout the year, subject to occasional severe though short-lived excursions. (f) The cost of raw material varies randomly throughout the year, where the estimated pdf at time t is normal with mean and variance /i(t) and cr^{t). The tails of this pdf are highly uncertain. 2. Formulating info-gap models, continued. Repeat the info-gap formulations in items l a - l f where the phrase "cost of raw material" is replaced with "costs of N different raw materials". That is, we now consider an uncertain vector function rather than an uncertain scalar function. 3. Uniform- and energy-bound info-gap models. Consider the following info-gap models for uncertain scalar functions u{t) defined on the domain 0
/

[u{t) - u{t)f

dt < a^ I ,

Uniform bound: U2{a, u) = {u{t) : \u{t) - u{t)\ < a } ,

a > 0

a > 0

(2.49)

(2.50)

(a) At any positive value of the uncertainty parameter a neither of the sets Ui{a, u) nor U2{ot, u), is contained in the other. For each set, find an element which it contains and which is not contained in the other set. With these results, explain the different uses or interpretations of these two info-gap models of uncertainty. (b) Despite the fact that neither of the info-gap models is included in the other, there are nonetheless an infinity of functions which belong to both. Consider the function: u{t) = S ( 0 + e " ^ *

(2.51)

C H A P T E R 2.

UNCERTAINTY

33

where A is a positive constant much less than unity. Show that: u{t)

e

Ui{a,Z)

for all

u{t)

e

U2{a,u)

for all

a > - ^ 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
a>0

(2.60)

Fourier ellipsoid bound: u(t) = u(t) + 2^Wn

COS - — + hn sm — = Z(t) -h J(t){t)

(2.61)

where c is the vector of uncertain Fourier coefficients and (j){t) is the corresponding vector of cosine and sine functions. A Fourier ellipsoid bound info-gap model is: U2{a,u) = {u{t) = u{t)-\-J(l){t)

: c^c < a^} ,

a > 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)
where VK is a real, symmetric, positive definite matrix. •^Problems marked with a double-dagger, J, contain more advanced material.

C H A P T E R 2.

UNCERTAINTY

35

Interval bound: U{a,u) = \u: [

rT""" I

lin

< a , n = l, . . . , i v l , I

a>0

(2.68)

J

Hint: A set ^S* is convex if: X ^ S and y ^ S

implies

7a: + (1 — 7)2/ G ^S

(2.69)

for a l l 7 G (0,1). 8. Following is a riddle posed and "solved" by Lewis Carroll [44]. Is his argument correct? If not, where are his errors? "A bag contains 2 counters, as to which nothing is known except that each is either black or white. Ascertain their colours without taking them out of the bag." Answer: "One is black, and the other white." Solution: "We know that, if a bag contained 3 counters, 2 being black and one white, the chance of drawing a black one would be 2/3— and that any other state of things would not give this chance. "Now the chances, that the given bag contains (a) BB, (/?) BW, (7) WW, are respectively 1/4, 1/2, 1/4. "Add a black counter. "Then the chances, that it contains {a) BBB, {(3) BWB, (7) WWB, are, as before, 1/4, 1/2, 1/4. "Hence the chance, of now drawing a black one, _1 ~4"

1 2 1 1 _ 2 '^2'3^4*3~3

"Hence the bag now contains BBW (since any other state of things would not give this chance. "Hence, before the black counter was added, it contained BW, i.e. one black counter and one white." 9. Three-box problem. A valuable prize is located in one of three boxes; the other two boxes are empty. We know that the probability is p^ that the prize is in the ith box. We choose a box, hoping to win the prize. The Master of Ceremony, who knows our choice and the location of the prize, opens one of the boxes, other than the one we have chosen, and shows us that it is empty. The question is: Do we have a rational basis for changing our choice, from the box initially selected, to the remaining closed box? 10. Modification of the 3-box problem. Suppose that we have imprecise probabilistic information about the location of the prize. Let p^ be the best guess of the probability that the prize is in the zth box, where Pi + P2 + P3 = ^^ However, the actual probabilities p — {pi,p2,P3) are uncertain, and are constrained to the info-gap model: U{a,p) = {p:

peV,

(pi-?i)' + (p2-?2)'<«'},

c.>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
min_ R{q, u) \ < r^ > \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


msix_ \\D{q,u) - D{q,u)\\\ \ueU{a,u)

< r^} J

(3.13)

J

The robustness is the greatest value of the uncertainty parameter up to which the decision does not dither by more than TC as the uncertain quantities fluctuate. (The definition of "|| • ||" depends on the details of the application.) When a is large the decision is stable with respect to the uncertainties. This means that the decision is steady over a wide range of values of the uncertain variables u. On the other hand, if a is small then the decision is fragile with respect to the uncertainty: small variation of u can cause the decision to change by at least TC- If S is large the decision is robust, insensitive to uncertainties, and the algorithm can be reliably used for making the decision. If a is small then the algorithm cannot be relied upon to yield consistent decisions. The robustness function in eq.(3.13) evaluates the performance of an arbitrary decision algorithm D{q,u) which may itself be an info-gap decision algorithm, or D[q^u) may be a probabilistic algorithm. Info-gap decision theory is employed here in a supervisory role, overseeing the performance of D{q^u). Since a{q^rc) includes the design vector q which specifies the structure of the decision algorithm D{q,u)^ the robustness function can be used to choose among possible modifications of D{q, u). In other words, the decision algorithm D{q,u) is embedded in the info-gap decision function 3(^,rc). We will explore this further in chapter 8 when we study the infogap supervision of learning processes. Let us identify the circumstances in which decision-robustness to uncertainty, a{q,rc), is a relevant measure of the correctness or validity of the decision algorithm D{q^u). Uncertainties may occur in the environment in which the system operates, as well as in the properties of the system itself. In both cases we are assuming that the uncertainty models are sufficiently "rich" to encompass a realistic representation of the actual eventuahties. Regarding environmental uncertainty this means that the info-gap model represents a realistic range of operating conditions. Regarding system uncertainties we are assuming that the uncertainty model includes a realistic representation of the system, even though the nominal model upon which the decision will be based may differ from more realistic models. Robustness to environmental uncertainty means that the decision is relevant to the full range of environmental contingencies, which is realistically represented by the uncertainty model. Robustness to system uncertainty means that the decision which is based on the nominal model differs by no more than Tc from the decision which would be obtained from a more realistic model, whose identity is not known but which is contained in the range of the infogap model of system uncertainty. In short, if the robustness S(g, re) is large, then the decision D{q, u) is unaffected by large error in the information (the uncertain quantities u). In this sense a large a{q^rc) warrants the validity

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

43

or reliability of the decision D{q,u). We will discuss warrant and credence for info-gap decisions in sections 13.3 and 13.4. We can also define an opportuneness function for decision algorithm D{q^u). Let Dj. denote a reference decision, for instance a complex algorithm which is more expensive or difficult to implement than the decision algorithm D{q,u). The opportuneness of D{q,u) is the lowest horizon of uncertainty at which the decision D{q,u) is no further than r^ from the reference decision: P{q,r^) = mm\a: [

I

min ^ \\D{q,u) - Dr\\] < r^\

\uel({a,u)

J

(3.14) J

where TW is a measure of decision error, and typically takes desirably small values. The decision algorithm D{q,u) is highly opportune — it efficiently exploits ambient uncertainty — if quite desirable values of r^ are attainable (though not guaranteed) even at very low levels of uncertainty. Thus, as usual for the opportuneness function, small values of l3{q^r^) are better than large values.

3.1.4

Multi-Criterion Reward

In eqs.(3.4) and (3.5), as well as in eqs.(3.10) and (3.11) and eqs.(3.13) and (3.14), a single reward function or decision algorithm expresses demands or desires for outcomes of a single type. In some situations there are multiple performance requirements, just as there may be multiple relevant outputs. In this case there will be a reward function, Ri{q,u), required and desired performance thresholds r^^i and rw,i, and a pair of immunity functions, c^i{Q^f^c,i) and /?^(^,rw,i), for each requirement. These immunity functions, ai{q,rc^i) and j3^{q,r^^i), may be combined in various ways. A number of methods have been developed in the context of expected-utility theory for leading a decision maker towards the definition of an acceptable mix of multiple performance criteria [98]. Some of these methods are adaptable to info-gap decision theory as well. Other ways of combining robustness functions to express either redundancy or uniqueness of distinct performance requirements are discussed elsewhere [14, section 3.6; 29]. The most direct and commonly encountered method is to evaluate the overall robustness as the greatest horizon of uncertainty at which all the performance requirements are satisfied. If there are / performance requirements, all of the form Ri[q^u) > 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)<^(g',rw)

(3.21)

The opportune-windfalling decision, §w(^w), minimizes the opportuneness function on the set of available decisions: $w(^w) = argmin^(g,rw)

(3.22)

qeQ

The two preference rankings, eqs.(3.19) and (3.21), as well as the corresponding the optimal decisions 5c(^c) and q^{r^), may be different. The resolution of this conflict between robustness and opportuneness rankings is usually simple: most decision makers simply concentrate on enhancing their robustness while guaranteeing a specified level of performance, and thus adopt the robustness ranking of eq.(3.19). The opportuneness function then becomes a secondary decision-support device useful in fine-tuning ones' choice between alternative options. This is not always the case, and methods for combining considerations of robustness and opportuneness will engage us in subsequent chapters.

3.1.7

Trade-Offs

Fig. 3.1 illustrates the assertion (which can be proven mathematically) that the robustness function of eq.(3.4), S(g,rc), decreases monotonically as the minimal required reward TC is increased: Tc > 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

dt
a>0

(3.30)

In many applications the performance requirement is that the absolute amplitude of vibration not exceed a specified value: \0{t)\ <6>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
g{t-T)-g{t),

q(t)

= e ^^^ I —,

(3.39)

(3.40)

t>T

(3.41)

sincc;H^ + cosc^d^ 1

Now the robustness function can be expressed as: ^

2J(9cu;2y^ - 2^^1^07(01

"^^'^^) =

(3.42)

:jm

if this is non-negative, and a{q, 9c) = 0 otherwise. Robustness 2

Robustness

3(9,6>c) 1 S{q,Oc)

\ A7\AAA^/ ,

00

10

20

Time, t [s] Figure 3.4: Robustness versus time for three values of the natural frequency a; = 1, 3 and 4 (bottom to top). Negligible damping: C = 0.01. 1 = JO^ = So. T = 5.

0

10

20

Time, t [s] Figure 3.5: Robustness versus time for three values of the damping ratio C = 0.03, 0.3, 0.5 (bottom to top). Fixed natural frequency ct; = 1. 1 = JOc = UQ. T = 5.

T h e robustness as a function of time is shown infig.3.4 for three values of the natural frequency uj and for very low damping, u increases as the square root of the stiffness of the beam, so the figure shows t h a t increasing the stiffness causes substantial enhancement of the robustness t o load-uncertainty. At LU = 1, the b o t t o m curve, the robustness is small and even vanishes occasionally, which indicates t h a t at those times the nominal response violates the performance requirement. At u; = 3 and UJ = 4 (middle

54

INFO-GAP DECISION THEORY

and upper curves) the robustness still oscillates but is positive throughout the displayed time range, a initially decreases sharply but falls more slowly later on. The transition from rapid to slow decrease occurs approximately at the end of the nominal step input, t = T = 5. Fig. 3.5 shows the robustness versus time for three values of the damping ratio C, at fixed natural frequency. The lowest curve, with C = 0.03, shows only slightly better robustness than the lowest curve in fig. 3.4 for which C = 0.01; (J = 1 in both cases. However, the middle and upper curves in fig. 3.5 are quite different from the behavior displayed in fig. 3.4. The trend in fig. 3.5 resulting from increasing the damping is marked by predominantly low a until the end of the nominal input at t = T = 5, and then rapid rise in a which stabilizes at a fixed value. The design concept underlying fig. 3.5 is dissipation of energy by damping, so that, after the transient passes, the energy which it introduced is eventually removed from the system and the angular defiection is contained. In contrast, fig. 3.4 is based on rigidity but with negligible energy removal, so that resonant excitation causes increasing response which decays only very slowly due to the very low damping. Which of these two design options is preferable depends primarily on the time frame of interest. Fig. 3.4 shows high robustness at short times (for cj = 3 or 4) while fig. 3.5 gives good performance at longer times.

3.2.2

Structural Reliability

One of the heaviest responsibilities carried by the engineering designer is to evaluate the reliability of a structure upon which human safety depends. Bridges, buildings, dams, aerospace, automotive and many other structures and devices fall in this category. While engineers are cautious and equipped with millenia of experience, the vagaries of environmental forces such as wind and earthquakes, unpredictable aging and wear of structural materials, as well as the pressures to reduce manufacture, construction and maintenance costs, present the decision maker with difficult choices. The robustness function provides the designer with a basic tool for deciding whether to accept or reject a proposed design. We consider the analysis of an extremely simple system in order to illustrate the methodology. More advanced discussion is found in [14, 17, 25, 96]. Robustness function and the design equation. We consider the prototype of a wide range of structures: the linear spring, for which the force / is linearly related to the deflection x: f = kx

(3.43)

It is necessary to choose (or to certify the choice of) the stiff'ness k such that the deflection x is acceptably small, subject to uncertain positive load / . The engineering requirement is that the deflection be no greater than a critical value: \x\ < X, (3.44)

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

55

where x^ is positive. The nominal load / is known jind positive, but the actual load / may deviate in an unknown way from / . This load uncertainty is represented by an interval info-gap model: U{aJ)

= [f:

|/-/j
a>0

(3.45)

This is an unbounded family of nested sets U{a, / ) , for which the uncertainty parameter a is unknown. The robustness function, defined in eq.(3.10), is the greatest value of the uncertainty parameter for which the performance requirement of eq.(3.44) is not violated. One finds the robustness by evaluating the maximum absolute deflection up to uncertainty a, equating this to the critical deflection, and solving for a\ f-\-a

max_ |x| = —-—

^^

a{k,Xc) = kxc — f

(3.46)

or a{k, Xc) = 0 if this is negative. The info-gap certification of the design (the choice of k) is based on requiring that the robustness exceed a minimum design value: a{q,r,)>a^

(3.47)

The choice of the threshold value of robustness. 3d, is based on calibration of the robustness, which we will discuss shortly. Applying the robustness requirement, eq.(3.47), we find that the stiffness must satisfy: k > ^^±1

(3.48)

XQ

One interpretation of this design requirement is that acceptable defiection is guaranteed if the actual value of the uncertainty parameter a is no greater than the robustness threshold Sd. That is. if k satisfies (3.48) then: a < Sd

implies

|x| < Xc for any load function / G U{a, f)

(3.49)

This shows the central importance of the choice of the threshold robustness 3d. Choosing the threshold robustness 3d. The threshold robustness 3d can be chosen in several different ways. The choice of 3d is not entirely a technical decision: it also involves qualitative evaluation of acceptable levels of risk which we will discuss in chapter 4. The approaches described here are attempts to calibrate 3d against various quantities whose levels of acceptability are readily acknowledged. It is usually desirable to combine several calibrations.

56

INFO-GAP DECISION THEORY

1. Prior experience. The robustness values a{q^rc) of existing highperformance structures can provide a means of identifying desirable values of the threshold robustness Sa2. Dimensional calibration. The horizon of uncertainty a, as well as the robustness function a{q,rc), may have physical dimensions. By normalizing a to a dimensionless form one can obtain some idea of how large a value of 3d is desirable. In the present example a and the nominal load / have^the same units: force or stress. Thus a/f is dimensionless. If a/f
3.2.3

Structural Reliability with Uncertain Probability

We again consider the linear spring, eq.(3.43), and require that the probability of failure not exceed Pc- Denoting the probability density function (pdf) of the load by p{f) and using the performance requirement in eq.(3.44), the probability of failure is:

p{f)df+ -oo

p{f)df

(3.50)

IJkxn kxc

Our probabilistic performance requirement is: Pi{v) < Pc

(3.51)

We suppose that we have an estimated pdf for the load, p{f). However, this density is uncertain, especially in its tails where failure occurs, so the evaluation of Pf(p) is also uncertain.'* Let U{a,p) denote an info-gap model for uncertainty in the estimated pdf of the load, such as described in section 2.5, p.26. ^Uncertain pdfs arise in many situations. See [26] for an example of assessing and managing financial risk with an uncertain pdf.

CHAPTER 3. ROBUSTNESS AND OPPORTUNENESS

57

The robustness of design k is the greatest horizon of uncertainty a in the estimated pdf, p{f), up to which all pdfs indicate acceptably low probabihty of failure: (3.52) a(/c, Pc) = max < a : I max_ Pf (p) < Pc I \peU{a,p) I A large value of S(fc, Pc) indicates that the estimated probability of failure of the system with stiffness k is immune to errors in the estimated pdf, while a low value of a(A:, Pc) indicates that the estimated rehability of the system is vulnerable to error in p{f). a{k,Pc) is the robustness (to uncertainty in the pdf) of the probabilistic estimate of the system reliability. Let us assume that the estimated probability of failure, Pf{p)^ is less than 1/2; certainly not an uncommon situation. In this case we find that the inner maximum in eq.(3.52) equals (1 -h a)Pf(p). Equating this to Pc and solving for a yields the robustness: 0 S(A:,Pc)

pm

ifPc
ifPf(p)
(3.53)

Robustness

S(A:,Pc)

Pf(p,/c)

PiipX)

Figure 3.6: Robustness curves for structural reliability with uncertain pdf, eq.(3.53). Eq.(3.53) shows the usual trade-off between robustness and performance: small failure probability, Pc, entails low robustness, S(A:,Pc), as illustrated in fig. 3.6. Furthermore the robustness is zero when the required probability of failure, Pc, is less than or equals the estimated probability of failure, Pf (p). Consider two different stiffnesses, k > k', so that the former has lower estimated probability of failure:

Pf{p,k)
(3.54)

This, by itself, is not a particularly strong recommendation for choosing k over k\ because the robustness is precisely zero for Pc = Pf{p,k). However, the robustness curve for k is steeper than for /c', implying that the

58

INFO-GAP DECISION THEORY

performance-cost of increasing the robustness is lower with k than with k'. This, combined with eq.(3.54), means that k is more rehable than k^ in achieving Pc, for any P^ > Pi{p,k).

3.2.4

Set-Points for Process Control

In the control of many manufacturing processes — forging, extrusion, chemical processing, automated milling and so on — one selects the values of control variables (temperatures, pressures, flow rates, processing times, etc.) in order to achieve desired values of quality features (product dimensions, hardness, purity, and so forth). A mathematical model of the process is used to select and regulate the control variables to keep the quality features within specified tolerances of desired values. Process models are typically inordinately complex and sensitive to innumerable and often unidentified disturbances. We will demonstrate the use of the robustness function to manage uncertainty in the process model while maintaining product quality. Model-parameter uncertainty. We will consider a single quality feature y which depends linearly on three control variables: temperature T, pressure P and duration D, which all must be positive. We denote the control variables with the vector q^ = (T, P, D) and the system model by: 3

y[q,m) = ^ m ^ g ^

(3.55)

where the model coefficients rrii are uncertain. The best estimate of the ith model coefficient is fhi, and a measure of the error of this estimate relative to the other estimates, e.g. a standard error, is Si. However, even though the estimate is based on extensive historical data, the estimate is highly uncertain because operating conditions change, raw materials vary, and the fundamental laws governing the process are more complex than the linear relation in eq.(3.55). We represent the uncertainty in the model parameters with the interval-bound info-gap model: U{a^m) — {m : \mi — frii\ < asi, i = 1, 2, 3} ,

a > 0

(3.56)

The performance requirement is that the quality feature fall within a specified interval: Vi < y{q,rn) < y2 (3.57) The robustness of set-point q is the greatest horizon of uncertainty a up to which all realizations of the system model, m, cause the quality feature y to fall in the performance interval: 3(g, 2/1,^2) = max {a : yi
for all m^U{a^m)}

(3.58)

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

59

This is a multi-criterion decision problem as discussed in section 3.1.4. The robustness can be expressed as: a{q,yi,y2)

=mma^{q,yi)

(3.59)

where: Si(g,yi)

=

max
^2(^,^2)

=

min_ y{q,m) \ > yi> \mel{{a,m)

max< a : [

J

max_ y{q,m) ] \meU{a,m)

J

(3.60)

J

(3.61)

J

The anticipated value of the quality feature is y{q, m) which equals fh q. The robustness is zero \i fh q does not fall in the performance interval [^1,^2]- If ^ q does satisfy the performance requirement then the robustness functions are: (^iKQ^yi) =

r (^-^2) s^ q - ( \ ^2 - ffi^q (^ no\ 0^2(9,^2) = r (3.63) s^ q where s^ = (51, S2, 53). If either of these expressions are negative then the corresponding ai is defined as zero. The robustness decreases as the interval [^i, ^2] becomes smaller, expressing the trade-off between robustness and performance. A set-point vector q which places the nominal quality, fh g, in or near the center of the performance interval is desirable. However, eqs.(3.62) and (3.63) show that robustness is also increased by reducing s^q. Since s is the vector of relative errors of the estimated model coefficients, this means that the redundancy between the control variables should be exploited so that greater values are chosen for variables with lower relative uncertainty. Eqs.(3.62) and (3.63) provide a quantitative basis for choosing the set-points and the performance interval. Model-parameter and functional uncertainty. We now extend this example to include, in addition to uncertainty in the coefficients m of the linear model, uncertainty in the functional form of the model. Specifically, the best available model is the linear expression in eq.(3.55), but it is suspected that more thorough understanding or study of the process would reveal a quadratic component. That is, the quality feature should be modelled as: y[q, m, C) = m^q + q^Cq but the matrix C is completely unknown.

(3.64)

60

INFO-GAP DECISION THEORY

The task then is to choose values for the control variables q, so as to guarantee that the quality feature y is within the performance interval [yi, 2/2], for the greatest possible uncertainty in both the linear-model coefficients m and the unknown quadratic coefficients C. A model of the info-gaps facing the designer is the following extension ofeq.(3.56): ZY(a,m,0) = {m, C : \mi — mi\ < asi, \Cij\ < a, V z, j } , a > 0 (3.65) The performance requirement is still the interval in eq.(3.57). The robustness is zero if the nominal quality, rfi q, lies outside this interval. If m g is acceptable, then the robustness is evaluated from eqs.(3.59)-(3.61) with the info-gap model of eq.(3.65). Now, unlike eqs.(3.62) and (3.63), we have:

^^(^'^^) = - ? ^ ^ T ^

^^^^'^^) = "^^TT^

^^'^^^

^^'^^^

where Q — Yl,ij^i QiQj- Since Q is positive (since all the control variables qi are positive) we see that the additional functional uncertainty reduces the robustness. In addition, the robust-satisficing design (which maximizes the robustness) will differ from the case with linear-model uncertainty alone, since now the term s^q-\- Q must be reduced, rather than only s^q.

3.2.5

Sequential Decisions

In some decision problems the decision maker is able to measure various features or symptoms and defer decision again and again until the measurements are conclusive. The legendary blind man trying to identify an elephant would do well to defer his decision until he has sampled several distinct features. The central question after each measurement is: how confident will we be if we decide now? Or, conversely, how much could be learned by measuring again? The statistical technique called 'sequential analysis' is the prototype of this type of decision problem [171]. We will use the robustness function to illustrate an info-gap approach, in the context of a very simple example, based on [16]. Let us suppose that we must distinguish between two situations: "normal" or "faulty", "healthy" or "ill", "go" or "no-go", etc. We will sample from a large number, N, of features in order to decide between them. However, the two situations are not uniquely determined by the feature values. Rather, the range of possible values for the feature vector is specified by a different info-gap model for each situation. To keep the example simple let us suppose that these info-gap models are families of A^-spheres: U{a,uo)

=

{u:u^u
a>0

(3.68)

C H A P T E R 3.

U{a,ui)

=

ROBUSTNESS AND OPPORTUNENESS

{u:

{U-1N)^{U-1N)
,

Of > 0

61

(3.69)

where liv is the column A^-vector whose elements are all Is. ui is the nominal feature vector for situation i, so So = 0 while ui — IN- Since the uncertainty parameter a is unbounded, any measured collection of features could arise from either situation. We will sample the first feature, then the second, the third, and so on, until we finally are ready and willing to decide. Let q(n) denote the vector of the n sampled features, for n = 1, ... ,N. qj{n) is the measured value of the j t h feature. q{n) is the decision vector for this problem since the decision to terminate the information-gathering process, as well as the choice of which situation prevails, both depend upon q{n). The question we have posed is: after the nth sample, how reliable is a decision based on the current information? Or, the converse question is: how much additional possible learning are we forgoing by ending the sample and making a decision? If a decision at this point is very reliable then, presumably, there is not too much left to be learned. We will return to a discussion of learning in chapter 8. After making n measurements we have the sample vector q{n). If we decide to choose between the two situations based on this sample vector, we would do so by comparing it against the two nominal feature vectors UQ and ui. Let \\q{n) — Ui\\ denote the Euclidean distance between q{n) and the sub-vector of Ui containing the corresponding nominal feature values:

\\q{n)-u,\\

=

.|Efe(^)-^Mf

(3.70)

\3 = l

A decision, when it is made, employs the following 'nearest-neighbor' rule: Decide "0" if and only if \\q{n) - Soil < \\q{n) - ui\\

(3.71)

That is, we choose the situation whose nominal feature vector is closest to the measured features. We now define the robustness function in the spirit of eq.(3.1). S[g(n)] is the greatest value of the uncertainty parameter a which is consistent with not making an erroneous decision based on employing the sample vector q{n) in the algorithm of eq.(3.71). The ambient uncertainty must be sufficiently great to enable the sample vector q{n) to arise from at least one of the infogap models. Also, as illustrated schematically infig.3.7, an error can occur at uncertainty a only if q{n) could come from both models. In the left-hand frame offig.3.7 we see that the decision based on the nearest-neighbor rule and sample vector q{n) is "0". At uncertainty ai this decision is in fact correct since q{n) cannot arise from U{ai,ui). However, if the ambient uncertainty is 0^2 as on the right-hand side offig.3.7, then the same sample vector could have arisen from situation " 1 " , while the nearest-neighbor decision remains the same as before. That is, if the ambient uncertainty is

62

INFO-GAP DECISION THEORY U{0L2,Ux)

l({ai,ui)

ai < a2 1

OL2\

U{a2,uo) \^{^V\ >v

\

•

Figure 3.7: Correct decision at uncertainty ai (left); possible error at a2 (right). large enough so that q{n) belongs to both U{a^uo) and ^/(a,Si), then the actual situation can differ from the decision according to algorithm (3.71). From these considerations it is evident that a[q{n)] is the greatest value of a such that q{n) does not belong to both U{a, UQ) and U{a, ui):^ a[g(n)] = max {a : q{n) ^U{a,Uo)

D

U{a^ui)}

(3.72)

U{a^Ui) is a sphere of radius a, so the sampled feature vector q{n) can come from U{a,Ui) if and only if: \\q{n) -UiW
(3.73)

Consequently, for the spherical info-gap models in eqs.(3.68) and (3.69), the robustness — the greatest error-free uncertainty — is simply the greater of the two distances between q{n) and the nominal feature vectors: a[q{n)] = max{||g(n) - Uo||, \\q{n) - ui

(3.74)

We now are ready to address the question which we posed earlier: how confident can the decision maker be if the current sample vector is used to make a decision? The question is qualitative so the best we can hope for is a plausible qualitative answer; definitive quantitative answers to qualitative questions simply don't exist. We will briefly consider plausible inference by analogy, to which we will return in detail in chapter 4. The level a of ambient uncertainty must be great enough for the observed sample vector q{n) to have arisen from at least one of the info-gap models. Consequently, for the spherical models of eqs.(3.68) and (3.69), the lowest possible value of a is: Q^min = min{||(7(n)-ixoll, \\q{n) - ui\

(3.75)

^We are making a slight notational transgression in eq.(3.72): q{n) is an n-vector while the info-gap models are sets of A'^-vectors. The meaning of "g ^ U{a,uo) nU{a,uiy^ in eq.(3.72) is that q{n) is not the n-sample vector of any element of the intersection.

C H A P T E R 3.

ROBUSTNESS AND OPPORTUNENESS

63

If the robustness 5[g(n)] — the greatest errorless uncertainty — is 'much greater' than this minimum uncertainty a^in^, then one can be fairly confident in a decision based on q{n). If a[q{n)] :$> amin then only an extraordinarily large deviation of the feature vector from its nominal value could lead to an erroneous decision. So the question becomes: what is a 'significant' increment in a? How much greater than amin niust a[q{n)] be, in order to be 'much greater' ? In chapter 4 we will consider several approaches to this problem. As a preliminary treatment of the matter let us consider the following slight generalization of the info-gap models of eqs.(3.68) and (3.69): U '{a,Ui) =
a>0

(3.76)

for i = 0 and 1, and where the center-points Ui are the same as before and w \s di positive constant, li w <1 then W{a, Ui) is a subset of ^ ( a , ^ ^ ) : W{a,Ui)

C U{a,u^),

a>0

(3.77)

This means that U^{a^Ui) constrains the uncertain feature vectors u more tightly than U{a,Ui) for every positive value of a. In this sense W{a, ui) is more informative than U{a^Ui). We will discuss informativeness of info-gap models repeatedly in this book. We will use eq.(3.72) to define the robustness function ct[q{n)\ for the info-gap models in (3.76) just as for the models of eqs.(3.68) and (3.69). These robustnesses are related as: a'[q{n)] = -^a[q{n)]

(3.78)

That is, enhancing the information (by using info-gap models W{a, Ui) with w < 1) enhances the robustness. Conversely, the increment: AS = 5*[^(n)]-a[^(n)]

(3.79)

is the amount of robustness which is required in order to compensate for the decrement of information entailed in using U{a, ui) rather than W{a, Ui). We will now use AS to address the question of what is a significant increment of robustness. We wish to know if the robustness, 3[g'(n)], of a given sample vector q{n) provides strong warrant for terminating the sample. In order to qualitatively evaluate the robustness function we must anchor our judgment in some entity which we understand qualitatively. Let us suppose that we are able to associate qualitative linguistic values to numerical values of it;, by indicating for instance that t^ = 0.7 generates a 'substantially' tighter and more informative info-gap model than w — l. In this case, the corresponding robustness increment, eq.(3.79), is 'substantial' or 'significant' hy analogy to the substantial change in information. An analogy is an inference whereby

64

INFO-GAP DECISION THEORY

it is concluded that things which are similar in some attributes will tend to be similar in other attributes as well. The conclusion is plausible, though not definitive, if the two sets of attributes are related, as for instance informativeness of an info-gap model of uncertainty and reliability of a decision employing that model. We now return to the comparison of the robustness a[^(n)] and the minimum uncertainty Qmin- Let AS in eq.(3.79) be a 'significant' increment of robustness derived from a significant increment in information. We conclude that S[g(n)] is 'substantially' greater than amin and, therefore, a decision based on q{n) is dependable, if: a[q{n)] > 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
-\- Wntn^

(3.86)

In other words, U{a^ t) is the set of t-vectors whose elements tn vary from their nominal values by no more than a fraction WnOi- If the Wn are all unity, then the maximum fractional variations of the task durations are all equal to a. Alternatively, if each Wn equals l/t^, then a becomes the maximum absolute variation of the task durations. Other choices of the Wn can express other fragmentary information about the relative variability of the task durations. We will consider a particular choice of the Wn- In any case, U{a^t) is the set of all task-time vectors whose uncertainty is no greater than the uncertainty parameter a. While we know the nominal durations tn-, we do not know the value of a, which expresses the unknown horizon of variable task durations. The robustness is the greatest horizon of uncertainty, a, which is consistent with timely completion of the project. Let q denote the project flow chart under consideration. The robustness function is evaluated as: S(9, ^c) = max la: I

inax_ T(i) < tc > \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.

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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

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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

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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.

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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
a>0

(3.95)

Robustness function. The decision vector q is chosen to guarantee that the change in the portfolio worth, R{q,u)^ is no less than a minimum critical reward r^ sometimes called a minimum attractive rate of return (MARR) [73]. The robustness of the portfolio investment q for critical reward TC is the greatest value of the uncertainty parameter a such that any vector u in U{a, u) results in a net worth R{q, u) which is no less than Tc This is precisely the robustness in eq.(3.4) on p.40. The least reward up to uncertainty a is readily found to be: min_ q^u — q^u — ayq^W'^q

(3.96)

Equating this minimum reward to the critical value Vc and solving for the uncertainty parameter a results in the robustness: (^{q^rc) ^

J^

V^w"

(3.97) Q

if this expression is non-negative. The robustness is zero otherwise. Robust-satisficing investment. The robust-satisficing strategy for choosing the investment portfolio is to select q to maximize S(g,rc). The choice of the portfolio is subject to many different possible constraints. For instance, some securities may be accessible only if some other securities are purchased as well. Or, the quantity of an security sold may be limited by the decision maker's holdings. In addition, overall budgetary constraints may limit the total purchasing power. For simplicity, we will consider only the last constraint. Let Q, the set of feasible investment vectors, be the set of all ^-vectors which exactly meet the budget, Q: N

J2Q^ = Q

(3.98)

1=1

where qi is negative if security i is sold, and positive otherwise. To represent this constraint more conveniently let 1 denote the A^-vector whose elements are all ones. The budget constraint is: q^l = Q

(3.99)

The robust-satisficing investment for critical reward TC is the vector q which maximizes S(^,rc), as in eq.(3.20) on p.45. To simplify matters we will consider a special case: the anticipated values of all the securities are the same, though their uncertainties may be different. That is, the nominal vector u is: S = i/ol

(3.100)

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INFO-GAP DECISION THEORY

where Uo is a known constant. With this simpUfication, the robustness in eq.(3.97) becomes: a{q,r,) =

" / j ^

(3.101)

where we have employed the budget constraint of eq.(3.99). Examining eq.(3.101) we see that the robust-satisficing investment QC^ which maximizes S(g, TC), is the vector which minimizes q^W~^q subject to the constraint in eq.(3.99). Using Lagrange optimization one readily finds the robust-satisficing investment vector to be: - -

^ l^Wl

Wl

(3.102)

This means that, when the nominal values of the securities are equal but their uncertainties are possibly different, the robust-satisficing investment in the ith security is proportional to the sum of the zth row of the uncertainty shape matrix W. The meaning of this becomes particularly transparent in the further special case that W is diagonal, so that the investment in the ith. security becomes:

The investment in an security is inversely proportional to its relative propensity for variation. In both cases, eq.(3.102) and (3.103), we see that the robust-satisficing investment in securities with equal nominal values is controlled entirely by the info-gap uncertainty, and is independent of the demanded critical return TCSubstituting the robust-satisficing investment Jc of eq.(3.102) into the robustness function of eq.(3.101) we obtain the maximal robustness: (^{QC^

re) =

{uoQ-r,Wl^Wl ^

(3.104)

(recalling that S(§c,?^c) = 0 if the right-hand side is negative.) Eq.(3.104) shows the trade-off between immunity to uncertainty (large a) and reward (large TC): the decision maker can confidently demand great reward only in exchange for low immunity against failure due to uncertain fluctuations in the security values. The robustness equals zero when the critical reward, Tc, equals the nominal, anticipated, value for the portfolio, UoQC o m p a r i n g portfolios. Now consider the choice between two different portfolios, each with its own set of securities, its own nominal values and its own ellipsoid-bound info-gap model of uncertainty.^ Assuming eq.(3.100) ^The two portfolios may contain different numbers of securities. This would require re-normalization of the uncertainty parameters of the two info-gap models so that the robustnesses of the two portfolios are commensurable.

CHAPTER 3. ROBUSTNESS AND OPPORTUNENESS

73

^ Portfolio 1 Robustness S(?c,rc)

Portfolio 2 r-x rc,i

re,2

Critical Reward, TC Figure 3.9: Robustness functions for two different portfolio investment alternatives.

holds, separately, for each set of securities, let Uo^i be the anticipated value of each security in the zth portfolio, and assume t h a t Uo,i < Uo^2- Likewise, let Wi and W2 be the shape matrices for the two uncertainty models, which are historical inverse covariance matrices. T h e maximum-robustness functions for the two portfolios are each described by eq.(3.104), as shown schematically in fig. 3.9 versus the critical reward TCT h e anticipated value for the portfolios are rc,i = Uo^iQ- Based on anticipated returns, portfolio 2 is more valuable t h a n portfolio 1: rc,2 > '^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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75

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.

C H A P T E R 3.

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ROBUSTNESS AND OPPORTUNENESS

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)
(3.115)

The robustness curves for choices x and x' are shown in fig. 3.12. 3(x^ ^c) intersects the E'c-axis to the right of S(x,£'c) because the best-estimated fidelity of x' is poorer than for x, eq.(3.115). However, eqs.(3.113) and (3.114) imply that the slope of S(x^ E^ is steeper than the slope oia{x, £'c), so these robustness curves cross. Crossing of robustness curves implies reversal of preference between choices x and x', where x' is less aggressive than x. Let us suppose that the value of robustness, Sx, at which the curves in fig. 3.12 cross is fairly low. If we are quite confident that the estimates G and ? are accurate, then we don't need much robustness, so 3x might be enough robustness and we would prefer choice x over choice x'. However, we are considering severe Knightian uncertainty: great error in G and ? is plausible and we need to choose a policy whose anticipated outcome is both acceptable and reliably achieved. Thus, if outcome-error E<^ in fig. 3.12 is good enough fidelity, and if Sd is great enough robustness, then our robust-satisficing preference is for x' over x. If an acceptable combination [E^, Sd) is not found on the

78

INFO-GAP DECISION THEORY

x'-curve, then we need to search for some other choice, x", whose robustness is adequate at acceptable fidehty. If no such x" exists, then no acceptable policy choice is available in the current state of knowledge. We either revise our aspirations, or do some data-hunting, or revise our model. This very simple example has illustrated the info-gap robust-satisficing version of Brainard's dictum, namely, calculate the optimum and then do less, or for what Blinder refers to as "a little stodginess at the central bank" [36, p. 12]. Fig. 3.12 shows that the optimal choice, x, is less desirable than the sub-optimal and less aggressive choice x' under severe uncertainty, because the latter more reliably yields acceptable outcomes (if E^ is adequate). While crossing of robustness curves as in this example is very common, it is not universal. It can happen that robustness curves do not cross, in which case the policy-selection stodginess disappears: the optimizing choice will coincide with the robust-satisficing choice. However, the caution remains in assessing what outcome can be considered reliable. Since the trade-off between robustness and outcomes is universal, the robust-satisficing policy maker will not anticipate (or depend upon) the best-estimated outcome because the robustness of this outcome is zero. Rather, by "migrating up" the robustness curve to an acceptable level of robustness, the analyst finds the corresponding outcome which can reliably be anticipated.

3.2.9

Search and Evasion

In a wide class of "tracking" problems an intelligent "hunter" tries to catch an intelligent "prey". Decision-theoretic interest in this problem arises when one side has only fragmentary information about the other side's strategy and intentions. We will use the robustness function to optimize the hunter's tracking strategy in a simple realization of this problem. The hunter and prey both move along a line, and their positions at time t are denoted x{t) and u(t) respectively. The hunter's initial position is x(0) — 0 while the prey is initially at a positive position u{fi). The hunter is able to measure the prey's position as it evolves in time, and the hunter's strategy is to move towards the prey with velocity proportional to the distance between them: '^^q[u{t)-x{t)\

(3.116)

where q, which represents the hunter's "eff'ort", is a positive constant which the hunter must choose before starting the chase. The hunter has only very limited information about the prey's technique of evasion. The hunter knows that the prey's typical speed is S^, and that the prey's speed deviates from S'by no more than a fixed constant, but this constant is not known. A slope-bound info-gap model represents this prior information:

W(a,?) = {u{t)

du{t) dt


a>0

(3.117)

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The hunter's mission is successful if, at a specified time T, the separation between them is no greater than a given value A: \x{T)-u{T)\

< A

(3.118)

The value of A represents the hunter's skill and technology. The hunter must choose the effort-coefficient q in eq.(3.116) to maximize the robustness to uncertainty in the prey's behavior, while also satisfying the performance requirement in (3.118). Evaluation of the robustness function requires an expression for the maximum value of \x{T) — u{T)\ for all evasion strategies u{t) up to uncertainty a. First we note that the solution of eq.(3.116) is: Xu{t) = q [ e-^('-")^(r) dr Jo

(3.119)

After some manipulation, including a partial integration, one finds that eq.(3.119) can be re-written as: Xu{t) - u{t)

=

-e-'^'uit)

- e-^' / (e^" ~ ^) ( ^ " ^) ^^

--(l-e-^M+5te-^* q

(3.120)

From this we find that the greatest absolute distance (up to uncertainty a) between hunter and prey occurs when the prey adopts the strategy of maximum velocity: u{t) = u{0) -i- (s-\- a)t. so that eq.(3.120) becomes: s — — l rv

max^ \xu{t) - u{t)\ = e-^*u(0) + -—— (l - e''^') ueu(ct,s) Q

(3.121)

The robustness function is obtained by equating this relation at time T to the critical distance A and solving for the uncertainty parameter o^, leading to: {A-e-^^u{0))q ^ a{q, A) = ^ ^ _ ^_qj. s (3.122) unless this relation in negative, in which case the robustness is zero. If the hunter employs effort q and technology A. then 3(g, A) is the greatest horizon of uncertainty at which the prey cannot evade the hunter with any speed-profile, whether constant or variable. From eq.(3.122) We see that: — ol

<

0 if «(0) < A

(3.123)

1^

>

0 if «(0) > 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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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) - <

iFs

{Be-Lpo)-l

ifBe>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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/ 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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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\
z = 0,1; j = 1, . . . , 4} , a > 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<^a: [

min^ L(X,p) > 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

92

INFO-GAP DECISION THEORY

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)=<(

^l^{q) 0

H^ else

c(gjj > 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)
Se(g,rc)

_c^ _ re

Po

if gpo

0

-coy/q>

(3.159)

else

T h e robustness, Se(9, TC), increases monotonically with increasing production volume ^, so t h e manufacturer will be inclined t o maximize production. However, t h e marginal increase in robustness decreases steadily with increasing q. ., V Vg

3e(g,rc) Robustness

\

^

n 0

rx R{q,c)

R{q\c)

Profit aspiration Figure 3.21: Robustness functions vs. profit aspiration for two different production volumes, eq.(3.159). Robustness curves for two different production volumes, q and 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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INFO-GAP DECISION THEORY

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(<^,2), and the corresponding robustnesses, eqs.(3.154) and (3.158), are ranked as 3e(g,TC) > 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{<^,c) if Z//i(a,c) C ^^2(0^,2), which suggests an absolute scale of information. This inclusion does imply an absolute ordering of extrema such as the robustness functions 3(g, TC) or S($c,'^c)- However, we have noted that info-gap models which are not nested one in the other are not commensurable in this way. But even info-gap models which are commensurable by inclusion may have very different implications when evaluated on the basis of performance criteria other than simple extrema. In short, the measurement of information is usually not absolute, but rather relative to an end use. Robustness and information-gap uncertainty. The 'robust rationality' which is developed in these examples is an alternative to the classical maximization of profit. This info-gap robust-satisficing analysis is based on a non-probabilistic quantification of uncertainty. The gap between what the manufacturer does know (a nominal cost function, in these examples) and what needs to be known in order to implement the classical analysis (the true cost function), is represented by an info-gap model. It is in the context of info-gap models that robust-satisficing provides a particularly natural decision algorithm. An info-gap model is an unbounded family of nested uncertainty sets, where the horizon of uncertainty determines both the level of nesting and the magnitude of the info-gap. The robust-satisficing decision procedure in eqs.(3.4) and (3.20) takes immediate advantage of the info-gap model structure to define a strategy which maximizes the decision maker's immunity to the info-gap. While this approach may appeal to some decision makers on its own merits as a criterion of optimality, it may also be a pragmatic necessity in those situations where lack of information prevents implementation of the more classical analysis. Furthermore, as we will show in section 11.4, the robust-satisficing strategy is a better bet than direct best-model optimization in a wide class of problems. Example 3 Robustness and opportuneness. Let us continue the production volume decision problem with the envelope-bound info-gap model

98

INFO-GAP DECISION THEORY

for uncertainty in the cost function, eq.(3.156) in example 2, and evaluate the opportuneness function. The greatest possible profit, for any cost function up to horizon of uncertainty a, occurs when the cost function falls on the lower envelope: max _ R{q,c) = qp{q) - [c{q) - aipiq)]

(3.160)

c{q)eUeia,c)

Equating this greatest possible profit to the windfall value TW and solving for the uncertainty parameter a yields the opportuneness function: MQ) - IC{Q) - a^(
=^

AiQ^ r„) = '"" + "^jj^^^"'^^^^^ (3.161)

unless this expression for ^^{q^r^) is negative, in which case P^iq^r^) — 0. The robustness function for this example, eq.(3.158), and the opportuneness function are an anti-symmetric pair, and are related as: X{q. rw) = -Se(g, re) + ^ ^ ^ ^

(3.162)

We can now explore how the robustness and opportuneness functions, /3g(g',rw) and 3e(9,rc), vary as the manufacturer's choice of the production volume q is altered. Robustness and opportuneness vfiay be sympathetic, but in this example they will sometimes be antagonistic. Differentiation eq.(3.162) with respect to q results in: ^ _ _ 9 3 _ rw - T c d0 dq ~ dq iPHq) dq

^^'^^^'

The quantity {r^ — rc)/'0^ is for all practical purposes positive, since the wildest dream for windfall reward r^ is doubtlessly greater than the critical survival level of reward TC. This means that the extrema of p and a will not occur at the same value of q unless dip/dq happens to vanish precisely at an optimal q. That is, we cannot have: f ^ O oq at the same value of q unless

and

^ = 0 dq

(3.164)

^ = 0 (3.165) dq a^s well. Furthermore, not only can we not simultaneously optimize both Pe{^^ ^w) and ae{q, TC), eq.(3.163) also implies that there can be some ranges of ^-values for which both functions have the same slope. Since "bigger is better" for a while "big is bad" for (3, this means that for some ranges of production volumes one can improve one of these functions only at the expense of degrading the other. In these ranges of q one can trade robustness

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for opportuneness or vice versa. On the other hand, this antagonism need not be universal. If, for instance, dip/dq is positive, then a range of g'-values over which S(^, TC) increases (which is desirable) will be a range of g'-values over which f3{q^r^) decreases (which is also desirable). We will explore the antagonism and sympathy of the immunity functions further in chapter 5.

3.4

X General Robustness and Opportuneness Functions

For most practical purposes eqs.(3.4) and (3.5) (or eqs.(3.10) and (3.11)) are sufficiently general and powerful assessments of the impact of pernicious and propitious aspects of uncertainty. In some circumstances, however, one needs a more general formulation of robustness and opportuneness. In this section we briefly address this need. These concepts are applied in sections 11.1 and 11.2. Robustness function. Let g' be a vector of actions made (or contemplated) by the decision maker, and let Q be the set of all allowed or available actions. Let ui, ... ,UN be info-gap variables with info-gap models Ui{a,ui), . . . ,UN{OI,UN)-> Q; > 0. Let IZ^ [q, Ui{a,ui), . . . ,UN{(^,UN)] be a reward algorithm which evaluates action q with respect to the sets Z//i(a, Si), . . . ^UN{<^I SAT), at fixed uncertainty a, and returns a real-valued assessment of the reward. The reward algorithm IZ^ is monotonically decreasing in the sets upon which it acts. That is, 7^* has the following property: A^CBi,

2 = 1, ...,iV,

implies

IZ^q.Ai,

...,AN]

>7^*[g,5l,

...,BN]

(3.166) We stress that the reward algorithm IZ^ is monotonically decreasing on the uncertainty sets, but we have assumed nothing about how IZ^ varies with the decision vector q. The most common motivation for requiring this monotonic decrease of the reward algorithm is that it thereby seeks the worst case on the uncertainty-sets in its domain. We will have more to say shortly about worst cases. Referring to eq.(3.4) we see that the expression: min_ R{q,u)

(3.167)

u^U{a,u)

is a special case of: 7^* [q, Ui{a,ui),

... ,1(N{(^,UN)]

(3.168)

The minimum in (3.167) decreases as the set U{a,u) grows, which is precisely the property of monotonic decrease required of the generic reward algorithm in (3.168).

100

INFO-GAP DECISION THEORY

The reward algorithm depends both on the decision maker's choice q and on the uncertain environment. In fact, the info-gap models themselves may depend on the action q, for instance by dependence of the centerpoints Ui o n q.

Let Tc be the least acceptable reward. TC is not a fixed value, but rather a parameter which the decision maker uses to explore options and preferences and, if necessary, ultimately chooses in light of the analysis of uncertainty. In accord with eq.(3.1), and as a generalization of eq.(3.4), the robustness function for action q is formally defined as the least upper bound of the set of a-values for which the reward is no less than TC : 3(g,rc) = max {a : 7^* [q, Ui{a,ui),

... ,1(N{O^,UN)]

> Vc}

(3.169)

Let A denote the set of a-values in this relation: A={a:

n,[q,Ui{oc.Ui),

... MN{oi.UN)] >r^}

(3.170)

If the set A is empty then we define a{q,rc) = 0. li A is not empty then a(g, Tc) is the least upper bound of A. The inequality 7^*[...] > Vc in eq.(3.169) is a selection rule for a satisficing procedure: action q is satisfactory at uncertainty a if the reward is no less than the critical value r^. The robustness of q is the greatest level of uncertainty at which q is satisfactory. Equivalently, 3(g, TC) induces a preference relation, based on robustness to uncertainty, on the set Q of available actions: action q is preferred to q' if the decision maker is more robust with q than with q'\ qy^q

if

a{q,r,)>a{q',rc)

(3.171)

which is precisely eq.(3.19) on p.45. In light of this preference ordering, a decision maker who is concerned both about reward and robustness has an incentive to choose the action which satisfices the performance and which maximizes the robustness to uncertainty. For any choice of the critical reward r^ the robust-satisficing action with info-gap modelsZYi(a, ui), . . . ,UN{OL, UN) is denoted $c(^c) and is defined as: qdvc) = argmax3(9,rc) (3.172) qeQ

which is the same as eq.(3.20). This decision algorithm is 'rational' in a sense employed extensively in economic and psychological studies. However, the central role played by the info-gap uncertainty should be noted: the preference-ordering is derived from the robustness function a{q,rc) rather than directly from the reward function 1Z^[...]. More specifically, the choice of ^c(^c) niay differ from the choice which maximizes the reward function, as we have seen in many examples. If "rationality" means choosing an action which maximizes the best-estimate of the outcome, as is assumed in much economic theory.

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then info-gap robust-satisficing is not rational. However, in a competitive environment, survival dictates rationality. In section 11.4 we will show that, for a wide range of situations, the robust-satisficer is more likely to survive than the direct optimizer. If "rationality" means selecting reliable means for achieving essential goals, then info-gap robust-satisficing is extremely rational. Optimization of the robustness in eq.(3.172) is emphatically not a worstcase analysis. In classical worst-case min-max analysis the decision maker minimizes the impact of the maximally damaging case. But an info-gap model of uncertainty is an unbounded family of nested sets: W(a, S), for all a > 0. Consequently, there usually is no worst case: any adverse occurrence is less damaging than some other more extreme event occurring at a larger horizon of uncertainty a. What eqs.(3.169) and (3.172) express is the greatest level of uncertainty consistent with satisficing to level r^ When the decision maker chooses the action q to maximize a{q, TC), what is maximized is the immunity to an unbounded ambient uncertainty. The closest this comes to "min-maxing" is that the action is chosen so that "bad" events (causing reward IZ^ less than TC) occur as "far away" as possible (beyond a maximized value of a). Opportuneness function. We now define a function which is the formal dual of the robustness function. Let IZ* [q^ ZYi(a,Si), . . . ^UN{<^^UN)] be a monotonically increasing reward function on the uncertainty sets. Instead of eq.(3.166), IZ* is real-valued and monotonically increasing on the sets upon which it acts: AiCBi,

i = : l , ...,Ar,

implies

7^*[^,^l, ...,AN]


...,BN] (3.173) As before, we have made no assumption about how IZ* varies with the decision vector q. Referring now to eq.(3.5) we see that the expression: max_ R{q,u)

(3.174)

uEU{a,u)

is a special case of: 7^* [q, Ui{a,ui),

... ,UN{oi,UN)\

(3.175)

The maximum in (3.174) increases monotonically as the set U{a^u) grows, as required of the generic reward algorithm in (3.175). As in eqs.(3.2) and (3.5), the opportuneness function for action vector q is the least uncertainty for which the increasing reward algorithm is no less than a windfall reward r^^: ^(^,^w) = min{a : 7^* [q, Wi(a,Si), ...

,UN{(^IUN)]

> r^}

(3.176)

Let B represent the set of a-values defined in this relation: B^{a:

7^* [q, Ui{a, Si), . . . MN{a. UN)] > r^}

(3.177)

102

INFO-GAP DECISION THEORY

If B is empty then we define P{q, r^) = (X). If S is not empty then (3{q, r^) is the greatest lower bound of B. Tw is the decision maker's "wildest dream" for reward, ^{q^ry^) is the least level of uncertainty, based on uncertainty models Ui{a,Ui) , . . . , UN{OL^UN)^ which is sufficient to entail the possibility of reward as large as Tw Reward r^ is not a rigidly fixed value, but rather a parameter with which the decision maker explores the possibilities for windfall inherent in the ambient uncertainty. In general, the range of Tw-values of interest exceeds the range of rc-values upon which the robustness function S(g, TC) is based. (^{^•> ^w) is the formal dual of the robustness a{q, TC) defined in eq.(3.169): the min and max operators are interchanged and the monotonicity of the reward functions is reversed: IZ^ decreases while Vf increases on nested uncertainty sets. While a{q^rc) is the greatest uncertainty which is consistent with satisficing at reward level TC, we see that (5{q,r^) has no connection to satisficing, but rather to what we have called "windfalling". The concept of satisficing expresses the decision maker's cautiousness: eq.(3.169) is designed to guarantee the achievement of a 'critical' or 'survival' level of reward. In contrast, eq.(3.176) expresses a propensity to gamble and an ambition to garner windfall gains from the ambient uncertainty. In short, the duality between the robustness and opportuneness functions reflects two antipodal attitudes to uncertainty: aversion and proclivity. Info-gap gambling is discussed in chapter 6. A cautious decision maker desires immunity to uncertainty and prefers a large rather than a small value of 3; hence the preference ordering of relation (3.171). On the other hand, a windfalling decision maker hopes that the ambient uncertainty will grant undreamed-of reward, so a small value of (5 is desired. Like S(g, TC), (3{q, r^) also induces a preference relation on the set Q of available options based on opportunity from uncertainty: action q is preferred to q' if q entails greater opportunity for the decision maker than q'\ q^oq'

if

^(g,rw)<^(g',rw)

(3.178)

which is identical to eq.(3.21) on p.46. q is preferred to q' if q exploits ambient uncertainty more efficiently than q'. That is, q ^o Q' if lower uncertainty is needed with q than with q' in order to entail the possibility of the same windfall reward. A windfalling-rational decision maker would choose the action which maximizes the opportunity from uncertainty, which is done by minimizing 13. For any choice of the windfall reward TW, the opportune-windfalling action with info-gap models Ui[a,ui), ... ,UN{(^,UN) is denoted $w(^c) and is defined as: $w(rw) = argmin^(g,rw) This is the same as eq.(3.22).

(3.179)

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Problems

Following is a list of problems in this section. 1. Static deflection of a cantilever, p. 103. 2. Static deflection of a cantilever: continued, p. 104. 3. Quality control, p. 104. 4. Stability of a platform, p. 105. 5. Stability of a platform: continued, p. 105. 6. Moments on a robotic arm. p. 106. 7. Environmental contamination, p. 106. 8. Satellite targeting, p. 107. 9. Simple harmonic oscillator, p. 107. 10. Project management, p. 108. 11. Project management: continued, p. 108. 12. Uncertain probabilistic task time, p. 109. 13. Control of a production system, p. 109. 14. Reliability of a milling process, p. 109. 15. Reliability of robotic motion, p. 110. 16. Planning machine initialization, p. 110. 17. EflScient fuel allocation, p . l l l . 18. Machine efficiency, p . l l l . 19. Planning for uncertain costs, p. 112. 20. Budget allocation, p. 112. 21. Budgeting the manager's time, p. 112. 22. Uncertain probabilistic profit, p. 113. 24. Customer satisfaction, p. 113. 25. Uncertain lotteries, p. 113. 1. Static deflection of a cantilever. In many problems the reward function can be expressed as a linear function of an info-gap uncertain vector. Here is an example. A uniform cantilever beam is subject to A'^ static point loads applied, in a single plane, perpendicular to the beam axis. The vector of loads is denoted: f = {h,...,fNf (3.180) The deflection of the free end of the cantilever, y, is linearly related to the loads by: y = k'^f (3.181) where /c is a column vector of known flexibility coefficients. The load vector is uncertain and belongs to the following Fourier ellipsoid bound info-gap model: na,7)={/:

(/-7)^H/(/-/)

< a n ,

a>0

(3.182)

where W is a real, symmetric, positive definite matrix. (a) If a is given, find the load vector / which results in the maximum end deflection and find the maximum end deflection. (b) The beam fails if the end deflection exceeds the critical value yc- What is the robustness function of the beam?

104

INFO-GAP DECISION

THEORY

(c) Now assume that a is a random variable with a known exponential distribution. Combine this information with the info-gap uncertainty about the load vector / to evaluate a hybrid robustness. 2. Static deflection of a cantilever: continued. In some problems the reward function can be expressed as a quadratic function of an info-gap uncertain vector. We modify problem 1 as an example. Instead of the deflection y in eq.(3.181), consider: r = fVf

(3.183)

r is proportional to an elastic energy of deformation. V is a known, real, symmetric, positive definite ^ a t r i x . Re-examine questions l a and l b for the following two cases: (a) / = 0. (b) / 7^ 0. 3. Quality control. A long, thin, cylindrically symmetrical surgical needle is finely milled to conform to its specified shape, t(x). The allowed deviation of the actual thickness t{x) from the designed thickness t{x) is ±D microns throughout the length L of the needle. The manufacturer guarantees the following two conditions: (i) The precise dimension of the needle will be verified at each end by direct measurement. Any needle for which ^(0) ^ t(0) or t{L) / t{L) will be rejected. This quality control measurement is completely accurate (or its accuracy is vastly greater than the allowed tolerance). (ii) The deviation of the slope of the surface of the needle from its specified slope is bounded, in an attempt to exclude bristles and dents. However, the value of the bound on the slope is unknown. (a) Formulate an info-gap model for the shape of needles produced to these specifications. (b) Construct the robustness and opportuneness functions for this manufacturing process. (c) The manufacturer now proposes to replace the first condition above with the following extended quality check: (i') The precise dimension of the needle will be verified at each end and at A^ equi-distant intermediate points by direct measurement. Any needle which deviates from the specified dimension at any measured point will be rejected. The second condition, (ii), remains valid. Construct the immunity functions and use them to choose the number N of measurement points. In particular, study the marginal utility of the A^th measurement. (d) i What modification of the initial information, and consequently of the info-gap model, would lead to a more meaningful opportuneness function? (e) i Modify the quality control specification in item (i) to consider error in the quality control measurement. Specifically, any needle for which |t(0) - t{0)\ > sD or \t{L) -t{L)\ > eD will be rejected, where 0 < e < 1. Now repeat questions 3a and 3b for the robustness function only.

C H A P T E R 3.

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

G

'

105

•

^ = ^

Rigid 'latform x = 0

Flexible Column Stream

Figure 3.23: problem 7.

Figure 3.22: Platform for problem 4.

Illustration for

4. Stability of a p l a t f o r m . A rigid beam-like platform is supported from below at its midpoint by a flexible column which is at elastic equilibrium when the platform is horizontal, as shown in fig. 3.22. The flexural stiflFness of the elastic column is k [Nm/radian] and it applies a restoring moment of force M — kO when the platform is tilted by 6 radians. The width of the platform is 2L [m]. The platform is loaded at its two ends by static forces F and G which are uncertain. That is, forces F and G belong to the following info-gap model of uncertainty: ^/(a,0) = {F, G : \F\ < a,

1^1 < a } ,

a>0

(3.184)

The platform is satisfactorily level if the angle of tilt at static equilibrium is never greater than the critical value 9^'. (3.185)

\e\ < Oc

The condition of static equilibrium requires that the moment of force at the midpoint vanish: 0 = F L - G ' L + /c6> (3.186) Determine the robustness and opportuneness functions of the platform. The decision vector is q = (/c, L)^. Study the variation of the immunity functions as these design variables are changed. Stability of a p l a t f o r m : c o n t i n u e d . We now modify problem 4 to consider uncertain distributed loads, f{x) [N/m], —L < x < L, on the platform. Evaluate the robustness and opportuneness for each of the following info-gap models for uncertainty in the load. (a)

Uniform-bound:

fix)

7 7TX • / c o s -


a>0

(3.187)

where / is a known constant. (b) Fourier ellipsoid bound: The uncertain part of the load profile is a truncated sine series: f[x) = / c o s — -h

E Cn sm —-— = / c o s —- + c (J{x)

(3.188)

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INFO-GAP DECISION

THEORY

where c is the vector of uncertain Fourier coefficients and a{x) is the vector of sine functions. The info-gap model is: ^Y(Q,7) = | / ( x ) = / c o s ^ - h c ^ a ( x ) : c^Wc
a>0

(3.189) where H^ is a known, real, symmetric, positive definite matrix. (c) Different nominal load. How will the answers to questions 5a and 5b change if the nominal load is: /(x) = / s m ^

(3.190)

6. M o m e n t s on a robotic arm. The angle of rotation of a robotic arm, ?/, varies according to the moments Mi applied at the joints according to: Y,k^^^

(3-191)

where ki is a known positive stiffness parameter. The moments Mi are poorly known. The best estimate of Mi is Mi which is positive. The fractional error of Mi is unknown so the vector of moments belongs to the info-gap model:

{a,M) == {M i^^: U{a,M)

Mr - Mr

< a , z = 1,2,3 L

a>0

(3.192)

Mr

(a) The angle of rotation must be at least ijc. What is the robustness, to uncertainty in the moments, of the robotic rotation? (b) The nominal moments Mi can be chosen subject to the constraint: Y^Mf=f^

(3.193)

where ji is known. What choice of the nominal moments maximizes the robustness? 7. Environmental contamination. The concentration of contaminant, / ( x ) , varies over a farmer's field as a function of distance x from a stream passing along one side of the field: N

/(a;) = 7 + ^ 0 n x " = / + 0^C

(3.194)

where (j) is the vector of unknown coefficients and ^ is the corresponding vector of powers of x. The nominal concentration, / , is a known constant. The stream is located at x = 0 and the far edge of the field is at x = L as in fig. 3.23 on p.l05. The quantity of contaminant which reaches the stream by the end of the season is: 9

M /

[l-(|)']/(x)dx

(3.195)

C H A P T E R 3.

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107

where /i and i> are positive constants which reflect absorption and transport properties of the soil, and which can be influenced in known ways by treating the field. The water in the stream is potable if the quantity of contaminant does not exceed the value ^c, but it is highly desirable that the quantity of contaminant not exceed the far lower value pw For the Fourier ellipsoid bound info-gap model specified below, determine the robustness and opportuneness functions and study their behavior as a function of the decision variables // and f. Discuss the choice of these variables. Consider the antagonism and sympathy of the immunity functions, a and (3. U{aJ)

= ^f{x)^J+(f^i:

(t>^W^
a>0

(3.196)

where H^ is a known, real, symmetric, positive definite matrix. 8. Satellite targeting, (a) A satellite is launched from point L directly at a stationary target located a distance D away at point T. The satellite moves in a single plane, but the slope of the satellite trajectory, with respect to the line LT^ varies in an unknown manner during flight. The satellite carries a pay load of photographic devices, which are just barely effective if the satellite-target distance is no greater than re as the satellite passes the target. The pay load is highly effective if the fly-by distance is as small as Tw Evaluate the robustness and opportuneness functions. (b) Now consider a slightly more complex situation. The satellite is initially launched at a slope so which is lower than the slope of the line from L to T. The slope of the satellite trajectory deviates in flight from so in an unknown manner. The mission fails if the satellite passes too far from or too near to the target. That is, failure occurs if the satellite-target distance is greater than rc,2 or less than rc.i as the satellite passes the target. The mission is highly successful if the satellite passes as near as TW from the target, but not too near, where Vd < r^ < rc,2- Evaluate the robustness and opportuneness functions. 9. Simple harmonic oscillator. The displacement x{t) of an oscillator in simple harmonic motion is: mx{t)-\-kx{t)

= u{t)

(3.197)

where the mass m and stiffness k are known but the driving force u{t) is uncertain. The energy of the oscillator is proportional to the square of the displacement. Failure occurs if the energy at a specified time T exceeds a critical value: x^{T) > Ec (3.198) The uncertainty in u{t) is represented by the following Fourier ellipsoid bound info-gap model: N

u{t) = u{t) + ^

(/^n sin ^

= u{t) -h (/)^cr(t)

(3.199)

108

INFO-GAP DECISION THEORY

where 0 is the vector of uncertain Fourier coefficients and a{t) is the vector of corresponding sine functions. The info-gap model is: U{a,u) = {u{t) = u{t) + 0^cr(t) : ^'^1^0 < a^} ,

a > 0

(3.200)

where VK is a known, real, symmetric positive definite matrix. Evaluate the robustness and opportuneness functions and illustrate their use in choosing the system parameters, m and k. Assume that the initial displacement and velocity of the oscillator are both zero. f22 = h

*^4L

0__^g}_^^}_L0 Figure 3.24: A 5-task project schedule for problem 10. 10. Project management. Consider the 5-task project shown in fig. 3.24. This project has two task-paths: Path 1: Path 2: In path 2, task 4 is initiated when task 2 is half finished, as indicated by /22 = 0.5.

The nominal task durations are: ti

t2=t5--

1,

ts

t4 ^ I - q

(3.201)

where q is a, parameter which the project manager is free to choose in the interval [0, 1]. q represents a valuable resource which must be allocated between tasks 3 and 4. The uncertainty in the actual task durations is represented by a fractionaluncertainty info-gap model:

U{a,t)

{'

tn

•tn\

< Q, n — 1,

a>0

(3.202)

'^n

The project must be completed within about 4 time units. Construct the robustness and opportuneness functions and demonstrate their use in choosing g. 11. Project management: continued. Consider a modification of problem 10. If the project completes in a duration T, then the "reward" is R{T). R{T) decreases as T grows to express the penalty associated with delayed termination. The project owner would very much like to earn reward as large as rw, and cannot tolerate reward less than Vc. Formulate and evaluate the robustness and opportuneness functions. Indicate the choice of the parameter q. Explain how the robustness and opportuneness functions can be used to choose aspirations re and r w Compare these results to the solution of problem 10.

C H A P T E R 3.

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109

12. Uncertain probabilistic task time. The probability that the project will be completed within the critical duration tc is: P = —— ^c -\-uq

(3.203)

where q is positive and controlled by the manager and u is uncertain: U{a, u) = {u:

\u-u\
a > 0

(3.204)

The customer demands that the task complete within duration tc with probability no less than Pc. Derive an explicit algebraic expression for the robustness of q. 13. Control of a production system. The performance of a production system depends on temperature T, pressure P and time 0 according to the relation: y = T-2P + 0/3 (3.205) Define the vectors x^ = ( r , P, 0) and ^^ = (1, - 2 , 1/3). Eq.(3.205) becomes: y = z^x (3.206) The system operator chooses operating values for T, P and ^, which will be denoted T, P and (9, or collectively: x^ = (T, P , 6). The actual implemented value of x can vary in an unknown manner from the selected operational value x. The uncertain deviation of x from x is described by an info-gap model: U{a, x) = < X : —^—-

I

< a,

i = 1, 2, 3 > ,

I ^O

a> 0

(3.207)

J

The product of the system is unsatisfactory if: y>yc

(3.208)

where t/c is a known value. (a) Derive an expression for the robustness of this production system, to uncertain variation of the system parameters. (b) Consider the choice of the operational value T, which must be selected in the interval 69°C
y{t) = 2 ] &n cos —

= 6 lit),

0
(3.209)

n=ni

where b is the vector of known Fourier coefficients and 7(^) is the vector of corresponding cosine functions.

no

INFO-GAP DECISION

THEORY

The actual height profile differs from y{t) in an uncertain manner: y{t) = y{t) + ^

6n cos ^

= y{t) + 6^7(^),

where 6 is a vector of unknown Fourier coefficients. represented by the following info-gap model: U{a,b)


n = n i , . . . , 712 / ,

0< f< T

(3.210)

Uncertainty in h is

a>0

(3.211)

The milling process fails if the cutting tool is too far above the planned height at the end of the run, t = T. That is, failure is defined as: y{T) - y{T) > D,

(3.212)

(a) Formulate an expression which defines the robustness function. (b) Derive an explicit algebraic expression for the greatest deviation of the actual height above the planned height, up to uncertainty a. (c) Derive an explicit algebraic expression for the robustness function. (d) Repeat parts (a)-(c) if the failure criterion is revised as follows. Failure occurs if the actual height deviates by more than Dc from the planned height at any time during the milling run. That is, failure is: \y{t)-y(t)\>D,,

0
(3.213)

15. Reliability of robotic motion. The arm of a robot moves in the {x,y) plane. The trajectory of the end effector as a function of time t is specified by: x{t) = c,{t + l), yit) = jfj (3.214) for t > 0. The coefficients ci and C2 are uncertain and the uncertainty is described by the following info-gap model: ^ ( a , c ) = {c: \c^-c^\
2 = 1,2},

a>0

(3.215)

The values of Ci are known and positive. The end effector must reach the following location at specified time t = T\ J = ?:(T+1),

V=7^^

(3.216)

The robot motion fails if the end-effector location at time T deviates from the desired location by more than a distance Dc. (a) Derive an expression for the robustness of the robotic motion. (b) What is the optimal value of T, from the point of view of robustness? 16. Planning machine initialization. A manufacturing machine runs continuously, and the longer it runs, the more it produces. However, the productivity, in terms of number of items produced per unit time, decreases the longer the machine runs. The total amount produced in time t is ^(t).

C H A P T E R 3.

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

HI

which is an increasing function whose slope decreases in time. The function g{t) is imperfectly known. An info-gap model for uncertainty in g{t) is: Uia,-^

= {g{t) : \git)-g{t)\

< ah{t)} ,

Q > 0

(3.217)

where ^{t) and h{t) are known functions. You must plan the production schedule for a total time of T hours. If the machine runs continuously for T hours its production will be g{T). You can plan to stop and initialize the machine periodically. After each initialization the machine restarts with its initial productivity, which is high. However, each initialization requires r hours. You must choose the number of restarts, assuming that the machine will run for the same duration after each initialization. The goal is to assure that the total production will not be less than re. (a) Derive an explicit expression for the robustness to uncertainty in the production function, when the machine is initialized n times during the T hours, assuming that the machine operates for the same duration after each initialization. (b) Now assume that: g{t)

=

bVi

(3.218)

h{t)

=

Vt

(3.219)

What is the robust-satisficing number of initializations? 17. Efficient fuel allocation. The distance which a vehicle can travel with quantity q of fuel is: /(q,c) = — ^ (3.220) 1 + cq^ where c is an uncertain parameter described by the following info-gap model: U{a,^

= {c:

\c-^
a>0

(3.221)

The vehicle must be able to travel at least a distance /c. (a) Find an explicit algebraic expression for the robustness of fuel quantity q to uncertainty in the coefficient c. (b) Find an explicit algebraic expression for the quantity of fuel q which maximizes the robustness. Compare this to the value of q which maximizes the distance based on the nominal value of c. 18. Machine efficiency. The efficiency of a machine is described by: h{q) = q+-

(3.222)

where g > 0 and c is uncertain and described by an info-gap model: Z^(a,2) = {c: | c - ? | < a c r } ,

a>0

(3.223)

(a) It is required that h{q) be no less than he. What is the robustness of the machine to uncertainty in c, for a given value of ql (b) The designer can choose q in the interval [^i, q2\ What choice of q do you recommend?

112

INFO-GAP DECISION

THEORY

19. Planning for uncertain costs. Consider a project with A^ tasks whose costs are c = (ci, . . . , Cjv)^ and which are initiated at times t = (^i, . . . , ^iv) • The total project duration is T. The money needed for each task must be available when that task is initiated, and will be funded by a loan at interest rate r. The loan will be repaid at the end of the project. The total project cost, including interest, is: N

C ( c , 0 = ^c.e'^(^-*^)

(3.224)

We have very limited information about the task costs. We have estimates from the past of the mean costs and the variances and co-variances of the costs. The historical mean vector and covariance matrix are Z and V. We will use the ellipsoid-bound info-gap model: Z^(a,?) = { c : ( c - ? ) ^ y - ' ( c - ? ) < a ^ } ,

a>0

(3.225)

(a) Derive an expression for the robustness, to uncertainty in task costs, of task-initiation times t. The manager aspires to keep the total project cost below the critical value Cf. (b) Now suppose that the covariance matrix V is diagonal with elements t'l, . . . , t'iv. Use the results of question 19a to indicate how one might wish to choose the task-initiation times t. 20. Budget allocation. Consider a project with A'' tasks. After budget allocation for basic operational needs of the A^ tasks, a quantity Q of money remains for allocation among the tasks, on the basis of their anticipated contribution to project profitability. The anticipated rate of return of each additional dollar allocated to task i is S^, for z = 1, . . . , A^. Thus an allocation q is anticipated to result in added revenue: N

i?(g,S) = ^ 9 ^ S ^

(3.226)

z= l

The allocation q has succeeded if the added revenue is no less than Re. However, the anticipated return vector u is highly uncertain. The actual rates of return, n^, are unknown, and the uncertainty in the vector of rates of return u is denoted by an info-gap model, U{a^ u). Consider the info-gap model: U{a,Z)^{u:

\u^-Z^\
1 = 1, ...,N},

a>0

(3.227)

where the uncertainty weights Wi are known positive numbers. Suggest how one might go about choosing an allocation of Q among the tasks, based on (a) the robustness function, (b) the opportuneness function. Assume that no allocations are negative, so g'z > 0 for all z. 21. Budgeting the manager's time. The project manager's time must be allocated among N tasks. The planned allocation is t = (ti, . . . , tiv)^. In practice, the actual time which the manager devotes to the N tasks will

C H A P T E R 3.

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

113

he t = (ti, . . . ,tN)^' The uncertainty in the manager's attention times is represented by the following info-gap model: U{aJ) = {t: {t-IfV{t-t)
a>0

(3.228)

where V is real, symmetric, positive definite and known. The utility from time-allocation t is y^t where y is a, known vector. It is required that the utility be no less than Uc. Develop an explicit analytical expression for the robustness to uncertainty in t. 22. Uncertain probabilistic profit. The profit from a new project is a random variable x which is distributed exponentially: P(x|A) = l - e ~ ^ ^ ,

x>0

(3.229)

The estimated value of A is A, but this estimate is highly uncertain: ^ ( a . A) = | A : A > 0, |A - A| < ax\ ,

a >0

(3.230)

It is required that profit at least Xc occur with probability no less than Pc^ FToh{x > Xc|A) > Pc

(3.231)

Derive an explicit algebraic expression for the robustness to uncertainty in A. 23. Project profitability. The profit from a project is described by: R{q,u) = qiu + q2

(3.232)

where q = (^1,^2) are positive numbers chosen by the manager. The parameter u is uncertain: U{a,u)^{u:

\u-u\
a>0

(3.233)

The profit must be at least Re. Profit as large as Pw would be wonderful. Derive expressions for the robustness and opportuneness functions. 24. Customer satisfaction. The satisfaction of the customer is measured as: S^qA + q^B

(3.234)

where q is controlled by the manager and A and B are uncertain: U{a, A,B) = [A,B : {A - Af + [B - B)'^ < a^\ ,

a >0

(3.235)

The requirement is that the satisfaction be at least ^c. Derive the robustness of decision q. 25. Uncertain lotteries.^^ Consider a lottery with two prizes whose values are V(, > Vs. The probability of winning the larger prize is uncertain. The best estimate of this probability is p and the info-gap model for p is:

U{a,p)=:lp:

0
^
a>0

^We will consider lotteries with uncertain probabilities in chapter 11.

(3.236)

114

INFO-GAP DECISION THEORY

(a) For any critical reward Vc, such as the cost of a lottery ticket, what is the robustness, to uncertainty in p, of winning at least Vc^ (b) Now consider a different lottery with prizes v'^ > Vs and estimated probability p of winning v'^. Furthermore, the estimated average prize is now greater: p v'^ -\- {1 — p )Vs > pve + (1 — p)vs. However, the smaller prize is now even smaller: v's < Vs. The uncertainty p is represented with the info-gap model of eq.(3.236). Under what conditions (e.g., with what values of Vc) will you prefer this new lottery? (Consider the crossing of the robustness curves of these two lotteries.)

Chapter 4

Value Judgments [T]he scientific method can teach us nothing else beyond how facts are related to, and conditioned by, each other. ... [Yet] knowledge of what is does not open the door directly to what should be. One can have the clearest and most complete knowledge of what is, and yet not be able to deduct from that what should be the goal of our human aspirations. Albert Einstein [66, pp.21^22]

It is sometimes difficult to decide that a given option or outcome is desirable. Knowing how desirable it is, or how much better than a given alternative, is often far more challenging. In chapter 3 we began our study of the quantitative assessment of a decision maker's options by developing the conceptual and mathematical basis of the robustness and opportuneness functions. We came to the understanding that robustness to the pernicious potential of uncertainty is desirable: bigger values of a{q, Vc) are better than smaller values. But how big is big enough? What is the marginal increment of desirabihty accruing from a marginal increment in S? These questions, and the analogous ones regarding the opportuneness function (3{q,r^), are quite formidable. The degree of desirability of an option — as opposed to the numerical value of the immunity function for that option — is a qualitative judgment based on personal or collective values. Enlightened technological society has long ago abandoned the cold and impersonal image of the analyst, engineer or technocrat who is detached from value judgments. The decision analyst cannot remain aloof from the context which generated the need for a decision, nor be indifferent to the underlying values. The analyst has a moral obligation to participate in the judgment of values which are inherent in the process of decision making, as well as an irreplaceable role to play. The analyst's skills and knowledge are essential to providing the basis for 115

116

INFO-GAP DECISION THEORY

reasonable and rational judgments of value. The need for value judgments is not unique to info-gap decision theory. Probabilistic decision theory shows the same need to calibrate quantitative assessments in terms of personal or social values. If P is the probability of success resulting from choosing a particular option, it is clear that "bigger is better" for P just as for the robustness a. But this does not determine what values of either P or 3 are acceptable. We may accept a probability of catastrophic failure in automobiles of 0.001 per year per car, and yet demand much lower probability in the air travel industry. The intuitive interpretations of P and a — establishing that "bigger is better" — do not determine criteria for acceptability. In both cases, only reference to considerations outside the decision theory itself leads to acceptance or rejection of a given numerical value. In this chapter we consider the calibration of the robustness function in terms of values whose meaning is expressed in natural, not mathematical, language. This calibration is, by its nature, approximate, intuitive and imprecise. But it is essential in linking quantitative decision analysis to the choices made by human decision makers. The same considerations can be applied to calibrating the opportuneness function but we will not elaborate on this. The simplest calibration of the robustness function is by prior experience. If a particular system or action, whose performance or outcome is known and trusted, has a robustness function with the value S*, then a similar system or action can be evaluated by comparing its robustness against 3*. Another common ad hoc calibration of the robustness function is by normalization of a to a dimensionless form. We discuss this in section 4.1. However, what we are really after is an explicit linguistic evaluation, in natural human language whose meaning we can relate to everyday values, of numerical increments in the robustness. The format for this is reasoning by analogy, which is described generically in section 4.2. In section 4.3 we apply this method to calibration of a in terms of consequences, where the severity of consequences is understood in natural language. One can also calibrate a by relating it to intuitively meaningful increments of prior information, as discussed in section 3.2.5 on p.63 and elsewhere [14, chapter 9]. In section 4.4 we discuss some aspects of the rationality of ranking the options in terms of the robustness function.

4.1

Normalization

The horizon of uncertainty a of an info-gap model of uncertainty U{a,u) typically has physical units. For instance, in the envelope-bound model of eq.(2.8) on p.22, a and the uncertain function u{t) have the same units. If u{t) is the uncertain temporal variation of the quantity of fertilizer deposited per dunam, with units of kilogram/dunam, then these same units must be

CHAPTER 4. VALUE JUDGMENTS

117

ascribed to the uncertainty parameter in order for the units to 'balance'. (The envelope-shape function ?/;(t) is dimensionless.) Alternatively, in the cumulative energy-bound info-gap model of eq.(2.4), the units of a equal the units of u{t) multiplied by the square root of time. If u{t) has units of pressure, Newtons/m^, then a has units Newtons^^/m^. The robustness function a{q^rc) is the least upper bound of a set of a-values, as seen in eq.(3.4) on p.40. Consequently the robustness function and the uncertainty parameter have the same units. Info-gap models can be defined in such a way that the uncertainty parameter is in fact dimensionless. An example is the envelope-bound model of eq.(2.9). In this model a is the fractional variation of the uncertain function u{t) with respect to the nominal function u{t). In this case a is dimensionless regardless of the units of u{i) and u{t) (which are the same). In this model the robustness a is the greatest tolerable fractional variation of the uncertain function, and is therefore also dimensionless. When a
(4.1)

then fluctuations which are just a small fraction of the nominal value entail the possibility of failure, suggesting substantial vulnerability to uncertainty. On the other hand, if a > 1 (4.2) then we understand that the system is resistant even to fluctuations which are much greater than the nominal function, which implies a sense of durability or reliability of the system. More generally, when the robustness function 3(g, TC) has physical units it can usually be normalized to a dimensionless form by judicious use of prior information. When this is possible, then values much less or much greater than unity can be interpreted as we have done with eqs.(4.1) and (4.2). For example, let us return to the cumulative energy-bound info-gap model of eq.(2.4), for which both the horizon of uncertainty and the robustness function have units of Newtonsy^/m^. Let T be a typical time constant of the system — e.g. the duration of an oscillation or of a load cycle. Then u{t)yT has the same units as the robustness and provides an approximate calibration of the robustness function, so a meaningful dimensionless form is: ^

^

(4.3)

Comparing this quantity against unity gives some rough intuitive calibration of the robustness.

4,2

Analogical Reasoning

We now consider reasoning by analogy, which will lead to a method for evaluating numerical increments of robustness in linguistic terms whose meanings

118

INFO-GAP DECISION THEORY

are known in the context of personal or collective values. We will also see that calibration by prior experience and by normalization are both in fact simple forms of reasoning by analogy. Given two alternative decisions, represented by the decision vectors qi and q2 which have different values of robustness 3(gi,rc) and S(g'2,^c), we wish to know how much better one is than the other. If 3(^i, TC) > a{q2,rc), we would like to know if option qi is really substantially better than option 92- That is, we seek a qualitative, linguistic evaluation of numerical increments in the robustness. We do this by identifying 'significant' increments in a different parameter and then relating these to changes in robustness resulting from the different options. Analogy, as a method of reasoning, is the inference that if two things agree with one another in some respects, then they will tend to agree in others as well. Reasoning by analogy is central to insightful thought. Analogy does not provide certain knowledge but, as Locke concludes, wary reasoning from analogy leads us often to the discovery of truths, and useful productions, which would otherwise lie concealed. [114, IV.xvi.l2, p.589] Quine explains that, in learning the meanings of words, "[w]hat makes insensible things intelligibly describable is analogy . . . ." [143, p. 14]. In discussing inventiveness in mathematics, Polya concentrates on "generalization, specialization, and the perception of analogies" and remarks that [tjhere is perhaps no discovery either in elementary or in advanced mathematics or, for that matter, in any other subject that could do without these operations, especially without analogy. [134, p.l7]. The perception of analogy is a guideline to insight and to an understanding of meaning, which is precisely what we seek: what does a numerical increment in the robustness function signify in qualitative terms? Calibration by prior experience is in fact a simple form of analogical reasoning. If two systems or actions are similar in structure and if prior experience indicates that the first is satisfactory, then the second will also tend to be satisfactory if it has about the same robustness as the first. This is not a proof, only an analogical plausibility. Likewise, normalizing the robustness function with respect to a typical value T] with the same units, enables interpreting the magnitudes of a by analogy. If 77 is a typical value corresponding to moderate resistance to failure, and if a is much greater than 77. then a is atypical or extreme and will tend to entail high resistance to failure. The analogy is, of course, imperfect, since it does not establish clearly how large a/r] must be. But analogies are by their nature almost invariably imperfect. Analogy is not proof, only suggested plausibility. But analogy is useful when attempting to interpret quantitative entities in qualitative linguistic terms, since words are at best only approximate and imprecise.

C H A P T E R 4.

119

VALUE JUDGMENTS

An elementary example of reasoning by analogy with robustness functions is the following. For any two quantities hi and /12, we will indicate that hi is 'significantly greater' than /12 with the notation: hi

(4.4)

>- /i2

The quantities hi and /i2 may be numerical entities like fluid pressures or monetary units, or they may be linguistic terms like 'reliability' having values such as 'low', 'moderate' etc. If hi = 'moderate' and /i2 = 'low' reliability, then (4.4) asserts that 'moderate reliability' is substantially greater than 'low reliability', the validity of which we understand from the meanings of these words.

Figure 4.1: Illustration of calibration by analogy, relation (4.5).

Figure 4.2: Illustration of calibration by reverse analogy, relation (4.6).

If (4.4) holds and if the robustness function a{h) increases with /i, then we conclude by analogy that a{hi) is significantly greater than 3(/i2)- Formally: hi >- /i2

and

a{hi) > 3(/i2)

analogically imply that

a{hi)

y S(/i2) (4.5) The analogy, illustrated in fig. 4.1, is that, since the /I's and the S's agree in ordinal ranking, then they should tend to agree in qualitative degree of ranking. The implication in relation (4.5) is not a precise logical deduction but rather a calibration or a plausible suggestion: the S-values are interpreted in terms of the /i-values so that a{hi) is "significantly greater than" 3(/i2). The validity of this calibration of robustness depends intimately upon the identity of the quantity h and upon the strength of its relation to a{h). Analogy is not proof, but rather explanation of plausibility. Relation (4.5) 'interprets' the increment in a in terms of the increment in h. Reasoning by analogy is not a precise logical process, and must be treated with the caution due to all appeals to rough natural language. However, it provides a method for relating a precise quantitative entity — the robustness function — to meaningful human experience.

120

INFO-GAP DECISION THEORY

It is an elementary m a t t e r to see t h a t one can also reason by reverse analogy, by which is meant t h a t if /ii is substantially greater t h a n /i2 and if a{h) decreases with /i, then 3(/i2) is substantially greater t h a n a{hi): hi

y

/i2

and

3(/i2) > a{hi)

S(/i2) >- a{hi) (4.6) As illustrated in fig. 4.2, t h e analogy is t h a t since t h e increment from hi to /i2 is a significant change, t h e increment from a{hi) to 3(/i2) is also plausibly a significant change. These changes are b o t h qualitatively "significant", though one is an increase and t h e other a decrease. Relations (4.5) and (4.6) are paradigms for analogical reasoning with a single independent variable, h. In calibrating t h e robustness by analogical reasoning with consequence severity we will employ two independent variables. Suppose t h a t t h e robustness depends on two variables, h and g. If hi is substantially greater t h a n /12 and if 3(/i,^2) increases with /i, then the analogical inference in (4.5) states t h a t a{hi,g2) is substantially greater t h a n 3(/i2,^2)' hi

>- h2

analogically imply t h a t

and

a ( / i i , ^ 2 ) > 3(/i2,^2)

(4.7)

analogically imply t h a t

(4.^

3(/ll,^2) >- 3(/l2,^2)

Now, if 3(/i2,^) increases with g, and if gi is a value such t h a t a{h2-,gi) = S ( / i i , ^ 2 ) , then gi is, reasoning analogically, substantially greater t h a n ^2T h a t is, together with (4.7) and (4.8), we conclude: ^{h2,gi)—

di{hi,g2)

and

gi > g2

analogically imply t h a t

gi y g2 (4.9)

These two inferences are shown schematically in fig. 4.3.

a{h2,g) a{hi,g2)

-A-

vs. g ^di(h,g2)

vs. h

S(/l2,^2)

g or h Figure 4.3: Illustration of calibration by analogy with two variables, relations (4.7) - (4.9).

In t h e next section we explore an application of t h e method of reasoning by analogy, always keeping in mind t h a t we are engaged in plausible reasoning not precise proof.

C H A P T E R 4.

4.3

VALUE JUDGMENTS

121

Calibration by Consequence Severity

We will use an example to explain how the robustness function is calibrated by analogical reasoning from the 'severity of consequences'. We will show how analogical reasoning is used to establish significant increments in robustness and to identify corresponding significant increments in a decision variable. Consider an environmental management problem in which we must determine a program of treatment for a region, perhaps applying fertilizer or pesticide, or harvesting trees, or allowing animals to graze or people to visit. We have a model which predicts the benefit accruing from any given treatment program. However, the model is not very accurate; the actual benefit may difi"er substantially from the predicted value. We need to determine the program of treatment so as to achieve no less than a specified benefit. The approach will be to choose the treatment program to maximize the robustness to uncertainty in the model, while attempting to guarantee that the benefit is no less than the specified minimal value. We first derive the robustness function and then we demonstrate its calibration by analogical reasoning from 'consequence severity'.

4.3.1

Robustness Function for Environmental Management

The region to be treated is represented as a line segment: the interval from X = 0 to X = 1. The density of treatment at point x in the region is q{x) and the resulting benefit is: R{q,b)=

[ b{x)q{x)dx

(4.10)

The term h{x) is the benefit per unit of treatment at x. The benefit function b{x) is uncertain. What is known about b{x) is that it varies with bounded slope around a known nominal function 6(x), and that the variation vanishes at the boundaries and is greatest in the center of the region. We use the slope-bound info-gap model of eq.(2.23) to represent the uncertainty in b{x): U{a, b) = I b{x) = b{x) + u{x) : u(0) = u{l) = 0,

du{x) < aip{x) dx a>0 (4.11)

The nominal benefit-model, b{x), and the envelope function, '0(x), are known. The minimal required benefit is r^ and the robustness to uncertainty in the model b{x) is precisely as in eq.(3.4): S(9,r"c) = max < a : niin_ R{q,b) I > TC > I \beU{a,b) J J

(4-12)

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INFO-GAP DECISION THEORY

Eq.(4.10), the benefit from treatment q{x) with uncertain benefit-model b{x)^ can be rewritten as: R{q,b)=

/ l{x)q{x)dx+ ^0 ^ -y '

u{x)q{x) dx

(4.13)

Jo

R{Q)

where R{q) is the nominal, error-free, benefit accruing from treatment program q{x), and u{x) in the right-most integral is the term whose slope is bounded in the info-gap model of eq.(4.11). Define the following function: /i(x)= /

q{x')dx'

(4.14)

so that dh{x) — q{x)dx. The right-most integral in (4.13) is found by partial integration to be: [^ / u{x)q{x)dx Jo

—

f^ |i f^ I u{x) dhlx) = u{x)h{x)\ — / Jo '0 io

du(x) h{x)—-—dx ^^ (4.15)

From this relation, together with eq.(4.13), we see that the least benefit, for any benefit model h{x) up to uncertainty a, is: min^ R{q, h) = R[q) -a [ \h{x)\ilj{x) dx beU{a,b) Jo

(4.16)

Equating this lowest possible benefit to the critical value TC and solving for a yields the robustness:

/o

\h{x)\^p{x)dx

provided that the nominal benefit R{q) is no less than the critical reward Tc; otherwise a{q,rc) = 0. We consider the following special case. The treatment intensity q is constant and the known nominal benefit b{x) and the envelope ip{x) are: b{x) = bo -{-bi siuTTX

and

ip{x) — sinyrx

(4.18)

We now find the robustness in eq.(4.17) to be: a{q,r,) = 7rbo + 2b,--^ or zero if this expression is negative.

(4.19)

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Robustness 3(g,rc,2) vs. q

S{q2,rc) vs. Tc I consequence ^c,2 severity ^c,l (low) (hi)

Figure 4.4: Robustness a(g, re) vs. re and vs. q, eq.(4.19).

4.3.2

Calibration by Consequence Severity

The robustness function a{q,rc) in eq.(4.19) is plotted versus two different variables in fig. 4.4. The decreasing straight line shows the trade-off between robustness a and reward r^ at fixed treatment intensity q: greater demanded reward is obtained in exchange for decreased immunity to failure. The monotonically increasing curve shows the variation of robustness a against treatment intensity g, at constant reward Vc'. the robustness to failure is enhanced by increasing the intensity of treatment, but the marginal return in robustness decreases as q grows. Two values of the critical reward, rc,i and rc,2, are shown in fig. 4.4. These values are ranked numerically: ^c,l > ^c,2

(4.20)

Not only are the r^i ranked numerically, but the decision maker's understanding of the meaning of the benefit model enables the association of qualitative values to these levels of reward. rc,i is substantially greater than rc,2, and attainment of reward rc,i constitutes a high level of reward as compared to rc,2- As in eq.(4.4), the qualitative ranking of these values of critical reward is denoted: rcA y rc,2

(4.21)

At fixed treatment intensity q2, the increment in robustness from rc,i to rc,2 is indicated in the figure as AS(rc). qi is a different treatment intensity, for which the change in robustness from q2 (at fixed TC) equals the change in robustness due to changing TC from low to high reward (at fixed q). That is, the increment in robustness from q2 to qi (at fixed TC) is Aa{q) as shown in the figure, which is the same as A3(rc): Aa{q) = Aa(rc)

(4.22)

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INFO-GAP DECISION THEORY

Reasoning by analogy as in relations (4.7)-(4.9), we conclude that treatment intensity qi is substantially greater than treatment intensity ^2* qi y 92

(4.23)

Stating the analogy explicitly: 1. The increment from ^2 to qi (at fixed TC) agrees in robustness with the increment from rc,i to rc,2 (at fixed q). 2. The increment from rc,i to rc,2 is a qualitatively significant change in reward. 3. Therefore the increment from g'2 to qi is, by analogy, expected to result in a qualitatively significant change in robustness. This analysis indicates the value of a significant or substantial increment of robustness. Furthermore, it establishes an increment in the treatment intensity which would plausibly enhance the robustness by a meaningful amount.

4.4

Rationality and Preference

The value judgments we have considered concern the qualitative assessment of increments in the robustness function. We have simply assumed that more robustness is better than less. There is no doubt that robustness to the adverse and pernicious potential of an uncertain environment is desirable. It seems reasonable that if some robustness is desirable then more robustness is even better. But when might this be false or misleading? In chapter 3 we repeatedly claimed that "bigger is better" for the robustness function a(g, TC). We explained in connection with eq.(3.19) on p.45 that a{q^rc) induces a preference relation on the set Q of available actions: action vector q is preferred to q' if the decision maker is more robust with q than with q' at the same level of critical reward Vc'. qyq'

if

a((7,rc) >S((7^rc)

(4.24)

We claimed that the 'rational' decision maker has an incentive to choose the option qc whose robustness is greatest from among all available actions which guarantee reward no less than TC'S(§c,^c) = max3((7,rc)

(4.25)

qeQ

However, the incentive to maximize robustness is not necessarily inviolable. Other considerations may 'rationally' overwhelm the desire for robustness. For one thing, robustness may conflict with the aim of exploiting emergent opportunities inherent in ambient uncertainty. In the examples in

CHAPTER 4. VALUE JUDGMENTS

125

sections 3.2.7 and 3.3 we showed that by enhancing the robustness ojae may diminish the opportuneness: the immunity functions a{q,rc) and (3{q,r^) may be antagonistic. As noted in relation (3.21) on p.46, the opportuneness function (5{q, r^) also induces a preference ranking on the set of available actions, and this ordering of options need not coincide with the robustnessordering of eq.(4.24). In addition, the 'robust rationality' of the preference relation (4.24) is subject to a limitation common to all rank-based decision schemes. It can be shown that choice by a preference relation such as >~ is equivalent to a 'consistency' condition on the decision maker's choices [150, p.11]. Roughly speaking, this consistency condition states that the decision maker's choice will not change if new but less desirable or other irrelevant options are included. In a strict sense this can never be objectionable: what is less desirable will not be selected. However, in the broader sense that new options may throw light on new aspects and implications of the decision problem, one sees that a human decision maker may 'rationally' change a choice when the situation is seen in a new perspective. In this way the addition of "irrelevant" options — ones which the decision maker would never choose — may nonetheless cause a change of choice, violating the consistency condition and the 'rational' preference ranking. Deviations from preference-rank rationality are well known in economic theory, where vast evidence has accumulated showing that a decision maker's choices can be influenced in many ways, such as by seemingly irrelevant details of how the decision problem is framed [7], [120, p.7], [150, pp.14-15]. What is particularly appealing about the ranking of preferences, like in eq.(4.24), is the orderly and deductive nature of the decision schemes which are induced. However, the 'rationality' of reaching a decision as the outcome of a linear chain of deductive reasoning has been forcefully challenged by James March [117]. Deductive reasoning, even when it is supplemented inductively by introducing sense data or other evidence, presumes a clearly posed problem and a well-defined framework of influence which do not always correspond to real human decision environments. This position is taken by John Dewey, who divides human decisionmaking into two rough and partly overlapping categories. The first is "routine, instinct, the direct pressure of appetite, or a blind 'hunch' ". In the second, action follows upon a decision, and the decision is the outcome of inquiry, comparison of alternatives, weighting of facts; deliberation or thinking has intervened. Considerations which have weight in reaching the conclusion as to what is to be done, or which are employed to justify it when it is questioned, are called "reasons". If they are stated in sufficiently general terms, they are "principles". [58, p.355] It is the latter type of decision-making which would be considered 'rational'

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INFO-GAP DECISION THEORY

and which can lead to decision rules such as eq.(4.24). But Dewey is not convinced that 'reasoned behavior', in a strict forwardlogical sense, can actually describe human behavior or should be held up as a paradigm of desirable decision-making. "The trouble" with logical deduction, according to Dewey, "is that while the syllogism sets forth the results of thinking, it has nothing to do with the operation of thinking." He goes on to explain: As a matter of fact, men do not begin thinking with premisses. They begin with some complicated and confused case, apparently admitting of alternative modes of treatment and solution. Premisses only gradually emerge from analysis of the total situation} The problem is not to draw a conclusion from given premisses; that can best be done by a piece of inanimate machinery, by fingering a key-board. The problem is to find statements, of general principle and of particular fact, which are worthy to serve as premisses. As a matter of actual fact, we generally begin with some vague anticipation of a conclusion . . . and then we look around for principles and data which will substantiate it or which will enable us to choose intelligently between rival conclusions. [58, pp.358-359] Dewey concludes that one must use "a logic relative to consequences rather than to antecedents, a logic of prediction of probabilities rather than one of deduction of certainties." [58, p.361] To a certain extent the 'robust rationality' of the decision scheme in eq.(4.24) does in fact concentrate on consequences rather than antecedents. The robustness 3(g, TC) assesses the attainability of specified goals: the degree of ambient uncertainty which can prevail without endangering the attainment of reward at least as great as TC. Furthermore, the robustness curve — a{q,rc) vs. r^ — is a tool for supporting the "operation of thinking" rather than only summarizing the "results of thinking". The decision maker iteratively chooses a goal (re) and an action {q) in light of the available immunity (a(g',rc)), while also choosing the immunity though considering the consequent reward and the burden of implementation. The robustness analysis induces a preference ranking which is not derived exclusively from the reward for performance. Rather, the robustness function enables the decision maker to look holistically at the decision problem, considering rewards and performance requirements, uncertainties and system properties, and thereby to judge the options for action. But nonetheless, value judgments accompany and ultimately control all responsible rational action. Returning to the quote with which we began this chapter: One can have the clearest and most complete knowledge of what ^My italics here.

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is, and yet not be able to deduct from t h a t what should be the goal of our h u m a n aspirations.

4.5

Problems

1. Employee productivity. An office manager commissioned a study of employee productivity p as a function of the number of vacation days per year, n. The results of the study are summarized in the following correlation, which shows that productivity increases with increase in the number of vacation days: p{n) = 0.25n°-^-h0.1 (4.26) The correlation is valid for n between 0 and 60 days. The productivity function is defined in terms of performance measures which are well known to the office manager, who can linguistically interpret the values of p as in table 4.1. p value 0-0.5 0.5 - 0.6 0.6 - 0.8 0.8 - 0.9 0.9 - 1.0

Interpretation low low-moderate moderate moderate-high high

Table 4.1: Qualitative interpretation of the productivity function, for problem 1. An employee who currently enjoys n days of vacation per year is requesting increased vacation time. What is the minimal increment in duration which is significant when evaluated from the perspective of the manager's understanding of productivity? Explain the role of analogical inference. What is analogous to what, and how strong is the analogy? 2. Employee productivity, continued. A study of employee productivity p is made as a function of two independent variables: salary s and the number of vacation days per year, n. Productivity increases with increase in both s and n: p{n,s) = (0.25n°-^ + 0.l)

(4.27)

The correlation is valid for n between 0 and 60 days and for s/smax in the interval [0.5, 1]. Given that salary si is significantly greater than salary S2, establish a significant increment in the number of vacation days. Explain the role of analogical inference. How many inferences are made? What is analogous to what, and how strong are the analogies? Consequence severity. The robustness function for a particular decision problem is: S(9,rc) = 2.5e-^^v^ (4.28) The decision variable q can be chosen from the interval [0, 5]. Undemanding, moderate and demanding values of the critical reward TC are 0.3, 1 and 2, respectively.

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The current design calls for g = 2 at moderate critical reward. How large a change in q is needed to produce a significant increase in robustness?

Chapter 5

Antagonistic and Sympathetic Immunities The battle between right and wrong is often difficult to resolve, but it is almost impossible to resolve the conffict between right and right. Martin Kaufman [97, p.86]

Decisions under uncertainty must be responsive to both adverse and favorable aspects of uncertainty. Info-gap theory uses the immunity functions to grapple with these two sides of uncertainty. The robustness function a(^, Tc) is the immunity against failure, and the opportuneness function /3(g, Tw) is the immunity to windfall reward. An aversion to failure will induce a decision maker to concentrate more on robustness than on opportuneness. Nonetheless, robustness must be balanced against agility. An excessive dependence on robustness against uncertainty can lead to lethargic decision-making and loss of opportuneness, just as cumbersome medieval body armor is inferior to warfare tactics based on the mobility of light infantry. The robustness function a{q,rc) is the immunity against failure: the greatest level of uncertainty which is consistent with always attaining no less than a specified critical level of reward. When a{q,rc) is large then great ambient uncertainty can be tolerated, which is clearly desirable. "Bigger is better" as far as a(g, TC) is concerned. The opportuneness function f3{q^r,^) is the immunity against windfall gain: the lowest level of uncertainty which can enable reward as large as r^^. If (3{q,r^) is small then wonderful unanticipated reward can be garnered even in the presence of low ambient uncertainty. Clearly a small value of /?(g, Tw) is desirable. "Big is bad" for P{q,ry^). 129

130

INFO-GAP DECISION THEORY

We indicated on p.48 that the robustness and opportuneness functions are "coherent" in the sense that, as the decision maker reduces the demand for critical survival-level reward TC or restrains the hope for large windfall gain Tw, both a{q^rc) and (3{q,ry^) indicate a more encouraging picture of the impact of uncertainty. 3(^, r^) gets larger as Vc is made less demanding, indicating greater robustness to uncertainty, while f3{q,r^) gets smaller as Tw decreases, meaning that windfall gains are more accessible as windfall aspirations are reduced. But the really important relation between robustness and opportuneness is expressed by the variation of a{q, TC) and f3{q^ r^) with the decision vector q rather than with the reward parameters TC and TW In example 3 on p.98 we saw that a change in the decision maker's action can produce a simultaneous improvement in both robustness and opportuneness, or it can improve one at the expense of the other. In this chapter we will explore this antagonism or sympathy between robustness and opportuneness.

5.1

Immunity Functions

In fig. 3.1 on p.47 we saw the monotonic decrease^ of the robustness function S(g,rc) versus the critical survival-level of reward TC- This steady decrease results from the inexorable trade-off between robustness and reward: greater reward is obtained only in exchange for reducing the immunity to uncertainty, if all else remains the same. ^{Qc{rc), Tc)

Critical Reward Figure 5.1: Optimal robustness function versus critical reward.

^(?w(^w), ^w)

Windfall Reward F i g u r e 5.2: O p t i m a l opport u n e n e s s function versus windfall reward.

The robustness curves in fig. 3.1 are shown for fixed action vector q. However, monotonic decrease of the robustness is observed also when, at each value of re, q is replaced by the robust-satisficing action §c(^c) as in ^Robustness functions vary monotonically with re, either increasing or decreasing depending on the sense of the reward function as expressed in either eq.(3.4) or eq.(3.10). Both monotonicities embody the same trade-off. We are now considering reward functions for which more reward is better than less, so the immunity functions are defined in eqs.(3.4) and (3.5) on p.40.

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fig. 5.1. The robust-satisficing action is the value of q which maximizes the robustness from among the set of available actions.^ Even though the robust-satisficing action $c(^c) varies with TC, the optimal robustness a{qc{rc),rc) nonetheless decreases monotonically with r^ Fig. 3.1 also shows a monotonically increasing opportuneness function /?(95 ^w) versus the windfall reward r^. This too is an unavoidable trade-off: great windfall gain is enabled only by allowing large ambient uncertainty. The opportune-windfalling action, q^^{r^), is the value of q which minimizes the opportuneness function on the set of available options, as in eq.(3.22) on p.46. The optimal opportuneness function, (3{q^{r^), TW), increases monotonically versus r^, as shown in fig. 5.2. As one reduces either demands or aspirations, TC or r^, the immunity functions predict a rosier picture: a{qc{rc)j Vc) increases with diminishing Tc while /^(§w(^w), ^w) decreases with diminishing TW Robustness and opportuneness functions are always coherent in this sense, as we noted ear her. However, we are presently interested in the correlation between the immunity functions as the action vector varies. A risk-averse decision maker will focus primarily on the robustness curve, fig. 5.1. The decision maker must choose where on this curve to operate: how much reward should be exchanged for immunity-to-uncertainty. The position along the robustness curve can be chosen by selecting the demanded critical reward TC which in turn determines the robust-satisficing action vector §c(^c)As the decision maker contemplates alternative values of TC and $0(^0)5 the following question also arises: how will the opportuneness function vary? As the chosen operating point moves up the robustness curve to greater immunity to failure, will opportunities for windfall grow or wither? That is, the decision maker wishes to know how the opportuneness function behaves as a function of the robust-satisficing action: Piqci^c), ^w)- This function is a 'cross-breed' between figs. 5.1 and 5.2, and it need not be monotonic with Tc. However, its slope at any point determines whether the immunity functions are antagonistic or sympathetic, as we will now explain, and as shown in fig. 5.3. If the opportuneness function, evaluated at the robust-satisficing action, increases with increasing r^ then the two immunities are sympathetic, as indicated by the increasing curve in fig. 5.3. In this case, reducing Vc causes a decrease in the opportuneness function which is desirable (since it decreases the immunity to windfall), and it also causes an increase in the robustness function, which is also desirable (increasing the immunity to failure). That is, a change in the robust-satisficing action which improves one function improves the other as well. On the other hand, if the opportuneness function, evaluated at the robust-satisficing action, decreases with increasing r^ then the two immunities are antagonistic, as indicated by the decreasing curve in fig. 5.3. ^We encountered the robust-satisficing action in eq.(3.20) on p.45 of section 3.1.6, as well as in eq.(3.172) on p. 100 of the more formal discussion in section 3.4.

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Antagonistic Critical Reward Figure 5.4: Unimodal opportuneness function versus scalar action.

Figure 5.3: Opportuneness functions versus critical reward.

Reducing r^ causes an increase in the opportuneness function and also an increase in the robustness function; the former is unwanted, the latter is desired. Either immunity function can be improved only at the expense of the other. We can summarize fig. 5.3 by saying t h a t the robustness and opportuneness functions are antagonistic at critical reward re and windfall reward r^ if:

<0

dr.

(5.1)

T h e immunity functions are sympathetic if this derivative is positive.

5.2

Reward-Coherent Action

In an a t t e m p t to understand when and why the robustness and opportuneness functions are antagonistic, we will consider a simple though fairly common special case. First, we suppose t h a t the action g is a scalar variable rather t h a n a vector. Next, we suppose t h a t the robust-satisficing action Jc(^c) increases with increasing Vc- T h a t is: dqcjrc)

>0

(5.2)

This is reasonable if the effort made in a t t e m p t i n g to gain reward is proportional to the action q: more effort is required if more reward is demanded. W h e n relation (5.2) holds we will say t h a t the robust-sat isficing action is c o h e r e n t with the critical reward at this value of TC . Reward coherence is not universal. T h e numerical value of the action q may have no direct relation to a degree of effort, q may be a geometrical location, or a time at which some event is initiated, and so on. If the robustsatisficing action §c(^c) decreases with increasing TC, t h a t is, the reverse of

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inequality (5.2), we will say that $c(^c) is anti-coherent with the critical reward at this TC^ Finally, we will suppose that the opportuneness function /^(g, r^) is unimodal, having a unique minimum as a function of g, as shown in fig. 5.4. The value of q at which I3{q,r^) achieves its minimum is the opportunewindfalling action, §w(^w)- This value of q divides the opportuneness function into two domains: [3{q^r^) has negative slope for q < 5w(^w) while l3{q^r^) has positive slope for q > §w(^w)With these three assumptions — scalar q, reward-coherent §0(^0)5 and unimodal l3{q^ rw) — we can readily see that the robustness and opportuneness functions are antagonistic at TC and TW if and only if: Qcirc) < §w(rw)

(5.3)

That is, a and P are antagonistic at rewards TC and rw if and only if greater effort is required to enable windfall reward rw than is required to guarantee critical reward re. When (5.3) holds, P{qc{rc), r^) is in its negative-slope domain, to the left of the vertical dashed line in fig. 5.4. Opportunity is improved by increasing $c(^c) which, because the action is coherent with Vc and a{qc{rc), Vc) decreases with increasing r^ causes a decrease in robustness. Hence the antagonism: either immunity function is improved only at the expense of the other. If the inequality in (5.3) is reversed then robustness and opportuneness are sympathetic: f3{qc{rc), ^w) is in the positive-slope domain, to the right of the vertical dashed line in fig. 5.4. Now opportuneness is improved by decreasing §c(^c) which also causes an increase in robustness. Any change in the action which improves one immunity function will sympathetically improve the other as well. If the robust-satisficing action is anti-coherent with the critical reward then the situation is reversed. Inequality (5.3) is necessary and sufficient for sympathetic immunity functions.

5.3

Vibrating Mechanical Contact

In this section we illustrate the antagonism of the robustness and opportuneness functions with a mechanical design example adapted from [17]. Consider an undamped single-degree-of-freedom linearly oscillating system with mass m, stiffness k and driving force / ( t ) , as in fig. 5.5. The displacement of the mass in response to input / ( t ) , from zero initial conditions, is: 1 f' Xf{t) = / /(r)sinu;(^-r)dr (5.4) muj JQ

where u'^ = k/m is the square of the natural frequency of vibration. The known nominal driving force is /(f), while the actual forcing function, / ( t ) , varies in an unknown manner within an envelope of known shape

134

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h

THEORY

m mass

spring

-fit)

x{t)

"ground"

Figure 5.5: Mass-spring system.

but unknown size. The info-gap model for this type of uncertainty is the following unbounded family of nested sets of input functions: ll{aj) = [f{t): \f{t)-m\
a>0

(5.5)

T h e envelope ^p{t) is a known dimensionless function. T h e horizon of uncertainty, a , has the same units as the forcing function f{t). Let us suppose t h a t large displacements are desirable for this system. In particular, the oscillator is used to drive a mechanism by establishing physical contact a t some instant to be chosen beforehand by the designer within the first half-cycle: from t = 0 t o t = TT/LO. Contact is made if, at the chosen instant, the position of the mass equals or exceeds h^ any margin a critical value Vc- Since the known nominal driving force f{t) is subject to unknown perturbations, the device is designed with the view t h a t the greater t h e displacement the better, b u t t h a t re is the least acceptable displacement. T h e reward function is Xf{t) where the excitation force / is unknown, the contact time t is the design variable, and the performance requirement is: Xf{t) > r c (5.6) T h e robustness function, which is the greatest value of the uncertainty parameter a consistent with not violating requirement (5.6), is readily found to be:

0

else

where xj and Xi/j a r e the response t o the nominal load and t o the load envelope function, respectively, according to eq.(5.4). T h e designer chooses the contact instant t within the first half-cycle by maximizing the robustness and satisficing the extension of the spring. This robust-satisficing choice, t, is: t(rc) = arg

m a x S(t,rc)

(5.8)

0
While Vc is the required displacement, it may be padded or trimmed to satisfy the designer's need for assurance against the unknown load perturbations. In like manner the designer may imagine a large displacement

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Tw, greater than TC, which would provide much greater protection against the uncertain load. However, r^ cannot be relied upon to occur and cannot be demanded as a design requirement, but can be viewed as windfall gain should it occur. The opportuneness function is the lowest level of uncertainty which must be tolerated in order to entail the possibility of displacement as large as r^. The opportuneness function is: ,-

xj{t) if xj<

P{t,r^)

Tw

(5.9)

else

0

The anti-symmetric relation between robustness and opportuneness in this example is illustrated by combining eqs.(5.7) and (5.9) to obtain: /3(t,r,

?(^c)

-S(t,rc) +

x^{t)

(5.10)

0.2

0.1

0.0

0.4

0.6

0.8

1.0

Critical Reward, r^ Figure 5.6: Robust-satisficing contact time t vs. critical reward re. Let us consider the specific case of an increasing linear uncertainty envelope '0(f) = ijjt and a constant nominal load f{t) — f. The robustness function becomes: S(t,rc)

(1 — cos (jjt)f — rcTnuj'^ {ujt — smu;t){ip/Lo)

(5.11)

After some manipulation one finds that the instant which maximizes the robustness must satisfy: Lut sin cut-\-2 cos ujt

rcTnuj

1 — cos (jjt

f

(5.12)

The function on the left decreases monotonically as t increases from 0 to n/uj. Since the robust-satisficing instant t is the solution of eq.(5.12), one learns from eq.(5.12) that t increases monotonically with increase in the critical displacement re, as seen in fig. 5.6form = l,u; = 10 and / = 100. In other words, the robust-satisficing action t{rc) is coherent with the critical reward re in the sense of eq.(5.2).

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T h e robust-satisficing contact time t maximizes the robustness to uncertainty, a ( t , Tc). However, it is desirable to have a small value for the opportuneness function j3{t,r^). It is therefore of interest to learn how the opportuneness function behaves at t h e robust-satisficing time t(rc), so we will examine /3{t{rc), r^). In particular, since t{rc) increases and S(t(rc), TC) decreases with increasing re, it may be hoped t h a t /?(t(rc), r ^ ) will increase with increasing TC- This would mean t h a t by choosing TC to augment a one thereby automatically diminishes /3, t h a t is, robustness and opportuneness would vary sympathetically. While this in fact occurs in many situations^ it does not happen in the present example where we will see t h a t a and /3 are antagonistic.

Force Uncertainty [kN] S(t(rc),rc) 0.4

0.6

0.8

1.0

Critical Reward, Vc Figure 5.7: Optimal robustness a(t(rc), re) and opportuneness function /3{t{rc), Tw) at robust-satisficing contact time t(rc), vs. the critical displacement

Fig. 5.7 shows the robustness and opportuneness functions as functions of the critical displacement. T h e values of the constant parameters are Tw = 2, -0 = 1, / = 100, m — 1 and uj = 10. In this figure we see t h a t the optimal robustness decreases with TC, as required by the trade-off between immunity-to-uncertainty and reward. Consequently, the immunity to failure is improved by choosing a smaller value of Vc, resulting in choosing an earlier contact time t. However, the figure shows t h a t the opportuneness function, evaluated at the robust-satisficing time t, also decreases with TC. But a small value of /3 is desired since /3 is the immunity to sweeping success. T h u s a and /? are antagonistic: either can be improved only at the expense of the other. We see from fig. 5.7 t h a t the robustness a varies less strongly t h a n the opportuneness function /3. Increasing t h e chosen value of TC will reduce the robustness but will improve the opportuneness even more. T h e designer may be willing to trade some robustness in achieving displacement TC in

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exchange for the possibiUty of the much larger displacement r^ — 2. For example, changing the critical reward from TC = 0.4 to TC = 0.6 changes the robustness from a — 1.055 to a = 0.759; a decrease of 28% in robustness. However, the opportuneness function decreases by 59% from /3 = 2.033 to /? = 0.831. While this decrement in robustness is undesirable and not negligible, the reduction in the immunity to large "windfall" deflections is far greater. Furthermore, at TC = 0.6 the level of uncertainty which enables windfall, /3 = 0.831, is close to the tolerable level of uncertainty, a = 0.759.

5.4

Multi-Tasking of Computer Jobs

Info-gap methods are particularly suited to strategic planning decisions where information about uncertainties is often extremely limited. A typical decision is the choice of a multi-job task-sequence architecture. We considered a project-management example of this sort in section 3.2.6. We will now examine the problem of structuring a multi-job computer program in a distributed-processing environment in which many programs are running in parallel. Each local job depends upon the computational resources of a central processor, direct access storage devices, and communication channels to and from these units. The problem is that since many jobs (from this program and from others) call on these resources, one cannot depend on their availability [115, pp.348-350]. On the one hand, the program will tend to be more rapidly completed if many jobs run in parallel, on the other hand, this parallelism reduces the reliability of task completion within an allotted duration because of increased waiting times due to competition for resources. We will illustrate the info-gap analysis of this situation. In particular, we will study the relation between robustness and opportuneness and the implications for the choice of the program architecture.

5.4.1

Formulation

The specific numerical example we will consider involves a program with six tasks which can be arranged in various architectures, and which draw on central resources which also service many other jobs. The jobs of the program in question can be arranged in M parallel paths in various ways as illustrated in fig. 5.8. The mth path contains km jobs, whose indices are denoted n ( l , m ) , . . . ,n{km,rn). For example, in the two-path architecture of fig. 5.8, these indices are: m = l, m = 2,

A:i=3, A:2 = 3,

n ( l , l ) = l, n(l,2) = 3,

n(2,1) = 2, n(2,2)-4,

n(3,1) = 6 n(3,2) = 5

.

. ^^^

The actual duration t^(i,rn) of job n{i, m) is unknown and in fact will vary from run to run due to variable demands on the processor from other parts of the system, and due to the varying inputs which drive the job. The

138

INFO-GAP DECISION THEORY

6 paths Jl

Ji

Je

J2

J2

h

J3

J3

JA

3 paths

JA

Jl

J2

Je

Jz

JA

J5

Jz

JA

Js

J5

2 paths

Je Jl

J2

Je

1 path

Figure 5.8: Four different path structures for six jobs.

vector of all the task durations is t. The total duration of the mth path is:

X^^n(i,m),

m = 1, . . . , M

(5.14)

The nominal duration of job n{i,m) in an M-path configuration is known and denoted tn(i^m),M- This nominal duration increases with the degree of multi-tasking due to increased waiting time. In the example to follow we will model the increase in nominal run time of an M-path structure as follows: n(i,m),A/

M" U

(5.15)

where 6 is a positive constant. The nominal duration of each task in an Mpath structure is M^ times greater than the nominal duration of the same task in a one-path structure. We will examine various values of 6, where a larger value implies greater multi-tasking delay. The vector of nominal job-durations in an M-path structure is denoted t{M). The fractional variation of the actual run time of each job relative to its nominal duration is unknown. However, different jobs have different and roughly known relative levels of variation. With this prior information we can formulate an interval uncertainty model in which the lower bound on

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each job duration, at any level of uncertainty a, cannot be negative'^:

U{a,t{M)) = U:

2 = 1, ...,A:^; m - 1 , ...,Af i ,

a>0

(5.16)

The "uncertainty weight" Wj^(^i^jy^^ expresses the relative variability of job n{i,m). A large value of Wn(i^m) amplifies the horizon of uncertainty, a, for job n{i,m) and implies a large range of uncertain variation of the job duration, while a small value of Wn(^i^rn) moderates the uncertainty. The performance requirement is that all M job paths finish within a duration tc'. max Cjn < tc (5-1'^) l<m<M

Note that a small value of tc is desirable: reward increases as tc decreases, so the robustness and opportuneness functions will be defined with eqs.(3.10) and (3.11) on p.40. The robustness of the mth path in an Af-path structure, with job duration requirement tc, is the greatest value of the uncertainty parameter a for which the path is guaranteed to complete on schedule: a ^ ( M , tc) = max
max Cm \ < tc > \teU{a,t{M)) J J

(5.18)

The overall robustness with M parallel paths is the robustness of the least immune path: S ( M , t c ) = min S^(Af.tc) (5.19) l<m
In a similar manner we can express the opportunity for sweeping success of the mth path in an Af-path architecture. The performance requirement is that the path must finish within the duration tc, but the aim of course is to finish as early as possible. "Windfall" gain in this case is path completion within a time t^ which is less than^he performance requirement tcThe opportuneness function for path m, (3^{M,t^), is the lowest level of uncertainty needed in order to enable the possibility (but not guarantee the certainty) of path completion as early as t^: /3^(Af, tw) - min <^ a :

min c^ < tw > \teU{a,t{M)) I I

(5.20)

"^This is nearly the same as the info-gap model in eq.(3.85) on p.67, except that here we have explicitly imposed the requirement that job durations cannot be negative. This is essential because we will evaluate the opportuneness function.

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INFO-GAP DECISION THEORY

I3^{M, tw) is the dual of am{M, in eq.(5.20). The opportuneness function is desirable. The opportuneness among the path opportuneness

tc): each 'max' in eq.(5.18) becomes a 'min' is the immunity to windfall, so a small value of the entire structure is the worst (greatest) functions:

d{M,t^)=

max ^^(M,tw)

(5.21)

l<m<M

We see that eq.(5.21) is the dual of eq.(5.19).

5.4.2

Deriving Robustness and Opportuneness Functions

We now need an expression for the maximum duration of the mth path in an M-path architecture, as a function of the uncertainty parameter a. From this expression we will derive the robustness-to-uncertainty of the path. The maximum run time of task n(z, m), up to uncertainty a, is: max

tn{i,m)

= [ l + ^n(2,m)<^] tnii,m),M

(5-22)

teUia,t{M))

Substituting this into eq.(5.14) yields, after some manipulation: max

Cm =

M^ V r „ ( i ^ ^ ) ^ i + a M ^ y'4(i,^),i^i;^(^,^) (5.23)

t^U{a,t{M))

^

^

'—Z^

'

'

Cm.M

-

^

'

fm,M

M'cmM+aM'^fmM

(5.24)

which defines the quantities Cm,M and fm,M^ which are the nominal and weighted-nominal durations of path m, respectively. Equating this maximum path duration to the critical time tc and solving for a yields the robustness of the mth path in an M-path architecture: a^(M,fJ^^-'f'"-'^,

m=l,...,M

(5.25)

am{M^tc) = 0 if the expression in eq.(5.25) is negative. The overall robustness is obtained by substituting am{M,tc) into eq.(5.19). We now derive the opportuneness function, which is a bit more complicated because task durations cannot be negative, as expressed by the complex lower bound in eq.(5.16). The shortest duration of job n{i,m), up to uncertainty a, is: ,

teu{a,t{M))

1

[

[ l ~ ^n(i,m)Q^] tn{i,m),M

0

li a < jjj—,

else (5.26)

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141

Let us rank the inverse uncertainty weights of path m: 1

1

(5.27)

from least to greatest, so that: 1

1

< ... <

w.n(ri,m)

(5.28)

^n{rk^,Tn)

Thus n{ri,m) is the index of the job whose inverse uncertainty weight, l/'"'n(n,m)) has the ith-lowest rank, for i = 1, . . . , km- For notational convenience let us define: ^= 0 (5.29) ^n(ro,m) mm Cr, t£U{oc,t{M))

0=r:

*-a 2-m)

^n(r3,m)

Figure 5.9: Schematic portrayal of the minimal path duration in eq.(5.31) (solid line) and of the solution for (3 in eq.(5.38) (dashed line). This path has 3 jobs. We can now express the shortest duration of path m at horizon of uncertainty a. In light of eqs.(5.14) and (5.26) it is evident that, if: 1

< a <

1 J — 1 , . . . , /C7^

(5.30)

then the lowest possible duration of the TTith path, up to uncertainty a, is:

min teU{a,t{M))

C^ = y ] [ l - ^n(n,m)Q^] tn{r,,m),M i=J

rj.m(a)

(5.31)

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INFO-GAP DECISION THEORY

which defines the terms Tj^rnic^)- The sum on the right includes all the tasks in path m whose minimal durations, at uncertainty a, are greater than zero; all other tasks in the path have zero minimal duration. From eq.(5.31) we see that minc^ versus a is a piece-wise linear decreasing function, whose slope becomes less negative at each successive value of a = l/t^n(r ,m)? ^is seen schematically in fig. 5.9. We note that: Tfe^.mf

^

) - 0

(5.32)

V^n(r^,„,m)/

Hence, if: a >

^^n{rk^

(5.33) ,m)

then the minimal duration of path m vanishes: mij

c^ = 0

(5.34)

teUia,t{M))

The opportuneness function for the mth path, /3^(M, tw), is the least value of a for which a path duration as short as t^ is possible. This is obtained as the solution for a of: mm

Cjn = iw

(5.35)

teU{a,t{M))

Let us define: To.m = r i , „ ( 0 )

(5.36)

As indicated by the dashed lines in fig. 5.9, we see that, if:

r„m ( — ^ — ) < tw < r , - i . ^ ( — ^ — ) , \^n{rj,m)

J

\'^n{r,-x,ni)

j - 1,..., /c^ (5.37)

J

then the opportuneness function for this path, /3^{M,t^), for a of: rjMa)=t^

is the solution (5.38)

This becomes: fJ^[M,t^)

=

E

km

T

_

^^ '^

•

J.

(5.39)

If, instead of relation (5.37), the following holds: <w>ri.„(0)

(5.40)

then: 0„,{M.t^)

=O

(5.41)

The overall opportuneness function is obtained by employing eqs.(5.37)" (5.41) ineq.(5.21).

C H A P T E R 5.

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143

A

6

1

0.8

/3(M,^w)

S(M,tc)

K

0.4

0.0

30

10 _J

30

300

-^ tc

Figure 5.10: Overall robustness and opportuneness functions with M = 1, 2, 3 and 6 paths and with strong penalty for multi-tasking: b = 2.

5.4.3

Results

Fig. 5.10 shows the robustness and opportuneness functions for each of the four program architectures appearing in fig. 5.8. The four decreasing curves are (3{M^t^) vs. t^ as defined in eq.(5.21) and eqs.(5.37)-(5.41). The uppermost decreasing curve is the opportuneness function for the 6path architecture (M = 6), and below it appear in sequence the results for the 3-path, 2-path and 1-path options, as indicated by the small numerals to the left of the curves. The value of the "windfall" program completion time tw varies from 10 to 30 time units, and the opportuneness values range from 0.0 to 0.97 as seen on the left-hand scale. The robustness functions also appear in fig. 5.10, based on eqs.(5.19) and (5.25). The M-value is to the right of each curve. The range of critical run times t^ is from 30 to 300 time units, and the robustness varies from 0 to 4.7 as shown on the right-hand scale. n

1

2

3

4

5

6

tn,l

3 2

5 1

5 2

6 3

8 1

9 1

Wn

Table 5.1: Nominal durations and uncertainty weights.

The nominal one-path job durations and the uncertainty weights for all the calculations in this section appear in table 5.1. The nominal durations range from 3 to 9 units, and the sum of the nominal run times is 36, which is the nominal duration of the 1-path architecture. The actual run times of the jobs may differ greatly from the nominal times, due to two types of uncertainties. Each of these computer jobs is driven by inputs which may vary from run to run. Some jobs are complex and quite variable, as expressed by a large uncertainty weight Wn- For example, w^ = 3 means that the uncertain range of run times for job J4

144

INFO-GAP DECISION THEORY

tends to be 3 times that of job J5 for which wr, = I. The second uncertainty in run times derives from the unknown and variable availabihty of the central resources — processing, data storage and retrival, and communication — upon which each job depends for implementation. The results in fig. 5.10 are obtained with a strong penalty for multi-tasking: 6 = 2 in eq.(5.15) so the nominal job durations for an M-path implementation are M^ times the 1-path nominal durations. Thus the nominal durations of the 6-path architecture are 36 times greater than the 1-path nominal times. The nature of these particular jobs enables complete freedom in sequencing,^ so they have been grouped in the four different structures of fig. 5.8. The 2-path nominal completion times are (3 + 5 + 9) x 2^ = 68 and (5 + 6 + 8) X 2^ = 76 units. The 3-path nominal completion times are 108, 117 and 99 time units. For the 6-path option the shortest and longest nominal run times are 3 x 6 ^ = 108 and 9 x 6 ^ = 324. These are indeed strong penalties for multi-tasking. It is therefore not surprising to find in fig. 5.10 that robustness to uncertainty decreases dramatically with increasing multi-tasking. In fact, in the M = 6 option the robustness is zero throughout, because the nominal duration of one of the paths (path 6, with ^6,6 = 324) exceeds the greatest tc value considered. At the other extreme, the 1-path architecture has no robustness to uncertainty at t^ = 30 because this is less than its nominal run time of 36 units. At tc = 300 however, the robustness of the 1-path architecture reaches 4.7. This is quite substantial robustness. Each job can exceed its nominal duration by 470w% (where w is the job's uncertainty weight) without entailing the possibility of time overrun for the program. The robustness curves in fig. 5.10, a{M, tc) vs. t^ imply the usual tradeoff between immunity and reward: large reward (which is expressed by guaranteed completion in a short duration tc) is obtained only in exchange for low immunity to uncertainty. We are used to seeing robustness curves which decrease versus the critical reward, while the curves in fig. 5.10 increase versus tc. The reason is that the critical reward increases as tc decreases: a short completion time tc is desirable. The opportuneness functions /^(M, t^) in fig. 5.10 show a similarly strong sensitivity to the multi-tasking penalty. A small value is desirable for the opportuneness function, since /^(M, tw) is the immunity to achieving program completion in as short a time as t^. We see that the 1-path option is twice as good as the 2-path alternative, and that the 3-path and 6-path options are progressively less opportune. We note also that the opportuneness and robustness functions are sympathetic: a change in the number of paths which improves the robustness also improves the opportuneness: the 1-path option is best in both robustness and opportuneness, the 2-path option is next, etc. It is instructive to consider the point of intersection of each robust^We have dealt with sequencing constraints and more comphcated flow charts in section 3.2.6.

C H A P T E R 5.

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ness curve with the corresponding opportuneness function in fig. 5.10.^ In the 1-path case this occurs at tw = 12 and t^ — 62 time units, at which ;^(1,12) = 3(1,62) = 0.46. The meaning of this intersection is that a level of uncertainty of a = 0.46 both enables completion as quickly as ^w = 12 and guarantees completion within the duration tc = 62 time units. /^(M,tw)

a(M,tc)

Figure 5.11: Overall robustness and opportuneness functions with M = 1, 2, 3 and 6 paths and moderate penalty for multi-tasking: b = 0.7737.

P{M,t^)

Figure 5.12: Overall robustness and opportuneness functions with M = 1, 2, 3 and 6 paths and low penalty for multitasking: b = 0.1464.

Fig. 5.11 again shows the robustness and opportuneness functions for each of the four architectures, now with moderate penalty for multi-tasking: b — 0.7737. With this value of b the nominal job-durations in the 6-path configuration are 6^'^^'^'' = 4 times longer than the nominal 1-path run times. The results are very nearly the same for all four configurations. With this level of penalty for multi-tasking, the time-saving from multitasking is nearly cancelled by the delays resulting from increased burden on the central resources. This is unlike the situation seen in fig. 5.10, where the stronger penalty for multi-tasking resulted in clearly preferable performance for the 1-path configuration. The curves for the 1-path configurations are in fact the same in both fig. 5.10 and fig. 5.11, since the nominal durations are identical in both cases. What has happened in fig. 5.11 is that 3(2, tc), 3(3, tc) and 3(6, tc) have improved, reaching and in some cases slightly exceeding the 1-path robustness curve. Likewise, the opportuneness curves /^(2,tw), P{3,t^) and /3(6,tw) have also improved and "collapsed" onto /3(l,tw)One result of this coalescence of the immunity functions is that now robustness and opportuneness are not strictly sympathetic, as they were in the case of strong multi-tasking penalty. While the opportuneness curves ^Because the left and right scales in fig. 5.10 are different, the actual point of intersection is not the point at which the plotted curves cross.

146

INFO-GAP DECISION

THEORY

are clearly separated in fig. 5.10, they now cross so t h a t at some values of windfall completion times t^ the opportuneness is reduced by increasing the robustness. However, since all four configurations are nearly the same, neither antagonism nor sympathy between robustness and opportuneness is particularly important. Fig. 5.12 shows the immunity functions once again, now for low multitasking penalty: b — 0.1464 so t h a t the nominal durations of the 6-path configuration are only 6^"^'^^'* = 1.3 times greater t h a n in the 1-path case. We now see clear preference for multi-tasking. The 1-path curves, S ( l , t c ) and /?(l,tw), are of course the same as before, but now the robustnesses are all greater with greater parallelism, and the opportuneness functions are all likewise lower. W i t h this low penalty for multi-tasking, the time gains outweigh the waiting times. We also again witness the sympathetic variation of the robustness and opportuneness functions.

5.5

Problems

1. Antagonistic or sympathetic immunities? The robust-satisficing action for a particular decision problem, 5c (^c), depends on the critical reward re as: 5;(rc) = 1.2rc' (5.42) The opportuneness function for this decision task is: ^(g,rw) = 3 r w ' / ' ( g - 2 ) '

(5.43)

Are the immunity functions sympathetic or antagonistic? What does this entail for the choice of the critical reward and the robust-satisficing action? 2. N e w approach. The concept of antagonistic and sympathetic immunities expressed in fig. 5.3 on p. 132 and subsequently in eq.(5.1) is based on the variation of the optimal robustness and opportuneness functions, S(5i;(rc),rc) and /3(5^(rc),rw), as the critical reward TC is changed. In this problem we explore a different approach. (a) We begin with a special case. The immunity functions of a particular decision problem are: S(g,rc) = — ^ i — re + (?2

(5.44)

^(9.rw) = -

(5.45)

^

where q\ and q2 are two decision variables to be assigned positive values by the decision maker. The critical and windfall rewards, r^ and Tw, are positive. By exploring partial derivatives, explain the sense in which the robustness and opportuneness are sympathetic with respect to variation of q\ and antagonistic for variation of q2.

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(b) From now on we consider general robustness and opportuneness functions S(g, Tc) and f3{q, TW) where q — {qi, . . . , 9iv)^ is a decision vector with N elements. For any decision variable qn, explain that a small increment in qn is desirable if: ^ ^ ^ > 0 oqn and undesirable if the sign is reversed.

(5.46)

(c) For any decision variable qn, explain that a small increment in qn is desirable if: ^ ^^i^l^<0 oqn and undesirable if the sign is reversed.

(5.47)

(d) For any decision variable qn, explain that robustness and opportuneness are sympathetic with respect to qn if: dS(q,r,)dP{q,r^) dqn dqn

^^

(5.48)

and antagonistic if the sign is reversed. (e) Return to part 2a and explain your results in terms of relation (5.48). (f) Discuss the interpretation of:

3. i N e w approach, continued. Consider a decision problem in which the reward function is bi-linear in the decision vector q and an uncertain vector u: R{q,u)=u^q (5.50) Large values of R{q, u) are better than small values. The uncertainty in u is represented by a Fourier ellipsoid bound info-gap model: U{a,u) = [u = u-^v\

v^Wv


,

a>0

(5.51)

where l y is a known, real, symmetric, positive definite matrix. (a) Derive the robustness and opportuneness functions. (b) Study the antagonism and sympathy of the immunity functions based on relation (5.48). (c) Repeat parts 3a and 3b with S = 0 and the quadratic reward function: R{q.u) = {u^qf

(5.52)

Chapter 6

Gambling and Risk Sensitivity If risk were exclusively of the nature of a known chance or mathematical probability, there could be no reward of risk-taking. Frank H. Knight [103, p.46] But history shows that logic launched from introspection alone lacks thrust, can travel only so far, and usually heads in the wrong direction. Edward O. Wilson [177, p.l05]

The immunity functions a{q,rc) and (i{q,r^) are the fundamental tools of info-gap decision theory, but they are not a complete characterization of the decision maker. In particular, the immunity functions are not by themselves sufficient to uniquely determine how a decision maker will choose from among available options, nor do the immunity functions establish unequivocal proclivity for or aversion to risk. 3(g', TC) and /3(g, rw) are decisionsupport instruments with which the decision maker evaluates the options, prior to choosing a course of action. A decision maker's immunity functions reveal something about risk sensitivity, but more importantly they enable the decision maker to explore the dimensions of risk and gambling, and thereby assist in making a choice. The lexical definition of risk is a "possibility of loss or injury", a "peril", a "dangerous element or factor" [172]. The etymological origin of the word 'risk' is unclear, though it may come from earlier words meaning cliff, rock or submarine hill [173]. It is evident that the intuitive pre-scientific concept of risk can be quantified in many ways. Methods for exploring risk sensitivity must be formulated cautiously, and they must be fiexible enough to accommodate the fluidity of human behavior under severe uncertainty. 149

150

INFO-GAP DECISION THEORY

Furthermore, one must sometimes be willing to employ distinct and even conflicting concepts in order to capture different aspects of the complex phenomena of risk. In this chapter we will explore several ideas of gambling and risk-taking from an info-gap perspective. We will explain how risks are explored, and how a decision maker uses the identification of risk sensitivity to support the decision-making process. In motivating the info-gap approach to gambling we must recall that info-gap uncertainty is highly unstructured. In contrast, the preponderance of probabilistic work on gambling and risk-taking is done with models of uncertain phenomena whose structure is that of a lottery: a set of events (usually finite in number) whose character is understood and whose likelihoods are known.^ 'Risk' has a rather broad connotation, but 'gambling' connotes a game of chance: coin-flipping, card playing, and so on. While the outcome of the toss of a fair coin is quite uncertain, the uncertainty is highly structured and well understood. There is a clear rule for calculating probabilities. We can accurately evaluate the likelihood even of rare events such as long runs of 'heads'. A probability model contains a wealth of information about events which, like rarities, may never have been observed but nonetheless come within the quantitative purview of the model. Probabilistic uncertainties are highly structured in the sense that the probability function imposes a systematic ordering of events — in terms of likelihood — on all the elements of the event space. Info-gap uncertainties are different. An info-gap uncertainty is a serious lack of knowledge, which is typical in strategic planning and design. The uncertainties have none of the well-ordered and carefully groomed appearance of a lottery. The event space is infinite and complex, sometimes made up of multi-dimensional functions. What is more, the rules by which the events occur are incompletely known. No measure functions are known for calculating frequencies of recurrence, or for otherwise systematizing the behavior of the events. When facing a highly unstructured uncertainty such as an info-gap, it is useful to adopt new attitudes towards gambling and risk-taking.

6.1

Preview

In a tentative and preliminary sense, info-gap risk appears either as low immunity to failure or as limited opportunity for windfall. Risks are expressed by a small value of the robustness function S or by a large value of the opportuneness function /3. In this chapter we will develop various extensions and combinations of these ideas. Risk sensitivity is manifested in a decision maker's choice from among accessible options. Or, turning it around, a decision maker is assisted in choosing among options by interpreting them in terms of perceived riski^We will consider the info-gap analysis of lotteries in chap. 11.

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151

ness. Sensitivity to risk is evaluated by comparing what is chosen against what could have been chosen and by evaluating these choices in terms of robustness and opportuneness. But choices are not simple, and risk sensitivity can be assessed in different ways reflecting different decision contexts. We explore perceptions of risk by querying the decision maker (hypothetically, if not literally) on selections between specified options. In this chapter we will consider three different types of choices which lead to three different reflections of risk sensitivity: 1. Given a robustness curve, where along the curve does the decision maker choose to operate? This analysis will focus on the robustness premium. (Sections 6.2 and 6.9.) 2. Given two robustness curves, which one is preferred, and how much reward would be willingly relinquished to move from one curve to the other? This discussion will concentrate on the reward premium in addition to the robustness premium. (Sections 6.3, 6.4 and 6.9.) 3. Given the alternative between the robustness strategy and the opportuneness strategy, which is selected? We will introduce the opportuneness premium in addition to the robustness premium. Furthermore, interpretation of a decision maker's choices in terms of risk neutrality or sensitivity will employ the antagonism or sympathy between the robustness and opportuneness functions developed in chapter 5. (Sections 6.5, 6.6 and 6.7.) The proflle of risk sensitivity which emerges from the decision maker's answers to these questions need not be entirely consistent. The response to one question may appear rather risk loving, even rash, while other choices may reveal an aversion to risk. The reason is not that the decision maker lacks logical coherency (though this may be the case), but rather that the different questions interrogate different facets of risk. There is no simple and pat answer to the query: "Are you averse to risk?", because risk is not monochromatic or uni-dimensional. We will have to keep track of different types of sensitivities. More importantly, our goal in the analysis of info-gap risk is exploratory rather than descriptive. The questions we have posed, and the methods developed in response, are intended to assist a decision maker in formulating preferences and making a decision. Section 6.8 will summarize and complete the discussion, followed by a recapitulatory example in section 6.9. Sections 6.10 and 6.11 will discuss more advanced and technical matters.

6.2

Risk Sensitivity and the Robustness Curve

The basic decision function in info-gap decision theory, and the most important characterization of the decision maker's relation to the uncertain envi-

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ronment, is the robustness function, a(g,rc). On p.47, in connection with fig. 3.1, we began our discussion of gambhng by describing the robustnessvs.-reward trade-off. We indicated that, for fixed action q, the robustness S(g, Tc) decreases monotonically with increasing critical reward TC. On p. 130 and with fig. 5.1 we explained that the same trade-off arises also when, at each value of TC, q is replaced by the robust-satisficing action §c(^c) which maximizes the robustness.

V risk \ ^ v e i se

^>.

,

risk \ ^ ^ neutral

Act

risk loving

I ^c,l

I

1 ^0,2

.

^c,3

Figure 6.1: A robustness curve: robustness versus demanded reward. Illustrating robustness premia, and risk aversion, neutrality and proclivity. To summarize, the robustness curve manifests the trade-off between robustness and reward. The decision maker's immunity to uncertainty decreases as greater reward is demanded, as illustrated schematically once again in fig. 6.1. Robustness curves characterize the decision maker's propensity to gamble: courageous or cautious depending on whether great or small demands are made on the environment, in face of specified vulnerability to uncertainty. The robustness-vs.-reward curve enables the decision maker to explore one aspect of risk, as we will now explain. How do we decide where a decision maker falls in the spectrum of risk sensitivity from aversion to proclivity? Conversely, how does a decision maker use perceived risk to select an operating point on a robustness curve? In fig. 6.1 we have indicated a range, from rc,i to rc,3, of available critical reward values from which the decision maker must choose. Any choice of an Tc-value entails implementing the corresponding robust-satisficing action vector §c(^c)- Of course, it is rare indeed that a decision absolutely must be made within a rigid context. If all of the specified options are repugnant the decision maker will look for some external alternative, for instance deferring the decision. The assessment of risk is always relative to a particular context, and it is usually important to evaluate risks with respect to different contexts. Suppose, however, that the decision maker has chosen to operate at critical reward rc,i when choosing within the range from rc,i to rc,3. $c(^c,i)

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will be implemented and robustness S(5c(^c,i),'^c,i) will be obtained. We would consider the decision maker to be risk averse in this context since the least demanding and most robust option has been chosen. The decision maker has relinquished all disposable reward in exchange for maximum immunity from the pernicious potential of the ambient uncertainty. The decision maker has chosen the action with the greatest available robustness premium, AS, defined as: AS = a(§c(rc,i),^c,i) - S(5c(rc,3),^c,3)

(6.1)

At the other extreme, imagine that the decision maker has chosen to strive for a reward no less than rc,3. This choice displays a proclivity for risk. This operating point is the most demanding which can be selected, and the decision maker has opted for this position while sacrificing the maximum amount of immunity to uncertainty. The robustness premium is zero for this choice from among the available options. Finally, a decision maker who chooses a moderate reward, rc,2, is expressing an intermediate or neutral attitude toward risk. This decision maker is positioned in the middle ground between great demands with low robustness, and low demands with high immunity. The robustness premium, AS' in fig. 6.1, is moderate from among the available options. When a decision maker asks these questions introspectively the result is an evaluation of options — rc,i, rc,2 and rc,3 — in qualitative linguistic terms of riskiness. This association between options and attributes is the basis for formulating preferences among the options. In this section we have formulated risk as a condition of high demanded reward and low immunity to uncertainty. A behavioral proclivity for risk is the willingness to exchange immunity-to-uncertainty for an increment in reward, as expressed by relinquishing a robustness premium on a robustness curve. These definitions are meaningful, but they are formulated with a specific type of information in mind: a single robustness curve. This is legitimate provided we allow ourselves to formulate other concepts of risk as well. In this way we accumulate a collection of answers to a collection of questions, which together provide perspective and depth of understanding.

6.3

Risk Sensitivity and Two Robustness Curves

A decision maker's attitude to risk is revealed in choosing between specified options. Conversely, a choice between options can be made after characterizing the options in terms of the perceived riskiness of each. In section 6.2 we interpreted a choice as risk averse, neutral or loving depending on the magnitude of the robustness premium of the decision maker's selection, relative to the premia of the other alternatives on a single robustness curve. We now consider a situation where a choice must be made between two alternative

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3(5c,z,r,

32($c,2,rc) 3l(§c,l5^c) vulnerable ri modest

r2

demanding

Figure 6.2: Maximal robustness versus demanded reward for two alternative options. Illustrating robustness and reward premia. strategies, each strategy being represented by a different robustness curve as in fig. 6.2. T h e Tc axis in fig. 6.2 is labeled 'modest' near the origin and 'demanding' away from the origin, and the a axis is labeled 'vulnerable' and 'immune' for small and large values of robustness, respectively. Option 2, the upper curve, is relatively immune to uncertainty (compared against option 1) even when making great demands on the environment, while option 1 is relatively vulnerable to uncertainty even with modest demands for reward. Consider the arrows rising from the point labeled r on the Vc axis. For this value of demanded reward, option 2 is relatively immune to uncertainty while option 1 is relatively vulnerable. There is a positive r o b u s t n e s s p r e m i u m for option 2 in comparison with option 1: A S = a2{qr2{r),r)

-

ai{q^^^{r),r)

(6.2)

A decision maker who is averse to risk will tend to prefer the upper robustness curve over the lower one since any value of critical reward can be sought with greater immunity to uncertainty. As in section 6.2, risk aversion is manifested in a preference for positive robustness premium. Now, however, t h e robustness premium is an increment between two robustness curves, rather t h a n an increment between two points on the same curve as in eq.(6.1). We note t h a t each robustness curve, Si{q^ | , TC) and S2($c 2^ ^c), has its own robust-satisficing action, q^ i{r) and §c 2(^)5 which may be different and may entail different costs. The strongly risk-averse decision maker will be willing to invest substantial resources in order to be able to implement option 2 rather t h a n option 1. A test for risk aversion or proclivity, and a basis for choosing between the two strategies represented in fig. 6.2, is based on the 'reward premium' which we now define.

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Consider the arrows stemming from the robustness a on the a axis of fig. 6.2. At this level of uncertainty, option 1 allows the decision maker to demand only a low guaranteed return, ri, while option 2 enables greater demanded return r2. There is a reward premium for option 2: (6.3)

Ar(a) = r2 — ri

A risk-averse decision maker would be willing to forfeit some of the Ar{a) units of reward in order to exchange option 1 for option 2. For instance, if implementation of q^ 2(^) is more expensive than q^ ^(r), the risk averse decision maker would accept the trade at some positive cost. A risk-loving decision maker would not agree to the forfeit, and a risk-neutral agent would be indifferent to the offer. What the risk-averse decision maker is buying with this exchange is the reward premium of option 2 over option 1. Of course, the decision maker doesn't really know how much reward to forfeit since the level of ambient uncertainty a is unknown, nor is the magnitude of the reward premium known since that also depends on a. What the decision maker is buying into is the schedule of greater rewards with option 2.

vulnerable cheap

costly

Figure 6.3: Maximal robustness versus robust-satisficing action for two alternative options. Illustrating robustness and commitment premia. It is instructive to consider the variation of robustness a{qc,rc) versus the robust-satisficing action $c(^c) (which for now we consider to be a scalar rather than a vector, for simplicity of exposition). For each choice of the critical reward TC there is a robustness-maximizing action 5c (^c), which we will assume to be unique. So, the rc-axis of fig. 6.2 can be transformed to a §c-axis, as in fig. 6.3. Sometimes the action can be interpreted as a commitment of money, time, space or some other valuable resource, for instance when the action is reward-coherent (p. 132). The action axis varies from 'cheap' at small values of §c to 'costly' at large values. Following the arrows from fixed robustness a, one finds q2 > qi which means that the

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decision maker using option 2 is willing to make a greater commitment than if option 1 is employed. There is a positive c o m m i t m e n t premiium for the more robust option. Conversely, following the arrows from fixed action q, option 2 enjoys greater immunity to uncertainty than option 1: there is a positive robustness premium.^ To summarize the discussion of figs. 6.2 and 6.3, we can say that greater robustness to uncertainty (of one curve over another) implies access to greater reward or a willingness for greater commitment of resources at the same level of ambient uncertainty. Equivalently, greater robustness enables the decision maker to demand greater returns from the environment. This is in contrast to the single-curve risk assessment in fig. 6.1, in which greater robustness entails lower reward. Risk is multi-faceted, and its evaluation involves different and sometimes confiicting considerations. However, in both the one-curve and the two-curve risk assessments, a preference for positive robustness premium implies an aversion to an uncertain environment, though the practical meaning of the preference is different. We now consider an example before continuing our study of the interpretation and assessment of risk.

6.4

Initial Commitment and Uncertain Future

In many development projects an investment at one time yields profit later on, subject to uncertain intervening occurrences. In this example we plan a production facility whose size is q and whose construction time L{q) depends on q. In addition, the construction time L{q) can also be either short or long at fixed plant size, depending on the choice between an accelerated or a moderate construction strategy. A different robustness curve is derived for each of the two construction strategies. The decision maker must choose not only the plant size q^ but also one or the other of the two construction durations, short or long. The decision maker's sensitivity to risk is central to the selection process, which is facilitated by considering the reward and robustness premia for these two options, which we introduced in section 6.3. Fuller and Gerchak studied a probabilistic version of this problem for a shale-oil plant [75]. The nominal profit function is the revenue corrected for time minus the construction cost: R{q) = qpe-'^^o^ - c{q) (6.4) p is the revenue ("downstream" in time) per unit plant size, 5 is the discount rate (an interest paid on money borrowed to finance the investment) and c{q) is the cost of a plant of size q. •^This discussion of fig. 6.3 is not completely precise since it is not true that greater robustness necessarily implies greater commitment. We study this further in section 6.11.

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At the initiation of the project, when the size q must be chosen, the unit revenue, p, of the products to be manufactured, as weh as the cost of money during the hfe of the project (as determined by 6) are quite uncertain. The estimated discounted revenue per unit of plant is: u{q) = pe-^^^^)

(6.5)

which is known. However, the actual discounted revenue per unit of plant, u{q), is quite uncertain. The uncertain profit function, as distinguised from the estimated profit in eq.(6.4), is: R{q,u) = qu{q)-c{q)

(6.6)

where u{q) belongs to an info-gap model of uncertainty with centerpoint u{q) from eq.(6.5). We will consider two different levels of prior information. One info-gap model is very meager, stating only that the actual unit revenue u{q) deviates from the estimated revenue u{q) by no more than a, but a is unknown. This uniform bound info-gap model is: U{a,u) = U{q):

\u{q) - pe'^^'^'^A < a\ ,

a>0

(6.7)

The info-gap model of eq.(6.7) implies that we know nothing about the rate of variation of the discounted unit revenue u{q) with plant-size q, and that u{q) may even be a function of unbounded variation. This may be unrealistic, and one may in fact have prior information which constrains the rate of variation of u{q). One form which this prior knowledge can take is spectral information expressed as uncertain bounds on Fourier coefficients. The uncertain function is represented: u{q) = u{q) + V^ [xn sinnTrg + yn cosnTrg]

(6.8)

n=ni

where the Fourier coefficients Xn and yn are uncertain. Let N — n2—ni-\-lhe the number of modes in the expansion oiu[q)^ and let a{q) and 7(g) be column A^-vectors of the sines and cosines, respectively, appearing in the sum in eq.(6.8). Let x and y be the column A^-vectors of Fourier coefficients. For convenience we define the 2A/'-vectors:

Now we can write eq.(6.8) more succinctly: u{q) = u{q)+z^T^{q)

(6.10)

A Fourier ellipsoid-bound info-gap model for uncertainty in the discounted revenue is: U{a,u) = {u{q) = u{q)^z^r]{q)'.

z^Wz
a>0

(6.11)

where VF is a known, real, symmetric, positive definite matrix and, as in eq.(6.7),S(9)=pe-^^(^).

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Uniformly Bounded Uncertainty

The lowest acceptable profit is r^ whose value is chosen by the project owners or managers. Vc may be negative if, for example, government bailout is an option. A larger positive value of re is chosen if hungry stockholders are demanding profitable return on their investments. In evaluating risks we consider TC as a reward parameter and explore the implications of different choices of r^.

^ Tc/a Figure 6.4: Maximal robustness curves for two values of the construction duration. For any value of TC, the robustness is the greatest horizon of uncertainty consistent with profit no less than Vc- Using the info-gap model of eq.(6.7) the robustness is found to be: S((7,rc

pe

-6L{q)

+ c(g)

(6.12)

The robustness decreases linearly with r^ for fixed plant size q. However, the robust-satisficing plant size gc(^c) varies with TC, SO the dependence of S(§c(^c)7'^c) upon Tc will be nonhnear though still decreasing. For concreteness consider a numerical example. Let c{q) — a^/q and L{q) = biq -\- bo, so the construction time, at fixed plant size q, increases with bi. The parameters of the estimated discount rate are S = 0.08 and Fig. 6.4 shows the optimal robustness a{qc{rc),rc) versus the critical utility Tc, for two different choices of the construction-time coefficient 5i. For long construction time the acceptable uncertainty (i.e. the robustness, a) is quite a bit lower than in the case of short construction time. The decision maker is substantially more vulnerable to uncertainty with long rather than short lead time. There is a large positive robustness premium AS for the short construction time, which would attract a risk-averse decision maker.

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Comparison of t h e robustness curves in fig. 6.4 also shows a substantial reward premium Arc for t h e short over t h e long lead time option. Any decision maker (either risk-loving or risk-averse) may be drawn t o the short-time option because of t h e greater guaranteed reward TC at any horizon of uncertainty a. However, a risk averse decision maker would opt for t h e short-time option even if forced to surrender some or all of t h e resulting reward premium ArcWe also note t h a t in fig. 6.4 t h e robustness and reward premia vary with Tc- At low values of t h e critical reward t h e advantage of t h e short construction time is less t h a n at very large and ambitious values of r^. These considerations can assist t h e decision maker to select a value of critical reward in those situations where r^ is not fixed by external constraints.

Hqi^rc)

a

6i=0.08

short lead time

61=0.25

long lead time -^qc{rc) R{q2,u)

Figure 6.5: Maximal robustness versus the robustsatisficing plant size, for two values of the construction duration.

R{qi,u)

Figure 6.6: Reversal of preference between two plant sizes.

For each value of t h e critical profit, re, there is a unique value §c(^c) of the plant size which maximizes t h e robustness. Fig. 6.5 shows t h e maximal robustness versus t h e robust-satisficing plant size for both t h e short and t h e long construction times. T h e decision maker is considerably more immune to uncertainty with t h e short construction time t h a n with the long construction time, manifested in t h e fact t h a t a substantially greater plant size can be chosen at t h e same level of ambient uncertainty. T h a t is, t h e commitment premium Aq for short construction times is substantial. Equivalently, with short construction time t h e decision maker adopts a greater plant size t h a n at long construction time without relinquishing immunity t o uncer-

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INFO-GAP DECISION THEORY

tainty. As before, the robustness premium for short construction time is large. We now briefly consider an example of preference reversal as discussed in section 3.1.8. The estimated profit with a plant of size q is R{q,u) in eq.(6.4). The robustness of this choice of q is given by eq.(6.12) which can be re-written as: _ _ R{q,u)-rc . . (^[Q^ re) =

(6.13)

We will consider the choice between two plant sizes, qi and q2j where: -\

A

R[qi,u) > R{q2,u)

uf

~\ ^

D/

and

R{QU^)

R{q2,u)

< Qi

(6.14) Q2

The first relation implies that, based on the best-estimate of the revenue per unit of plant, p, and the discount rate, S, plant-size qi should be preferred over ^2- However, the second relation in (6.14) implies that the robustness curves for these two choices cross in the positive quadrant, as shown in fig. 6.6. If the decision maker requires profit in excess of rx or does not require robustness greater than a x , then qi is preferred over q2 because the robustness- and reward-premia are positive for qi. On the other hand, option q2 is preferred by both premia if profit less than rx is acceptable or if robustness in excess of 3x is required. The preference between these two options, based on the robustness and reward premia, changes depending on the decision maker's requirements.

6.4.2

Ellipsoidal Fourier Uncertainty

As before, the robustness function is the greatest horizon of uncertainty such that the profit is no less than TC. We now use the Fourier-ellipsoid infogap model of eq.(6.11) rather than the uniform-bound uncertainty model in eq.(6.7). After some manipulation one finds the robustness function to be: Sfiq, re) =

, ^u(g'^c)

__

^g^5-^

where au(9, Vc) is the robustness for the uniform-bound uncertain, eq.(6.12). The diff'erent information encoded in the Fourier info-gap model entails possibly different robust-satisficing action and diff'erent robustness- and reward-premia.

6.5

Risk Sensitivity, Robustness and Opportuneness

A decision maker's sensitivity to risk is assessed by observing the behavior which is chosen from among specified alternatives. Conversely, the decision

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process is catalyzed by exploring the risks perceived by the decision maker. In section 6.2 we found that one facet of risk sensitivity is revealed by the position on a specified segment of a single robustness curve at which the decision maker chooses to operate, illustrated in fig. 6.1. In section 6.3 we explored risk sensitivity by comparing the robustness curves of two alternative options in figs. 6.2 and 6.3. This two-option analysis was illustrated in section 6.4. We now consider a diff"erent aspect of risk sensitivity. We get a new glimpse at a decision maker's attitude to risk, and the implications for formulating a decision, by exploring the choice between strategies based upon robustness or opportuneness. The decision maker could choose a robust-satisficing action qdvc), which maximizes the robustness function 3(g,rc) while guaranteeing reward no less than r^ Or the opportune-windfalling action $w(^w) could be selected, which optimizes the possibility of reward as great as TW by minimizing the opportuneness function /?(g, Tw). The robustness strategy is an attempt to guarantee "survival", while the opportuneness strategy aims at achieving the decision maker's "wildest dreams". The first approach, robustness, is characteristic of a riskaverse attitude, while the latter is attractive to a risk-loving decision maker. Nonetheless, things are not so simple, and we will have to take account of the sympathy or antagonism between robustness and opportuneness. Consider a pair of reward values, (rw, TC), where r^ is a large windfall reward and TC is a lower critical survival level of reward. A decision maker has, of course, many lines of action open. However, concentrating on optimal actions at either of these two values of reward, the decision maker could choose either to implement the robust-satisficing action §c(^c) or the opportune-windfalling action gw(^w)Let us first consider the implications for risk sensitivity if the decision maker chooses the robust-satisficing option, qdrc), rather than choosing the opportune-windfalling option, Jw(^w), recalling that our conclusions are tentative until we consider the antagonism or sympathy between the immunity functions. Choice of the robustness rather than the opportuneness strategy is an expression of aversion to risk. But we can say more than that. Compare the robustnesses of the two decision alternatives from which the decision maker must choose: a{qc{rc),rc) and S(§w(^w)5 ^c)- By definition, the former is the greatest robustness obtainable while guaranteeing reward no less than r^ so it can be no less than the latter: S(§c(^c),rc) > a{q^{r^),rc)

(6.16)

This inequality asserts that there is a non-negative robustness premium for choosing $c(^c) rather than §w(^w)A 3 = S(§c(rc), re) - a{q^{r^), r^)

(6.17)

This premium attracts the risk-averse decision maker to the robustness strategy, and the magnitude of the premium tells something of the decision

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maker's risk sensitivity if the robustness strategy is selected. This robustness premium is different from those calculated earlier, eqs.(6.1) and (6.2). We are now comparing the robustnesses of action vectors which are optimal under either a robust-satisficing or an opportune-windfalling strategy. We can state the significance of the robustness premium in eq.(6.17) more precisely. If the decision maker chooses $c(^c) rather than §w(^w)7 and if the resulting robustness premium is large: S($c(rc), re) > 5(§w(rw), ^c)

(6.18)

then the decision maker is evidencing great risk aversion since great robustness to uncertainty together with limited reward is selected, rather than opportunity for great reward with hmited robustness. Still considering the robustness option, if the decision maker chooses qc{rc) rather than $w(rw), and if the resulting robustness premium is small: S(gc(rc),re) > S($w(rw),re)

(6.19)

then the decision maker shows only slight aversion to risk: while the robustness strategy $c(rc) was chosen, small changes in parameters or other conditions could presumably cause a switch to Jw(rw) since the difference in robustness is small. Now we will interpret the decision maker's risk sensitivity if the opportune-windfalling option, §w(rw), is chosen rather than the robust-satisficing option, $c(rc), keeping in mind that we must still consider whether the immunities are antagonistic or sympathetic. The decision maker's selection of the opportuneness strategy over the robustness strategy indicates a proclivity for taking risks. Again, we can go further. Consider the opportuneness function for each of these decision alternatives: /3($w(rw),rw) and /3(§c(rc),rw). By definition, the former is the least immunity to achieving reward as great as TW, SO it can be no larger than the latter: fe(rw),rw) < %c(rc),rw) (6.20) The opportuneness premium for choosing $w(rw) rather than gc(rc) is: A ^ = 0{Urc). rw) -

fe(rw),

r^)

(6.21)

A large value of this premium is what draws the risk-loving decision maker to the opportuneness option: the large windfall reward r^ can be achieved at lower ambient uncertainty if $w(rw) is implemented than if qdrc) is implemented. If the decision maker chooses 5w(rw) rather than $c(rc), and if the resulting immunity to windfall is much lower than the alternative (i.e., large opportuneness premium): fe(rw),rw)

< ^($c(rc),rw)

(6.22)

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then the decision maker is evidencing great risk proclivity since a substantial opportuneness for large reward has been chosen, rather than the more modest but more highly protected robustness strategy. The decision maker has chosen to optimize on the propitious potential of uncertainty rather than to optimize the resistance to pernicious contingencies. The risk that is preferred is the unknown array of possibilities inherent in ambient uncertainty. The decision maker is a gambler at heart. Recall, however, that we have yet to consider the import of antagonistic immunities. Let us continue with the opportuneness option. If the decision maker chooses §w(^w) rather than $c(^c) and if the resulting opportuneness is only slightly better than the alternative: %w(rw),rw) < ^(§c(rc),rw)

(6.23)

then the decision maker shows only slight proclivity for risk: while the opportuneness strategy §w(^w) has been selected, small changes could presumably cause a switch to qc{rc) since the opportuneness premium is small. What we have so far overlooked is the possibility of sympathy or antagonism between robustness and opportuneness, which we discussed in chapter 5. A conscientious and subtle decision maker will not make such an oversight. If robustness and opportuneness are sympathetic, meaning that both can be enhanced together, then a decision maker who chooses to concentrate on the robustness strategy may not be risk averse at all, since a highly opportune strategy can be implemented at the same time that one protects against the pernicious potential of uncertainty. Likewise, a choice of the opportune-windfalling strategy does not necessarily imply a prochvity for risk if the immunities are sympathetic. On the other hand, if robustness and opportuneness are antagonistic, meaning that one can be improved only at the expense of the other, then opting for robustness over opportuneness is strong evidence for risk aversion since the decision maker is explicitly relinquishing opportuneness in favor of robustness. This is especially true when the robustness premium is large as in eq.(6.18). In the same way, a preference for the opportuneness strategy under antagonistic immunities, especially with a large opportuneness premium as in eq.(6.22), is clearly characteristic of risk proclivity because the decision maker is exchanging immunity-to-failure for opportuneness of large gain.

6.6

Risk-Neutral Line

In the previous section we were able to interpret the choice of the robustness or the opportuneness strategy as risk averse or risk loving, respectively, subject to reservations if the immunity functions are sympathetic. Now we ask: what is risk neutrality in the robustness-vs.-opportuneness context? When will the decision maker be indifferent between the two strategies?

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rc=U

Risk-neutral line

Figure 6.7: rc-vs.-rw plane showing the risk-neutral line. While we were not able to predict risk averse or risk loving behavior, in the case of risk neutrality we can make a prediction, which we will refer to as the risk-neutral line. The risk sensitivity of a decision maker, as expressed by a preference for either the robustness or the opportuneness strategy, can be mapped on a Tc-versus-rw plane as in fig. 6.7. The relevant domain is below the straight line labeled TC = r^. In this domain the windfall reward r^ exceeds the critical reward TC. The decision maker's strategy choice — robustness with $c(^c) or opportuneness with $w(^w) — could be identified at each point (^w^'^c), thereby mapping out regions of risk aversion and risk affinity. The curved line in fig. 6.7 shows the locus of points at which the robustsatisficing and opportune-windfalling actions are the same: Qc{rc) = $w(rw)

(6.24)

Along this curve the robustness and opportuneness premia, eqs.(6.17) and (6.21), are both zero. The decision maker will be behaviorally indifferent in choosing between the two decision strategies: the action is the same, and the immunities are the same, whichever strategy is chosen. On this curve the decision maker is operationally risk neutral in the robustness-vs.opportuneness sense. The risk-neutral curve is identified from the immunity functions without querying the decision maker. We will consider an example in section 6.7. Since risk neutrality is the axis around which risk aversion and proclivity revolve, we will be able to tentatively deduce something about risk sensitivities (but let's recall E.O. Wilson's warning quoted at the opening of this chapter). In fig. 6.8 we consider an arbitrary point P on the risk-neutral line in the Tc-versus-rw plane. The vertical line through P is the locus of points of fixed windfall reward, so this is the locus of points at which the optimal opportuneness function /9(§w(^w),^w) is constant. As we move down this line we are traversing a range of decreasing re-values, just as we move from ^c,3 to rc,i in fig. 6.1. There is a positive robustness premium, in the sense

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^^w

Figure 6.8: Variation of the immunity functions around the risk-neutral hne. of eq.(6.1), for points below P compared to points above P , while the opportuneness function /3($w(^w)5^w) is constant along this vertical line. We would expect a risk-averse decision maker, in deliberating between a point on the risk-neutral line and a point below it, to prefer the latter. In other words, a risk-averse decision maker will tend to prefer Pi over P : P i ^av P

(6.25)

Likewise, P would be preferred over a point directly above the risk-neutral line by a risk-averse agent.

Windfall Reward Figure 6.9: Optimal opportuneness function versus windfall reward.

Figure 6.10: Possible nontransitivity of preferences.

The optimal robustness function S(§c(^c):^c) is constant along the horizontal line through P in fig. 6.8 since Vc is fixed. Moving to the right on this horizontal, the optimal opportuneness function /3(§w(^w),^w) increases (increasing immunity to windfall gain), since TW is also increasing. That is, by moving to the right on the horizontal in fig. 6.8 we are moving up the opportuneness function of fig. 6.9 towards values of greater windfall gain obtainable only at greater ambient uncertainty. The risk-loving decision maker is drawn to the right of the risk-neutral line by the possibility of large gain, even though great uncertainty must be accepted.*^ Conversely, a ^This is the opportuneness-analog of the single robustness curve analysis in section 6.2.

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risk-averse decision maker will tend to prefer P over P2 in fig. 6.8: P ^av P2

(6.26)

We can summarize our discussion of fig. 6.8 as follows. The risk-neutral line is the locus of points at which robust-satisficing and opportune-windfalling action vectors are identical and along which the robustness and opportuneness premia are zero. Points on the rc-versus-rw plane which lie below P on the risk-neutral line will tend to be chosen over P by risk-averse decision makers, while points to the right of P will be selected, rather than P , by risk-loving agents. We see in this analysis that risk sensitivity is not absolute, but relative to the specific options considered by the decision maker. In fig. 6.10, P and Q are two points on the risk-neutral line. The point R is to the right of P , so eq.(6.26) suggests that P will tend to be preferred over P by a risk-averse agent: P ^av R (6.27) However, R is below some other point Q on the risk-neutral line in fig. 6.10. Thus eq.(6.25) suggests that R is preferred over Q by a risk-averse decision maker: R ^av Q (6.28) But this analysis does not tell us how a risk-averse decision maker would choose between points P and Q. We should not necessarily expect that the preferences in relations (6.27) and (6.28) imply a risk-averse preference for P over Q. Predicting a transitivity of this sort requires a stronger structuring than can be supported by decision-making behavior under infogap uncertainty. A choice between P and Q can be made by comparing them directly. The decision maker is indiff'erent between the robustness and opportuneness strategies at each of these points. However, one could compare these two points on a single robustness curve containing them, as described in section 6.2. P corresponds to a lower value of critical reward TC than Q, so the optimal robustness at P exceeds the optimal robustness at Q. We cannot predict, however, whether or not the positive robustness premium of P over Q will induce the decision maker to select the former over the latter option. One could also compare P against Q by examining their positions on a single opportuneness curve, as suggested in footnote 3 here. This might lead to a different assessment of the relative merits of these two options.

6.7

Pure Competition with Uncertain Cost

We will illustrate the previous discussion with a simple production volume example with uncertain costs, like the example in section 3.3. Consider a static production situation in which the manufacturer must choose the quantity q which will be produced, and let us assume this to

C H A P T E R 6.

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167

equal the quantity sold. p{q) is the price per item, which depends on the quantity produced, and u{q) is the total cost of producing q items. The reward function is the profit: R{q) — qp{q) — u{q). Suppose that the demand function p{q) is known exactly, but that the cost function u{q) varies in an unknown manner within an envelope of unknown size. The estimated production-cost function u{q) is known, and the shape of the envelope of uncertain variation of the actual cost function changes in a known way with q. The envelope-bound info-gap model is suitable to describe this incomplete prior information: U{a, u) = {u{q) : \u{q) - u{q)\ < ai;{q)} ,

a>0

(6.29)

u{q) and ip{q) are known. '0(g) determines the shape of the envelope within which u{q) varies in an unknown way, while the unknown horizon of uncertainty, a, determines the size of the envelope. The robustness and opportuneness functions for production volume q are: ^/

X

QP{q) -reV^(9)

u{q)

, . ^^x

a(g, Tc) is the greatest uncertainty which guarantees reward no less than Tc, while P{q^r^) is the least uncertainty which enables reward as great as Tw These immunity functions display the usual trade-offs. In eq.(6.30) the robustness to uncertainty decreases as the demanded reward TC increases. In eq.(6.31) the immunity to windfall becomes greater as the desired windfall reward TW rises. As a specific case let the envelope function in the info-gap model of eq.(6.29) be linear: i/j{q) — q. The estimated total cost function will represent dis-economies of scale at large production volume in the following way: u{q) = uiq - U2q^ (6.32) where 0 < ^ < 1 and ui and U2 are positive and such that the estimated cost u[q) is positive. The estimated cost per unit, u{q)/q, increases with increasing volume q. Consider pure competition, so the demand function which the firm sees is independent of the firm's production volume: p{q) — P07 where po is constant and no less than Ui. The immunity functions now are: S(g,rc)

=

PQ - ui - —-V U2q^~^ q

(6.33)

P{q^ ^w)

=

U1-P0 + — - U2q^~^

(6.34)

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INFO-GAP DECISION THEORY

The robust-satisficing production volume is obtained by maximizing a(g, Vc on g, resulting in the following robust-satisficing production volume: lA

§c(rc) =

(6.35)

(1-0^2

The opportune-windfalling production volume, q^w{r^)^ minimizes P{q,r^). However, since immunity functions cannot be negative, §w(^w) is the solution of f3{q, Tw) = 0, or: (6.36)

(Po -Ui)q-\-U2q^

The curve of risk neutrality in the (rw, ^c) plane of fig. 6.7 is defined as the locus of points at which the robust-satisficing and opportune-windfalling solutions coincide: Qc{rc) = ?w(r'w) (6.37) This is obtained by substituting §c(^c) from eq.(6.35) for q in eq.(6.36), leading to the following expression for the risk-neutral line: 1/^

^w = {PO -Ui)

(1-0^2

+1-^

(6.38)

•C=0.1^^C=0.5

Po -ui

Po -ui Figure 6.11: Risk-neutral lines for various dis-economies of scale. Fig. 6.11 shows risk-neutral curves for three levels of dis-economy of scale, as determined by <^ in eq.(6.32), with U2/{po — Ui) — 1- When ^ is small the penalty for large production volume is greater than when ^ is large. In other words, a large value of ^ implies greater economy of scale than a small value. The risk-neutral line shifts to the right — to greater values of windfall gain Tw — as the dis-economy of scale becomes less severe. With mild penalty for large volume, ^ = 0.5, the line of neutrality runs roughly along Tw — 6rc. At the other extreme, ^ = 0.01, the risk-neutral line is very

C H A P T E R 6.

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169

nearly TC = TW for TC < 1, and thereafter becomes essentially flat. From our discussion of eq. (6.26) and fig. 6.8 we understand that points to the right of a risk-neutral line are preferred by a risk-loving agent over risk-neutral points. The curves in fig. 6.11 are thus interpreted to mean that, as economies of scale improve (as ^ increases), the decision maker comes to view options as risk neutral which previously were seen as risky. As economies of scale improve, the decision maker favors ambitious projects (large V^/TC) which formerly were rejected.

6.8

Interim Summary

We now summarize and round out our discussion of the three info-gap methods for assessing a decision maker's sensitivity to risk and for using this sensitivity to assist in choosing between options. These methods correspond to the three questions listed in section 6.1: we can query the decision maker about the choice of an operating point on a single robustness curve; we can ask the decision maker to select between two robustness curves; and we can offer the decision maker a choice between a robustness or an opportuneness strategy. 1. Single robustness curve. When the decision maker is confronted with a single robustness curve, as in fig. 6.1, risk aversion is expressed by preference for an option with large robustness premium, eq.(6.1): A S = 3 ( § c ( ^ c , m ) , ^c,m) - S ( $ c ( ^ c , n ) , ^c,n)

(6.39)

where rc,m < '^c,n- The risk averse decision maker opts for a point at the upper end of the robustness curve.^ In this context, a risk-loving decision maker exchanges robustness premium for large critical reward, and prefers to operate at the lower end of the robustness curve. We can complete the discussion of section 6.2 by considering the opportuneness premium. The decision maker is asked to compare the option with critical reward rc,m against the alternative option with greater critical reward Tc^n- Consider the opportuneness premium in this comparison of robust-satisficing actions §c(^c,n) and Jc(^c,m)A^=^(§c(rc,n),rw) -^($c(rc,m),rw)

(6.40)

A large value of A/3 implies that §c(^c.m) is more opportune than §c(^c,n)If the immunities are antagonistic, then immunity to failure is increased (increased robustness) only at the expense of increasing the immunity to windfall (decreased opportuneness). Antagonistic immunities imply that a large robustness premium is accompanied by decreased opportuneness for "^Whether the upper end of the robustness curve is at the right or left end of the range of rc-values depends on whether the robustness curve is monotonically decreasing or increasing. See the discussion on p.47.

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INFO-GAP DECISION THEORY

windfall gain. That is, when AS in eq.(6.39) is large then AP in eq.(6.40) is small or even negative. The decision maker (with antagonistic immunities) who opts for large robustness premium is not only forgoing large critical reward Tc, but also diminishing the opportuneness for windfall, which further emphasizes the aversion to risk. On the other hand, if the immunities are sympathetic — robustness and opportuneness improve together — then a large robustness premium entails improved opportuneness for windfall gain. That is, when A 3 is large then A;^ is also large. Choosing a large robustness premium simultaneously selects for greater opportuneness, masking the decision maker's risk sensitivity. 2. Two robustness curves. When the decision maker is asked to choose between two robustness curves representing alternative strategy options, as in fig. 6.2, risk aversion is again expressed by a preference for the option with large robustness premium, eq.(6.2): A a = 32($c,2(^),^) - S i ( ? c , i ( ^ ) . 0

(6.41)

which is the difference in robustness between the two curves at the same value of critical reward, unlike eq.(6.39). The decision maker's risk aversion can be further measured by the willingness to relinquish part or all of the reward premium, eq.(6.3), in exchange for using the more robust strategy. We can again consider the opportuneness premium which accompanies the choice between strategies represented by two robustness curves: A ^ = ^ i ( $ c , i W ' ' - w ) -^2(9c,2W,^w)

(6.42)

If the immunities are antagonistic then large robustness premium A 3 entails small or negative opportuneness premium A/^, so the risk-averse decision maker diminishes opportuneness for windfall while enhancing robustness against failure. Sympathetic immunities allow the decision maker to "save the cake and eat it too", so selection of large robustness premium leaves the decision maker's risk sensitivity ambiguous. 3. Robustness and opportuneness strategies. Finally, the decision maker can be offered the choice between robustness and opportuneness strategies. The risk-averse agent will prefer the former, especially if the robustness premium, eq.(6.17), is large: A 3 = 3(§c(rc), Vc) - 3(§w(^w), re)

(6.43)

As before, an aversion to risk is accentuated by antagonistic immunities and ameliorated by sympathetic immunities. The risk-loving decision maker will opt for the opportuneness-maximizing strategy, especially if the opportuneness premium, eq.(6.21), is large: A^=^(§c(rc),rw)-fe(rw),rw)

(6.44)

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171

The influence of antagonistic or sympathetic immunities is the same as before. Risk neutrahty is characterized by an indifference between the robustness and opportuneness strategies, which is inevitable when the robustsatisficing and opportune-windfafling action vectors are the same, as on the risk-neutral hue in fig. 6.7.

6.9

Risk Assessment in Project Management

We now illustrate the assessment of risk based on the one- and two-curve methods summarized in section 6.8. We continue with the 16-task project plan introduced in section 3.2.6 and shown in the flow chart of flg. 3.8 on p.65. In table 3.2 on p.69 we saw that the path robustnesses Srn increase as the allowed project duration tc increases. This is the usual trade-off of immunity-to-failure against reward: large robustness is obtained only by relinquishing short project duration.

Figure 6.12: Robustness function for a 16-task project.

Figure 6.13: Robustness functions for two different implementations of a 16-task project.

In fig. 6.12 we plot the continuous variation of the overall project robustness versus the allowed project duration. These results employ the nominal durations tn and uncertainty weights Wn of table 3.1, so the curve includes the results of table 3.2. The robustness vanishes for a project duration of 17 time units, indicating extreme vulnerability and hence high risk at this value of tc- The robustness increases linearly from tc = 17 to tc = 19, and throughout this interval the critical task-path is path 3. Beyond this value the uncertainty-critical path becomes path 1 and the robustness increases at a somewhat lower slope. This transfer of criticality from the nominalcritical path (#3) to a different uncertainty-critical path (#1) was observed also in table 3.2. The robustness reaches a = 0.30 at a project duration of tc = 24.7 time units. At this value of tc the project can tolerate 30% overruns in the

172

INFO-GAP DECISION THEORY

durations of each task without endangering the timely completion of the project as a whole. (Some tasks, of course, can be prolonged even more without project failure.) Values around tc — 25 are substantially less risky (S !^ 0.3) than the values around tc = 17 (3 ?^ 0.0). If these risks are too large, as assessed by the robustness curve in fig. 6.12, the planner can look for ways to improve the immunity to failure without lengthening the total project duration. We consider one example, based on accelerating the implementation of particularly lengthy tasks. At short project duration the uncertainty-critical path is path 3, containing tasks 1, 5, 6, 7, 8 and 16. From table 3.1 on p.69 we see that tasks 6 and 8 have rather long nominal durations: 6 and 5 time units respectively. Consultation with the individuals who are responsible for these tasks reveals that the nominal durations could be reduced by employing alternative technologies, though these methods are new and entail greater uncertainties and additional costs. It is decided to evaluate the robustness curve if te is reduced from 6 to 4 time units and tg is reduced from 5 to 3. To roughly capture the augmented uncertainty in the new technologies required to achieve these task-time reductions, the uncertainty weight w^ for task 6 is increased from 2 to 3, and w^ is increased from 1 to 2. All other nominal durations and uncertainty weights remain the same as in table 3.1. In fig. 6.13 we show the new robustness curve based on accelerating tasks 6 and 8, together with the robustness curve from fig. 6.12 based on the original task durations. As expected, the critical path at short tc-values, path 3, shows enhanced robustness. Robustness vanishes at tc = 15 rather than at 17 time units as in the original plan; at tc = 17 the robustness now is a — 0.10, and at a project duration of tc = 24.7 time units the robustness is a = 0.37 rather than 0.30 as in the original plan. If short project durations, around tc = 17, are necessary, then it is clear that the accelerated plan is substantially less risky than the original plan. Comparing the original and accelerated robustness curves in fig. 6.13 we see that the eff'ort required to shorten tasks 6 and 8 results in a reward premium of about two time units. At any level of ambient uncertainty the planner can demand a project duration which is 2 time units shorter with the accelerated plan, without relinquishing robustness. The decision maker can evaluate the cost of the acceleration against the gain in reducing the project duration which is expressed as this 2-unit reward premium.

6.10

X More on the Robustness Premium

We now continue our discussion of the robustness premium introduced in section 6.3 to compare two alternative options represented by distinct robustness curves. In particular, we will explore the relation between the robustness premium and the reward and uncertainty inherent in the decision. We will show that the robustness premium increases with the amount

C H A P T E R 6.

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of information available to the decision maker, and that it also increases with the reward which the decision maker stands to gain. The third conclusion will be that the loss of robustness resulting from increasing uncertainty can be offset by increasing the reward. To make these assertions precise we need the following theorem, whose proof is found elsewhere [21]. Theorem 1 Given two alternative options with the same set Q of available actions, with uniformly continuous reward functions Ri{q,u) and R2{q^u), with info-gap models of uncertainty Ui{a^u) andU2{o^^u), and with finite robustness functions Si{q,r) and a2{q,r). If: {l + 6)Ri{q,u)
(6.45)

for some 6 > 0 and for all q e Q and all u in the range of the info-gap models, and if: U2[{l^e)a,^ C Ui[a,u) (6.46) for some £ > 0 and for all a > 0, then: ( l + £ ) a i ( g , r ) < a 2 b , ( l + ^)r]<S2(g,r)

(6.47)

with strict inequality on the right if S > 0, unless both robustness functions are identically zero. This theorem asserts that if option 2 is more rewarding (eq.(6.45)) and is based on a more informative info-gap model (eq.(6.46)), then option 2 is more robust (eq.(6.47)). Eq.(6.47) implies a non-negative robustness premium between option 2 and option 1: ai{q,r) 0 or e > 0, then the robustness premium is strictly positive: ai{q,r)
174

INFO-GAP DECISION THEORY

Corollary 1/1

The robustness premium increases with information.

More specifically, if two options have the same set of available actions, the same uniformly continuous reward function, Ri{q,u) = R2{q,u) in eq.(6.45), and nested info-gap models as in eq.(6.46), then the robustness premium of option 2 with respect to option 1 increases as U2{OL^ U) becomes more informative with respect to Ui{a,u). From eq.(6.47) we see that the robustness premium is bounded as: A a = 32((7,r) - 3 i ( g , r ) > - — - 3 2 ( g , r )

(6.50)

This lower bound increases with e. As the info-gap model of option 2 becomes more informative, the advantage of option 2 over option 1 is manifested in a greater robustness premium. Example 1 We can illustrate this with the Fourier-bound info-gap model of the example in section 6.4. Let UN{OL,U) denote the info-gap model of eq.(6.11) on p. 157 with A^ pairs of trigonometric functions, and with the 2N X 2^" diagonal shape-matrix: W — dmg{wi, . . . ,WN, wi, ... ,WN)

(6.51)

As TV increases we add new terms but retain the earlier ones. As N increases the info-gap model becomes more inclusive, less restrictive and less informative: UN{a,u)

C W7v+i(a,S)

(6.52)

From eq.(6.15) the robustness function for info-gap model UN is: a^(g, re) =

.

1

^

(^^-6L{q)

rc + c{q)

pe-'^^^^ - ^ - ^ ^

Y Z ^ n = l Wn ^

V So

(6.53) ^

which defines the quantity SQ which is the robustness with the uniformbound info-gap model, eq.(6.12). The robustness premium OUAN over UN+I is: AS

=

Siv(g,rc)-Siv+i(g,rc)

(6.54) So

(6.55)

/ The robustness premium for information, AS, is positive but diminishing with increasing dimension (and hence decreasing informativeness) of the info-gap model.

C H A P T E R 6.

A^ AS So

GAMBLING AND RISK SENSITIVITY

1

2

3

5

10

0.29

0.13

0.077

0.039

0.015

175

Table 6.1: Fractional change in robustness premium versus informativeness of the info-gap model.

For example, if the shape parameters Wn are all equal, then the robustness premium of UN over UN-^-I varies with A^ as in table 6.1, based on eq.(6.55). The first column shows that the robustness premium of the extremely restrictive (hence informative) info-gap model with only a single mode {N = 1), compared against the next most informative model {N = 2), is 29% of the uniform-bound robustness, SQ. However, the robustness premium diminishes rapidly with decreasing information, and is only 3.9% at TV = 5 and 1.5% at TV = 10. • The second special case of theorem 1 can be interpreted to mean that, other things being equal, an option's robustness premium is increased by the reward to be obtained. The second special case can be expressed as: Corollary ^ 1

The robustness premium increases with reward.

That is, if two options have the same set of available actions, the same info-gap model of uncertainty (lAi{a,u) — U2[ot,u)) and ranked uniformly continuous reward functions as in eq.(6.45), then option 2 has a positive robustness premium with respect to option 1, which results from eq.(6.47) with e = 0 and 5 > 0, so that: Si(g,r) < S2(g',r)

(6.56)

Finally, theorem 1 can be used to show that reward can be used to compensate for uncertainty. Corollary 3/1

Uncertainty is offset by reward.

More precisely, given the conditions of theorem 1, there is a new reward function R\{q,u) where R\{q,u) > Ri{q,u) and such that, if Ri{q,u) is replaced by R\{q,u), then: S2((7,r) < a\{q,r)

(6.57)

where S^(g, r) is the robustness function of option 1 based on reward function R\{q.,u). In other words, the greater uncertainty of option 1 (relation (6.46)) can be offset by increasing the reward from Ri{q, u) to R\{q-, u), thereby causing the robustness of option 1 to exceed the robustness of option 2 (that is, reversing relation (6.48)).

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INFO-GAP DECISION THEORY

6.11

X Robustness Premium and Resource Commitment

In fig. 6.3 on p. 155 we examined a situation in which a positive robustness premium entails a positive commitment premium. That is, the figure depicts a situation in which the more robust strategy enables the decision maker to commit greater resources in the search for greater reward. However, it is simply not true that greater robustness always entails greater commitment, even if the meaning of the decision variable q implies that large values of q entail large commitment of a valuable resource. The following theorem and its corollaries establish a relationship between the robustness premium and the size of a decision maker's commitment of resources [21]. In particular, by comparing the robustness functions of two strategies we will be able to determine which strategy leads to greater commitment of resources. Theorem 2 Consider a decision maker faced with two strategy options with the same set Q of available actions, and whose robustness functions Si(^,r) and a2{q^r) both reach unconstrained maxima on Q. The actions are represented by an N-vector variable q whose elements are denoted qji, n — 1, . . . , A^. Suppose that, for some value r of the critical utility, both robustness functions are positive, smooth and continuous for all q ^ Q, that a2{q^ r) is uni-modal in qn at this r for some particular n, and that Si(g, r) and a2{q,r) at this r are related as: 3i(g,r) = (j){q)a2{q,r)

(6.58)

for all q e Q, where (j){q) is a positive real function. Then, for this value of r, the nth component q^ ^(r) of the robust-satisficing action vector for option 1 is less than the nth component §2n(^) ^/ ^^^ robust-satis ficing action vector for option 2 if and only if (t){q) is decreasing in qn at qi (r). Likewise, ^i^^(r) > 52,n(^) ^/ ^^^ ^^^2/ '^f ^{Q) ^<^ increasing in qn at qi{r). That is: ?i,n(0

= >

Q2,n{^) if and only if

—-^ oqn

~

0

(6.59)

The basic import of theorem 2 is that greater robustness premium, by itself, does not necessarily imply greater robust-satisficing action. If (j){q) in eq.(6.58) is less than unity, then option 2 has a positive robustness premium with respect to option 1. However, the theorem establishes that this does not guarantee that option I's actions will take smaller magnitude than those of option 2. This by itself is not surprising since the action vector q may contain decisions, such as the choice of a geometrical dimension or the time of a meeting, for which 'big' does not imply any subjectively greater commitment. Hence, the interesting part of theorem 2 is that it estabhshes

C H A P T E R 6.

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a simple criterion by which to determine when positive robustness premium does imply greater values of the optimal actions. We elaborate this in the following corollaries. Corollary 1/2 With the conditions of theorem 2 and also that ai{q,r) is uni-modal in Qji at this v, the robust-satisflcing actions of the two options ave ranked according to the logarithmic derivatives of their optimal robustness functions: ?i,n(0

^

?2,n(^)

if and only if

0

=

d\na2{q,r) dqn

(6.60) q=qi{r)

Equivalently: Qi,n{^)

=

52,n(0

if and only if

0

=

d\nai{q,r) dqn

(6.61) q=q2{r)

If an option's robustness function achieves unconstrained maxima on the set of available actions, then the logarithmic derivatives of the robustness, dina{q,r)/dqn^ equal zero at the robust-satisficing action. Consequently, we can roughly paraphrase corollary 1/2 as follows: greater logarithmic derivative of the robustness implies greater robust-satisficing action. ,

L

/

a{q^ r) a

1

>yfci=0.08

^\

/ ^ 1

\6i=0.25

Figure 6.14: Robustness versus the action, re/a = 25. Example 2 To illustrate the use of corollary 1/2 let us continue the example in section 6.4.1 (p. 158) with the same numerical formulation. Fig. 6.14 illustrates that the robustness functions are indeed uni-modal, as required by theorem 2. (Note however that the lower function is not concave. Concavity, which is a stronger condition than uni-modality, is not required by theorem 2.) Fig. 6.15 shows the logarithmic derivative of the robustness function versus the lead-time parameter bi. (Recall that the construction lead time increases linearly with 6i.) The three curves are for different values of the critical utility re/a. For each curve, the derivative is evaluated at the robust-satisficing action 5c for lead-time parameter 6i = 0.08. So of

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INFO-GAP DECISION THEORY

d\na{q,r)/a q{bi=0.08)

-0.02 h

/a=35

Figure 6.15: Logarithmic derivative of the robustness at the robust-satisficing action for bi = 0.08, versus 6i, for various values of Vc/a.

course t h e derivatives vanish at bi = 0.08. (The robust-satisficing actions at bi = 0 . 0 8 for these three values of TC are QC = 34.11, 44.27 and 53.21 for Tc/a = 15, 25 and 35, respectively.) We see from fig. 6.15 t h a t t h e derivatives are negative for longer construction lead time {bi > 0.08), and positive for shorter lead time. T h e interpretation of this sign trend is provided by corollary 1/2: t h e robustsatisficing action (optimal plant size) decreases with lead time. For example, eq.(6.61) states t h a t QI < ^2 if and only if 0 > 9 1 n 3 i / 9 g at Q' = 52- ^^ ^P" tion 2 entails a lead time coefficient of 6i = 0.08 while option 1 entails a different lead time, then negative derivatives of option I's robustness, evaluated at option 2's optimal action, imply lower optimal plant size for option 1. Likewise, positive derivatives imply greater optimal plant size for option 1. It is also interesting t o note t h a t , at large lead times, t h e magnitude of the derivative increases with the critical utility. T h a t is, demanding greater return ultimately forces much smaller investment when the lead time is large, because of t h e decision maker's aversion t o risk. • We have already explained t h a t theorem 2 shows t h a t a positive robustness premium by itself does not imply increased robust-satisficing action. However, t h e following corollary establishes a condition which, in addition to t h e robustness premium, does imply ranking of t h e robust-satisficing actions. C o r o l l a r y 2 / 2 With the conditions of theorem 2, positive robustness premium implies greater robust-satisficing action if and only if the robustness premium is increasing in q at the less-robust optimum. That is, given that

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SENSITIVITY

179

(j){q) < 1, then: 9i,n(0

6.12

=

^2,n(0

if and only if

dqn

q=qi{r)

^ 0 < (6.62)

Problems

1. A n investment decision. Consider an investment problem in which you must choose the dollar amount q to be invested in a commodity. The monetary return is uncertain but proportional to q: (6.63)

R{q,u) — uq

where u is the rate of return: dollars per dollar invested. The uncertainty in u is represented by an interval-uncertainty info-gap model: u :

——— < a > ,

1 ^ 1

(6.64)

a > 0

J

where the anticipated rate of return is w = 1.35. The uncertainty parameter a is the unknown maximum fractional variation of the rate of return. (a) Construct the robustness and opportuneness functions. (b) Use your personal monetary preferences and a plot of the robustness function versus the critical reward, TC, to choose a realistic and desirable value of Tc, in dollar units. (c) Plot the robustness and opportuneness functions versus the size of the investment, q. Are they antagonistic or sympathetic? (Hint: problem 2d on p. 147.) Discuss the implications for assessing the riskiness of the investment. (d) Another investment program offers a lower anticipated rate of return u = 1.2, for the same uncertainty model, eq.(6.64). In compensation, the reward function has an automatic bonus of Uo = $100: (6.65)

R{q,u) — uq-\- Uo

Construct the robustness function and compare it to that of the original investment to evaluate your preferences between the two programs, and to choose a critical reward. (e) Now consider a combination of the two programs. The decision vector \sq — (g'l, ^2)^ where qn is invested in program n, whose rate of return is ttn, for n = 1, 2. The total return is: R{q, u

(6.66)

u q-\- Uo

where the uncertainty in u is: U{a,Z) = [u: y

Un

h " ^ """I < a. 7 1 ^ 1 , 2 ! . \ Un \ J

a > 0

(6.67)

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INFO-GAP DECISION

THEORY

The anticipated rates of return are u = (1.35,1.20)^. The total investment is constrained to Q dollars, all of which may be exploited: gi + 92 < Q

(6.68)

where Q = $1000 and qi and q2 are both non-negative. i. Evaluate the robustness function, S(q',rc), and determine a reasonable level of critical reward, reii. i Determine the robust-satisficing investment vector, Jc(^c). Construct the optimal robustness function, a(qc{rc),rc), and determine a reasonable level of critical reward, Vc. 2. Antenna design. Consider the design of a radio antenna bearing static loads. The end displacement x[m] depends on the uncertain load vector u[N] and the flexibility vector /[m/N] as: X = u^f

(6.69)

where / is to be chosen by the designer and the uncertainty in u is represented by an ellipsoid-bound info-gap model: U{a,u)-:-

{u = u + v: v^Wv


,

a>0

(6.70)

where H^ is a known, real, symmetric, positive definite matrix. (a) Construct the robustness and opportuneness functions, given that small end-displacements are desirable. (b) Propose a calibration of the robustness function. (Suggestion: normalization.) (c) Using engineering judgment of the severity of displacement of the end of the antenna, the calibration of the robustness from part 2b, and a plot of the robustness versus the critical displacement Xc, choose a realistic and desirable value of Xc • (d) Suppose that a single material parameter q controls the entire flexibility vector / which is to be chosen: / = qfo

(6.71)

where fo is known. Plot the robustness and opportuneness functions versus the material parameter q. Are the immunity functions antagonistic or sympathetic? (Hint: consider problem 2d on p. 147.) Discuss the implications for assessing the riskiness of the designer's choice of q(e) A different design concept for the antenna is not sensitive to the enddisplacement, but fails if the following quadratic term exceeds a critical value: N

E = J2y'lfn = u'^F\

(6.72)

n==l

where we have defined F as the diagonal matrix whose diagonal terms are the elements of the flexibility vector / . The uncertain load is described by eq.(6.70) but now we assume that the nominal load, S, vanishes.

C H A P T E R 6.

G A M B L I N G AND R I S K SENSITIVITY

181

i. J Let Ec denote the critical value of E at which the new design fails. Show that the robustness function is: (6.73) where A* is the greatest eigenvalue of the matrix:

ii. t Let £^w be a highly desirable upper limit for E, where ^w < EcShow that the opportuneness function is not:

P(f,E^) = J^

(6.75)

where A* is the least eigenvalue of E. What is the opportuneness function? iii. Assume that the flexibility vector is controlled by a single material parameter g, as in eq.(6.71). Compare the robustness functions of the two design concepts, as functions of q, and discuss your preferences between the designs. iv. J Repeat the questions of part 2e without assuming that the nominal load, S, vanishes. 3. Project management. Consider the five-task project schedule shown in fig. 3.24 on p. 108 and discussed there in problem 10. (a) Construct the robustness function. (b) The project should be completed in about 4 time units. Use your personal preferences for project completion time, and your understanding of the robustness function, to choose a satisfactory and plausibly attainable project duration, tc(c) The introduction of new technologies allows the nominal duration of task 2, t2, to be shortened. Explore the variation of robustness with t2 and explain how the choice of this parameter influences the riskiness of the project. (d) Repeat part 3c with the overlap parameter /22. (e) Study the effect of combined variation of t2 and /22. (f) t Construct the opportuneness function. (Caution: since task durations cannot be negative, the info-gap model of eq.(3.202) on p. 108 must be modified, as mentioned in footnote 6 on p.67. Consider the info-gap model in eq.(5.16) on p. 139.) (g) Study the riskiness of this project by comparing the robustness and opportuneness functions. (Are they sympathetic or antagonistic?) 4. Comparing two decision strategies. Consider a decision problem in which an uncertain scalar function u{t) is described by the uniform-bound info-gap model:

U{a,u) = L{t):

I

l^^i^Z^Ual,

I ^W I

J

a>0

1.76)

182

INFO-GAP DECISION

THEORY

In other words, the uncertainty parameter a expresses the unknown maximal fractional variation in u{t). Whatever the details of the decision problem are, the optimal robustness as a function of the critical reward is found to be: S(5;(re),rc) = 0.7e-^^/'

(6.77)

which is valid for critical reward TC in the interval [1, 3]. (a) The decision maker has examined this optimal robustness curve, and determined that the greatest reward which can responsibly be demanded is about Tc = 2. Consequently, the recommended action is 5i:(2). What does this reveal about the risk sensitivity of this decision maker. (b) New technologies are now available which entirely alter the analysis of the decision problem. The optimal robustness function with these new methods is: S(?c(rc),rc) = 3 - ^ i ^

(6.78)

which is valid for TC in the interval [1, 4]. i. Why might you expect the same decision maker now to choose the robust-satisficing action for critical reward TC = 3.2? ii. Upon examining the new situation, the same decision maker in fact chooses the robust optimal action for critical reward TC = 2.7. Interpret this behavior. (c) X The decision maker is able to choose between the old technology, whose implementation entails the robustness function of eq.(6.77), or the new technology with robustness given in eq.(6.78). However, the greater robustness of the new method incurs greater costs. At any value of the critical reward TC , each technology has a different robustsatisficing action Jc(^c), and the estimated costs of these actions are different. Specifically, the estimated cost of implementing the robustsatisficing action for the old and the new technologies are r i ( r c ) and r2(rc), respectively. How could the decision maker use these cost estimates to assist in choosing between the two technologies? 5. Risk-neutral line. Consider a decision problem for which the immunity functions are: S(g,rc)

=

ge-*^"^

(6.79)

^(g.rw)

=

(grw)' + -

(6.80)

where q > 0. (a) Find the robust-satisficing action 5^(rc) and the opportune-windfalling action 5w(^w). (b) Construct the risk-neutral line.

CHAPTER 6.

GAMBLING AND RISK SENSITIVITY

183

(c) Consider three points, P , Q and R in the re-VS-TWplane, as in fig. 6.10 on p. 165. Points P and Q are on the risk-neutral line constructed in part (b) at TW-values of 4 and 5, respectively. Point R is positioned in relation to P and Q as in fig, 6.10. Discuss plausible preferences between: Pand R Q and R P and Q

Chapter 7

Value of Information We live on an island of knowledge surrounded by a sea of ignorance. As our island of knowledge grows, so does the shore of our ignorance. John A. Wheeler [174, p.20] [T]he process of formulating and adopting goals creates a dependence of value on belief, simply because goals are more attractive when they are feasible. Glenn Shafer [158, p.222]

As the biological name implies, Homo sapiens quite naturally accumulates and processes information in a much more sophisticated manner than other animals. For our species, contemplation is just as basic as any of the physiological instincts. But "new ideas are commonplace, and almost always wrong" [177, p.60]. There is a substantial gap between raw data — just knowing things — and useful understanding. The systematic acquisition, evaluation and application of information is costly. Useful information is valuable both because its possession confers power and because its acquisition requires effort. Valuating information is central to the management of uncertainty. We have touched on this in previous chapters and now we will organize the subject. Two ideas are at the heart of our treatment. In section 7.1 we define the informativeness of an info-gap model of uncertainty, mentioned in earlier sections. In section 7.2 we define the demand value of information which is closely related to the reward premium discussed in section 6.3. In sections 7.3 and 7.4 we illustrate these ideas with an engineering decision analysis of the cantilever subject to uncertain static loads. In section 7.5 we briefiy consider the gathering of information as a strategy for enhancing reliability in project management. 185

186

INFO-GAP DECISION THEORY

The value of information depends strongly on how one perceives the ambient uncertainty. In traditional theories of the economics of information this dependence appears in the use of probability distributions [93]. In infogap decision theory the demand value of information is formulated with the robustness function. In section 7.6 we use the opportuneness function to formulate a windfall cost of information which expresses the loss, due to an increment of information, in unexpected but potentially favorable reward. An example is developed in section 7.7 which compares the demand value and the windfall cost of information.

7.1

Informativeness of an Info-Gap Model

An info-gap model of uncertainty, U{a, S), a > 0, is an unbounded family of nested sets. For any value of the horizon of uncertainty, a, the set U{a, u) delimits a range of possible variation of the uncertain function u(t). As a grows, so does the horizon of uncertainty. Information is the complement of uncertainty, so it is useful to define the informativeness of an info-gap model in terms of the horizon of uncertainty. More specifically, one info-gap model, Ui{a,u), is more informative than another, U2{oi^u)^ if the former constrains the uncertain variation more tightly than the latter. When the info-gap models have the same center point, S, the relation of "greater informativeness" of Ui over U2 can be expressed by set inclusion: Ui{a,u)

C U2{oL,u),

for all a

(7.1)

For example, consider two envelope-bound info-gap models: Ui{a,u)

=

{u{t):

\u{t)-u{t)\
U2{a,u)

=

{u{t):

\u{t)-u{t)\
a>0 a>0

(7.2) (7.3)

where the known envelope function '0(t) satisfies 0 < '^(t) < 1. These info-gap models satisfy relation (7.1): Ui{a,u) constrains the uncertainty more tightly than U2{oi,u) for any value of a. Every element of Ui{a,u) belongs also to U2{OL,U), but the latter set contains functions missing from the former. At any fixed horizon of uncertainty, a, a decision maker faces less unknown variation when Ui(a, u) is warranted rather than U2{oi^ u). In this sense, Ui{a^u) is more informative than U2{oC',u). Informativeness, defined as set inclusion in (7.1), is independent of the size, a, of the info-gap model. If Ui{a,u) is contained in ZY2(o;,S) for some value of a, then the inclusion occurs for all values.^ When inclusionbased informativeness holds, it holds at all horizons of uncertainty and is a property of the basic structure of the info-gap models. ^This is readily proven from the 4 axioms of info-gap models in section 2.6.

C H A P T E R 7.

VALUE OF INFORMATION

187

The inclusion in (7.1) refers explicitly to info-gap models centered on the same point, u. Info-gap models with different center points, Ui{a,ui) and U2{oL-> '^^2), ^i 7^ ^2, have different frames of reference and their relative informativeness is not meaningfully expressed by inclusion. But \iUi{a,u) is more informative than IA2{OL^U) at some center point S, then this is true whenever these info-gap models are centered together, regardless of the specific choice of the center point. Again, informativeness is a structural property of the uncertainty models. Set inclusion does not provide a universal rule for ranking the informativeness of info-gap models of uncertainty. Two info-gap models may simply have different information, even when centered on the same point. In terms of set inclusion, info-gap models may intersect without either of them containing the other. Two models may have a non-empty intersection at uncertainty a\ Ui{a,u) n ZY2(a,S)/0 (7.4) and yet neither is contained in the other: U2{a,u)-Ui{a,u)^^

(7.5)

Ui{a,u)-U2{a,u)^%

(7.6)

while also:

As an example, let U2{oi, u) be the uniform-bound model of eq.(7.3) and re-define Ui{a,u) as the following Fourier elhpsoid-bound info-gap model: Ui{a,u) - {u{t) = u{t) ^-f^(t){t) : -^^W-f < a^} ,

a > 0

(7.7)

where (t){t) is the A^-vector of sine functions: 0^(0 = sinriTrt,

n = l , ...,A^

(7.8)

and W is a diagonal matrix whose nth diagonal element is 1/n. Relation (7.5) holds because the function: u{t)=u{t)-^a

(7.9)

belongs to U2{o^->u) but not to Ui{a,u). But similarly, for any 1 < n < A', define the function: u{t) =u{t)-\-ay/nsmn-Kt (7-10) This function belongs to Ui{a,u) but not to U2[oi,u), which shows that relation (7.6) holds. These info-gap models represent different, but partially overlapping, domains of uncertainty.^ •^Another example appears in problem 3 of chapter 2.

188

7.2

INFO-GAP DECISION THEORY

Demand Value of Information

The basic question we ask is: what is the worth of information? The answer has two parts. We wih assess the usefulness of information as the robustness premium which it entails. We will also determine a fair price for information as the increment of additional reward which the decision maker can confidently demand when the information is exploited. This will turn out to be the reward premium which the information induces and will be recognized as what economists call a 'demand value'. Both the demand value and the robustness premium depend on the ambient uncertainty or on the decision maker's choice of critical reward. We must first establish that: information enhances robustness. More specifically, if Ui{a,u) is a more informative info-gap model than U2{oL,u), in the set-inclusion sense of eq.(7.1), then Ui{a,u) has a positive robustness premium with respect to U2{OL^U). We can understand this result intuitively, without appealing to rigorous proofs which are available elsewhere [18, 19]. Let Si(6?, Tc) and a2(^, TC) denote the robustness functions for info-gap models Ui{a,u) and U2{OL,U), using the same reward function R{q,u) and the same set of available actions Q in both cases. By definition, the robustness S2(g,rc) is the greatest uncertainty consistent with obtaining reward no less than TC, when using U2{oi-, u). That is, the lowest reward R{q^ u), for any function u in U2{oi2,u), is no less than r^- But Ui{a^u) is a subset of ^2(0^,2). Thus the lowest reward R{q,u) on Ui{a2,u) is also no less than Tc, and will usually be greater than r^. In fact, the set Ui{a, u) can expand at least as far as a = a2{q-, Vc) (and usually beyond) before the least reward dips below the critical value r^. This means that the robustness for Wi(a, u) is no less (and usually greater) than the robustness for U2{ot^u). What we have found is the following: Ui{a,u)

C U2{oi,u)

(7.11)

32(g,rc) < Si(g,rc)

(7.12)

imphes that: In other words informativeness, eq.(7.11), implies a non-negative (and usually a positive) robustness premium, eq.(7.12). The usefulness of the information is, in part, the robustness premium which accompanies the information. Now we come to the second aspect of the worth of information: how much should we be willing to pay for this information? What is a fair price? It is evident that, as in fig. 6.2 on p. 154 and again in fig. 7.1, a positive robustness premium AS entails a positive reward premium Arc. We have concluded that information is valuable in the sense that it enhances the robustness to uncertainty. In fig. 7.1 we show two robustness curves where the upper curve corresponds to a situation of greater information. We have

C H A P T E R 7.

VALUE OF INFORMATION

189

Immunity to Uncertainty

Robustness AS^ premium More info. ai(g, TC) Less info. S2(g','rc) Reward

— •

Arc Demand value

Figure 7.1: Two robustness curves: before acquiring information (lower) and after (upper), based on [23]. "reached" the more comfortable and protected situation in the upper curve after investing resources in the acquisition of information. The question is: what price should reasonably be paid for this increment of information? The demand value of the information needed to move from the lower to the upper curve in fig. 7.1 is the increment of additional reward which can be demanded after acquiring the new information. The horizontal arrows in fig. 7.1 show the additional reward which the decision maker can now confidently demand as a result of enhancing the information. This is a "demand value" in the economic sense that the decision maker would be willing to pay any quantity up to, but not more than, this increment in reward in exchange for the information. The demand value of information is expressed explicitly in the units of the reward by which the project itself is evaluated. We note that the demand value depends on the level of ambient uncertainty: where on the robustness curve the decision maker is operating. Information has no absolute value; its value is relative to the environment. In particular, we see from this pair of curves that information is of relatively lower worth at a low level of demanded reward and at a high level of robustness. The robustness premium of information, Aa, expresses the increment of immunity resulting from the added information, and is shown in fig. 7.1 by the vertical arrows. The robustness premium speaks directly to the issue of immunity, but it lacks the economic overtone of the demand value. The robustness premium, like the demand value, depends on the operating point at which the decision maker is functioning. The demand value — the reward premium Arc — is the economic worth of the information required to replace the info-gap model U2{ot,u) by Ui{a,u). But in order to calculate Arc one must know the form of

190

INFO-GAP DECISION THEORY

both of these uncertainty models; neither robustness function, Si(g,rc) nor ^2(9,^0), can be evaluated without exphcit knowledge of the corresponding info-gap models, U2{ct, u) and Ui{a, u). The 'acquisition' oiUi{a, u) entails the verification rather than the mere identification of that info-gap model of uncertainty. We are considering the following scenario. The decision maker is confident, before acquiring the new information, that the uncertain variation is no greater than ^2(0^,^). The decision maker is contemplating investing resources to verify a more refined understanding, namely Ui{a, u). The decision maker is speculating in the following way: "Suppose that we were convinced that Ui{a, u) is true rather than the informationally weaker assertion that only U2{OL,U) holds." The decision maker is considering the imphcations of a state of the environment which can be modelled but which is not presently known to hold true. That is, the mathematical structure of Ui{a^u) is known, but the decision maker cannot yet assert that its tighter description truly represents the environment. We are not considering the very different situation where an information-dealer tells the decision maker: "I have data in my pocket which should interest you. How much will you pay me for revealing it?" Finally, we note that we can calculate a demand value and a robustness premium regardless of whether or not one of the info-gap models is strictly more informative than the other. Let 3i(g,rc) and a2{q-,r^) be the robustness functions from two diff'erent constellations of information, each with its own info-gap model and reward function. These robustness curves may or may not cross when plotted versus the critical reward. Nonetheless, the demand value at uncertainty a, for decision model T with respect to decision model '2', is the difference: Arc(a) = rc,i - rc,2

(7.13)

where the critical rewards rc,i and re,2 satisfy: Si((7,rc.i) = a = 32(g,rc,2)

(7.14)

If large reward is better than small reward then Arc (a) will be positive or negative if the first model is more or less valuable than the second.*^ Furthermore, the sign of Arc(Q^) may change with a since the robustness curves may cross. In like manner, the robustness premium at critical reward re, of decision model ' 1 ' with respect to model '2', may be either positive or negative, and is defined as: A3(rc) = 3i((7,rc) - a2[q,r^) (7.15)

7.3

Uncertain Loads on a Cantilever

We used the cantilever to illustrate the info-gap analysis of robustness in section 3.2.1. We again employ the same structure, subject to uncertain •^The sign is reversed if small reward to preferred to large reward.

C H A P T E R 7.

VALUE OF INFORMATION

191

static loads, to illustrate the demand value of information. Consider a cantilever of length L. Let x denote position along the beam, where the clamped end is at x = 0 and the free end is at x = L. Static forces are applied in a single plane and perpendicular to the beam axis, where u{x) is the unknown force density, in units of Newtons/meter, at position X. The deflection profile resulting from this load is v{x) which, for small deflections, obeys the following differential equation: EI^^=u{x)

(7.16)

where E is Young's modulus for the material and / is the area moment of inertia of the cross-section. We assume that EI is constant along the length of the beam. The boundary conditions at the clamped end are: ^(0) = v\0) = 0

(7.17)

where the prime denotes differentiation with respect to x. At the free end the boundary conditions are: v'\L) = v''\L) = 0

(7.18)

The performance criterion is that the deflection at the free end must not exceed a critical value, Vc'. \viL)\ < V, (7.19) Information about the uncertain load proflle is accumulated as a spectral constraint on u{x). The bandwidth of spatial frequencies composing the load profile is determined, and the uncertainty in the Fourier coefficients is represented with an ellipsoid-bound model. The info-gap model for uncertain load profiles whose least and greatest spatial modes are ni and 712 is:

Un,,712{o^ 1 u) = f^u{x) =u-\--f^(pn,,712{x) ' 7 ^ 7 < ^ ^ < ^ ^ } .

»>0

(7-20)

where the nominal load, u, is known and constant and (/)„!,n2(^) is a vector of sine functions: / niTvx . (ni-hl)7rx . n27rx\ (Pnu7i2W = ( s m — — , sm , . . . , sm—-— 1

(7.21)

The informativeness of the info-gap model Uni,7i2{^',^)i i^ ^he sense of the set-inclusion eq.(7.1), is determined by the bandwidth. A tighter bandwidth implies stricter constraints on the possible load profiles, and hence greater information about the environment. If: rii < mi

and

m2 < n2

(7.22)

192

INFO-GAP DECISION THEORY

with at least one strict inequality, then:

Umi,m2{^^^) is a more informative info-gap model than Uni,n2{^->^)The solution of eq.(7.16), in response to an uncertain load profile u{x) = S + 7-^(/>ni ,712(^)5foi"the clamped-free boundary conditions in eqs.(7.17) and (7.18), is:' ^^ . .

^

x^ x^ 24

Lx^ , L^x^\ Lx^ L^x^\ 6 4 J

x^ x^ ^y^ jnL^ 2 ^-^ ^ 7nir n=ni

Y

S

/ r \ 2

/ r \ 3

nn

x"^ 6

I L \ \niT J

( L \ , Kn-K J

n2

^

n=ni

^

^

nnx L

n=ni

We are particularly interested in the end deflection, for which eq.(7.24) simplifies considerably: EIv{L) ML^

_ ~

1 1 Y> /_1 8 ^ S ^ ^" I (n7r)3

=

i + -7^C 8

(-1)" 3n7r

(7.25)

(7.26)

li

where the elements of the vector C are defined in eq.(7.25). The quantity EIv/uL"^ on the left of eq.(7.25) is a dimensionless deflection. In order to calculate the robustness function we must evaluate the greatest deflection for any load profile up to horizon of uncertainty a. Using Lagrange optimization the result is: max ^

EIv{L) - ^

_\ __r7¥7 = -+aVC^C

(7-27)

Denote the dimensionless form of the maximum allowed deflection v^ as:

Equating the maximum normalized deflection, eq.(7.27), to the normalized performance requirement, p^, and solving for a leads to the robustness for this info-gap model: ^'^ - '

Sv'C^C

(7.29)

C H A P T E R 7.

VALUE OF INFORMATION

193

The performance requirement is t h a t the normahzed end deflection be no greater t h a n pc, which means t h a t reward increases as pc decreases: a small value of pc is more desirable or demanding t h a n a large value. In eq.(7.29) and in fig. 7.2 we see t h e usual trade-off" (even though the robustness curves have positive rather t h a n negative slopes): the robustness Sni,n2 increases as the reward decreases (as pc increases).

*-mi ,7712

More informative Robustness a ni,n2

a

Less informative - • Reward pc Pc,m

Pc,n

Figure 7.2: Robustness curves for more and less informative info-gap models. Robustness functions versus critical deflection are shown in fig. 7.2 for values of m^ and rii satisfying relations (7.22), so the corresponding info-gap models are nested as in eq.(7.23). T h e horizontal arrow shows the demand value, Apc{a)^ of t h e more informative uncertainty model. This demand value is calculated as in eqs.(7.13) and (7.14). T h e critical rewards pc,m and pc,n are evaluated from: ^mi,777,2 vPc,mj — Oti —

T h e demand value oiUmi,m2 ^PC(Q^)

0!.ni,n2\Pc,n)

(7.30)

with respect to V(ni,n2 is:

=

(7.31)

Pc,m-Pc,n 7722

=

a

1E n=mi

\

Ec^

(7.32)

The demand value of an informative info-gap model, in units of pc, is negative because a small value of p^ is desirable. We also see t h a t the magnitude of the demand value increases as the ambient uncertainty, a, rises: an informative model is more useful in the presence of great uncertainty. T h e demand value in eq.(7.32) is shown in table 7.1 for a range of m2 and 712 values and with mi — ui — 1 and a = 1. Moving from left to right on each row we see t h a t Apc gets better (more negative) as the difference n2 — m2 increases. This means t h a t the worth oiUi^rn2 increases in comparison

194

INFO-GAP DECISION THEORY

\ n2 = I 7712 =

2 3 4 5

1

0

2 -0.0084 0

3 -0.013 -0.0045 0

4 -0.015 -0.0067 -0.0022 0

5 -0.017 -0.0082 -0.0037 -0.0015 0

Table 7.1: Demand value of an informative info-gap model, eq.(7.32), for mi ni = 1 and a = 1. with Ui^n2 ^s t^^ latter info-gap model becomes less informative than the former. The reverse is true as we move down each column: the worth of Wi,m2 decreases as it becomes more similar to ^/i,n2- These increments of value are expressed in units of the performance requirement of the system itself, Ape or, equivalently, At'c. The decision maker presumably has a sound qualitative understanding of the significance of increments in these parameters, which enables the assessment of the significance of increments of information.

7.4

Cantilever: Simple and Complex Info-Gap Models

The set-inclusion criterion for informativeness, eq.(7.1), captures only one aspect of the full gamut from 'simple' to 'complex' decision models. Nonetheless, in connection with eq.(7.13) we explained that a demand value can be calculated for any pair of decision models. The sign of this demand value will be constant at all levels of uncertainty only if the info-gap models are nested. Otherwise, the sign will change indicating a different preference for one or the other of the models in diff'erent circumstances. We will illustrate this by comparing the decision model of section 7.3 against a nominally simpler model. The 'complex' decision model represents uncertain distributed loads with the Fourier ellipsoid info-gap model of eq.(7.20). We will suppose that spatial modes from the 5th to the 10th make up the uncertain load functions, so ni = 5 and 712 = 10. The 4th-order differential equation for beam deflection, eq.(7.16), is used, leading to the robustness function of eq.(7.29). We will denote this robustness by 3c which, for this specific info-gap model, becomes:

3e = 26.46 (p, - M

(7.33)

The 'simple' decision model assumes that the loads are concentrated at the mid-point of the beam. Instead of eq.(7.25), the end deflection resulting

C H A P T E R 7.

VALUE OF INFORMATION

195

from a point force / at the mid-point is: v(L) - ^ ^ ^

(7.34)

The fractional variation of the point load is uncertain, so the info-gap model for variation in / is:

/ - /

U{aJ)={f:

^ ^ <

/

a y, —

" V2

Q>0

(7.35)

where the nominal point load / corresponds to the nominal uniform load density u in eq.(7.20): / = SL (7.36) The normalization by \/2 in eq.(7.35) is chosen to coordinate the scales of value of the uncertainty parameters in this info-gap model and the Fourier ellipsoid model in eq.(7.20). We want U{a, f) and Uni,n2{^->'^) ^^ be comparable in size. In eq.(7.36) the least upper bound of the squared fractional variation of the load is: / .

7\ '

„2

(7.37)

V / Evaluating the analogous quantity for Uni,n2{^^^) ^^ find:

n=ni

<

v2

Y

(7.39)

The info-gap models U{a, f) and Uni,n2{^j'^) ^^"^ different, but the magnitudes of their uncertainty parameters are comparable in terms of the horizons of uncertainty of these models. The robustness function for the simple decision model is found by evaluating the maximum deflection at uncertainty a, equating this to the critical deflection v^^ and solving for a. One finds: Ss = 12pc - 1

(7.40)

where pc is the dimensionless critical defiection in eq.(7.28). The robustness functions of the 'simple' and 'complex' decision models, eqs.(7.40) and (7.33), are depicted in fig. 7.3. These curves cross at the point (0.16, 0.92) in the pc-y^-ot plane. The intersection of the robustness curves means that there is no unequivocal demand-value preference for one or the other of these two decision models. The 'simple' model is more robust for

196

INFO-GAP DECISION

THEORY

Robustness

'complexY y^ 'simple' 0.92

^ Reward Pc 12

0.16

Figure 7.3: Robustness curves for 'simple' and 'complex' decision models.

normalized critical reward pc less t h a n 0.16 and for ambient uncertainty OL less t h a n 0.92. Consequently, the demand value of the 'simple' model over the 'complex' model is negative (preferable) in this range. However, for Pc > 0.16 and a > 0.92 the situation is reversed, and the 'complex' model is preferable to the 'simple' model in terms of robustness premium or demand value. In comparing the robustnesses of these two decision models we are assuming t h a t the underlying uncertainty parameters of the respective infogap models are commensurate. As in the present case, this assumption is justified when each uncertainty parameter has the same meaning and scale of value as a measure of variation. In the spectral-bound model, eq.(7.20), a is the fractional variation of the force-density spectral amplitudes. In the interval-uncertainty model, eq.(7.35), a is the fractional variation in the magnitude of the point force. In each case the total force nominally apphed to the beam is the same, / = SL, and the scales of value of the uncertainty parameters have been coordinated as explained earlier.

7.5

Gathering Information in Project Management

An important strategy for enhancing reliability in project management is to reduce uncertainty by gathering information. We will demonstrate this by continuing the example in section 3.2.6. Increased information can be expressed as reduced uncertainty weights, Wn, in the info-gap model U{OL^ t) in eq.(3.85) on p.67. Fig. 7.4 shows the five p a t h robustnesses versus the uncertainty weight of activity 6, with tc — 21. The plotted values of Sm Sit WQ = 2 correspond to the values in the b o t t o m row of table 3.2 on p.69, and are evaluated from

C H A P T E R 7.

197

VALUE OF INFORMATION

Robustness

Robustness m= 5

0.24J \^ m

m

%

"^ X

0

0.5

^^ 1

0.171

2

Uncertainty weight WQ Figure 7.4: Path robustnesses, Sm, versus the uncertainty weight of task 6, WQ.

1

^_>.

1

2

Uncertainty weight W2 Figure 7.5: Path robustnesses, Sm, versus the uncertainty weight of task 2, ii;2.

eq.(3.92). From the 6th column of the participation matrix F , eq.(3.81), we see that three paths depend on activity 6: paths 2, 3 and 4. In fig. 7.4 we see that these three paths show increased robustness as w^ decreases. However, path 1 (the lowest curve) remains constant and is always the most vulnerable to uncertainty. In other words, gathering information about task 6 does not enhance the reliability of the project as a whole. We can influence the 'critical' path, # 1 , by gathering information about activity 2, whose uncertainty weight in table 3.2 is 102 = 2. In fig. 7.5 we show the robustnesses of paths 1, 2 and 3 versus W2, still with t^ = 21. Only the first path varies, as we understand from the participation matrix, eq.(3.81). The robustness of path 1 increases by about 40% as W2 decreases from 2 to 0, and the critical path becomes # 3 for W2
7.6

Windfall Cost of Information

Up to now we have evaluated increments in information from the perspective of robustness to uncertainty. Since the robustness and opportuneness functions express complementary immunities to uncertainty it is not surprising that we can develop an analogous, though conceptually distinct, interpretation of increments of information by examining the opportuneness function. In section 7.6.1 we formulate a windfall cost incurred by using information to replace a less informative info-gap model U2{oi, u) by a more informative model ZYi(a, u). In section 7.6.2 we compare the demand value and windfall

198

INFO-GAP DECISION THEORY

cost of information. In section 7.7 we consider an example.

7.6.1

Formulation

The trade-off between certainty and windfall reward, displayed in fig^S.l on p.47 and again in fig. 7.6, states that the opportuneness function f3{q,r^) increases monotonically with increasing "wildest dream" level of reward Tw This proposition embodies the trade-off between the propensity for favorable surprises and the extent of certainty about the environment. The decision maker must accept great ambient uncertainty in order to entail the possibility of large, favorable and unexpected windfall outcomes. The monotonic increase in the opportuneness function is illustrated schematically in fig. 7.6 for two info-gap models, Ui{a,u) and U2{oi^u), where the former is more informative than the latter, in the set-inclusion sense of eq.(7.11) on p. 188. In fig. 7.6 the upper opportuneness function is based on the more informative info-gap model, which expresses the fact that immunity to windfall reward increases with informativeness. That is, the prospect for surprising favorable outcomes diminishes if one uses a more informative info-gap model. This can be stated formally as follows. Ui{a,u)

C U2{a,u)

(7.41)

^2fer-c)<^i(g,rc)

(7.42)

implies that: Since a small value of the opportuneness function is more desirable than a large value, this conclusion shows the decrement in windfall opportuneness accruing from an increment in information. As we see in fig. 7.6, employing information so as to replace U2{ot, u) by Ui{a,u) results in a decrease in the windfall reward TW which can be hoped for at any given level of ambient uncertainty a. This decrease, /S.r^{a), is the (non-negative) cost in windfall reward incurred by employing the information, and it is in units in which the outcome of the decision is measured. The windfall cost is evaluated as: Arw(a) = rw,2 - rw,i

(7.43)

where r^^^i and r^^^2 are defined by: ^i((7,rw,i) = c^ = ^2(
(7.44)

/^r^{a) is the decrement, which results from the augmented information, in the potential windfall value which could accrue from a decision. Alternatively, Arw(Q^) is the least amount which a thoroughly "windfalhng" decision maker would be willing to accept as compensation for employing the increment of information, since by using the information the decision maker must accept a decrease in potential windfall revenue at least as large

C H A P T E R 7.

199

VALUE OF INFORMATION

Ambient Uncertainty More info. Certainty

f^i{q,r^)

Less info. I32{q,r^)

Windfall reward, r^ r^A

rw,2

Figure 7.6: Opportuneness function versus windfall reward for two different levels of information. Illustrating the windfall cost of information /S.r^{a).

as Arw(Q;) if the level of ambient uncertainty is a. We will refer to Arw(a) as the windfall cost of the information required for going from U2{a^u) to

Ui{a,u). The trade-off between certainty and windfall reward, shown in fig. 7.6, is for fixed action vector q. However, the monotonicity of the opportuneness function holds also for the opportune-windfalling action $w(^w) defined in eq.(3.22) on p.46, with a corresponding value of windfall cost. For many decision makers the robust-satisficing action ^cC^c), defined in eq.(3.20) on p.45, is of more relevance than the opportune-windfalling action $w(^w)- In this case, a pertinent evaluation of the windfall cost is based on the robust-satisficing action. This is evaluated by comparing /^2(9'^w,2) against Pi{q,r^,i) in eq.(7.44) with q = ^c,2(^c)- This windfall cost is incurred by improvingU2toUi, based on the action which maximizes the robustness with the prior info-gap model, ZY2.

7.6.2

Discussion

Prudence or rashness. In comparing the demand value of information Arc against the windfall cost Ar^, we note that the former ^epends on a(^, Tc), the robustness function, while the latter derives from /5(^,rw), the opportuneness function. a{q^ TC) expresses the greatest uncertainty which is consistent with a guaranteed reward at least as large as Vc- That is, a{q, TC) is based on satisficing to a reward level TC, SO a{q, Vc) is of natural interest to a cautious decision maker, though one can express a propensity to gamble by choosing a large value of r^. l3{q,r^) is the lowest level of uncertainty

200

INFO-GAP DECISION THEORY

which enables a windfall at least as large as r^. f3{q, r^) has nothing to do with satisficing but rather with 'windfalling', and expresses the potential for great reward and would attract an ambitious gambler. Nonetheless, the decision maker can choose a small value of r^ in P{q,r^) to express a cautious tendency. The demand value of information, Arc, is the most that a decision maker would be willing to pay in order to obtain information which will increase the immunity to uncertainty. The windfall cost, Ar^, does not have as standard an economic interpretation as the demand value. One interpretation is that the windfall cost of information is the least that a gambler would accept in payment for using information which curtails the windfall potential which titillates the imagination. Alternatively, Ar^ is the decrement in the potential windfall value which could accrue from a decision which results from augmented information. Windfalling rather than satisficing will tend to appeal to the gambler, and an economic scenario which would attract /^-thinkers rather than athinkers is the public offering of stocks in a speculative venture. The firm making the offer may well withhold information which would suggest a narrow range of uncertainty, even if the information would also suggest a slightly higher anticipated return, on the grounds that prospective investors are seeking speculative gains and only wide margins of uncertainty contain the potential for large reward. Gambling is sometimes thrust from the pale of economic behavior, but Arrow has noted several ways in which "gambling is reconcilable with a general predominance of risk aversion", one of which is that the decision maker may be risk averse in one context and not in another [5, p. 148]. Windfalhng, as modelled by the opportuneness function f3{q, TW), combined with satisficing with the robustness function 3(g, TC), provide complementary means by which the decision maker can satisfy diverse propensities for dealing with uncertainty. Windfall and opportunity costs. The windfall cost is somewhat like an opportunity cost in the standard economic sense. In a guns-vs.-butter trade-off the opportunity cost of a gun is the amount of butter which must be forgone in order to obtain the gun. The windfall cost is the amount of potential windfall reward which must be forgone in exchange for the immunity-to-uncertainty resulting from the information. A difference between the windfall cost and the ordinary opportunity cost is that the former is a constriction of reward at fixed uncertainty, while the latter is a constriction of consumption at fixed budget. The maximum windfall reward is reduced by Ar^^{a) at uncertainty a, but greater windfall is still possible if the ambient uncertainty exceeds a. This is nonetheless analogous to the classical fixed-budget cost constraint: a dollar spent on guns is a dollar not spent on butter at a fixed budgetary level. One can of course always look for another dollar outside of the budget. The analogy between windfall and opportunity costs can also be viewed

C H A P T E R 7.

VALUE OF INFORMATION

201

from a different perspective. The windfall cost results from the trade-off between robustness-to-uncertainty and opportuneness-from-uncertainty as information varies. This uncertainty trade-off is precisely^ guns-vs.-butter trade-off, but expressed in units of uncertainty (a and /3) rather than in commodities. Augmenting robustness causes a decrease in windfall opportuneness and vice versa: the trade-off curve between guns and butter appears now on the axes of windfall opportuneness (/?) vs. robustness (3). When robustness improves (3 increases) by augmenting information, then windfall potential deteriorates (/? also increases).

7.7

Initial Commitment and Uncertain Future: Revisited

We now demonstrate the calculation of the demand value and windfall cost of information, by continuing the example in section 6.4, adapted from the study of probabilistic risk aversion in the choice of tar-sand oil plant size by Fuller and Gerchak [75]. The decision maker must choose the size q of the production plant, for which profit is evaluated from the uncertain unit revenue u{q) and the construction cost c(g), eq.(6.6) on p. 157: R{q,u) = qu{q) - c{q)

(7.45)

The unknown discounted unit revenue u{q) belongs to the following infogap model, which is slightly different from eq.(6.7): U^{a,u) = {u{q) : \u{q) - pe'^'^^^A < -] , a > 0 (7.46) L I \ w) The uncertainty weight w modulates the horizon of uncertainty a, so that the informativeness of the info-gap model increases with increasing w^ which may be chosen subjectively or based on variational data. Thus, W2 < wi imphes that Uwi is more informative than Uuj2' Furthermore, given the info-gap model Uu)2 ^ the decision maker may determine the demand value and the windfall cost of the information needed to reduce uncertainty and up-grade the uncertainty model to Uuj^. Note that the uncertain revenue u{q), as well as its estimated value S(g), both depend on the action chosen by the decision maker. The lowest acceptable profit is Vc. The robustness function based on uncertainty model U^ is: S.(9,rc) = fpe-^^(') -

'-1±^)

yj

(7.47)

The robustness decreases linearly with Vc for fixed plant size q. However, the robust-satisficing plant size $c varies with r^ so the dependence of the optimal robustness, a{qc,rc), upon TC will be nonhnear.

202

INFO-GAP DECISION

THEORY

For concreteness consider the same numerical example studied in section 6.4.1. Let c{q) — a^fq and L(g) = 6ig -h ^o- T h e parameters of the estimated discount rate are b = 0.08 and pe-^^^ - 3a. T h e robust-satisficing action is obtained by solving da^/dq — 0 for q, which leads to the following transcendental equation in q\ pSbie

-6{biq-\-bo

-3/2

+ rcq- 2

(7.48)

This equation does not depend on the uncertainty weight w^ indicating t h a t the robust-satisficing action is independent of the degree of informativeness of the info-gap model. This however does not imply t h a t new information would not alter the decision maker's action. T h e robustness curve changes with improvement of the information, as is seen in fig. 7.7. It is from the robustness curve t h a t the decision maker chooses a trade-off between immunity and reward, by selecting the critical reward TC, and thence the robust-satisficing action $c(^c)- Since the robustness curve changes with addition of information, the choice of TC and of qdrc) may change as well. Arc/a

a. .($c(^c),rc)/a

Figure 7.7: Optimal robustness versus the critical utility for two values of w.

Figure 7.8: Demand value of information versus the critical utility.

Fig. 7.7 shows the optimal robustness S^^(5c(^c)5'^c) vs. the critical utility Tc for two diff'erent values of the uncertainty weight w. T h e monotonic decrease in a is the expression of the trade-off between immunity and reward. Since Uu)=i is more informative t h a n UW=Q.7, we are not surprised to find t h a t the robustnesses are ranked according to: Oin = 1 > 0^^=0.7

(7.49)

as seen in the figure. Fig. 7.8 shows the demand value of the information required to replace Uw=o.7 by Uw=i^ based on the robust-satisficing actions for these two infogap models. This is calculated from fig. 7.7 as illustrated schematically in fig. 7.1 on p. 189. More precisely, for any value of r^ use the lower curve in fig. 7.7 to define S* as: a* = a^=o.7(5c'^c) (7.50)

C H A P T E R 7.

VALUE OF INFORMATION

203

and then use the upper curve to define r* from: S* = S ^ = i ( g ^ , r * )

(7.51)

The demand value plotted in fig. 7.8 is: Arc = K - r,

(7.52)

Figure 7.8 shows non-monotonic variation of the demand value of information as a function of the decision maker's willingness to gamble on the critical reward, re. While the propensity for uncertainty increases with the choice of re, the demand value of the information does not increase continuously but rather attains a maximum at an intermediate value of r^ T h e demand value of the information eventually vanishes, as the two robustness functions together converge to zero. T h e opportuneness function P{q, rw) for the uncertain profit R{q, u) and info-gap model U^ is:

h{q.r^)

= {^-^^^±^ - pe-^^(^)) w

(7.53)

We will evaluate the windfall cost at a fixed robust-satisficing action, $c(^c)- W i t h fixed action, the opportuneness function increases linearly with the windfall r w Also, with W2 < wi, we see t h a t P^^ < (3^^, as expected from the trade-off between certainty and windfall. Referring to eqs.(7.43) and (7.44) and using eq.(7.53), one finds the windfall cost at uncertainty level a and based on a fixed robust-optimal action $c(^c) to be: Arw = aqc{rc)

,

Wi > W2

(7.54)

W1W2

At fixed W2 one sees t h a t A r ^ has a positive first and negative second derivative in Wi. This means t h a t the windfall cost increases monotonically with enhanced information, but the marginal windfall cost decreases continuously.

7.8

Problems

1. I n f o r m a t i v e n e s s a n d d e m a n d value of i n f o r m a t i o n . The following two optimal robustness functions are derived for the same decision problem with different info-gap uncertainty models: ai{q,,Arc),rc) ^

^

=

O.Te"'''^/' 15

(7.55) (7.56)

(a) What can you say about the relative informativeness of the info-gap models underlying these robustness functions?

204

INFO-GAP DECISION

THEORY

(b) Evaluate the demand value of one of these models with respect to the other. Which is more valuable, and by how much? How does the value vary with ambient uncertainty? 2. Financial investment. Investment can be made over the time interval 0 < i < 1 at a rate of q{t) dollars per day. The return at the end of the investment period is: q,u) = / u{t)q{t)dt u R{q,u) Jo

(7.57)

The investor must choose the investment rate function q(t), while the rate of return, u{t) dollars per dollar invested, is uncertain and represented by an envelope-bound info-gap model: U{c The estimated rate of return is: u{t) = ^

(7.59)

where 6 is a known constant. Not much is known about the actual rate of return u{t) over the period of this investment. However, it is generally agreed that the envelope of uncertainty, ip{t), increases linearly in time: V;(t) = 0.5 + t

(7.60)

The research department suggests that only modest resources are needed to verify or refute the assertion (put forward by promoters of the investment program) that the uncertainty envelope is in fact constant: i>{t) = 1

(7.61)

(a) Assume that the rate of investment q is constant. Evaluate the robustness function for each uncertainty envelope. (b) Based on part 2a, determine the demand value of the improved infogap model. Do you recommend adopting the research department's proposal? At what cost? Assuming what level of ambient uncertainty? (c) Now allow the investment rate to vary linearly in time: q{t) = qit + qo

(7.62)

where qo and qi are decision variables. Evaluate the robustness function for each uncertainty envelope. (d) With this linear investment strategy, determine the demand value of the improved info-gap model. Do you now recommend adopting the research department's proposal? At what cost?

C H A P T E R 7.

VALUE O F INFORMATION

205

(e) J The total investment must not exceed the budget allotment of Q dollars: /

q{t)dt
(7.63)

Jo Find the robust-satisficing linear investment strategy and evaluate the optimal robustness function for each info-gap model. Determine the demand value for the improved model. 3. Search and evasion. Review the search and evasion example in section 3.2.9 on p.78. The hunter is offered a surveillance program aimed to test the following Fourier ellipsoid bound info-gap model for the uncertainty in the prey's evasion capabilities. The prey's position function is described as: N

u{t) = u{t) + ^

cn COS ^

= u(t) 4- c^7(0

(7.64)

n= l

where the known nominal position of the prey is: u{t) =^7t-\-Uo

(7.65)

The proposed uncertainty model for u{t) is: U{a,u) = [u{t) = S + c^7(t) \ JWc
,

a >0

(7.66)

where Vl^ is a known, real, symmetric, positive definite matrix. Evaluate the robustness function for this info-gap model, determine its demand value and compare with the robustness function developed in the original example. Should the hunter fund the surveillance program? At what cost and at what levels of ambient uncertainty?

Chapter 8

Learning [Tjhey know enough who know how to learn. Henry Adams, [1, p.314] To me, logic and learning and all mental activity have always been incomprehensible as a complete and closed picture and have been understandable only as a process by which man puts himself en rapport with his environment. It is the battle for learning which is significant^ and not the victory. Norbert Wiener, [176, p.324]

8.1

Learning and Deciding

When we decide that 'This bird is an albatross' or 'That snark was a boojum' we can be said to have learned something. More generahy, decision under uncertainty is a process of sorting out a vague and unclear situation: we learn something about the world. The decision is the content of what we have learned. But what is learning, and how do the more general aspects of learning pertain to decision processes? Learning can be thought of from (at least) three distinct perspectives: content, function or structure. When emphasizing content one can ask: what has been learned? What is the semantic content of the information which has been acquired? When emphasizing the function of learning one concentrates on the effect which learning has on subsequent behavior. Finally, when emphasizing structure one focuses on the process by which learning occurs. One studies the question: what concepts and cognitive structures have governed and facilitated the learning process. 207

208

INFO-GAP DECISION THEORY

In studying a decision theory we focus primarily not on the content or the function of specific decisions, but rather on the structure of the decision process. Of course, both the meaning and the behavioral implications of a decision influence the process of reaching the decision. Nonetheless, the enormous range of topical applications covered by a single theoretical framework is evidence that a decision theory is more general than a specific decision event. In hke manner, Piaget's theory of intelligence de-emphasizes the content of intelligent thought and focuses on the structure and processes underlying thoughtful behavior [79, p. 16]. Another taxonomy of learning, also characteristic of Piaget's theory of intelligence, is based on the distinction between the immediate acquisition and processing of data, as opposed to the modification of the cognitive structures governing the learning process itself. Classification decisions are common examples of immediate learning, while the gradual evolution of an individual's intellectual repertoire is achieved by modifying cognitive structures. One's intellect is in dynamic and evolving equilibrium with the environment. Information is processed on both local and global scales: immediate operative decisions are made, while also the mind's overarching conception of the environment is altered [79, p. 172; 131]. The key to the maturation of the intellect — and to the improvement of a decision process — is the analysis of error. Rousseau emphasized the importance of error in learning: "If he never makes mistakes he will never learn anything thoroughly." [149, p.78]. This chapter is motivated primarily by the structural aspects of learning as they provide a metaphor for the dynamic process of improving the decision procedure. The central questions will be: how do we decide that a decision algorithm is defective and can be improved? What constitutes evidence against the validity of a given decision model? How is this evidence exploited to improve the decision process? While this may seem a circuitous route to actually improving the decision algorithm, it is in fact the "long road which in eff"ect is short". There is no universal format for decision. One gropes along in the shadow of incomplete and uncertain information. The best compass is falsification: a method of exclusion in the search for improvement. In other terms, we will use infogap decision theory to criticize and improve the structure of the specific decision algorithm. We are considering a two-tiered decision situation: an algorithm is making operative decisions, while info-gap decision theory is hovering over the algorithm in a supervisory role. The higher-level decision that we concentrate on is directed towards enhancing the decision algorithm itself. In section 8.2 we will consider the info-gap supervision of a classifier. Both levels of learning are represented in this example. At the immediate operative level the decision algorithm assimilates and processes information which enhances the decision maker's picture of the environment. However, the classifier is not perfect. Its fundamental limitation will be seen to reside in the inadequacy of the info-gap models upon which the classification deci-

C H A P T E R 8.

LEARNING

209

sions are based. Learning in the long-range and strategic sense is achieved by exploiting new data to improve the uncertainty models, thereby modifying (and hopefully improving) the structure of the decision algorithm itself. In section 8.3 we consider a learning problem which does not involve classification. Again info-gap decision theory supervises a decision process, improving it by updating the info-gap model of uncertainty upon which the decision process depends.

8.2

Info-Gap Supervision of a Classifier

8.2.1

Robustness of a Classifier

We consider a decision problem in which a quantitative measured vector u is employed for selecting one from among a number of classes. In a medical diagnosis, for instance, u may contain the results of hematological measurements, biochemical analyses, bacteriological culture-counts. X-ray and tomographic assessments, and so on, while the classes to be distinguished are distinct pathological conditions. While strategic structural modification of the decision process may entail revision of the classes themselves, the operative decision algorithm assumes that any measured vector arises from one and only one class. The difficulty of course is that no class produces a unique vector of measurements. In fact, even in medical diagnoses for which extensive population data exist, the range of sub-population variations in disease manifestation, combined with lack of information about the patient, can result in quite gross fluctuations of response for a given pathology. Consequently, the range of u-vectors for class n is represented by an info-gap model of uncertainty:^ Un{0i,Un),

a > 0.

The class index n can be a taxonomically informative vector rather than a single number. For instance, if the medical diagnosis is 'acute osteomyelitis of the right second metatarsus' then n — (730,02,1) which is an extension of the standard medical diagnostic classification of this condition. Using taxonomically informative index vectors enables one to define a measure of distance between index vectors as an indication of the degree of qualitative diff'erence between the corresponding classes. We will denote this distance measure by || • ||. A classification algorithm is denoted as a function C{u) whose argument is the measurement vector u and whose function value is the index vector n of a class. Thus 'C{u) = n' represents the decision that measurement u arose from class n. Furthermore, if i^ is a measurement vector arising from class m, then ||C(2x) — m|| is a quantitative and meaningful assessment of the error in the classification. ^A probabilistic version of this example would consider estimated probability densities for u in each class, with info-gap models for the uncertainties in these pdfs.

210

INFO-GAP DECISION THEORY

The robustness function for algorithm C is the greatest horizon of uncertainty at which the error in the classification is no greater than r^^ a{C,rc) — max
max niax_

||C(u) — n||

< TC >

(8T)

When S(C, Tc) is large, then the error in identifying each class is no greater than Tc, even at large uncertainty. Also, the usual trade-off between robustness a and classification error TC will be observed. A small value of TC is more demanding of the classifier than a large value, meaning that 'reward' increases as Vc decreases. The trade-off is expressed by the fact that, as the reward is increased (by reducing TC), the robustness S(C, TC) is decreased. The robustness 3(C, TC) of any classifier C{u) can be calibrated in terms of the asymptotic robustness inherent in the info-gap models of uncertainty, which we now explain.

8.2.2

Asymptotic Robustness

An info-gap model U{a,u) is 'unbounded' if, for any vector u, there is some sufficiently large horizon of uncertainty a such that u belongs to the info-gap model at uncertainty a: u e U{a,u). Most of the commonly encountered info-gap models are unbounded. For example, the ellipsoidal info-gap model: U{a,u) = {u: {u-ufW{u-u)
a>0

(8.2)

is unbounded. For any vector u there is an a such that the ellipsoid U{a,u) contains this u. If the info-gap models which represent the class uncertainties in the classification problem are unbounded, then a measured vector u could, in principle, have arisen from any of the classes. This is the root of the classification ambiguity from the info-gap point of view. The asymptotic robustness of a classification problem is the greatest value of the uncertainty parameter for which the info-gap models are all disjoint: 3oo=niax{Q;: Un{oL,Un) n Um{(^^Urn) = ^ for all m ^ n}

(8.3)

The asymptotic robustness is independent of the classification algorithm C{u), and depends only upon the structure of the info-gap models. If the info-gap models accurately represent the ambient uncertainty, and if the actual level of ambient uncertainty is less than 3oo, then any measured vector u is consistent with one and only one info-gap model. If the ambient uncertainty is less than the asymptotic robustness then, in principle, it will be possible to correctly identify the provenance of any measured vector. For example, a classifier which performs an exhaustive search of the sets ^n(3oo,2n) will find one and only one set containing u.

C H A P T E R 8.

LEARNING

211

A different classifier C{u) may fail to correctly classify a given measurement, but no classifier can have a robustness, with classification error fc = 0, which is greater than the asymptotic value. In other words, Soo is the least upper bound of the error-free (re = 0) robustness of any realizable classification algorithm C{u): S(C,0)
(8.4)

Any classifier, no matter how sophisticated, will encounter the ambiguity of multiple membership once the level of uncertainty exceeds the asymptotic robustness. Conversely, there is at least one classifier (the exhaustive search algorithm) which has an error-free robustness equal to the asymptotic value.

a > ac

Ui{a,ui) Figure 8.1: Intersecting info-gap models, illustrating ambiguous classification for

The info-gap classification ambiguity is illustrated in fig. 8.1. The uncertainty sets of two info-gap models are plotted for an a-value greater than the asymptotic robustness, so they necessarily intersect. Any measurement vector u belonging to the intersection could have arisen from either model, however nothing marks the provenance of u more specifically than that. No algorithm can invariably choose correctly between these two alternatives. Consider, for instance, the measurement u indicated by the dot in fig. 8.1, belonging to the intersection of these two info-gap models. Any classifier C{u) can be "fooled" by feeding it this measurement, once from Ui{a,ui) and once from U2{a,U2). The classifier must reach the same decision both times, so it must err once. Soo defines the limiting level of uncertainty beyond which info-gap classification ambiguity occurs for any realizable algorithm. Consequently, we can use Soo to calibrate the robustness of classification algorithms. If the robustness of an algorithm C{u) is much less than the asymptotic robustness then the algorithm could be improved. If, on the other hand, S(C, 0) is only slightly less than Soo, then this classifier is performing about as well as can be demanded with respect to the info-gap models which are being used. The asymptotic robustness is a property of the info-gap models alone, which in turn depend on the information available to the decision maker.

212

INFO-GAP DECISION THEORY

If the uncertainty models can be improved, then the asymptotic robustness will change, and the expectations from classification algorithms will change. The process of updating the info-gap models by exploiting new information is the basis for a structural change in the classification algorithm. This 'structural learning' is the focus of this section.

8.2.3

R o b u s t - O p t i m a l Classifier

We begin by specifying a classifier whose robustness (at zero classification error, TC == 0) equals the asymptotic value. Such a classifier, which is not necessarily unique, is called a robust-optimal decision algorithm. No algorithm can have greater robustness at re = 0. The pivotal importance of updating info-gap models, in the structural learning of a classification task, will be evident from the structure of this robust-optimal classifier. We first define the gap function of an info-gap model ZY(a, u). For any vector u (in the space of measurements) the gap function T{u) is the lowest value of the uncertainty parameter a at which U{a,u) contains u\ T{u)=m:m{a: ueU{a,u)}

(8.5)

T{u) is the smallest horizon of uncertainty at which the uncertainty model U{a^u) accommodates the measurement u. Because of the nested structure of info-gap models, u belongs to U{a,u) for all a > T{u). When T{u) is small, the measurement u is compatible or coherent with the model U{a, S), in the sense that u can arise from U{a,u) without invoking extraordinary circumstances. On the other hand, if T{u) is large then the measurement u is incoherent with the model: u can arise from U{a, u) only under extraordinary conditions deviating widely from the norm, u. Of course, what constitutes 'small' or 'large' values of the gap function requires qualitative calibration of the uncertainty parameter, either in terms of the asymptotic robustness or as discussed in chapter 4. Given a collection of info-gap models, we denote the gap function of uncertainty model Un{<^,Un) by Tn{u). Consider the following classification algorithm: C{u) = n if Tn{u)
(8.6)

This classifier assigns the measured vector u to any info-gap model whose gap function for u is no greater than the gap functions of all the other models. The assignment is unique if strict inequality holds in all the gapfunction comparisons in (8.6). C{u) chooses the provenance of u as the info-gap model which accommodates the measurement at the lowest level of uncertainty. This is a "nearest neighbor" decision algorithm, whose closest probabihstic analog is the class of maximum likelihood decision algorithms. If strict inequality does not hold in all the gap-function comparisons in (8.6) for a measured vector u, then this classifier will not uniquely identify the origin of this u. An instance of this ambiguity is shown infig.8.2.

C H A P T E R 8.

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Any li-vector situated on the vertical chord midway between these two symmetrically structured info-gap models will, of necessity, have the same gapfunction value for each model. The classifier of eq.(8.6) will assign this u to both models. (Or, some arbitrary tie-breaking rule may be used to select one or the other model, but this rule must then sometimes err.) a > S(C,0)

Ui{a,ui)

U2{ot,U2

Figure 8.2: Intersecting info-gap models, illustrating ambiguous classification for a>S(C,0). Note also that the a-value at which the models in fig. 8.2 are plotted must necessarily exceed the error-free robustness of the algorithm precisely because the sets intersect. No algorithm can always successfully classify events which occur at an a-level which allows set intersection. That is: Ui{a,ui)

r\U2{(y,U2) 7^ 0 implies that

a > a(C,0)

(8.7)

This relation is understood by considering the measurement vector marked with a dot and labeled u' in fig. 8.2. While this vector belongs to both models at this level of uncertainty, for an a' which is only slightly less than a one will find that: u'eUi{a\ui)

while

u'^U2{a\u2)

(8.8)

This means that the gap-function value for Ui{a, ui) is less than for U2{ot, U2): Tt{u')
(8.9)

Thus u' is never assigned to U2, even though u' could arise from U2 because u^ G U2{cx,U2) for sufficiently large a. We again see that the asymptotic robustness is the least upper bound for the robustness of all realizable algorithms, as asserted in relation (8.4), because SQO is the greatest a-value at which the info-gap models are disjoint. It can now be appreciated that, for unbounded info-gap models, the error-free (re = 0) robustness of the classifier in eq.(8.6) equals the asymptotic robustness for these info-gap models: 3(C,0) The outline of a proof of eq.(8.10) is presented in section 8.2.4.

(8.10)

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INFO-GAP DECISION THEORY

8.2.4

t A Proof

In light of relation (8.4), in order to prove eq.(8.10) it is sufficient to show that: Soo<S(C,0) (8.11) Let ^oo denote the set of a-values in eq.(8.3) whose least upper bound is the asymptotic robustness 3oo- Likewise, let Ac represent the set of Q;-values in eq.(8.1) whose least upper bound is the robustness of algorithm (8.6) at zero classification error, 3(C, 0). That is: -4oo

=

{Oi ' Un{0L,Un)

H Um{Oi,Um)

Ac

— [oi : C{u) = n

= ^

for all m ^ 77,}

(8.12)

for ah u G Un{ot, Un) and for ah n] (8.13)

Choose an arbitrary a* belonging to ^ooa* G ^oo

(8.14)

For any u such that: ueUn[a\un)

(8.15)

we conclude, from the definition in eq.(8.5), that: Tn{u) < a *

(8.16)

But since a* belongs to the set ^oc we see that Wn(<^*,^n) is disjoint from Ujnic^'^^Um) for any m ^ n. Hence, in light of (8.15): U^Um{ci\Um)

(8.17)

r ^ ( ^ ) > a*

(8.18)

Consequently:

From relations (8.16) and (8.18) we see that: ^n{u) < Tm{u)

for any

m ^ n

(8.19)

which implies that: C{u) = n

(8.20)

a* G Ac

(8.21)

Aoc C Ac

(8.22)

which in turn implies that:

It is now evident that:

This implies (8.11) and completes the proof.

C H A P T E R 8.

8.2.5

LEARNING

215

Robust Severe Tests of T r u t h

We have established that the classifier of eq.(8.6) is a robust-optimal algorithm. We need search no further in trying to improve upon it: no other algorithm can have a greater error-free robustness, assuming the underlying info-gap models are correct. The task of improving the performance of the classifier thus devolves onto the task of updating the uncertainty models. The current info-gap models may reflect quite well the structure of the ambient uncertainty. We don't want to tamper blindly with the uncertainty models. The question then becomes: what constitutes evidence against a given set of info-gap models? Once we know how to identify erroneous models we can proceed to the search for better ones. The methodology which we will develop has a two-sided irony. The falsification of an info-gap model will be based on a criterion for truth, while the search for better (that is, truer) models will be based on a concept of error. In this subsection we will discuss the concept of a severe test of the truth of a proposition. This test will be the basis for judging uncertainty models, rejecting (and then updating) those which fail the test. Then, in the next subsection, we explore the updating of info-gap models based on a measure of the extent of error of a model. Truth and falsity are complements: exclusion of one invokes the other. Let us suppose we have "learning data'': a collection of measurement vectors ui^ . . . , UK of known provenance. That is, measurement Uk is known to have arisen from class n/^, for /c = 1, . . . , X. If the learning data are all reasonable (none is extraordinary or pathological with respect to its class of origin), then we can reasonably hope to correctly classify each of the measurements. Since we know the classorigin of each measurement, we know that correct classification according to eq.(8.6) requires:

rn,M
(8.23)

^OY all k = 1, ... ,K and for all i ^ k. So, still under the "reasonable data" assumption, if:

rn.(^fc)
(8.24)

for any k and for any i ^ k^ then (8.24) is evidence against one or both of the info-gap models Un, (o^, u^) and Urik i^^ ^nk)In other words, if the learning data are all reasonable, then we should modify the info-gap models so that all the learning data are correctly classified. Conversely, a collection of info-gap models is accepted (as provisionally true) to the extent that only extraordinary data are incorrectly classified. Re-stating this in terms of hypothesis testing: the info-gap models will have passed a "severe test of truth" if they correctly classify all but extraordinary data. Conversely, we reject a collection of info-gap models if they do not pass this test.

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INFO-GAP DECISION THEORY

We are subjecting a proposition to a severe test if an erroneous inference concerning the truth of the proposition can result only under extraordinary circumstances. Mayo has developed an account of scientific inference based on learning from error and severe tests of hypotheses which, while employing probabilistic thinking, is quite readily adapted to info-gap decisions. Mayo's probabilistic formulation [121, p.445] can be slightly modified, as follows, to express the basic intent of a robust severe test: It is learned that an error is absent from a proposition to the extent that a procedure of inquiry with high robustness for detecting the error if and only if it is present nevertheless detects no error. In deciding whether to update a collection of info-gap models, the proposition we are testing is that the models are realistic. The severe test has been successfully passed if all but extraordinary and highly irregular measurements are correctly classified by the robust-optimal classifier based on those uncertainty models. Furthermore, once we reject an info-gap model which has failed a robust severe test, we proceed to modify the model so that it would pass the test. For further discussion of robust severe tests, and their relation to probabilistic inference, see [16, 20].

8.2.6

U p d a t i n g Info-Gap Models

Let us suppose, as before, that we have "learning data" in the form of K measurement vectors whose classes-of-origin are known: measurement Uk from class n/^, k = 1, . . . , X. The uncertain variation in the measurement vectors of each class is represented, to the best of our present knowledge, by an info-gap model: Un{oL,Un) for class n. Also, the gap function, eq.(8.5), for the nth info-gap model is denoted Tn{u). Now suppose that the robust-optimal classifier, eq.(8.6), fails to correctly classify one or more of the points in the learning set, as in relation (8.24). That is, the classifier has failed the severe test of the validity of the info-gap models, so that at least some of these uncertainty models are judged to be neither true nor realistic. What is to be done? Three types of corrective action can be taken. First, it may be that the basic conception of the event classes is erroneous and must be revised. For example, there may be an event class which is not represented by any infogap model, or the reverse, that a coherent class is artificially bisected by two uncertainty models. Second, it may be that the structure of the info-gap models of uncertainty must be altered. For instance, interval-bound models may need to be altered to ellipsoidal-bound models. Finally, it may suffice to modify some of the parameters which determine the specific realizations of the uncertainty models, such as centerpoint vectors or shape matrices.

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We will consider the last option, since it generally is most suited to the severely limited information characteristic of info-gap decisions. Let q represent the vector of info-gap parameters which can be updated in an attempt to improve the classification capability of the robust-optimal algorithm. That is, q contains parameters of the info-gap models Un{oL, Un), such as centerpoint vectors or weighting matrices. We will formulate an empirical robustness function which depends on the update parameters q as well as on the learning data. This empirical robustness will provide the basis for updating the info-gap models. For any classification algorithm C{u), the empirical robustness function ae(^, Tc) is an estimate of the greatest value of the uncertainty parameter a for which C{u) makes classification errors no greater than r^- The empirical robustness is evaluated from the performance of C{u) in sorting the data of a learning set (iii,ni), . . . , {UK^TIK)- Measurement Uk comes from class n^, and the gap function for the corresponding info-gap model Uuk (<^7 ^uk) is r^^ {uk)- A gap function is a specific value of the uncertainty parameter a (see eq.(8.5)), so the empirical robustness function is estimated from the gap functions evaluated on the learning data: Se(
:

\\C{uk) - rikW < r^ \iTnk{uk) < ^nAui), A: = l, . . . , X ,

2 = 1, . . . , K }

(8.25)

(3e(g',rc) = 0 if the set defined in (8.25) is empty.) Roughly speaking, Se(g',rc) is the greatest "uncertainty horizon" within which the learning data are all classified with error no greater than Vc- More precisely, 3e(g, r^) is the value of the greatest measured gap function, F^. (i/^), for which all measured vectors Uk (including Ui) whose corresponding gap functions are no greater than F^. (it^), are classified with error no larger than r^The condition ^Tn^i^k) ^ T^.(li^)' is equivalent to:

Thus we see that all measurements in the learning set which satisfy: Uk el(nk{Se{q,rc),Unj,)

(8.27)

are classified by C{u) with error no greater than TC; there may be at least some other data points that are classified with greater error. The error-free (re = 0) empirical robustness function is: Se(^,0) = m a x | F n , ( w O :

C{uk) = Uk if Fn^(i^fc) < ^miui)^ /c = l, . . . , K ,

2 = 1, . . . , K }

(8.28)

In particular, for the robust-optimal classifier in eq.(8.6), the error-free empirical robustness function can be expressed as:

218

INFO-GAP DECISION THEORY

j^k,

/ c - 1 , . . . , i ^ , i = l, ...,ir}(8.29)

An empirical robustness function can take any non-negative value. If the data are very unusual, or if the info-gap models are very unrealistic, then Se(g, 0) will be small: only very few (or perhaps none) of the learning data will be correctly classified. However, ae{q,0) can be small for an utterly opposite reason as well: if the data are highly compatible with the info-gap models then only a low level of uncertainty will be needed to envelope the data within the uncertainty models. Likewise, the empirical robustness may be large for either of two contrary reasons. ae{q,0) can be large if the info-gap models poorly reflect the actual ambient uncertainty, because then the uncertainty models may become greatly distended before "capturing" even some of the data. Or, on the contrary, if the uncertainty models have high fidelity to the uncertain phenomenon, then ae{q,0) wih be large when C{u) successfully classifies extreme data. We see that the "bigger is better" aphorism which we have applied repeatedly to robustness functions (encountered for the first time in section 3.1.1), is not entirely appropriate for the empirical robustness function when 3e(^,rc) is used for updating uncertainty models. A large value of ae(g, Tc) may be either desirable or not, depending on why the large value is obtained. The decision vector q of Se(g',rc) contains parameters of the uncertainty models, and we cannot automatically assume that q should be modified so as to increase the empirical robustness. In fact, when the empirical robustness is used as a decision function for updating uncertainty models, a combined increase-and-decrease strategy must be employed. This will become evident in the example of section 8.2.7.^ Special treatment is needed in decisions which are aimed at updating uncertainty models. This is evidence that, in tampering with uncertainty models, we are touching a very fundamental situation, diff"erent in character from other decisions. All decisions from uncertain evidence (as distinct from deductive decisions like mathematical proofs) involve judgments of the pragmatic truth of a proposition, as explained in the discussion of severe tests of truth in section 8.2.5. Also, a severe test of truth depends intimately upon the assessment of uncertainty. To roughly paraphrase that discussion: a proposition is (pragmatically) true to the extent that only highly irregular or extreme evidence would contradict the proposition. In order to apply a severe test of truth one must be able to assess degrees of uncertainty. This is where the robustness function comes into play in info-gap decision theory, or the probability function in statistical decisions. When the decision is •^This is an example of a decision vector which modifies the uncertainty models, as first mentioned on p.45.

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directed at the truth or falsity of the uncertainty model itself, the severe test crumbles from within because we no longer know how to assess degrees of uncertainty, and we can no longer rely on large robustness as a clear and constant guide. We have in fact already encountered something similar. In comparing the robustnesses of two different decision models we noted that the uncertainty parameters of the respective info-gap models must be commensurate (p. 196). The more robust decision model is preferable provided that the calibration of the uncertainty parameter does not change across the comparison.

8.2.7

Plantar Pressures in Metatarsal Pathology

We will illustrate the updating of two info-gap models for the simplified representation of uncertain dynamic morphology of the foot during walking. The pressures occurring at specific locations in the foot change substantially during each heel-to-toe gait cycle. Pressures vary considerably between individuals, but the stability and comfort of normal walking depend on the magnitudes of these pressures. In addition, numerous orthopedic pathologies are accompanied by abnormal pressure distributions. The pressure distributions for any given pathology are characteristic, but also show significant variation, depending both on the severity of the pathology and on individual characteristics. A common pathology is the occurrence of a bunion on the first joint of the great toe. This pathology, known as hallux valgus, is characterized by an enlargement of the angle between the 1st metatarsus and the 1st phalanx. This alteration of the relative positions of these bones can result in modified stress distributions during walking. Prolonged excess pressure on the 1st metatarsal-phalangeal joint can result in inflammation and infection. Surgery is required to rectify extreme cases. In our simplified example we will consider the uncertain variation of the pressures under the first and fifth metatarsal-phalangeal joints, though in practice the pressures induced elsewhere are also important. We will consider only two info-gap models, one for the normal condition [n = I) and one for a pathological state (n = 2). The info-gap models will be denoted hin{(y-^Un)- The goal is to update the pathological info-gap model based on very hmited data. For instance, the initial info-gap model for the pathological state may represent the variation of plantar pressures in the afflicted population at large, while we are treating a specific group such as the members of a sports club. We wish to update the pathological info-gap model to represent the peculiarities of this group. In the normal condition the pressures under the first and fifth joints are typically equal, though of course large individual and activity-related variation is observed. In the pathological condition, both the magnitude of the pressure and the range of its variation under the first joint are enhanced

220

INFO-GAP DECISION THEORY

relative to the fifth. Let u be the vector containing the first- and fifth-joint plantar pressures normahzed to the subject's body weight. The information about the uncertain pressure variations is quantified in the following ellipsoidal info-gap models: Ui{a,ui)

= {w = S i - h 0 : (/>^0
U2{a,U2)

=

{u = U2 + (p: (l)^W(l)
a>0 a>0

(8.30) (8.31)

where 1^ is a positive definite diagonal matrix determining the shape of the uncertainty ellipses in the pathological state. It is the shape matrix W which we will update using learning data. The nominal pressures in the two states, relative to average body weight, are:

"^ = ( 0 ' ^^ = (a8)

(«-^2)

The gap function for measurement u, defined in eq.(8.5), is the least a for which the uncertainty model is consistent with the measurement. In the present case, the gap function Ti{u) for healthy-state uncertainty model Ui{a^ui) is simply the Euclidean distance from i^ to Si: Tiiu) = yj{u-ui)^{u-ui)

(8.33)

The gap function T2{u) for the pathological uncertainty model U2{oL-,Un) is the weighted Euclidean distance: V2{U) = yj{u - U2YW{U - U2)

(8.34)

Given learning data, one can use these gap functions to evaluate the empirical robustness of the robust-optimal classifier. In our example we have K pressure vectors, i^i, . . . , UK^ all from class-2 (pathological) subjects. Using eq. (8.29), the empirical error-free robustness of the robust-optimal classifier is evaluated as: 3e(g,0)=max{r2(ufc): T2{uk) <Ti{uk),

A:-l, ...,X}

(8.35)

If the set in this relation is empty then we define 36(^^,0) = 0. In our example, the parameters to be updated are the elements of the diagonal shape matrix, ^o q = W. The open circles infig.8.3 represent six learning-data points from (hypothetical) pathological subjects. The solid dots denote the centerpoint vectors, ui and U2 from eq.(8.32), of the healthy and pathological uncertainty models, respectively. We will evaluate the empirical robustness function for this data, eq.(8.35). The update parameters are the diagonal elements wi and W2 of the diagonal

C H A P T E R 8.

5th tarsal pressure

Se(u;i,0)

0

0

1 1.3 1st tarsal pressure

Figure 8.3: Nominal measurements (•) ui and U2, and learning data (o) ui, ... ,UK-

221

LEARNING

0.3

0.6

Figure 8.4: Empirical robustness function Se(w^i,0) vs. shape parameter wi.

shape matrix W. T h e goal is to u p d a t e U2{oi,U2) by varying t h e shape of the ellipse. T h e area of an ellipse defined by (p^Wcj) = 1 is -KjyJw\W2' Since it is the shape and not t h e area of this fundamental ellipse which we wish to optimize, we will vary wi and adjust W2 according to W2 = l/wi. Fig. 8.4 shows the calculated empirical robustness function versus the shape parameter wi. T h e jagged features of this curve arise as follows. Only t h e pathological gap function T2{uk) varies with Wi] the healthy-state gap function Ti{uk) does not change with W. Along the segment from a to b only three of t h e six d a t a points are correctly sorted. T h e empirical robustness is decreasing from a to 6, which is an improvement in the infogap model ^ 2 , since it (and its gap function r 2 ) are adapting themselves to the three measured pressure vectors which are successfully sorted. T h a t is, as wi increases from a to 6, the info-gap model becomes more compatible with the correctly-sorted data. At point h there is a discontinuous j u m p to point c in t h e empirical robustness as suddenly an additional d a t u m is correctly classified. T h e "capture" of a new d a t u m constitutes an improvement in the gap function, though it is now manifested as an increase in t h e empirical robustness function, while from a to 6 we understood t h a t r 2 was improving because a© was decreasing. T h e "capture" of the new d a t u m , at the transition from b to c, causes Se to increase because the "horizon of uncertainty", within which all classifications are correct, has grown. From c to d the uncertainty model readjusts to the foursome of points (below and to the right of U2 in fig. 8.3) which are correctly classified. T h e empirical robustness decreases along this segment just as it did from a to b. At point d there is a discontinuity in the slope of t h e empirical curve. W h a t has happened is t h a t a different captured d a t u m (from among t h e same foursome) has become t h e critical point: the particular vector Uk

222

INFO-GAP DECISION THEORY

which, in eq.(8.35), has the maximum gap-function value. From point d to point e the uncertainty model is deteriorating: the model is becoming less harmonious with the correctly classified points. If the curve were continued to greater values of Wi the classifier would eventually "lose" successively more and more of the correctly sorted data. Of course, these "losses" would appear in the Sg-vs.-if i curve as sudden drops in the empirical robustness, though the general profile would be a rising saw-tooth shape. From this discussion we see that the optimal update of the pathological info-gap model, ^2(0;, 1^2), is a value of shape matrix W which minimizes the empirical robustness function. At point d in fig. 8.4 we find Wi = 0.38 and W2 = l/wi — 2.61. This corresponds to a semi-axis ratio of the ellipses in the family of uncertainty sets equal to \/w2lw\ — 2.62, the horizonal axis being the long axis of the ellipse. However, we must be careful not to conclude that a global minimum of the empirical robustness is necessarily the optimal update. There will be points along the curve (not shown in fig. 8.4) where Sg = 0 because no measured points are classified. Such points are global minima, but clearly they do not correspond to optimal updates of the info-gap model. It is necessary to consider the causes of variation of the empirical robustness, as we explained on p.218. In our discussion of the example we concluded that both increase and decrease of Sg, under different circumstances, can be interpreted as desirable changes. In short, U2 is updated by minimizing the empirical robustness (rather than maximizing the robustness as we are accustomed to from other decision problems), but not necessarily to the global minimum. Nor are all local minima, such as point h in fig. 8.4, optimal updates of the info-gap model.

8.3

Acoustic Noise

Not all decisions can be usefully described as classifications, and not all strategic learning can be analyzed as the info-gap supervision of a classifier. In this section we consider a very different learning situation. We all recognize acoustic noise when we hear it. Nonetheless, a generic characterization of sound fields constituting "noise" is difficult. Despite the difficulties, recent years have witnessed a growing number of commercial appliances for active suppression of acoustic noise. A feedback circuit is employed wherein the amplitudes of an array of loudspeakers are adjusted to minimize the squared acoustic pressures on an array of sensors. Applications have been developed for a number of low-frequency noise sources such as duct-borne noise from air conditioning units, automobile or turbofan engine noise, road noise, and others [52]. A highly successful commercial unit is a noise-cancelling headset which uses feedback to suppress acoustic disturbances in the vicinity of the ear. The proximity of sensor and loudspeaker enables quite satisfactory performance.

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Suppression of noise is more difficult in large acoustic cavities subject to complex and variable noise sources, where the sensor and actuator arrays are sparse compared to the spatial volume to be controlled. In such applications it is essential to model and manage the uncertain variation of the acoustic field. In this section we will consider the updating of an info-gap model for the uncertain spatial variation of acoustic pressure.

8.3.1

Empirical Robustness

Let f{x) represent the acoustic pressure at position x in the cavity. The sound field f{x) varies from case to case, or over time. We sample the noise field at discrete points and obtain only partial information about the entire continuous field. In other words, we do not know f{x) completely; this lack of knowledge of the sound field is the main uncertainty which the acoustic controller must handle. A noise suppression system must be optimized with respect to the unknown noise field. For instance, a controller is good if the total residual sound energy, after suppression, is small regardless of what noise field f{x) actually occurs in practice. Such a controller is robust with respect to the uncertainty in the sound field. On the other hand, if small variation in the sound field can cause substantial residual energy, then the controller is fragile with respect to uncertainty, and this is undesirable. We can never know the total residual energy because we are able to sample at only a discrete number of points. Consequently, both the design of the acoustic suppression system and the evaluation of its performance depend intimately upon the model of the uncertainty in the noise field. The variation in the acoustic field can be quite dramatic. Consider for instance the fluctuation in the ambient noise in the passenger compartment as a car moves from city street to highway, from open flat terrain to tunnel or bridge, to slopes and curves of various grades, and so on. The range of variation, and the diversity of factors influencing the noise field in the passenger compartment, are enormous. These variations motivate the need for updating the info-gap model in real time with limited data. Let U{a, f) be an info-gap model for the severely uncertain variation of the acoustic field f{x). We wish to use measured acoustic pressures to evaluate this uncertainty model, and to update it if necessary. This evaluation and revision is the "strategic learning" in this example. Typically, this learning must occur with limited measurements from a small number of sensors. Let us suppose we have measured acoustic pressures ui, . . . , UM at positions xi, . . . ,XM' These positions may all be different, or measurements may have been repeated at different times at some of the sensors. The basic question we ask is: do these measurements provide evidence against the info-gap model U{a, f)? Then, if the info-gap model is indicted, how do we improve it?

224

INFO-GAP DECISION THEORY

If a measurement is "reasonable", if it is neither extraordinary nor highly unusual, then it should be compatible with the uncertainty model. Conversely, if U{a, f) is incompatible with some of the observed pressures, except for highly irregular ones, then the info-gap model must be revised. As in section 8.2, we use the gap function to define an empirical robustness which assesses the compatibility of uncertainty-model and measurements. Let q be the vector of parameters or properties o^ the info-gap model which can be modified in attempting to improve W(a, / ) , if necessary. The gap function is defined in eq.(8.5). In this example we evaluate the gap function for the mth datum {xm.Um) as: T{um) = min \a : f{xm) = Um for some f{x) e U{a, f)\

(8.36)

A measured acoustic pressure is "reasonable" if it is not excessively large. The uncertainty model Z//(a, / ) is compatible with the measurements if the gap functions of "reasonable" measurements are also not large. We combine these two considerations in the following empirical robustness function: Se(g,rc) = m a x | r ( i x ^ ) :

\un\ < r^ \iT{un) < T{um), m = 1, . . . , M , n = 1, . . . , M | (8.37)

Se(^, ^c) is an estimate of the greatest horizon of uncertainty within which all the measurements are no larger than r^ • Equivalently, this is an estimate of the greatest level of uncertainty, assessed on the basis ofU{a,f), which is consistent with acoustic pressure no greater than r^ Seiq^Vc) assesses the fidelity of the model to the measurements, and is the basis for updating the info-gap model. The empirical robustness in eq.(8.37) is quite similar, both conceptually and structurally, to eq.(8.25) for the empirical robustness of the robustoptimal classifier. All data which satisfy: UmeU{ae{q,rc)J)

(8.38)

also satisfy: \um\
(8.39)

There may be other data whose amplitude is greater. The empirical robustness, ae(g, TC), can be large for either of two opposing reasons, as we discussed on p.218. Se(g',rc) can be large because the info-gap model ZY(a, / ) is a true and accurate characterization of the acoustic uncertainty, and is highly compatible with all the data, even extreme measurements. Or, contrarily, Se(g,rc) may be large because U{a,f) is so erroneous that extremely large gap-function values are needed to capture even a few data points. The empirical robustness ae{q,rc) depends upon the update parameters q of the info-gap model. However, we see that, as in our discussion

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in section 8.2.6, we cannot update the info-gap model by indiscriminately optimizing the empirical robustness. Because we are modifying the uncertainty model, and thereby tampering with the essence of the robust severe test, we must understand why 3e(g,rc) is changing as we modify q. Only in this way can we sensibly update the uncertainty model.

8.3.2

Updating the Acoustic Uncertainty Model

We will discuss a simple illustration of updating an info-gap model for acoustic uncertainty. Consider a one-dimensional acoustic cavity whose domain is 0 < X < 1. For instance, this could crudely represent a cross-section of the passenger compartment of an automobile. The acoustic pressure field f{x) is the fluctuation around the undisturbed ambient pressure. The precise structure of the sound field is of course complex. However, its general character may be roughly uniform across the cavity, or biased predominantly to the left or to the right, depending on the surrounding noise sources. Furthermore, the shape of the field may "slosh around" fairly rapidly as ambient conditions change. We thus need to be able to adapt the uncertainty model quickly, based on very few measurements. The following envelope-bound info-gap model is useful in this situation: U{a,0) = {f{x):

\f{x)\
a>0

(8.40)

We will consider linear envelope functions of the form: i^{x) ^q(x-^^+l

(8.41)

These functions all cross the point (0.5,1), and the slope is determined by the slope parameter q which must be between —2 and +2 to assure that the envelope is non-negative over the domain of x. When the acoustic disturbance is coming predominantly from the right {x > 1/2), then q is positive; acoustic fields distorted to the left are represented by q < 0. A small number of measurements will be used to update the info-gap model by modifying the slope parameter q of the envelope function jp{x). Two sensors are available, one near each end of the cavity: at x = 0.05 and at X = 0.95. Fig. 8.5 shows the empirical robustness, evaluated from eq.(8.37) with six measured pressures, three at each side of the cavity. (Pressures at X = 0.05: 0.07, 0.21, 0.33. Pressures at x - 0.95: 1.83, 1.97, 2.15.) The empirical robustness is evaluated with a critical value of re = 2.5, so all six measurements are captured. The left branch of the calculated robustness curve, from a to 6 in fig. 8.5, shows decreasing robustness as the slope increases from g = 0 to ^ = 1.63. The info-gap model is improving from a to b as it becomes more consistent with the measurements. The right-hand branch, from b to c, shows degradation of the uncertainty model, which is

226

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Figure 8.5: Empirical robustness function Se(g, 2.5) vs. slope parameter q.

Figure 8.6: Six pressure measurements (o) and the updated envelope function V^(x) with q = 1.63.

becoming less compatible with the data. We conclude that at this value of acceptable noise, r^ = 2.5, the most compatible uncertainty model is obtained with the slope parameter g = 1.63. Essentially the same result is obtained with any value of Vc which is sufficiently large to capture at least one of the right-side measurements. On the other hand, for r^ < 1.83, only the left-side data are captured, the empirical robustness curve rises steadily with increasing g, and the optimal ^-value is zero. However, this generates an info-gap model which is not at all compatible with the data, since the gap-function values for the right-side data are quite large. Fig. 8.6 shows the optimal envelope function, ip{x) with q = 1.63, together with the six measured pressures. The positive slope of the envelope clearly reflects the predominately right-sided pressure field. However it is evident that this envelope function is not simply a parametric best-fit to the data. In addition, there is no presumption that the actual pressure fields are linear. On the contrary, it is to be expected that the pressure variation across the cavity is complicated and variable. This ?/^-function is optimal in that it generates the envelope-bound info-gap model of eq.(8.40) which is most compatible with the measured pressures, in terms of their gap function values.

8.4

Summary

We have studied several examples of improving an info-gap decision algorithm by modifying the underlying info-gap uncertainty model. Modification of the uncertainty model is based on evidence which impeaches the

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model. This two-stage process — impeach and improve — begins with the recognition of error, followed by adaptation of the info-gap model to augment its fidelity to the evidence. Two basic functions bear the burden of this process: the gap function T{u) and the empirical robustness function Se{q, TC), both of which depend intimately upon the info-gap model U{a^u). These functions are combined with a collection of measurements whose uncertainty is purportedly represented by the info-gap model. The gap function T{u) expresses the degree of compatibility between the uncertainty model U{a, u) and the measurement u. The empirical robustness estimates the horizon of uncertainty within which the info-gap model captures some fraction of the measurements, where greater or lesser fractions may be chosen by the analyst. As such, ae(g^, Tc) assesses the fidelity of the uncertainty model to the measurement set as a whole. The gap function and empirical robustness are central both to testing the validity of the uncertainty model and to modifying it if necessary. In the classification problem in section 8.2 the uncertainty model is tested by applying the robust-optimal classifier (based on the gap function) to a set of learning data of known provenance. The info-gap model is judged to be realistic if only extraordinary data are mis-classified. In the uncertain acoustic-field problem in section 8.3 the severe test of the infogap model is absorbed directly into the updating process: if the empirical robustness cannot be improved, then the model is adequate. The decision vector q of the empirical robustness function Se(g', Vc) represents those features of the info-gap model which can be modified. 3e(g', r^) is used to update the uncertainty model by answering the question: what changes would make the info-gap model more compatible with the available evidence? A datum is compatible to the degree that it belongs to the uncertainty model at a low horizon of uncertainty; a highly incompatible datum is on a far horizon, q is modified so that 3e(9, TC), the horizon within which a substantial amount of the evidence falls, is as small as possible. In fig. 8.4 on p.221 the info-gap model improves from point a to point 6, and then again from c to d, and deteriorates from d to e. Similar behavior is seen in fig. 8.5 on p.226. The logic of this procedure for improving the info-gap model is to view the measurements (or some fraction of them) as reasonable (not extraordinary), and to demand that the uncertainty model account for those measurements by capturing them at the lowest possible horizon of uncertainty. This is similar to the logic of maximum likelihood estimation of the parameters of a probability distribution. Given measurements, we ask what probability distribution has the greatest density of probability at those measurements? Both maximum likelihood estimation and info-gap learning optimize an uncertainty model by enhancing its fidelity to measurements. Where the info-gap procedure differs from the maximum likelihood method (in addition to the different structure and meaning of the uncertainty

228

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THEORY

models) is t h a t the empirical robustness depends not only on the model parameters q but also on the performance parameter r^. T h e updating of the info-gap model has an inherent element of openness to judgment by the analyst. T h e reward-for-performance expressed by r^ is used to select those d a t a of greatest relevance. A somewhat similar freedom exists in statistical methods for identifying and excluding outlying data. This judgment is an integral part of the info-gap analysis, and is evaluated directly in terms which can be meaningfully related to the bottom-line use of the info-gap model: performance of subsequent decision-making.

8.5

Problems

1. U p d a t i n g a centerpoint. ti is a scalar parameter whose uncertainty we wish to represent with a uniform-bound info-gap model: U{a, Z) = {u:

\u-u\
a > 0

(8.42)

We will use learning data to test this info-gap model and update it, by changing the centerpoint S, when necessary. (a) Construct the gap function T{u) for the info-gap model defined in eq.(8.42). (b) Given measured values ixi, . . . , UK^ explain the rationale of the following empirical robustness function: oteiu^ Tc) = max

jrK)

< r c , n = l, ...,K\

(8.43)

(c) The current accepted value of the centerpoint oiU[a^u) is u — 2.00. Use the following measured values of u to update the info-gap model: 1.73, 1.87, 1.92, 2.04, 2.10. How does the updated centerpoint depend on the critical value TC? (d) X Derive a general expression for the optimal estimate of the centerpoint, based on the empirical robustness function, given K measured values of u. 2. U p d a t i n g an uncertainty envelope. u{i) is a scalar function whose uncertainty we wish to represent with an envelope-bound info-gap model: ^ ( a , S ) = {ii(t) : \u{t)-Z{t)\
{)
a>0

(8.44)

We are confident of the centerpoint function u{t)^ and we will use learning data to update the shape of the uncertainty envelope '0(^). (a) Construct the gap function T{u) for the info-gap model defined in eq.(8.44). (b) Given t^-functions ui{t)^ . . . ,UK{t) which are based on data recorded throughout the interval 0 < t
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(c) Expand the envelope function xp{t) in a truncated Fourier series: N

V'(i) = ^ c „ c o s ^ = c ^ 7 ( i )

(8.45)

n=l

Develop a method for updating the envelope-function coefficient vector c based on the empirical robustness function. (d) Repeat parts 2a-2c if the observed i^-functions are not known in their entirety, but only sampled at A^ instants. 3. Updating a shape matrix, i^ is a vector whose uncertainty we wish to represent with an ellipsoid-bound model: U{a,u) = {u=-u-^v:

v^Wv


,

a>0

(8.46)

We are confident of the centerpoint w, and we will use measured it-vectors to update the shape matrix W. We will consider only diagonal shape matrices. (a) Construct the gap function r{u) for the info-gap model defined in eq.(8.46). (b) Given measured vectors ui, . . . ,iiK, construct and explain an empirical robustness function. (c) i Develop a method for updating the diagonal shape matrix W^ given measured vectors ixi, . . . ,iiK, and based on the empirical robustness function. Suggestion: seek increments in the diagonal elements of the shape matrix W which: (1) minimize a critical gap function and (2) keep the ellipsoid volume constant. The volume of an ellipsoid with semiaxis lengths £i, ... ,iN is: N/2

JL i—1

where r(-) in eq.(8.47) is the ordinary gamma function, not the gap function. 4. A linear classifier. An uncertain vector u is thought to arise from one or the other of two sources. Ui{a,ui) and U2{OL^U2) are info-gap models for li-vectors from each source: Un{oi,Un) = {u = Un+v:

v^WnV
,

a>0

(8.48)

for n = 1 or 2, where Wi and W2 are known, real, symmetric, positive definite matrices. Consider the following linear-regression algorithm for classifying it-vectors between these two uncertainty models: C{u) ^{ ; ^^^ ^ ^ \' 2^

- ;^ - < ^ otherwi otherwise

(a) Construct the error-free robustness function for this classifier.

(8.49) ^ ^

230

INFO-GAP DECISION THEORY

(b) Assume we have learning data, wi, ...^UK^ of known provenance. That is, for each vector Ui^ we know that it is of type n^, where Tii equals either 1 or 2. Construct the error-free empirical robustness function. (c) X Suggest reasonable choices for the parameters of the classifier: (j) and A. (Hint: consider the shapes and positions of the ellipsoids as determined by the shape matrices Wi and W2 and the centerpoint vectors ui and U2.) (d) X Let the dimension oi u he N — Q and adopt the following initial realizations of the info-gap models: Zi = 0, 1^1 = /,

^2 = (1, . . . , 1)^, W2 = diag(l, . . . , 6)

(8.50)

Generate synthetic test data (which need not match the initial info-gap models) and use it to update the initial uncertainty models. Should you update the classifier as you proceed?

Chapter 9

Coherent Uncertainties and Consensus [Wjhat an adequate epistemology must begin by acknowledging is the thoroughgoing plasticity of human understanding, and among the things it must be able to account for is the rationality not just of minor but of wholesale changes in the form that understanding assumes. Paul M. Churchland [47, p.88]

Complete consensus is hardly ever a realistic aim. The more practical question is: how much can opinions differ and still enable cooperative or coordinated action? In the context of info-gap uncertainty this question focuses on: how much and in what ways can assessments of uncertain possibilities differ and still allow for agreement? More precisely still: in what ways and to what extent can decision makers' info-gap models differ if they must agree on their preferences among the available options? These questions can all be inverted to address the resolution of differences: how should perceptions of uncertainty change in order for preferences to coincide? In section 9.1 we characterize increments of information which leave a decision maker's preferences unchanged. To do this we introduce the idea of the coherence of distinct info-gap models of uncertainty. We show that if two info-gap models are sufficiently coherent then the robustness-based preferences will be the same, for any choice of the reward function. Conversely, behaviorally significant information — which causes a modification of preference — must break the coherence between old and new uncertainty models. In section 9.2 we evaluate the coherency of several pairs of info-gap models. Section 9.3 contains an example of the evaluation and interpretation of coherence functions. 231

232

9.1

INFO-GAP DECISION THEORY

Preference Preservation Under Altered Information

Let us consider two decision makers seeking agreement on a course of action. These individuals agree on the performance requirements as expressed in a reward function, and they agree on the nominal or typical configuration of the system and its environment. Where they differ is in their perceptions of the possible variabilities of the system and environment, or of the possible errors of their nominal models of these entities. In short, each person has one, or several, info-gap models of these uncertainties, but they do not have the same uncertainty models in all cases. How much can these info-gap models differ without precluding agreement on a course of action? It is useful to think, hypothetically, of either decision maker's uncertainty models as derived from the other's by some abstract increment of information. In this way, we can phrase the question as: what increment of information can distinguish one decision maker from the other without precluding agreement on preferences? Conversely, how large an increment of information, and of what type, would close the door on cooperative action? In this section we will discuss a theorem and two corollaries which specify conditions under which a decision maker's preferences (based on the robustness function) are preserved when presented with an increment of information, as expressed by a new set of info-gap models of uncertainty. A decision maker's preference ranking of the available options depends upon a reward function and also upon knowledge (and ignorance) as expressed by info-gap models of uncertainty. A different decision maker with different knowledge and distinct info-gap models may have different preferences. However, we are able to establish a very specific condition on the increment in information between the two decision makers which leaves their preferences the same, for any choice of the reward function shared by the two decision makers. We now develop this result. Definition 1 Consider two collections of info-gap models Xi{a,ci), ..., X]S[{a^^N) andyi{a^Zi), ..., yN{o^->^N) defined in the same space. Real functions g{a) and h{a) are lower and upper coherence functions, respectively, for these info-gap models and centerpoints if: 0 < g{a) < h{a)

(9.1)

and if for alii — 1, ... ,N and for all o; > 0: X,{g{a),Zi)

C y,{a,Zi)

C X,{h{a),Z,)

(9.2)

We can understand the coherence functions g{a) and h{a) in the following way. Because of eq.(9.1), Xi{g{a)^Zi) is necessarily a subset of Xi{h{a),Zi) by the nesting axiom of info-gap models (axiom 1 on p.31). The lower and upper coherence functions g{a) and h{a) are such that

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233

^i{g{a)^Ci) is small enough to be a subset of 3^i(a, Q ) , while the expanded set ^i{h{a)^Ci) contains yi{a^Ci). If ^(a) and h{a) are close together then A!i{g{a)^Ci) and 3^j(a,Q) are similar in shape and size, which is the sense in which they are coherent. Furthermore, if the coherence functions g{a) and h{a) are equal then the inclusions in (9.2) become an identity: A:,{g{a),c^)=y,{a,d^)

(9.3)

which means that A'^(a,Q) is simply a scaled version of

3^^(Q;,Q).

Definition 2 h{a) is a minimal upper coherence function for info-gap models ^i{a,Ci) and 3^^(a,Q), i = 1, ... ,N, if h{a) is an upper coherence function for these info-gap models and if h{a) is no greater than any other upper coherence function H{a) for these info-gap models: h{a) < H{a)

for all

a > 0

(9.4)

A maximal lower coherence function is similarly defined as the greatest of all lower coherence functions. Referring to eq.(3.169) on p.100, let ax{q,rc) be the robustness function based on a reward algorithm 7Z^[q, A'i(a, ci), . . . , XN{O^^CN)] which is monotonically decreasing in the sets in its domain, as defined in eq.(3.166). Likewise, let o^yiq^rc) be the robustness function based on the same reward function IZ^, but with the info-gap models 3^I(Q;,CI), . . . , 3^Ar(Q;, C/v). 3 A ' ( 9 , '^'C) and ay{q^ TC) are the robustness functions of two decision makers. We can now establish several relations between the robustness functions a;\: and ay and the coherence functions g{a) and h{a). In particular, we will show how coherence of the info-gap models of two decision makers is related to agreement of the decision makers' preferences. All proofs appear in section 9.4. Theorem 1 / / the info-gap models ?ii{a,Ci) and yi{a,Ci), i = 1, .. - ,N, have lower and upper coherence functions g{a) and h{a), respectively, where h[a) is minimal and continuous and unbounded^ in a, and if the robustness functions ax{q,rc) and ay{q,rc) exist for reward algorithm IZ^, then the robustness and coherence functions are related as: 9 [oty[q, re)] < ax{q^ r^) < h [ay{q, re)]

(9.5)

The coherence functions g{a) and h{a) depend upon the structure of the info-gap models Af^ and 3^i, i = 1, ....N, but not upon the reward algorithm 7^*[...]. The robustness functions a^ and ay depend on both the uncertainty models and the reward function. Theorem 1 establishes relations between ax and ay which hold for any reward function which satisfies relation (3.166) and for which ax and 3 ^ exist. ^These assumptions concerning h(a) — that it is continuous and unbounded — could be related to more elementary properties of the info-gap models involved, but this is unnecessary for our purposes.

234

INFO-GAP DECISION THEORY

As explained in connection with definition 1, ii g{a) and h{a) are close, then the A!i{g{a),Ci) and the yi{a,Ci) are similar and the knowledge they represent is highly coherent. This coherence is manifested in the coincidence of preferences for the two decision makers, as asserted in the following corollaries to theorem 1. Let >:^ and >ry denote the preference relations induced by SxiQ^^c) and Sy{q,rc), respectively, according to (3.171). Corollary 1 Let the info-gap models Af^(a, Q ) and 3^i(a, Q)^ ^' = 1, • - - ^N, have robustness functions ax{Q, T^) and ay{q^ TC) for reward algorithm IZ^, and lower and upper coherence functions g{a) and h{a), respectively, where h{a) is minimal and continuous and unbounded in a. If: h[ay{q',r,)]
q ^y q' and q hjt q'

Corollary 1 asserts that if the two coherence functions are close enough to satisfy (9.6) for actions q and q', then Sx and Sy induce the same preferences for these actions. Fig. 9.1 illustrates why relation (9.6) implies a closeness of the coherence functions. h{ay) Qipty) distant M

•ay

Figure 9.1: Schematic illustration of close and distant coherence functions. The coherence functions on the left satisfy relation (9.6) and the pair of functions on the right do not. The following special case of corollary 1 is of particular interest. Corollary 2 / / the info-gap models Xi{a,Zi) and 3^i(a,c^), i = 1, ... ,N, have robustness functions ax{q^ ^c) ^^^ ^y{^^ ^c) for reward algorithm IZ^, and lower and upper coherence functions g{a) and h{a) which are equal and

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if h{a) is minimal and continuous and unbounded in a, then the robustness functions induce the same preferences. That is, with these conditions: q ^

q'

if and only if

q^

q'

(9.8)

The main importance of theorem 1 and its corollaries is that if the Xi{a,Zi) and the 3^^(a,Ci) are sufficiently coherent, then preferences with the Xi are the same as preferences with the 3^^, for any choice of the reward function 1Z^[...]. An increment of information between two decision makers is expressed as a difference between the info-gap models Xi and yi. A behaviorally significant increment of information must break the coherence between the X^ and the 3^^. Unless the increment of information significantly reduces coherence, the preferences are the same for the two decision makers and the new information has no behavioral impact for a robust-satisficing decision maker, irrespective of the nature of the reward function. It has been widely noted that behavior is the result of how a decision maker processes information, not of the raw information itself. A decision maker is not handed an info-gap model of uncertainty as such. Rather, an info-gap model expresses the decision maker's reaction to and processing of information. Theorem 1 and its corollaries show that new information which is processed so as to be coherent with previous information will not lead to a change in preference or, for a robust-satisficing agent, will not lead to a change in behavior. This is true regardless of how the decision maker evaluates reward, since theorem 1 and its corollaries hold for any reward function satisfying relation (3.166). People sometimes 'force' new information to conform to their previous perceptions. Other people very readily modify their attitudes and inclinations when presented with new facts or knowledge. Theorem 1 and its corollaries specify the change in the perception of uncertainties which nonetheless does not alter the action of a rational decision maker.

9.2

Examples of Coherent Uncertainties

In this section we illustrate some aspects of the previous discussion by evaluating the degree of coherence of several pairs of info-gap models. Example 1 Uniform-bound info-gap models. Consider the following pair of uniform-bound info-gap models: X{a,Z)

=

{x{t):

\x{t)-c{t)\
a>0

(9.9)

y{a,^j

=

{x{t):

\x{t)-c{t)\<(t)a},

a>0

(9.10)

where ip and 0 are known positive constants. We can readily find the coherence functions for these uncertainty models.

236

INFO-GAP DECISION THEORY

The left-hand inclusion in (9.2) requires that any x{t) which satisfies: \x{t)-d{t)\<Mc^)

(9.11)

must also satisfy: \x{t)-c{t)\

<(l)a

(9.12)

From this we find that the lower coherence function is constrained by: 0
(9.13)

In similar fashion we find that the upper coherence function must satisfy: h(a) > -a

(9.14)

g{a) = h{a) = | Q

(9.15)

We see that is both a maximal lower and a minimal upper coherence function, so the condition of corollary 2 is satisfied. We conclude that these uncertainty models will always lead to the same preferences, regardless of what reward function is employed. (Provided that the same reward function is used in conjunction with each uncertainty model.) In fact, as expressed in eq.(9.3), JY is just a scaled version of y-. A'( J a , ? l = K a , 2 )

(9.16)

The magnitudes of the robustness functions will differ for these two uncertainty models, but the preference rankings which they induce will be the same. Example 2 Envelope-bound info-gap models. Consider the following pair of uncertainty models for scalar functions defined on a domain V: PC{a,^

= {x{t),teV:

\x{t)-c{t)\<^P{t)a},

a>0

(9.17)

y{a,Z)

= {x{t),teV:

\x{t)-c{t)\<(t){t)a},

a>0

(9.18)

where ^(t) and (/)(^) are known positive envelope functions. Proceeding as in example 1 we find that the lower and upper coherence functions must satisfy: 0 < g{a)

< a | | |

Ha) > c M

(9.19) (9.20)

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237

where the inequahties must hold for all values of t in the domain D. A pair of maximal lower- and minimal upper-coherence functions are therefore: qia)

=

amm—;—

(9.21)

nia)

=

ctmax—-—

(9.22)

For example, let the domain of definition be the interval: V = [0, T]

(9.23)

and consider linear envelope functions with positive coefficients: ^{t) = V^o + V^it,

(p{t) = 00 + 0it

(9.24)

Let us define the following parameter, which expresses a degree of nonproportionality between the envelope functions, (l){t) and ip{t): A =

(9.25)

- 0o'0i

T/^O^I

A is the determinant of the matrix of coefficients ipi and (j)i. The envelope functions are proportional: (9.26)

Ht) = sm

for some positive constant s, if and only if A = 0. The extremal coherence functions, eqs.(9.21) and (9.22), can be expressed as: g{a)

=

aV^o

h{a)

=

g{a) + a

g{a)

=

h{a) + a

h{a)

=

> if

A > 0

(9.27)

y

A
(9.28)

TA

TA if

aV^o

From this we see that the distance between the maximal lower and minimal upper coherence functions is: h{a) -g{a)

=

aTlAl V^o(V^o + V^i^)

(9.29)

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INFO-GAP DECISION THEORY

This difference — the incoherence of the two info-gap models — increases steadily from zero to a finite value, as the length T of the interval increases. The greatest diflterence is bounded: lim [h{a)-g{a)]

= ^

(9.30)

The incoherence of envelope-bound info-gap models, with linearly increasing envelopes, increases with the duration of the uncertain functions but is always finite. The incoherence depends primarily on the duration T and on the degree of non-proportionality between the envelopes as expressed by the determinant A. If the extremal coherence functions are equal, then, relying on corollary 2, we see that the two decision makers will have identical preferences, regardless of the choice of the reward algorithm. If the coherence of these info-gap models is not complete, then corollary 1 suggests (though cannot guarantee) that the possibility of coincident preferences rises as the incoherence, h{a) — g{a), is reduced. Example 3 Fourier ellipsoid-bound info-gap models. Consider the pair of Fourier ellipsoid-bound uncertainty models which diff'er only in their shape matrices: P(:{a,Z) =

{x{t) = c{t) -f u^-f{t) : u^Wu < a^} ,

a > 0 (9.31)

3^(a,2)

{x{t)=d{t)

a>0

=

+ u^-f{t):

u^Vu
(9.32)

where j{t) is a vector of trigonometric functions and W and V are known, real, symmetric, positive definite matrices with the same orthonormal eigenvectors Ci, . . . , (^AT, but with diff'erent eigenvalues, /ii, . . . , /xyv and Ai, . . . , Aiv, respectively. These shape matrices can be expressed in spectral form as: TV

TV

W = Y.^-^-^n

and

n=l

F = ^A,CnCj

(9-33)

n=l

The lower coherence function g{a) is defined by the left-hand inclusion in (9.2), which states that any element x{t) belonging to A!{g{a),c) must also belong to 3^(a,c). This arbitrary element can be represented as x{t) = c{t) + u^^{t) where the uncertain coefficient vector u can be expanded in the basis vectors Ci, • • •, CN •

Membership of x{t) in A'(^(a), c) is equivalent to: u^Wu
(9.35)

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239

Using eq.(9.34) and the expansion of W in eq.(9.33) this becomes: N

Y,0l^in<9\a)

(9.36)

Likewise, since the C,n are also eigenvectors of U, membership of the same x{t) in y{a,'c) requires that:

Y.elK
(9.37)

It is readily shown that the extrema of the sum in eq.(9.36), subject to the constraint that the sum in eq.(9.37) equals a^, occur at the following N values: opt

Yl^n^^^

= a^Y^,

n=:l,...,7V

(9.38)

-^^

n=l

From this we see that the lower coherence function must conform to: 0
(9.39)

y l
A similar argument leads to the following constraint on the upper coherence function: h{a)>aj

max ^

(9.40)

The last two relations lead to the following expression for the difference between the maximum lower and minimum upper coherence functions: h{a)-g{a)=a(J

max ^ - J mm \\J l
^ )

\l l
(9.41) ^

^

These two decision makers have different perceptions of the ambient uncertainties, as expressed in the different shape matrices of their info-gap models of eqs.(9.31) and (9.32). The function in eq.(9.41) expresses the degree of incoherence of their perceptions, as it bears on the possibility that their preferences will nonetheless agree. Example 4 Fourier ellipsoid-bound info-gap models, continued. We now consider two ellipsoid-bound models of different dimension, where all of the modes of one model are included in the other. y{a^c) is the same as eq.(9.32), and F is an A^ x A^ shape matrix whose eigenvalues are Ai, . . . , AAT. The modes contained in y{a, c) are represented by the A/"-vector 7(t) of trigonometric functions.

240

INFO-GAP DECISION THEORY

However, the other uncertainty model contains M modes, where M > N\ ;f(a,2*) = { x ( t ) = r ( t ) + u V W :

u^Wu
a>0

(9.42)

7(t) makes up the first A^ elements of 7*(0, and the remaining M — N modes are unique to A'(a,c*). Likewise, c{t) fills the first N terms of c*(/;). Also, l y is an M X M shape matrix where the upper A^ elements of the first N eigenvectors of W are the same as the eigenvectors of V. The eigenvalues of W are denoted /ii, . . . , /J,MTheorem 1 and its corollaries do not apply to this pair of info-gap models since they are defined in different spaces: the A^ modes in y{a,c) are represented by an A'-dimensional subspace of the space representing the M modes in A'(a,c*). However, we can extend 3^(a,c) in such a way as to circumvent this problem. Let F* be an M X M matrix formed from V by appending M — N diagonal terms all equal to 1/e, with zeros elsewhere in these new rows and columns. The M ~ N new eigenvalues of y*, A^^+i, . . . , AM, all equal 1/e. Define the augmented version of 3^ as: K K 2 * ) = { x ( t ) = r ( t ) + ^ V W : u^V''u
a>0

(9.43)

By choosing e > 0 and arbitrarily small we see that 3^*(Q;,C*) becomes arbitrarily similar to y{a^c): the elements I^AT+I, ...^UM^ which are the amplitudes of the newly introduced modes, are vanishingly small. Theorem 1 and its corollaries now apply, and the incoherence of 3^* and A', expressed by h{a) — g{a), is given by eq.(9.41), where the extrema are sought on all M eigenvalues: h{a) ~ g{a) = a(J

max ^ ~ J

\ \ l<m<M

Xm

min y l<m<M

^

)

(9.44)

Xm J

Since l/e is very large, the M — N new eigenvalues of F*, AAT+I = • • • = AM = 1/^5 will not infiuence the maximum in this expression, which will be found from among the first TV eigenvalue pairs. The minimum, however, will be proportional to e and hence essentially zero. We see then that the incoherence of these two Fourier ellipsoid models is controlled by the N modes common to them both, where this results from the fact that the modes of one model, y, all appear in the other model, Af. It is evident that this technique of extending the smaller matrix, as we have done by enlarging F to F*, would not yield useful results if each infogap model has modes which are not contained in the other model. In this case both shape matrices would have to be extended, thereby introducing arbitrarily large eigenvalues to both models. This extension would indeed bring the pair of models within the purview of theorem 1, and eq.(9.44) would describe the incoherence. However, this expression would obtain an arbitrarily large value. Theorem 1 and its corollaries would no longer assist us in exploring the possibility that, for arbitrary reward algorithm, the two decision makers will agree on their preferences. It may still be true, however, that agreement would be reached for some specific reward algorithms.

C H A P T E R 9.

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COHERENT UNCERTAINTIES AND CONSENSUS

241

Principal-Agent Contract Bidding

The problem. An employer (the 'principal') wishes to offer a contract of employment to a prospective employee (the 'agent'). The agent's activity will generate income rj. The agent's wages w{rj,q) depend on the income according to some formula encoded in the decision vector q. The choice of q is the essence of the agreement which must be reached between employer and employee. In a bidding situation the principal hopes to make an offer, entailing a specification of q, which will be acceptable to the agent, and yet which will also assure adequate profit to the employer. The income rj depends on the effort £ which the employee will expend on the job, as well as on other unknown and uncontrollable factors. The prior knowledge of the parties is that the income is related to the employee's effort, s, according to: 7/ = 7£ (9.45) where this relation, and the value of 7, are known to both parties. However, the accuracy of this knowledge is uncertain. The employee declares an intended effort e^- The employer may be skeptical, that is, uncertain, about whether or not this declared intention truly reflects the future effort to be exerted by the employee. Likewise, the employee may be overly optimistic (or even just a bit deceptive) and hence uncertain as well about what effort will in fact be expended. Furthermore, the degree of disparity between effort and effectiveness is unclear to both parties. In short, the income is uncertain due to the uncertain effort and to additional unknown factors. The income and the effort are uncertain to both the employer and the employee, though their perceptions of these uncertainties are different. The employer would like to know, before entering negotiations, or before actually specifying a class of wage formulas w{r],q), that mutually acceptable agreement with an employee is plausible. A strong element of plausibility would be provided by coincidence of employee and employer preferences on the available g'-vectors, despite their different perceptions of the uncertainties. This does not entirely guarantee that agreement will be reached, since the parties may disagree about the required minimal level of reward. That is, the preferences generated by a robustness function S(g, TC) depend on the critical reward r^ whose selection by the two sides may differ. Nonetheless, if the robustness functions for principal and agent generate identical preferences at any given value of TC, then this is a firm basis for seeking a final agreement. Uncertainty models. Let us suppose that the employer's uncertainty about the income r] and employee effort e is represented by the following info-gap model: y{a, £d) = I (r/, e) : \lLZ^ I -f£

< a.

\E^M Ed

< ^,1, }

a > 0

(9.46)

In other words, t h e employer's uncertainty about the income, 7/, varies as

242

INFO-GAP DECISION THEORY

an interval of unknown size about the nominal value 75, where e is the unknown effort exerted by the employee. In addition, the employer's uncertainty about the employee's effort e varies on an unknown interval about the declared value Cd- y{a,ed) can be written equivalently in the following manner, which will be useful in comparing with the employee's uncertainty model. y{a, Sd) = {(r/, e) :

75(1 - a) < r] < 75(1 -h a) £d{l -a)<e<

6d(l + c^)} ,

Q; > 0 (9.47)

The employee has a similar uncertainty model, with the single difference that the uncertain interval for the employee's effort is asymmetrical with respect to the same centerpoint. In particular, the interval is shifted to lower values: A'(a, Sd) = {{v. ^) :

7^(1 - c.) < ^ < 7^(1 + ex) ^d(l -i^a) <e < Sdil + c^)} ,

(9.48) a>0

where -0 > 1 is a value chosen by the agent to reflect an inclination to implement lower-than-declared values of effort e. Alternatively, ^ < 1 curtails the lower side of the ^-uncertainty interval, and reflects an inclination for greater-than-declared effort by skewing the uncertainty interval to higher values. If-0 = 1, then A!{a,Sd) and y{a,£d) are identical. Coherence functions. We now calculate the coherence functions for the info-gap models A'(a,ed) and y{a,ed)' The lower coherence function, g{a), must satisfy the left-hand inclusion in eq.(9.2). This imposes two pairs of constraints on g{a)^ one for 77 and one for e. For rj: -f£{l - g{a)) >-f£{l - a)

and

75(1 + ^(a)) < 7^(1 + a )

(9.49)

ed{l-^ g{a)) < ed{l + a)

(9.50)

For e: 5d(l-V^^(a))>5d(l-a)

and

From these constraints, together with the fact that ?/^ > 1, we find that the lower coherence function must conform to: 0 < g{a) < -

(9.51)

The upper coherence function, h{a). must satisfy the right-hand inclusion in eq.(9.2). The constraints on h{a) for 77 are: 75(1 - / i ( a ) ) < 75(1 - ck) and

75(1 +/i(a)) > 7^(1 + a)

(9.52)

and £d{l ^ h{a)) > ed{l ^ a)

(9.53)

For £: £d{l - ^h{a)) < £d{l - a)

C H A P T E R 9.

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243

From these constraints and V^ > 1, the upper coherence function must satisfy: h{a) > a (9.54) The maximal lower and minimal upper coherence functions are: g{a) = — and

h{a) — a

(9.55)

where -0 > 1 because the agent anticipates the possibility of expending less than the declared effort ^dIf the agent anticipates ^rea^er-than-declared effort, so ^ < 1, the extremal coherence functions become: OL

g{a) = a

and

h{a) — —

(9.56)

The lower and upper coherence functions are necessarily different in both cases if i/j ^ 1. That is, an incoherence between employer and employee uncertainty models arises if the employee's anticipation of the effort to be invested differs from the employee's declared value of e^- The incoherence of the info-gap models in both cases is: Ma) - 5(a) = a ^ ^

(9.57)

We conclude from theorem 1 that the principal and the agent may have different preference orderings on the wage formula resulting from their mismatched perceptions of the uncertainties. However, from corollary 1, there can be substantial agreement if tjj is not too large. Furthermore, this is true regardless of the particular structure of the wage formula (provided the reward algorithm decreases monotonically in the uncertainty sets, which is a very lenient requirement). The value of the incoherence in eq.(9.57) gives a quantitative means of assessing the impact of inaccurate disclosure by the agent. Final agreement is still not completely guaranteed, even when preferences on q agree, since dispute may arise over the required minimal level of reward r^. We know from the extremal coherence functions of eq.(9.55) iov ip > 1, and from the inclusions in (9.2), that the info-gap models in eqs.(9.46) and (9.48) are related to one another as: ^ ( ^ ' ^ d ] C y{a.e^)

C A'(a.c-d)

(9.58)

Furthermore, we see from eq.(9.5) of theorem 1 that the coherence functions establish the following relation between the robustness functions of the two parties: -ay{q.rc)

< ax{q.r^)

< ay{q.r^)

(9.59)

244

INFO-GAP DECISION THEORY

This ordering of the robustness functions of the principal {ay) and the agent (S^') holds for any wage formula: the principal always preceives greater robustness due to the non-disclosure of the agent's inclination for lower-than-declared effort. (The situation is reversed ii ip < 1.) However, the difference in perceived robustness is small if ip is close to unity. These inequalities show the importance of true disclosure by the agent, but also give an indication of how large a non-disclosured inclination can be without seriously affecting the potential for agreement between principal and agent.

9.4

t Proofs

Lemma 1 A minimal upper coherence function h{a) is non-decreasing. That is: ai < (12 implies h{pi\) < h{a2) (9.60) Proof of lemma 1. Let a i < 0^2 so, by the nesting axiom: y^{a2.C^)Dy^{auC^) Since h{a) is an upper coherence function for A'i(a,Q) and l,...,Ar: A'^(/i(ai),Q) D ^^(^1,^)

and

(9.61) ^^^(Q;,^),

i =

A'i(/i(a2), Q ) 5 3^i(c^2,Q) 5 3^i(Q;i,Q) (9.62)

Define the function:

{

h(a)

,

a ^ ai (9.63)

h{a2) ,

a = ai

We see from the right-hand side of (9.62) that H{a) is an upper coherence function for ^i and yi, i = 1, ..., N. Now suppose that: h{a2) < h{ai)

(9.64)

If (9.64) holds, then h{a) is not a minimal upper coherence function for these info-gap models since H{ai) < h{ai). This contradicts the supposition of the lemma that h{ai) is minimal, so (9.64) is false and h{a) is nondecreasing. Proof of theorem 1. Referring to the definition of the robustness function in eq.(3.169), let ^4^- denote the set of a-values whose least upper bound is the robustness S;f(g, TC). Likewise, let Ay be the corresponding set of a-values for Sy{q,rc). The sets ^ A ' and Ay are bounded since the robustness functions are assumed to exist. In fact. Ax and Ay are simple intervals with zero as a greatest lower bound and ax and ay as a least upper bound, since IZ^ is monotonic as in relation (3.166) and since the sets of an info-gap model are nested by a.

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(1) Right-hand inequaUty. If ap(:{q,rc) < h{0) then it is evident that ^^'(^^^c) ^ h{ay{q,rc)) since 0 < Sy{q,rc) and h{a) is minimal and hence non-decreasing by lemma 1. Now suppose that SA'(^, ^C) > h{0). Since Ax is an interval, there is an a G A;t: such that a > h{0). Now define the set A^:: A;^ = {a:

a> h{0)

and

a G Ax}

(9-65)

Since Ax is the "upper part" of Ax, and since ax is the least upper bound of Ax^ it is evident that: o^x{q,rc) = max^A'

(9.66)

Ax C h[Ay]

(9.67)

We will demonstrate that: where a function operating on a set means that the function acts on each element of the set to produce a new set. This inclusion implies the righthand inequality. Since h{a) is assumed to be continuous and unbounded, we see that for any a G Ax there is an a* such that a — h{a*). That is, h~^{a) exists. For any a G Ax7^*[g,A'l(a,^l), ...,A';v(c^,?iv)]>re

(9.68)

Likewise, a G Ax and a — h{a*) imply that: ^^{c^.C,)Dy^{a\^C^)

(9.69)

since h is an upper coherence function for these info-gap models. Since 7^* is monotonically decreasing (relation (3.166)), combining the last two relations results in: 7^*b,3^l(a*,?l), ...,3^^(a*,?7v)]>rc

(9.70)

Hence for any a G Ax there is an a* such that a = /i(a*) and such that a* G Ay. This is stated more directly by (9.67). Since the robustness functions are the least upper bounds of the a-sets Ax and Ay, this implies the right-hand inequality in the theorem. (2) Left-hand inequality. Let a G Ay, which means that: 7^* b , y i ( a , ? i ) , . . .,yN{a,CN)]

> re

(9.71)

By lower coherence of the info-gap models (left-hand inclusion in (9.2)) and by the monotonicity of IZ^ (relation (3.166)) we find: 7^* b , A ' i ( ^ ( a ) , ? i ) , . . . , XNig{a),CN)] > r,

(9.72)

Thus a G Ay implies that g{a) G Ax from which we conclude that: g [Ay] C Ax

(9.73)

246

INFO-GAP DECISION

THEORY

Since the robustness functions are the least upper bounds of the a-sets Ax and Ay, this imphes the left-hand inequality of the theorem. P r o o f of c o r o l l a r y 1. Since g{a) and h{a) are lower and upper coherence functions they are, according to (9.1), ordered as: g[ay{q,r,)]
(9.74)

Combining this with supposition (9.6) of the corollary results in: h[ay{q\r,)]
(9.75)

Since h{a) is minimal it is non-decreasing (lemma 1), which means t h a t : Sy{q\rc)

< ay{q,rc)

(9.76)

which by (3.171) implies: q ^

q'

(9.77)

Theorem 1 implies t h a t : g [ay{q, TC)] < ax{q, r^)

and

ax{q\

r^) < h [ay{q', TC)]

(9.78)

Combining relations (9.78) with supposition (9.6) leads to: ax{q,r,)
(9.79)

Thus, by (3.171), we conclude: q b . q'

(9.80)

which completes the proof. P r o o f of c o r o l l a r y 2. Since g{a) and h{a) are equal and h{a) is minimal they are non-decreasing (lemma 1). Since g{a) = /i(a), theorem 1 implies: ax{q^r,) = h[ay{q,r,)] (9.81) So, since h{a) is non-decreasing: axiq.Vc)

> ax{q',rc)

if and only if

ay{q,rc)

> ay{q .r^)

(9.82)

Thus ax and ay induce the same preferences on Q.

9.5

Problems

1. Environmental protection. You represent an interest group in public hearings concerning the impact, on a local estuary, of fertilization on farms along the river feeding the estuary. The hearings are to determine the maximum number of days of fertilization, gi. and the smallest distance from the estuary at which fertilization is permitted. q2. It is generally agreed

C H A P T E R 9.

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247

that the impact on the estuary of fertihzation along the river depends on a function of the form: R{q,u)=

I

I

Jo

J q^

h{x,t)c{x,t)dxdt

(9.83)

c(x, t) is the rate of fertihzer appHcation, in units of kilogram per meter of river-front per day. /i(x, t) is a model of impact on the estuary per kilogram of fertilizer. c(x, t) is highly uncertain and h{x, t) is in considerable dispute. All parties agree that little is known about the variation of the fertilization rate function c(x,^); farmer's habits, as well as agricultural advice, are far from uniform. While it is clear that only finite quantities of fertilizer can be applied, there is no consensus concerning optimal strategies, either on rate of application or spatial distribution. Furthermore, different experts have proposed widely different impact models h{x,t). In short, the product h{x,t)c{x,t) is severely uncertain. (a) The review committee will use an info-gap decision model to choose qi and q2. The firm which produces the fertilizer acknowledges that the function h{x, t)c(x, t) is unknown, but is playing down its variability. Members of the interest group which you represent are clammering for the review committee to base its analysis and decision on explicit recognition of the great uncertainty in h{x,t)c{x,t). Is the dispute over uncertainty relevant to the committee's decision? In particular, does the robustness ranking of preferences on ((71,92) vary among the info-gap uncertainty models which the committee can reasonably be expected to adopt? (b) After long deliberations the committee has decided that, given the great uncertainties, it simply cannot responsibly decide; the robustness of any available option is too low. It has been resolved to acquire data and verify a more informative envelope-bound info-gap model for the uncertainty in h{x, t)c{x, t), with a linear-envelope function. It appears that the fertilizer producer will advocate a modest slope for the uncertainty envelope, while your clients will favor a less conservative model. Assuming the two parties will each promote a different linear-envelope uncertainty model, are you headed for a confrontation? (c) Experts for each party are now haggling over the precise values of the coefficients of the linear-envelope uncertainty function, in an attempt to close their differences before presenting separate recommendations to the review committee. How close do the envelope functions have to be to avoid a fight in the committee which will select (9i,g'2)? (d) An independent group has criticized the decision to adopt a linearenvelope-bound info-gap model, claiming that this is too restrictive and does not correspond to real variations. They advocate a quadratic envelope with substantial curvature. Would you expect the robustness ranking of preferences on (91,^2) to differ between a linear- and a quadratic-envelope-bound model?

248

INFO-GAP DECISION

THEORY

Cantilever design. A design team is charged with selecting a conceptual design for a cantilever structure subject to static loads applied along the beam and perpendicular to the beam axis. Considerable uncertainty accompanies the load profile, and two conflicting envelope-bound info-gap models seem reasonable, based on alternative interpretations of the extant data: Un{aJ)

=

{f{x)

fix) -

fix) < aipnix) >

a>0

(9.84)

for n = 1 or 2, and 0 < x < 1. The envelope functions for these two uncertainty models are: (9.85)

V^i(x) X^

1p2ix)

(9.86)

It would be possible, by studying the uncertain loading process, to resolve the differences between these info-gap models. Such a study would be expensive and time consuming. Do you deem it necessary for the successful and unambiguous completion of the design team's task? Financial investment. You must choose between two investment programs, where the projected profits depend upon a vector u, whose five elements are all uncertain, some more than others. Two separate research teams have studied the uncertainties associated with ix, in preparation for recommending an investment, and they each propose the following interval info-gap model: U{a,u) =
< aWn, n = 1 . . . , 5 > ,

a> 0

(9.87)

However, each team proposes a different vector of uncertainty weights, wi, . . . ,i(;5, specified in table 9.1.

[ J^i Team 1 1 Team 2 0.7

W2

1^3

W4

1 1.5

2 3

0.5 1

w^ 1 2

Table 9.1: Uncertainty weights for problem 3.

(a) Do you expect the investment recommendations of the two teams to be the same? (b) You recognize that the uncertainty weights of the two teams differ substantially, and you have decided to direct the teams to invest further effort in refining their info-gap models. Which elements should they concentrate on, if the goal is for the teams to agree on the investment decision?

Chapter 10

Hybrid Uncertainties Two is better than one. King Solomon [105, 4:9] If youWe got it, Raunt it! "Super" Joe Namath

The goal of this book has been to develop the implications of info-gap uncertainty for decisions with severe lack of information. We have used infogap models almost exclusively to represent uncertainty. Nonetheless, one sometimes has information which is amenable to probabilistic representation. Such information is valuable and should be exploited. Often, however, the probabilistic information is insufficient to cover all facets of the problem, in which case one can combine it with an info-gap model to create a hybrid decision algorithm. This has been recognized for a long time [69] and we have encountered examples of this in previous chapters. In this chapter we discuss several generic approaches to combining probabilistic and info-gap models of uncertainty. In section 10.1 we examine a Poisson process whose events entail uncertainties which are represented by an info-gap model. In this way we represent situations in which complicated and incompletely understood events recur randomly and independently in time. In section 10.2 we reverse the situation and consider an imperfectly known probability distribution embedded in an info-gap model. Examples of this from conservation biology and queueing theory are discussed in sections 10.3 and 10.4. Finally, in section 10.5, we let the info-gap horizon of uncertainty a be a random variable with a known probability distribution. 249

250

INFO-GAP DECISION THEORY

10.1

Info-Gap Uncertainty in a Poisson Process

The rate of recurrence, over a long period of time, of acute events such as earthquakes, hurricanes, economic disruptions, and so on, is a common and often quite rehable statistical datum. The rate of recurrence of distinct and unrelated events is constant over time, or nearly so, for many transient turbulent disturbances. When this is true, the Poisson distribution Pn{0) accurately represents the probability of n incidents in a long duration 6. The probabilistic Poisson model captures the long-range distribution over time (or space) of discrete and independent events. In contrast, the local temporal (or spatial) details of the event may be inordinately complex and poorly understood. For instance, the precise temporal and spatial waveform of an earthquake is complicated and site-specific. The same can be said of hurricanes and other such anomalies. Probabilistic models require an enormous amount of information in order to describe the uncertain dynamics within an individual incident, while info-gap models are well suited to the task. Clearly, the two approaches should be integrated. The hybridization of the Poisson process with an info-gap model can be done in many diff"erent ways, depending on the specific problem at hand. In this section we consider one rather general prototype of a hybrid Poisson/info-gap analysis. Other examples can be found elsewhere [15, 17, 92]. Problem formulation. We consider a linear dynamical system with state vector x{t) and subject to uncertain transient inputs u{t). Considering the transients as distinct, discrete and uncorrelated events, the distribution of their recurrence over a long period of time is represented by a Poisson distribution. For the short duration of a single transient, however, the uncertain variation of the input function u{t) is represented by an info-gap model U{a, u), a >0. Each transient occurrence u{t) induces a tiny increment of damage 5u which accumulates in the system over time.^ The system fails when the accumulated damage reaches a critical value Ac- The degree of damage in a single event depends on the input function u{t), while the number of events, n, in a long duration depends on the Poisson randomness of the event recurrence. We cannot construct a purely info-gap robustness function since failure depends, in part, on the probabilistic random variable n. Likewise, we cannot calculate a probability of failure since failure depends partly on the info-gap uncertainty of u{t). We must hybridize. The system behavior is modelled as: — = Ax{t) + Bu{t), dt

x{0) = 0

(10.1)

^We will not consider annealing, though this can be done. See section 8.4 of [14].

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251

where A and B are known constant matrices.^ The uncertain variation of the input vector u{t) is represented by an integral energy-bound info-gap modeh W(a,S) = | ^ ( t ) : /

[u{t)-u{t)fW[u{t)-u{t)]dt
a>0

(10.2) where VF is a known, real, symmetric, positive definite matrix. The uncertain deviation from the nominal load, u{t) — u{t), must eventually die away since the integral uiU{a,u) is bounded. The Poisson distribution expresses the probability of exactly n transient events in a long interval of duration 6\ P^{0)=iMlf^,

n = 0, 1, 2, . . . (10.3) n! which is a discrete normalized probability distribution on n with a single known parameter, A. The small increment of damage resulting from a single event is a power function of a linear combination of state variables at time t after onset of the transient: Su = 7[Vxuit)]'' (10.4) where 7 and /i are known, positive constants, and '0 is a known projection vector. Xu{t) is the state vector in response to input vector u{t). Maximal increment of damage at horizon of uncertainty a. The solution of eq.(10.1), for input u{t) and zero initial conditions, is: Xu{t)=

[ e'^^'-^^Bu{T)dr

(10.5)

We can express the projection of the diff"erence between the actual and the nominal state vector as: ^ ^ [x^{t) - x~it)] = f C^{t - r)VFi/2 [U(T) - U{T)] dr

(10.6)

where we have defined the vector: C^(t) = ij^e^'BW-'/^

(10.7)

Using first the Cauchy inequality, and then the Schwarz inequality, one finds that the maximum of the projection in eq.(10.6), as u{t) varies on the set hi{a^u)^ is: max^ ^ ^ [xu{t) - x~{t)\ =aJ ueU(a,u)

f C^(r)C(r) d r V Jo Z{t)

^See [13] for consideration of damage-induced variation of the system model.

(10.8)

252

INFO-GAP DECISION THEORY

which defines the known function Z{t). Combining eqs.(10.4) and (10.8), the maximum increment of damage in a single transient event, up to uncertainty a, becomes: max_ Suit) = 7 [ip^x-{t) -f aZ{t)]^ (10.9) uEU{a,u)

Robustness function. We begin by evaluating the robustness to n transient events. Failure occurs in n events if the cumulative damage exceeds the critical value Ac. The robustness to n events, Srx(Ac), is the greatest value of the uncertainty parameter a such that failure cannot occur in n events: Sn(Ac) = max < a : In max_ Su{t) < Ac > [ Y ueu{a,u) J J

(10.10)

Using the expression for the greatest increment of damage in a single transient, eq.(10.9), we find the robustness to n transients to be: a^{l^,)^'^-^LJ^_JL^^

n=l,2, . . .

(10.11)

or an(Ac) = 0 if this expression is negative. The number of events, n, is a Poisson random variable whose distribution is given by Pn{0) in eq.(10.3). Thus the n-event robustness an(Ac) is also a discrete random variable whose distribution is known. A concise and useful combination of the info-gap and probabilistic elements is the randomization of an(Ac) on Pn{0), where ^ is a long-term duration. That is, we average the robustness on the set of n-values which can lead to failure: n = 1, 2, — Explicitly, the Poisson average of the info-gap robustness is: ^

S(A,)

=

oo

-__^^S„(Ac)P„W

(10.12)

n=l

We have obtained an info-gap robustness which has been probabilistically randomized on the Poisson distribution. Like any robustness function, 3(Ac) provides a basis for exploring preferences and making decisions. In particular, the randomized robustness depends on decision variables q (embedded in the matrices A and JB, for instance) which can be chosen, in designing the system, so as to maximize the robustness. To stress the dependence of the randomized robustness S(Ac) in eq.(10.12) on the design variables q and on the critical level of damage Ac, we will write it as a{q, Ac). We can define a robust-satisficing choice of the decision vector $c(Ac) just as in eq.(3.20) on p.45: §c(Ac) = argmax3(g, Ac)

(10.13)

qeQ

where Q is the set of available decisions. Both robustness functions, a{q, Ac) and a($c(Ac), Ac), display the usual trade-off" of immunity versus reward.

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253

Optimizing t h e probability d i s t r i b u t i o n of an(Ac). Eq.(10.13) is not the only hybrid analog of the basic info-gap optimization in eq.(3.20). Another approach is based on choosing the design variables so as to maximize the probability of large S^(Ac) values. As noted earlier, by viewing Sn(Ac) as a random variable driven by the Poisson random variable n, we see that these two quantities have the same probability distribution, Pn{0). Furthermore, an(Ac) depends on the decision vector g, which we emphasize by writing 3^(9, Ac). We can combine our knowledge of the distribution of n, with our understanding of the dependence of 3^(g', Ac) on the properties of the system, to maximize the probability of large robustness values. We now define a robust-satisficing decision which may be different from the value calculated in eq. (10.13). Let Sd be a desired or demanded value of robustness. 3d may be based on value judgments which determine the required level of immunity to failure, as discussed in chapter 4. A robust-satisficing decision vector q{ad) is one which adjusts the n-event robustnesses 3^(g',Ac) so as to maximize the probability of those 3^(9, Ac) which exceed the demanded value 3d. That is, g(3d) is chosen as:

?(Sd) = argmax qeQ

E

Pr.{0)\

(10.14)

\3^(g,Ac) > 3d

This hybrid probabilistic/info-gap strategy is distinct from eq.(10.13), yet still analogous to the purely info-gap robust-satisficing strategy in eq.(3.20).

10.2

Embedded Probability Densities

We now consider a situation in which a random vector u is characterized by a probability density p{u) which is itself uncertain. The uncertainty in p{u) is described by an info-gap model. Specific examples are discussed in section 3.2.13 and [12, 26]. Formulation. We are considering a system for which the random vector u is an input to the system, or an internal or environmental property of some sort. The response of the system under the influence of uis Xu- The system fails if a particular condition is violated, for instance if a function f{xu) exceeds a critical value: f(xu) > X,

(10.15)

The probability density function p{u) is imperfectly known. The best estimate is p{u), though the actual density function, p{u), may differ substantially. The uncertain variation of the probability density is represented by an info-gap model, U{a,p), a > 0. For any specific probability density p{u) we can calculate the probability

254

INFO-GAP DECISION THEORY

of failure: Pf{p) = Fvoh{f{xu)>x,\p)

=

f

p{u)du

(10.16)

f{Xu)>Xc

We want the probability of failure of the system to be no greater than a critical probability Pc'. PdP) < Pc (10.17) In particular, we can make various decisions, for instance in designing the system, aimed at assuring acceptable probability of failure. Since the probability density p{u) is uncertain we cannot calculate the exact value of the failure probability. What we can do is use the info-gap model of uncertainty in p{u) to evaluate the robustness, to uncertainty in p{u)^ of the estimated failure probabihty. This robustness function is the greatest value of the info-gap uncertainty parameter, a, such that any density function up to horizon of uncertainty entails an acceptable probability of failure: 3(Pc) = max < Q^ : ( max_ Pf (p) ) < Pc >

(10.18)

This is an ordinary info-gap robustness function, where the probabilistic uncertainty, p{u)^ has been embedded in the info-gap uncertainty model U{a,p). We can use the robustness function to choose system properties, or to make other decisions, which enhance the robustness to our uncertain probabilistic knowledge of the random variation of u. Example. Let us consider a one-dimensional linear system whose dynamics are represented by eq.(lO.l) where A and B are scalars. The input u{t) is constant and equal to u during 0 < t < T and zero elsewhere. The amplitude i^ is a random variable with probability density p{u). The best estimate of the probability density, p{u)^ is a zero-mean normal distribution with variance cr^. The evidence for this normal distribution is quite good up to about k standard deviations, beyond which the fractional deviation of the actual from the nominal density is known to vary. This information can be encoded in an info-gap model in various different ways, one of which is: U{a,p) — < p{u) :

p{u) > 0, / p{u)du = 1, \p{u) —p{u)\ < ap{u) if \u\ > ka p{u) = cp{u) ii\u\
a > 0 (10.19)

where c is a normalization constant for each density p{u). The system fails if the response at time T, the end of the load interval, exceeds a critical value: \xu{T)\>x, (10.20)

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255

where: uB (e^^ - 1) Xu{T) = :^=^^ ^

(10.21)

The probabihty of failure, given density p{u), is: Piip) = P r o b ( | x „ ( T ) | > X e | p ) = Prob(|w|>»7Xe)

(10.22)

where we have defined:

As in eq.(10.17), we desire the probability of failure to be no greater than the critical value Pc- Since p{u) is imperfectly known, we will calculate the robustness of the failure probability to the info-gap uncertainty in the probability density. For simplicity we will assume that the weighted failure threshold, rjXc, falls outside the normal range of u: (10.24)

r]Xc > ka

To evaluate the robustness function we must find the maximal failure probability, up to info-gap uncertainty a. The maximum on the upper tail is: /•CXD

ix_ / max peu{ Q;,p)

/>00

p{u)du

=

JrjXc

/

p{u){l + a)du

(10.25)

"J T}Xf.

=

(l+a) 1_$

Cf)

(10.26)

where $(•) is the standard normal probability distribution function. The maximum on the lower tail is the same, so: max_ Pf(p) = 2(1 + Q^) 1 - *

(^)l

(10.27)

We have assumed that a is small enough so that this is no greater than one. This is assured, for some non-negative a, if the nominal density, p{u), entails acceptable probability of failure, which requires that: 1_$

m


(10.28)

The robustness function is evaluated according to eq.(10.18) by equating the maximum in eq.( 10.27) to the critical probability, Pc, and solving for a: 3(Pc) 1_$

m

(10.29)

This is the info-gap robustness, to uncertainty in p{u), of the estimated probability of failure Pf(p). 3(Pc) is based on an info-gap model for the uncertainty in p{u).

256

10.3

INFO-GAP DECISION THEORY

Example: Endangered Species

In this section we consider a specific example of the following generic problem. The decision maker must choose one action from among a suite of alternatives. The utility resulting from the action depends on the state of the world, which is uncertain. Furthermore, even if the state of the world were known with certainty, the actual outcome of the action is unclear and hence its utility is also uncertain. But nonetheless a choice must be made. The approach which we explore is to formulate the expected utility of the action, and then to embed this expectation in an info-gap analysis, where the robustness to uncertainty guides the choice of action. This example is based on [145]. The Sumatran rhinoceros is a severely endangered species. Only a small number of reproducing breeding pairs exist, and the prospects for the future of the population are dim. The reasons for the decline of the population are unclear, but the four contending factors are decimation by poaching, loss of habitat, demographic accidents resulting in low fertility, and disease. The probabihties of these four "states of the world" have been roughly estimated. The actions which conservation authorities can adopt are translocation of the population to a new region, extension or supervision of the current reserve, or captive breeding. Only one alternative can be adopted due to budgetary constraints. Expected utility. The utility of an action is assessed as the probability that the population will persist for 30 years. We denote the utility of action i, given that the state of the world is j , by Vij. The probability that the world is in state j is pj. The expected utility of action i is: V^{V,P)=J2^^JPJ

(10.30)

The optimal action, i*, from the perspective of expected utility theory, is the action which maximizes the average utility: i* = SiVgindixVi{v,p)

(10.31)

i

2* is the action which, on average, has the highest utility (greatest probability of survival), based on the values of Vij and pj. Uncertainty. The probabilities pi and Vij are highly uncertain. The best estimates are p^ and Vij^ but the fractional errors of these estimates are unknown. We will use info-gap models to represent these uncertainties. The probability distribution pj is normalized on the states of the world, j , and each value must be non-negative and no greater than unity. In contrast, the probability distribution for the utility of action z, f^j, is not normalized on j . It could be that ^ Vij < 1, which occurs if all the survival probabilities are very small (the plausible scenario). Or, it could

C H A P T E R 10. HYBRID UNCERTAINTIES

257

be that ^ • Vij > 1, which occurs if all the survival probabilities are very large. Consequently, the only constraint on the Vij^s is that they must be non-negative and no larger than one. Consequently, the info-gap models for uncertainty in p and v are: V{a,p)^[p:

^ P , = l, J

(10.32)

max[0, (1 - a)pj] < pj < min[l, (1 + a)p^], i = 1, 2, . . . } , a > 0

V{a,v)

=

{v: max[0, {1 - a)vij] < vij < min[l, {1 i-a)vij], i - 1 , 2, . . . , j - 1 , 2, . . . } ,

a>0

(10.33)

These are interval-bound info-gap models, adapted to the specific case of probabilities. Robustness. Given estimates p and v of the probabilities, we can estimate the expected utility of action i as Vi{p, v) as in eq.(10.30). For any other choice of the probabilities, p and v^ the expected utility is Vi{p,v). Since the estimates p and v are very uncertain, we are not confident that the utility which should actually be expected to result from action i equals ^i(p? ^)- That is, we have every reason to believe that, for many realizations of j9 and f, and especially for the true values: Vi{p,v)^V,ip,v)

(10.34)

Let Vc be the lowest level of expected utility (least average probability of persistence of the species for 30 years) which we are wilUng to accept. The robustness of action i, to uncertainties in the probabilities pj and Vij, is the greatest horizon of uncertainty a up to which adequate expected utility, K , is obtained for any realization of the probabilities: 3(i, K ) = max < a:

min

y^%]^j 1 > K /

(10.35)

The robust-satisficing action, i{Vc), maximizes the robustness and satisfices the expected utility at the value Vc'. ^(K) = argmaxS(i, K)

(10.36)

i

The robust-satisficing action, i{Vc), is very likely to be diflPerent from the action which maximizes the best-estimate of the expected utility, z* in eq.(10.31). Example. Table 10.1 shows the best-estimates of the probabilities, p^ is the estimated probability that factor j is causing the decline in population.

258

INFO-GAP DECISION THEORY

State Poaching Loss of habitat Demographic accidents Disease Expected utihty

[p^

1 ^'ij 0.1 0.3 0.3 0.1 0.5 0.05 0.1

1 0.1 1 0.095

V2j

V3j

0.25 0.2 0.09

0.9 0.2 0.01

0.1 0.14

0.4 0.195

Table 10.1: Estimated probabiUties. Actions z = 1, 2 and 3 are: translocation, new reserve, and captive breeding. Vij is the estimated probability of persistence of the population for 30 years, for action i in state j . The bottom row of table 10.1 shows the best-estimates of the expected utility, Vi{p^v)^ for each action, according to eq.(10.30). From these results we see that, based on the best-estimates of the probabilities, the most promising action is captive breeding, so z* = 3. The 2nd choice is extending or supervising the reserve, z = 2, and the least attractive alternative is translocation, i = 1. However, we know from eq.(3.25) on p.49 that the robustness of bestestimated expectations is zero. That is: 0 = 3(1,0.095) = S(2,0.14) = 3(3,0.195)

(10.37)

Even infinitesimal errors in p or t' may result in lower-than-anticipated probability of persistence.

0.20

0.15

h.

, ( 0 . 2 3 , 0.12)

0.10

0.05

3(i,K)

0.00 0.00

0.25

0.50

0.75

1.00

Figure 10.1: Robustness curves for actions 1, 2 and 3. Robustness curves for the three available actions are shown in fig. 10.1. These curves show the usual trade-off between robustness and utility. Fur-

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259

thermore, the curves cross, indicating reversal of preferences. In particular, the nominal optimal action, i* = 3, is the most robust for 0.12 0.23. If the fractional errors in the estimated probabilities can plausibly exceed 23% then action 3 cannot be relied upon, and action 2 is more dependable. Nonetheless, one notes that unfortunately the values of expected utility which can be reliably anticipated are all very low, and no action promises great confidence in the survival of this species. Either better information is needed, or different, more drastic measures must be taken.

10.4

Example: Serving an Uncertain Queue

We now consider a different info-gap analysis of uncertainty in a probability distribution. In many queueing problems a human or automated "server" must deal with the random arrival of "clients" which may be people or work orders of various sorts. Clients don't want to wait for service, while valuable server capacity is wasted if the server idly waits for clients. How large a server capacity should be chosen if the probability distribution of client-arrival is uncertain? To consider a very simple situation we will suppose that the server works only during the day, and that clients accumulate only during the night. The server can handle up to r clients per day, and the number of clients waiting for service in the morning, n, is a Poisson random variable whose probabihty distribution is: ^n(A) = ^ - ^ , n - 0 , 1, . . . (10.38) n! The best estimate of the average number of clients per day, A, is A. However, this estimate is based on past experience, while the future may be quite different, so the actual rate is highly uncertain. The following fractional-error info-gap model represents uncertainty in A: U{a, A) = {A : max[0, (1 - a)A] < A < (1 + a)x\ ,

a > 0

(10.39)

The system designer wishes to choose the server capacity, r, so that the probability of failing to serve 52 or more clients on the day they arrive is small. But it is also desired that the probabihty be small that the server could have handled 5i or more than the number of clients who actually arrived. These probabilities can be expressed with the Poisson distribution, assuming independent arrival at a constant average rate. The server can handle up to r clients in one day, so the probability of not serving S2 or more clients on a single day is:

n=r-\-S2

260

INFO-GAP DECISION THEORY

The probability of unused 1-day capacity for handling 5i or more clients is: (10.41)

7ruc(r,A)= ^ F n ( A ) n=0

The total loss function will be the sum of these two probabilities: r-\-S2 — l

TTe{r, A) = 7Tuc{r, A) + 7rns(r, A) == 1 - e ^

^

n=r—si + l

A^ ^

(10.42)

For instance, if 5i = 52 = 1: X \r e-^A

7r^(r,A) = l - P , ( A ) = l

(10.43)

The performance requirement is that the probability of loss, 7r£(r, A), not exceed the critical value e. The robustness is the greatest horizon of uncertainty up to which this satisficing requirement is guaranteed: S(r, e) = max < a :

I

niax_ 7r^(r, A) ] < £

V^^^(^'^)

(10.44)

/

Let M{a) denote the inner maximum in eq.(10.44). M{a) increases as a increases because the U{a,X) are nested sets. The robustness, S(r,e), is greatest a at which M{a) < e. Thus a{r^£) is greatest solution for a of M{a) ~ e diS indicated in fig. 10.2. In other words, M{a) is the inverse of M{a) =6

if and only if

S(r, e) = a

(10.45)

A plot of M{a) vs. a is the same as a plot of e vs. S(r, e). M{a)

0 S(r, e) Figure 10.2: Illustration of the calculation of robustness.

^i{r,X)

A Figure 10.3: Schematic illustration of 7r£(r, A).

Consider performance requirements Si = S2 = 1, so 7r£(r, A), specified in eq.(10.43), is a convex function as in fig. 10.3. Thus M{a) is obtained by

C H A P T E R 10.

261

HYBRID UNCERTAINTIES

choosing A as: r

(l + a)A

\ max[0, (1 - a)A]

if A > r

(10.46)

if r > A

Hence: A < r, a> 1 1

M{a)

A < r,

a
(10.47)

r
1

¥^. The best estimate of the probabiUty of failure is: (10.48) r\ From eq.( 10.47) we find that M(0) = 6 * , so the robustness of the nominal loss function vanishes, S(r, e*) = 0, and only worse (larger) probability of loss has positive robustness. We now consider a numerical example in which the best estimate of the average number of clients is A = 3. The performance requirements are 5i = 52 = 1 so the loss function, 7r£(r, A), is the probabihty of one or more un-served clients or one or more un-used units of server capacity. Fig. 10.4 shows robustness curves for server-capacities r = 1, 2, . . . ,5, showing the trade-off between aspired failure probability, e, and immunity to uncertainty in the average client arrival rate. e'=7Te{r,X) = l -

3(r, e) 0 0.2 0.4 0.6 0.8 1 Figure 10.4: Robustness curves for A = 3 and r = 1, 2, ..., 5. si = S2 = 1. The loss function at the estimated number of clients, 7r^(r, A) in table 10.2, is the ^-intersect of the corresponding curve in fig. 10.4. We want 7r^(r, A) small, so, based on the best-estimate of the client-arrival rate. A, our preferences on values of r are (subscript 'n' denotes 'nominal'): 3 ~ n 2 )-n 4 >-n l ) - n 5

(10.49)

262

INFO-GAP DECISION THEORY

r Server capacity 1 2 3 4 5

7Ti{r,X)

Nominal loss function 0.85 0.78 0.78 0.83 0.90

Nominal preference 3 ~n 2 3^nl 4^nl

Table 10.2: Nominal loss function for different server capacities.

Robust-satisficing preference 3 crosses 2 3 crosses 1 4 crosses 1

Table 10.3: Comparing nominal and robust-satisficing preferences.

Now consider the preferences based on the robustness curves in fig. 10.4. An r-value whose curve is further to the right has greater robustness. The following robust-satisficing preferences are observed: (10.50)

3 ^r 4 ^r 5 2 >-r 1 >^r 5

(10.51)

The robust-satisficing preferences in eqs. (10.50) and (10.51) are consistent with, but weaker than, the nominal preferences in eq.( 10.49). The robust-satisficing preferences are weaker than the nominal preferences because the robustness curves cross, implying reversal of preference. The differences between nominal and robust-satisficing preferences are shown in table 10.3. For instance, compare r = 2 and r = 3 in fig. 10.4. For £ < 0.9 :

3(3, e) > S(2, e)

=>

3 >-r 2

For £ > 0.9 :

5(2, e) > S(3, e)

=>

2 ^r 3

Nominally: 3 ~n 2. This example has illustrated that the analysis of info-gaps in a probability distribution can substantially alter design choices.

10.5

Probabilistic Info-Gap Parameter

The horizon of uncertainty, a, of an info-gap model U{a, u) often represents a physically meaningful entity, like the energy of a seismic event or the quantity of seasonal precipitation. The info-gap uncertainty relates to the temporal or spatial distribution of the phenomenon: the uncertainty in the seismic waveform or in the rain profile during the season. In contrast to the fragmentary information about these functions, there may be sufficient information to model a itself as a random variable whose probability density is known. An example from the field of water resources management is found

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263

in [92] and from mechanical shock dynamics in [15]. We consider a different, generic, example here. Consider the linear dynamical system in eq.(lO.l) whose uncertain input u{t) belongs to the info-gap model U{a,u) in eq.(10.2). As in section 10.1 we suppose that the system is damaged by the load u{t) according to 5u in eq.(10.4). However, unlike section 10.1, the loads are not recurrent, but rather severe enough to reach the critical failure-level of damage Ac in a single event. Furthermore, the uncertainty parameter a has a known probability density p(a), with cumulative probability distribution function P{a). The info-gap robustness function is, as usual, the greatest value of a for which failure cannot occur. Adapting eq.(lO.lO) to the present circumstance, this becomes: a{q, Ac) = max {oi.[ max_ 5u{t) < Ac > [ \ueu{oc,u) J J

(10.52)

where q is the vector of decision variables. Failure can occur only if the info-gap uncertainty parameter is no less than the robustness, S(^,Ac). On the other hand, failure need not occur even if a exceeds a^q^ Ac), since the load may be propitious. We are unable to precisely calculate the probability of failure since we do not know the probability distribution of the info-gap-uncertain input function u{t). However, knowing the robustness, 3(g', Ac), and the probability distribution of a enables us to evaluate an upper bound for the probability of failure Pf. Pi < Prob [a > a{q, Ac)] = 1-P

[a{q, Ac)]

(10.53)

We can construct a robust-satisficing decision which, precisely as in eq.(3.20), maximizes a(g, Ac) on the set Q of available decision vectors q. Furthermore, since the cumulative probability distribution P{a) is a monotonically increasing function, a value of q which maximizes a{q, Ac) also maximizes P{a), That is, since dP{a)/da > 0, and because: dP[aiq,A,)] dq

^ dP[aiq,A,)]daiq,A,) da dq

^^^^^

we conclude, using eq. (10.53), that a robust-satisficing action q (which brings a{q, Ac) to an unconstrained maximum) also minimizes the upper bound on the probability of failure.^ In light of these considerations we can define the robust-satisficing action: g=:maxF[S(g,Ac)] (10.55) qeQ ^In section 11.4 we will explore in greater depth the relation between robustness and probability of failure.

264

INFO-GAP DECISION

THEORY

This defines the same action q as in eq.(3.20). However, the hybridization of probabilistic and info-gap uncertainties provides an additional (and possibly different) calibration of the available options and of the associated increments of robustness. If g' is a sub-optimal value of the decision vector, then a measure of its sub-optimality is the decrement in the upper bound on the probability of failure entailed by replacing q with q. More generally, we can replace the strictly info-gap preference ordering of eq.(3.19) by: qyq'

if

P [ 5 ( g , A c ) ] > P[a{q\A,)]

(10.56)

In other words, referring again to eq.(10.53), option q is preferred to option q' if the upper bound on the probability of failure with q is lower t h a n with q'. This ordering of preferences will precisely agree with the strictly info-gap ordering of eq.(3.19). However, the preference ranking in (10.56) provides a probabilistic calibration of the relative merits of the options.

T h e richness of the alternatives which become available when a modicum of probabilistic information can be combined with an info-gap uncertainty model clearly vindicates the wisdom of b o t h King Solomon and Joe N a m a t h !

10.6

Problems

1. Surface treatment. Many design and decision problems involve surface treatments of various sorts: metallic coatings to prevent corrosion, submarine coatings to reduce sonar reflection, tribological treatments to reduce or to enhance friction. In this problem we examine a diff"erent surface treatment problem. You are the forest manager of a large arboretum. The park is subject to occasional infestations by parasites, and it is your responsibility to establish a maintenance strategy of spraying and other treatments. These treatments are needed both to nurture the forest and to minimize the cumulative damage during the parasite season. The treatment policy is expressed as a function q{x) over the domain of the park. q{x) is the intensity of treatment at point X. A specified total quantity of material Q must be applied to the park as a whole; the question is how to distribute this material. Parasite infestations recur randomly and independently in time, with an average rate of A per season. However, the infestation density, u{x) per dunam, is never uniform throughout the park. Some areas are known to suffer more than others, and some areas display wider fluctuations in the density of infestation than other areas. The damage to the park from a single bout with the parasite is: S{u) ( . )== // q{x)u{x)dx

(10.57)

The park is in danger of not recovering if the cumulative damage from all the infestations of the season exceeds the critical value of Ac- Develop an

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265

expression for the robustness to the uncertainty in spatial distribution and temporal recurrence of parasite attacks. 2. Non-linear spring. An asymmetric strain-hardening non-linear spring has the following relation between displacement x and force / : / = /cix-h/c2X^

(10.58)

Both coefficients, ki and /c2, are positive. The spring is permanently damaged if the magnitude of the displacement exceeds Xc, and you require the probability of failure not to exceed Pc. The magnitude of the force is uncertain and described by an incompletely known probability density. Our best probabilistic model is that the probability density of the force / is uniform on the interval [0, F], where F is a known value. However, there is evidence that rare events of greater force can occur. That is, there is a tail of the probability distribution for / > F , but the shape and magnitude of this tail is entirely unknown. In other words, the probability density is: , „. I constant, P(^) = | variable;

0< f
/-.r^ rr^\ ^''-''^

The value of F is known, but the behavior of p{f) for F < / is entirely unknown. Derive an expression for the robustness of predicting the probability of failure of the spring to uncertainty in the loading force. 3. Static deflection of a cantilever. Re-examine problem 1 on p. 103 in chapter 3. The horizon of uncertainty a in the info-gap model is a random variable with a known exponential distribution. Combine this information with the info-gap model of uncertainty about the load vector / to evaluate an upper bound on the probability of failure. 4. Simple harmonic oscillator. Consider again problem 9 on p. 107 in chapter 3. The horizon of uncertainty a has a known normal distribution. Evaluate an upper bound on the probability of failure.

Chapter 11

Robust-Satisficing Behavior The optimal sin problem. Standard notions of intelligent choice are theories of strict morality. ... Values and actions are to he consistent. Contrast that perspective with a view, somewhat more consistent with our behavior (as well as some theology), that there is such a thing as sin, that individuals and institutions sometimes do things even while recognizing that what they do is not what they wish they did, and that saints are a luxury to be encouraged only in small numbers. James G. March [10, p.51] Optimization works in theory but risk management is better in practice. There is no scientific way to compute an optimal path for monetary policy. Alan Greenspan, Financial Times, 27-28 August 2005.

The anomalies, foibles and even paradoxes of human decision-making under uncertainty provide fascinating reading and much delight (as well as anxiety) to behavior theorists [133, 166]. While the dominant paradigm asserts that decision makers optimize the attainment of their preferences, extensive evidence indicates that agents facing severe uncertainty will robustsatisfice rather than optimize. In this chapter we consider several empirical examples. In sections 11.1 and 11.2 we use info-gap decision theory to describe the Ellsberg and Allais 'paradoxes', which are empirical results challenging the classical theory of expected utility. We show that both the Ellsberg and the Allais observations are consistent with info-gap robustsatisficing behavior. In section 11.3 we apply the info-gap robust-satisficing paradigm to the model of Arrow-Pratt risk aversion as it is developed in 267

268

INFO-GAP DECISION THEORY

the expected-utility framework. In section 11.4 we show that, in a broad range of situations, robust-satisficing is a better bet — even without knowing the relevant probability distributions — than the direct optimization strategy. Finally, in section 11.5, we consider an extended example, and present an info-gap robust-satisficing explanation of the equity premium puzzle in financial economics.

11.1

The EUsberg 'Paradox'

Mas-Colell et al [120, p.207] explain the Ellsberg paradox [70] in terms of perceptions of uncertainty which have a natural formulation with info-gap decision theory. Ellsberg's observation, as adapted by [120], begins with two urns, R and H, where R contains a well shaken mixture of 49 white and 51 black balls, while H contains an unknown mixture of 100 white and black balls. Two balls are chosen randomly, one from each urn, and their colors are not revealed. The decision maker must choose one of these balls in each of two experiments. In the first experiment the decision maker wins $1000 only if the selected ball is black. Most participants choose the ball from R, suggesting that their subjective probability for a white ball from H is greater than 0.49. In the second experiment the decision maker wins $1000 only if the selected ball is white. Again most respondents choose the ball from R even though they are aware that the probability of a white ball from R is precisely 0.49. Ellsberg's 'paradox' is the decision maker's anomalous disregard for the fact that the subjective probability for white balls is presumably greater for urn H than for urn R. We can formulate these two experiments as choices between two pairs of lotteries, and then analyze the preferences of a robust-satisficing decision maker. In the first experiment the agent must choose between the following two lotteries: $1000 (black)

$0 (white)

(Choose i?-ball)

Li:

Pi

==

(0.51

,

0.49)

(Choose H-ha.\\)

L\:

Pi

==

(l-TT

,

^)

(11.1)

where the decision maker's nominal perceived probability of a white ball from H is 7T.

The probabilities of the lottery Li in eq.(ll.l) are precise numbers, and the participant may supply a numerical value for TT in L[. However, the participants do not perform numerical calculations but rather make subjective probability judgments. When a participant is presented with the probabilities Pi = (0.51, 0.49) the subjective perception of the meaning, for instance, of the 0.02 probability-difference between winning and losing may not be entirely clear. The respondent assimilates the precise numbers

C H A P T E R 11.

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269

as some more or less vague assertions about likely gain or loss. We model this uncertain perception by supposing that the participant translates a precise lottery p-^ = {pn^ P12) into an info-gap model of probability vectors, C{a^Pi) where the centerpoint Pi is {pn, Pu)At any given level of uncertainty a, the participant's perception of a lottery such as Li is represented by a set of lotteries C{a,Pi). One interpretation is that the participant would respond identically to any lottery in C{a^Pi), at least up to some unknown value of a; the participant does not distinguish, subjectively, between different but similar probability vectors. The perceived probabilities in both lotteries, Li and L[^ are uncertain and are represented by info-gap models C{a,Pi) and C'{a,p[). Roughly speaking, C\a,p[) is a more inclusive set than C{a,Pi), reflecting the fact that the probabilities of lottery Li (urn R) are clearer or less uncertain to the decision maker than the probabilities of lottery L[ (urn H). The fractional-error info-gap model is a good candidate to represent uncertain perceived probabilities. Let V2 denote the set of all mathematically legitimate probability vectors pi = (pn, ^12)- One realization of the decision maker's two info-gap models is: (11.2) I

I

Pii

I

I

I

Pii

I

J (11.3) J

where h > 1 implying that C\a^p[) is an expanded and shifted version of C{a^Pi). C\a,Pi) is shifted relative to C{a,Pi) because p^^ is different from Pi-

Let li be a positive number denoting the agent's known utility of winning $1000, and suppose the utility of winning $0 is zero. Thus the expected utility from either lottery in eq.(ll.l) is upu where pu is the agent's uncertainly perceived probability of choosing a black ball. The decision variable for choosing between lotteries Li and L[ is q:

{

1

for choosing Li

—1

for choosing L[

(11.4)

Suppose the agent wishes the expected utility to exceed a critical value Tc while also protecting against uncertainty in the perceived meanings of the lottery probabilities. This agent is a robust-satisficer, and we evaluate the robustness function as in eq.(3.169) on p. 100: a(9,rc)

=

H

max <j a : | —-—

(11.5) nim_ upn peC{a,pi)

-\-

mm_ upu ^

7l^[qXict,Pi)X'{ct,p[)

peC'(a,p[)

> r^ J

270

INFO-GAP DECISION THEORY

W h e n the decision maker chooses lottery L i , in which case ^ = 1 so the second minimum vanishes, the robustness is the greatest horizon of uncertainty a for which the least value ofupu exceeds r^ A similar interpretation applies when choosing the other lottery {q = —1). T h e robustness function in eq.(11.5) is very easily evaluated. Let / i i ( a ) and fi[{a) denote the first and second inner minima in eq.(11.5). Both /xi(a) and /ii(o;) decrease monotonically as a increases, because the uncertainty sets C{a^Pi) and C\a,p[) become larger as a increases. W h e n g = 1, the robustness is the greatest value of a at which iii{a) = TC- T h a t is, fii{a) is the inverse of 3 ( 1 , TC): /ii{a)

— Vc

if and only if

(11.6)

3(l,rc) = a

In other words, a plot of / i i ( a ) vs. a is the same as a plot of TC VS. 3 ( 1 , rc)^ Likewise, when g = —1, a plot of fi'i{a) vs. a is the same as a plot of TC vs. 3 ( - l , r c ) : p[{a) = Tc

if and only if

3(—l,rc) = a

(1L7)

One readily finds the following expressions for these //-functions: /ii(a)

(11.8)

max[0, (1 max[0, (1

ha)up[^]

(11.9)

Robustness S((7,rc) 1

upn

^Pn

Critical utility, TC Figure 11.1: Robustness curves for the first pair of lotteries, eq.(ll.l).

up2i

up2i

Critical utility, r^ Figure 11.2: Robustness curves for the second pair of lotteries, eq.(ll.lO).

Robustness curves based on eqs.(11.6)~(11.9) are shown in fig. 11.1. T h e robustness of choice ^ = 1 (lottery Li) is greater t h a n the robustness of lottery L'^ for any attainable critical utility. This is because p^ = 0.51 is

C H A P T E R 11.

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271

at least a bit larger than p[i = 1 — TT (recall that TT evidently exceeds 0.49), and because h > 1. The robust-satisficing agent will choose lottery Li over L[. Now consider the second experiment in which the decision maker must choose between the following two lotteries: $0 (black)

$1000 (white) (Choose i?-ball)

L2.

P2

=

(0.49

,

(Choose H-hal\)

L',:

P2

==

(TT

,

0.51) 1

(11.10)

-TT)

where the decision maker's nominal perceived probability of a white ball from i7 is TT as before and, from the results of the first lottery, it would seem that n > 0.49. The perceived probabilities are again uncertain and are represented by the info-gap models £(a,P2) ^^^ >C'(a,P2) i^ eqs.(11.2) and (11.3), with new center points P2 and p^2- This again implies that the probabilities of lottery L2 (urn R) are clearer or less uncertain to the decision maker than the probabilities of lottery L2 (urn H). The robustness of choice q is defined in eq.(11.5) with appropriate replacement of probabilities. The corresponding /x-functions are: A^2(<^) = /X2(«) =

max[0, {I — a)up2i] m8ix[0, {1 - ha)up2i]

(H-H) (11-12)

These relations are used to plot the robustness curves for lotteries L2 and L2 in fig. 11.2. These robustness curves cross because, though /i > 1 as before, but now P21 = TT which is greater than P21 =0.49. However, the difference between P21 ^^d P21 is likely to be small so the intersection occurs at very low robustness, especially if h is very large. Most decision makers may be expected to require robustness to uncertainty in excess of the robustness at which the curves cross, so the robust-sat isficing agent will again tend to choose lottery L2 over L2. In conclusion, there is nothing paradoxical in the decision maker's choice of urn R in both lotteries. In each case the agent is sat isficing the expected utility while maximizing the robustness to uncertainty.

11.2

The AUais 'Paradox'

The factors which can cause a person to alter a choice of action are numerous and diverse, ranging from rational conversion to forced coercion, and include doubt, error, deception and simple caprice. In this section we examine a classic example of seemingly paradoxical alteration of choice under uncertainty, and show how info-gap decision theory can describe this behavior pattern.

272

11.2.1

INFO-GAP DECISION THEORY

Formulation

The following illustration of the Allais paradox is based on [120, p. 179]. A lottery has three prizes with monetary values Vi = $2.5 x 10^, V2 = $0.5 x 10^ and V3 = $0. A lottery is specified by L = {pi^ P2^ Ps) where Pn is the probability of winning the prize valued at VnThe decision maker is twice offered a choice between two lotteries. The first pair of lotteries between which the decision maker must choose is: Li = (0, 1, 0)

and

L[ = (0.1, 0.89, 0.01)

(11.13)

The second pair of lotteries between which the agent chooses is: L2 = (0, 0.11, 0.89)

and

L^ = (0.1, 0, 0.9)

(11.14)

It is often found that people who prefer Li over L[ also prefer L2 over L2. However, this pair of choices is not universal, and for some people Li y L[ and L2 y L'2^ as well as other combinations of preferences. There are two distinct and essential elements of the observation made by Allais and subsequently by many others. First, and not surprisingly, human behavior is diverse: substantial fractions of the polled groups typically respond differently. Second, a fraction of respondents, those for whom Li >- L[ while L2 y L2, violate the fundamental axiom of independence of von Neumann-Morgenstern expected-utility theory [120, 125]: the bilinear additivity of personal utilities and outcome probabilities. Note that: L2 = L i + 0

and

L^ = L ; + 0

(11.15)

where (j) = (0, -0.89,0.89). Hence the preference Li >z L[ entails the preference L2 h L'2 if the preferences are based on maximizing expected utility. Thus, an agent who chose Li over L'^, but then preferred L'2 over L2, has displayed an axiom-violating alteration of preferences from the point of view of expected-utility theory. A decision theory must account for both of these behavior patterns: diversity and non-additivity. We will see that decision based on the info-gap robustness function does precisely this, while also providing an intuitively plausible model for human decisions under uncertainty. The first idea which underlies the info-gap explanation of Allais' observation is that different individuals focus on different uncertainties. The second idea which underlies the info-gap analysis is the hypothesis that a decision maker's behavior can be modelled as satisficing the expected utility while maximizing the immunity to uncertainty. That is, the decision maker (implicitly) robust-satisfices the choice of a lottery. We consider robust-satisficing behavior in two different situations. In section 11.2.2 the agent has uncertain perceived probabilities associated with each lottery, which leads to reversal of preference as observed in some of Allais's subjects. In section 11.2.3 the agent is uncertain about the utilities

CHAPTER U .

ROBUST-SATISFICING BEHAVIOR

273

of the lottery prizes, which results in behavior consistent with maximizing the expected utility as observed in other of Allais's subjects. Clearly an agent can be uncertain about both probability and utility, generating yet other behavior.

11.2.2

Probability Uncertainty

We first consider a situation in which the decision maker's uncertainty focuses on the subjective interpretation of the probabilities of the prizes. The probabilities of the four lotteries in eqs.(11.13) and (11.14) are precise numbers. However, as discussed in connection with the Ellsberg lotteries on p.268, the participants make personal probability judgments rather than numerical calculations. In the lottery L\ ~ (0.1, 0.89, 0.01), the perception of the meaning, for instance, of the 0.01 probability of gaining nothing may not be entirely clear. We model this uncertain perception by supposing that the participant translates a precise lottery L = (p^, ^2^ Ps) i^^o an info-gap model of probability vectors, £ ( a , p ) , where the centerpoint p is (P'i5 P2'> Ps)' For each lottery, Li, L[, L2 and L2, the decision maker has an info-gap model of probability uncertainty: Ci{a,Pi), C[{a,p[), £2(0^^P2) and C2{a,P2). For the first pair of lotteries, eq.(11.13), we suppose that Li is perceived more clearly and with less uncertainty than L[ because the latter entails more confusing details. Thus the info-gap model for Li must be "tighter" than for L[. Let V3 denote the set of all mathematically legitimate probability vectors pi = (j^n, pi2, Pis)- The fractional-error info-gap model, like eqs.(11.2) and (11.3), is: PieVs:

Pi i Pi

PieVs:

Pi

Pu < a , i = 1, 2 ^ a>0

(11.16)

Pu
(11.17)

i

i

z== 1, 2 I , a>0

p'\ i where h > 1 implying that C[{a,p[) is an expanded and shifted version of £i(a,pi). The situation is reversed in the second pair of lotteries, eq.(11.14), where 1/2 is more intricate and confusing that L2. Now the corresponding info-gap models are: ^2(0^, P2)

=

SP2^V3'

P2 i

P2z


z = 1, 2 I , a > 0 (11.18)

P2 i

^2i^.P2)

=

\P2^V3:

P2 i

PL

< a, z = 1, 2 I , a>0

(11.19)

p'2 i

where again h > 1 so now C2{a,P2) is an expanded and shifted version of ^2{(^^P2)-

274

INFO-GAP DECISION THEORY

The decision variable for choosing between lotteries L^ and L^ is g^:

{

1

for choosing Ln

—1

for choosing L^

(11.20)

The decision maker wishes to satisfice the expected utility at the critical value Tc while also protecting against uncertainty in the perceived meanings of the lottery probabilities. The robustness function, as in eq.(11.5) and eq.(3.169) on p.lOO, is: Snferc)

=

(11.21)

f / l + <7 max s <^ • —~—

. ~T 1-^ niin_ u p -{-

• ~T\^ inin_ u p \ > Tc

n4q,Cn{a.Pn)X'n{c^^p'n)]

where u is the decision maker's personal utility vector which, in this section, is fully known. When the decision maker chooses lottery L^, in which case q = 1 so the second minimum vanishes, the robustness is the greatest value of the uncertainty parameter a for which the least value of u p is satisfactory. A similar interpretation applies when choosing the other lottery (9 = - i ) This robustness function is readily evaluated, as in section 11.1. Let fin{(^) and ^^(o^) denote the first and second inner minima in eq.( 11.21). As explained in eqs.(11.6) and (11.7), iin{o^) is the inverse of ari(l,^c) while /i^(a) is the inverse of 3n( —1, TC). The utilities of high, moderate, and zero reward are Si, U2 and S3. We will suppose that Ui and S2 are positive and 1/3 = 0. The //-functions for the first pair of lotteries, eq.(11.13), are obtained by choosing pn and pi2 as small as possible at horizon of uncertainty a\ /ii(a)

=

IJi'i{a) =

max[0, (1 - a)uiPii] + inax[0, (1 - a)u2Pi2]

(11-22)

max[0, (1 - / i a ) S i p ( J 4-max[0, {1 - ha)u2p[2] i^^'"^^)

Similarly, the //-functions for the second pair of lotteries, eq.(11.14), are: /i2(o;)

=

max[0, {1 - ha)uiP2i]+rndix[0,

(1 "" ^<^)^2P22] (11-24)

//2(c^)

=

max[0, (1 - a)uiP2i] + niax[0, (1 - 0)1/2^22]

(11.25)

The robustness curves for the first pair of lotteries, eq.(11.13), based on eqs.(11.22) and (11.23), are shown in fig. 11.3. We have assumed that: u^p[ < u^p^

(11.26)

which implies that the expected-utility-maximizing agent would prefer Li over L[. The robust-satisficing agent would agree, since /i > 1, implying

C H A P T E R 11.

275

ROBUST-SATISFICING BEHAVIOR

Robustness Si(g,rc) 1

\ g = l

(^'l)

\

\

U p^

U pi

U^P2 U^JP2

Critical utility, fc

Critical utility, Vc

Figure 11.4: Robustness curves for the second pair of lotteries, eq.(11.14).

Figure 11.3: Robustness curves for the first pair of lotteries, eq.(11.13).

t h a t choice of Li is more robust to uncertainty in the perceived probabilities t h a n choice of L[ as seen in fig. 11.3. Robust-satisficers and optimizers agree on the choice, but for different reasons. T h e robustness curves for the second pair of lotteries, eq.(11.14), based on eqs.(11.24) and (11.25), are shown in fig. 11.4. In light of eqs.(11.15) and (11.26) we conclude t h a t : U P2
P2

(11.27)

which implies t h a t the expected-utility-maximizing agent would prefer L2 over L'2 (in order to be consistent with expected-utility theory's independence axiom). The robust-satisficing agent would now not necessarily agree. The robustness curves cross at utility-aspiration r^ and at robustness a^. Since the anticipated values of the expected utility, u p2 and u P2, are nearly the same, and especially if h is substantially greater t h a n unity, the curves cross at low robustness and high aspired utility. Lottery L'2 is more robust to uncertainty in the perceived probabilities t h a n L2 for utility-aspiration less t h a n r^ and at robustness greater t h a n Sx- T h e robust-satisficing agent will choose lottery L'2 unless very little robustness is needed or very great utility is aspired to. We can summarize our discussion of the Allais paradox, so far, as follows. We have considered an agent whose uncertainty is limited to the perceived probabilities of the lotteries. Li and L'2 are both simpler lotteries (they have fewer digits) t h a n L'^ and L2. T h e info-gap models reflect this. Ci{a^Pi) is an expanded and shifted version of Ci{a,Pi). Likewise, €2(0^,P2) is an expanded and shifted version of €2(0^,P2)T h e robust-satisficing agent prefers Li over L'^ because Li is more robust

276

INFO-GAP DECISION THEORY

(fig. 11.3). The expected-utility-maximizing agent agrees in this choice but for a different reason: the best-estimate of the expected utility is greater for Li. The robust-satisficing agent will tend to prefer L'2 over L2 (fig. 11.4), again because L'2 is more robust than L2 (except at very high aspiration). The expected-utility maximizer will disagree, thus conforming to the independence axiom. We see that the info-gap robust-satisficing paradigm explains the preference reversal observed by Allais.

11.2.3

Utility Uncertainty

We now consider a decision maker whose focus of uncertainty is entirely on the utility vector u, while the import of the probability vectors is perfectly clear. When the decision maker is presented with a lottery for prizes Vi, V2 and V3, the agent's uncertainty concerning the personal utilities of the prizes is represented by an info-gap model U{a,u), a > 0, where u is the agent's nominal utility vector. We consider the ellipsoid-bound uncertainty model, which is the following unbounded family of nested sets of utility vectors: U{a,u) = [u: {u-u)^W{u-u)
^ ^ ^

(^^-2^)

where VF is a known, real, symmetric, positive definite matrix which determines the shape of the ellipsoids of uncertain utility vectors. We use eq. (11.20) to denote the decision variable for choosing between lotteries L^ and L^, which are specified in eqs.(11.13) and (11.14). When presented with a choice between lotteries L^ and L^, the decision maker wishes to choose the more rewarding lottery, but is unable to unambiguously assess the expected utility from either lottery because of the uncertainty in the utility vector. What the agent (implicitly) does is to choose the lottery which has a satisfactorily greater level of expected utility with maximal robustness to uncertainty in the personal utilities. That is, the agent robust-satisfices the utility difference between the lotteries. We use the following expression for the robustness function for choice q between lotteries Ln and L^, where r^ is the smallest acceptable difference in expected utility: Sn(<7,rc) = max< a : I

min_ u^ [q {p^ - Pn)] ] ^ ^c \ \ueUia,u)

J

(11.29)

)

where p^ and p^ are the probability vectors of lotteries Ln and L^ respectively. The minimum in eq. (11.29) is: min^ u^ [q {p^ - p!^)] = qu^ {p^ - Pn) - (^ J{Pn ~ Vn) ^'^ {Pn - Pn) (11.30)

C H A P T E R 11.

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277

where ^^ is defined in this relation. From this we see that the robustness function in eq.(11.29) is the least upper bound of the set of a-values for which the following inequality holds: Q^^ {Pn - Pn) - < n > T^

(11.31)

Define A = u {p^ ~p'n)^ which is the anticipated utility-premium of lottery Ln over L^. The robustness of choice q, with required utility-premium r^ is: S(g,rc)

2A_ ^n

=

0

if re < gA " -^

(11.32)

else

The robustness-maximizing agent will choose q so that qA > 0. That is, the robust-satisficing decision maker will make the following lottery selection: Ln h L'^

if and only if

u^p^ > u^jp'^

(11.33)

In light of eq.(11.15) this means that the robust-satisficer will agree with the expected-utility-maximizer in both lottery selections. We can summarize the situations studied in sections 11.2.2 and 11.2.3 as follows. A robust-satisficing decision maker whose uncertainty is limited to the perceived probabilities will usually display the preference-reversal and axiom-violation displayed by some of Allais' subjects. On the other hand, a robust-satisficing decision maker with uncertainty only about the utilities of the prizes will behave just like an expected-utility-maximizer. In short, the robust-satisficing decision paradigm, combined with diversity of the agents' uncertainties, is consistent with Allais' famous experiments.

11.3

Info-Gap Analysis of Expected-Utility Risk Aversion

Attitudes towards risk can be assessed in various ways using info-gap decision theory as we demonstrated in chapter 6. Here we will follow the tradition of expected-utility theory [4, 139] and characterize a decision maker's aversion to or proclivity for risk by the choice between a bet and a certaintyequivalent outcome. In sections 11.1 and 11.2 we discussed several examples of the info-gap analysis of choices between lotteries. Those results can be directly applied to illustrate attitudes towards risk. We will see that the expected-utility interpretation of risk sensitivity can be incorporated into an info-gap decision model. Consider prizes of value wi > W2 > w^ where W2 = Pwi + (1 — ^ ) ^ 3 and 0 < P < 1. These prizes can be won in either of the lotteries: L:

p

L' '. p

=

(P, 0, 1 - P ) ^

(11.34)

=

(0, 1, 0)^

(11.35)

278

INFO-GAP DECISION

THEORY

where P and 1 — P are the probabihties of winning wi and ws respectively. L is a gamble between high and low gain, while L' is a sure bet on an outcome which is the mean of the extremes. Utility

Utility u{w)

u[w

Wi

W2

Ws

Wi

W2

Ws

Reward, w

Reward, w

Figure 11.5: Convex estimated utility function.

Figure 11.6: Concave estimated utility function.

Suppose t h a t the decision maker's perception of uncertainty focuses entirely on the utility vector and is modelled with the ellipsoidal info-gap model of eq.(11.28). Suppose also t h a t the agent is a robust-satisficer. Then, from eq.(11.33), the gamble, L, is preferred over the certainty-equivalent option, L', if and only if the utilities hes on a convex curve as in fig. 11.5: Ly

L'

if and only if

Pu{wi)

+ (1 - P)u{ws)

> u{w2)

(11.36)

where u{wi) is the decision maker's estimated personal utility from prize Wi. Likewise, if the estimated personal utility curve is concave as in fig. 11.6, then eq.(11.36) implies t h a t the robust-satisficing decision maker is risk averse and will prefer the sure bet, V, over the gamble, L. This robustsatisficing agent, who is uncertain only about utility, displays traditional expected-utility-maximizing risk behavior. Now consider a decision maker who is uncertain only about the meaning of the probabilities p and p' in the lotteries L and L' of eqs.(11.34) and (11.35). T h e former, more comphcated, distribution p is less clear t h a n the latter, certainty-equivalent, distribution p'. Let the fractional-error infogap model £2(0^,^) in eq.(11.18) denote the agent's uncertain perceptions of the distribution p. Likewise, let C2{a,p') in eq.(11.19) denote the less uncertain perceptions of p ' . C2{(^'P) is an expanded and shifted version of C2{a,p'). T h e decision maker wishes to robust-satisfice the expected utility, u p, where u is known and only the probability distribution p is uncertain. T h e

C H A P T E R 11.

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279

robustness of choice q between lotteries L and L', with critical utility TC, is a(g,rc) defined in eq.(11.21). Suppose that u p>u p\ so that relation (11.36) holds and the utilitymaximizing agent would prefer L over L' as illustrated in fig. 11.5. However, the robustness curves cross at utility-aspiration TX and robustness Sx ^ as in fig. 11.4. If S pis not too different from u p ' , or if /i is substantially greater than one, then TX will be large and 3x will be small, both motivating the robust-satisficer to choose L' over L, in contrast to the utility-maximizer with the same utility function. The robust-satisficing decision maker may appear 'paradoxical' or 'irrational' if the behavior is modelled from the perspective of expected-utility theory, but perfectly rational if modelled with the info-gap robust-satisficing paradigm. From the info-gap perspective it is not only the shape of the utility curve which determines risk orientation but also the interaction between the utility curve and the uncertainty perception. The robust-satisficer with uncertain probabilities and convex utility will tend to choose the certaintyequivalent lottery V rather than the gamble L which the utility-maximizer with the same utility function would choose.

11.4

Probability of Success

Evidence from competitive biological and economic systems, as well as from the Allais 'paradox' which we just discussed, seems to suggest that robustsatisficing may be, in some rather general situations, a better bet than direct optimization. In this section we will discuss a theorem which indicates why that might be true. This helps to understand why satisficing behavior is so commonly observed [51, 155]. Darwin [54, chap. 12] observed that invasive species introduced to a new region can dominate successful endemic species due to the newcomers' superior fitness. Simon [161, p.69] has pointed out that this illustrates that evolutionary success is a measure of comparative advantage rather than universal optimality. The search for "comparative advantage' is a satisficing strategy (don't optimize, just beat the competition), suggesting that biological systems may evolve by balancing performance against robustnessto-info-gaps rather than by optimizing their functionality. The outcome of Darwinian evolution is the survival of the more fit over the less fit, not necessarily of the most fit, illustrating the importance of comparative advantage rather than optimality. Darwin [54, p.378] also pointed to the "great fact" that similar habitats in the Old and New Worlds have "widely different . . . living productions!". Optimization would tend to produce similar phenotypes under similar constraints; robust satisficing produces diversity due to the added degrees of freedom associated with performance-sub-optimality. The analysis of foraging by animals from numerous diverse taxa seems to show that sub-optimal but robust decision strategies are preferred over

280

INFO-GAP DECISION THEORY

performance-optimal decisions [42]. Various economic puzzles, in particular the home-bias paradox [31] and the equity premium puzzle (which we will discuss in section 11.5), seem to be explained by supposing that economic agents employ robust-satisficing rather than optimizing strategies. In the present section we develop a simple example which illustrates why, in generic terms, robust-satisficing may be more successful than direct optimization. We illustrate a common situation in which the preference among available actions may differ between robust-satisficing and direct optimization, and in which the probability of success of the robust-satisficing strategy is never less than, and sometimes greater than, the probability of success of the optimization strategy.

11.4.1

Robust-Satisficing and Direct-Optimizing

Consider a choice of one from among / options or alternatives. Let Vi denote the value obtained from choosing option 2, where large value is better than small value. In this example we will compare the robust-satisficing strategy, which maximizes the robustness and satisfices the outcome, against the direct optimization strategy which chooses the most valuable anticipated outcome. The value Vi is highly uncertain and possibly varying in time, so that historical evidence is of limited utility. The best estimate of the value of option iisvi. For instance, this might be an historical mean, perhaps over a limited time window, and perhaps with temporal lag. Since things change, or since the long-range mean deviates greatly from the mean on short time intervals, this estimate is a poor indication of the true value which will accrue from option i the next time a choice is made. Let di denote an estimate of the variability of Vi, for instance, an historical standard error of the estimate. We have no more information on the future value of option i. Should we select the option whose best-estimated value is greatest (direct optimization), or should we robust-satisfice: select the option with adequate estimated value and maximal robustness? Which strategy has greater probability of success? The answer to these questions depends on the details of the decision problem and on the precise definition of 'success'. We will now demonstrate a class of problems for which the robust-satisficing strategy has probability of success as large or larger than the direct optimization strategy. This example is an idealization of a range of situations such as foraging by animals, biological evolution, economic competition, medical decision-making, and many other areas. One plausible info-gap model for uncertainty in the estimated value Vi, given the available information, is the following unbounded family of nested sets of values: U{a,Vi) = {vi : \vi-Vi\

< aai} ,

a>0

(11.37)

C H A P T E R 11.

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281

U{a, Vi) is the set of values Vi whose deviation from the estimated value, Vi, is no greater than ±aai. The horizon of uncertainty, a, is unknown, so the info-gap model is not a single set but rather a family of nested sets. The set U{a^Vi) becomes more inclusive as a increases. The direct optimization is the choice of the maximal best-estimate: i^ = argmaxi;^

(11.38)

i

The rationale for choosing option i^ is that, given our best estimates of the values, this is the most promising option in the sense that it has the highest estimated value. The large value of i;^* is desirable. But vi* is only an estimate of the value of option z, and this estimate is likely to be substantially wrong. An additional reason that large -zJ^* is attractive is the implicit assumption that, since vi^^ is large, then the actual value of option z* is also large even if Vi* errs. This of course is not necessarily true. Let Vc be a critical level of value needed for "success" or "survival". For instance, a foraging animal requires at least v^ calories in order to survive the night. Or, a brokerage firm must earn a rate of return no less than Vc in order to attract clients away from the competing firms. An option whose value exceeds v^ is acceptable, while an option with value less than Vc is not. Reliable achievement of at least Vc is desirable since shortfall below v^ means "failure" or "death". The robustness question for option i is: how wrong can the estimated value Vi be without jeopardizing the achievement of value no less than Vc from option i. Mathematically, the robustness of choosing option i is: a{i,Vc) — max < a : [

niin_ Vi\ >Vc\ \vieU{oc,V^)

J

(11.39)

J

The robustness function a(z,t'c) induces a preference ranking on the options. Option i is preferred to option j if critical value v^ is more reliably achieved with i than with j : i)-j

if

a{i,v^)>a{j,v^)

(11.40)

Note that the preference ranking of options may depend on the critical value v^. The robust-satisficing option, at critical value v^., maximizes the robustness: i(fc) = argmaxS(i,i;c) (H-^l) i

The robust-satisficing choice i{vc) is a Pareto efficient trade-off between robustness to info-gaps, 3, and critical value Vc- It is clear that i{vc) may differ from the best-model optimal choice i* in eq.(11.38). To evaluate the robustness let us denote the inner minimum in eq. (11.39) by ii{a,i). The robustness is the greatest value of a at which fi{a,i) > v^

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THEORY

T h e function //(a, i) decreases monotonically as a increases because the uncertainty sets U{a, Vi) are nested by a. Thus the robustness is the greatest value of a at which iJ^{a,i) = Vc. In other words, iJi{oi^i) is the inverse of the robustness function a{i,Vc)From the info-gap model of eq.(11.37) we find t h a t fi{a,i) = Vi — acji. Equating this to v^ and solving for a yields the robustness:

a(i,Vc)

if Vi > Vc

= 0

(11.42)

else

T h e robustness is positive if the estimated value of the option, Vi, exceeds the critical value Vc. If Vi < Vc then even the best-estimate does not satisfy the value requirement, so the robustness to uncertainty is zero. This robustness curve is illustrated in fig. 11.7.

S(j, '^c)

Figure 11.7: eq.(11.42).

Figure 11.8: Comparison of two options: reversal of preferences.

Robustness curve,

At the lower right end of the robustness curve in fig. 11.7 we see t h a t the robustness is identically zero when the critical value, Vc, equals the estimated value of the option, Vi. T h a t is, the reliability of the best-estimate of the value of option i is zero as expected from eq.(3.25) on p.49. At the upper left end of the curve we see t h a t robustness as large as Vi/ai entails zero critical value. T h e negative slope of the curve indicates t h a t robustness to uncertainty trades-off against critical value as discussed in section 3.1.7. Now consider two alternative options, whose robustness curves are shown in fig. 11.8. T h e parameters are such t h a t : Vi > Vj

and

— < -^ ai

(11.43)

(Jo

Roughly speaking these relations mean t h a t option i has higher estimated value t h a n option j , but i also has larger estimation-uncertainty t h a n j . T h e two conditions in eq.(11.43) are necessary and sufficient for the robustness

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curves to cross in the positive quadrant, as in fig. 11.8. It is clear that there are an infinity of choices of the parameters which satisfy eq.(11.43), as weh as an infinity of choices which do not. In the latter case one curve will lie above the other. The conditions in eq. (11.43) constitute the basic dilemma facing the decision maker in this problem. On the one hand option i promises higher value than option j , but on the other hand the data which indicate this preference for i over j are less reliable for i than for j . This means that the value could plausibly be worse with i than with j . The decision maker's dilemma is to balance estimated values against reliability of the estimates. Stated differently, crossing of robustness curves imphes that preferences will change as a result of altered demand for robustness, a, or altered critical value, Vc. If the estimated values are completely accurate then option i has greater value than option j and therefore i is preferred over j . Even if the estimates err by just a little, i is still preferred over j . Fig. 11.8 shows that this preference remains justified if the estimated values err no more than axFor any horizon of uncertainty less than a x the robustness curve for option i lies above the curve for option j , so z is preferred over j . On the other hand, when the horizon of uncertainty exceeds ax the robustness curve for option j lies above the curve for option i. Hence option j is preferred to option i. Equivalently, consider the critical value at which the curves cross, VxFig. 11.8 shows that i is preferred to j if value in excess o^ Vx is required for survival, while j is preferred to i otherwise. To summarize, the robust-satisficing and direct-optimization choice between options i and j differ if and only if all of the following conditions hold: Vi > Vj

and

— < —

and

Vc < Vx

(11.44)

They agree otherwise.

11.4.2

Satisficing and Survival

How will this appear in a competitive environment? Which option-selection strategy would fare better if the (real or simulated) world is governed by frequentist probabilities? Which strategy has greater probability of success? Option i succeeds if its value is no less than the critical value: Vi > v^Let Fi{') denote the cumulative probability distribution function of the value of option i. The probability of success for option i is: P,{i)

=

Fvoh{v^ > V,) = I - F^{v,)

(11.45)

There is an infinity of probability distributions of Vi and Vj such that the probability of success is greater with the option preferred by the robustsatisficing strategy, i{vc) in eq.( 11.41). If the conditions in eq.(11.44) hold

284

INFO-GAP DECISION THEORY

then this option is rejected by the direct-optimizing strategy, z* in eq.(11.38). Otherwise these two strategies agree and therefore have the same probabihty of success. To see that this is true we need the idea of a standardization class of probabihty distributions. Let f{x\q) be the probabihty density function (pdf) of a random variable x, where g is a vector of parameters of the pdf. The mean and variance of x are jix and cr^, and the standardized random variable is y = {x — ^^x)/^x and its pdf is g{y). f{x\q) defines a class of pdfs as the parameter vector q varies. If g{y) is independent of q then this is a standardization class. That is, if all the standardized random variables in the class have the same pdf, then this is a standardization class. Standardization classes are quite common, the normal, uniform, and exponential distributions all being examples. The standardized distribution g{y) may belong to the standardization class, as in the normal and uniform cases, but this is not necessarily true, as illustrated by the exponential case. Now let us suppose that Vi and Vj both belong to the same standardization class, where the cumulative probability distribution function of the standardized random variables is G{y). Now the probability of success for option i is:

Ps(i)

= Fvoh{v^>v,) = Fvoh('^^^^>^^^^^)

(11.46)

=

l_Gr!^^Jj

(11.47)

=

l-G[-3(2,^e)]

(11.48)

where eq.(11.48) results from eqs.(11.47) and (11.42) if Vc Psij)

if and only if

S{i, v^) > a{j, v^)

(11.49)

So, if the values of the various options all belong to the same standardization class, regardless of what that class is (normal, uniform, exponential, or whatever), then the robust-satisficing strategy is a better bet than the direct-optimization strategy if the conditions in eq.(11.44) are met. The strategies coincide otherwise and then have the same probability of success.

11.5 11.5.1

The Equity Premium Puzzle: A Solution Introduction

People around the world invest in both risk-free assets like government bonds as well as in highly risky assets like the stocks of commercial ventures. Risky assets are attractive because, on average, they off"er a higher rate of

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return than risk-free assets. Various "capital asset pricing models" (CAPM) attempt to explain the size of the premium which must be paid in order to attract investors to the risky assets [38, 50]. The equity premium puzzle is that the level of risk aversion which must be attributed to investors, in order to explain observed premia for risky assets, is far greater than required to explain investment behavior in other domains [106, 122]. In this section we discuss an info-gap robust-satisficing explanation. Investors cannot confidently maximize their future utility from an investment because of the severe uncertainty in future asset returns. The challenge facing investors is to decide whether adequate returns are sufficiently reliable. If not, then the resources can be invested elsewhere. We model this behavior by assuming that investors choose the investment so as to satisfice the utility and maximize the robustness against uncertainty in the future returns, rather than to maximize the total discounted utility. The crux of the matter is the treatment of uncertainty. We use an info-gap model to represent the investor's Knightian uncertainty about future returns: an unmeasurable and non-probabilistic epistemic gap between known past returns and unknown future returns. The info-gap formulation presumes that the investor knows the past returns, believes that future returns may deviate greatly from past experience, and that reliable probabilistic models of these deviations are unavailable. An info-gap model uses an unbounded family of nested sets to represent possible payoffs. No measure functions are involved. The investor has little confidence that past returns reflect future behavior reliably. The investor's central robustness question, which motivates our decision model, is: how wrong can the current estimate of future returns be, without jeopardizing the attainment of a specified level of utihty? The answer to this question generates preferences on options, without requiring probabilistic information. We present a multi-period two-asset implementation which provides insight into the equity premium puzzle. In section 11.5.2 we formulate the investment model, the info-gap model of uncertainty in future returns, the robustness function and the robust-satisficing decision strategy. In section 11.5.3 we derive asset-pricing relations which are the info-gap generalization of the Lucas asset-pricing model. In section 11.5.4 we derive an expression for the equity premium and discuss robustness, consumption growth, and risk aversion. In section 11.5.5 we discuss an info-gap concept of stationarity. A methodological summary and discussion of the results are presented in section 11.5.6.

11.5.2

Dynamics, Uncertainty and Robustness

We consider discrete time, t = 0, 1, . . . ,T. Q is the total consumption at time step t and u{ct) is the utility from this consumption. We assume that

286

INFO-GAP DECISION THEORY

u{ct) is continuous and that the marginal utihty is positive: u'{ct) > 0. We consider two assets, one a risky stock (i = 1) and the other a risk-free bond (z = 2). The generahzation to more than two assets is straightforward and would not entail any substantive alteration of our conclusions. Xit is the quantity of asset i held between t and t + 1 by a representative agent and can be either positive or negative. The holdings at time t are Xt = (xit, X2t)^• The holdings throughout the time horizon are X = (xo, X i , . . . , XT-I)- X is chosen by the investor. Pit is the ex-dividend price of asset i at time t, where pt = {pit^ P2t)^dit is the dividend of asset i at time t. The prices throughout the time horizon are p — (po, Pi, • • •, PT-I)Vit = Pit + dit is the payoff of asset i at time t, where yt = (yit, y2tV- Vo is known, yu is risky and uncertain while y2t is risk-free and known at t = 0 for all t > 0. The uncertain payoffs are ^(i) = (^n, . . . ,2/IT)- Vit can be either positive or negative. (Requiring non-negative payoff would alter our analysis only at robustness in excess of 100% of anticipated returns, which we will see is not of practical interest for understanding the equity premium puzzle.) The budget constraint is: ct+pjxt = yjxt-u

t = 0, . . . ,T

(11.50)

The initial endowment, x_i and yo, are known at time 0. The investment in the last step is zero: XT = 0. Short sells are allowed but Ct cannot be negative. The choice variables are XQ, . . . , X T - I , which determine the consumptions through the budget constraints. The payoff of the risky asset, yu^ is uncertain for t > 0, and y^ is the best known estimate of yu at time 0. We assume that the anticipated payoff is positive: y^ > 0. yio is known. The investor views y^ as a best but rough estimate of future payoffs, without knowing how wrong this estimate will turn out to be. We use the unbounded fractional-error info-gap model to represent uncertainty in the risky-asset payoffs:

y{(^.y) = {viD • \yit-yu\


a>o

(11.51)

y{a^y) is the set of risky payoffs yu^ for t = 1, . . . , T , whose fractional deviations from the anticipated payoffs y^ are no greater than a. The fractional error, a, is unknown, so the horizon of uncertainty is unbounded. This info-gap model is not a single interval, but rather an unbounded family of nested payoff intervals. The discounted utility up to time T is: T

^(x,i/) = ^ / ^ * i / ( Q )

(11.52)

t=0

The investor desires to choose the asset holdings x throughout the time horizon so as to attain discounted utility no less than u. That is, the investor

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287

wishes to satisfice the discounted utihty at the value u: U{x,y) >u

(11.53)

u can be thought of as a 'reservation utihty'. The investment wih be pursued if the investor has adequate confidence in achieving adequate reward. The r o b u s t n e s s to uncertainty in the payoff's, of holdings x with utilityaspiration u, is the greatest horizon of uncertainty a up to which all payoffs yield at least the desired utility: S(x, u) = max la: [

min _ U{x, y) \ > u> \yii)ey{a^y) J J

(11.54)

The set of a-values in this definition is empty if U{x,y) < u, meaning that holdings x do not attain utility u with the anticipated payoffs y. In this case we define a{x,u) — 0 and we say that utility aspiration u is 'infeasible'. Any other u is 'feasible'. We will be interested in robust-satisficing investments: those which satisfice the discounted utility and maximize the robustness: x{u) = diiguidiX a{x,u)

(11.55)

where the maximum on x is subject to the budget constraint with nonnegative consumption.

11.5.3

Asset-Pricing Relation

Consider the time horizon t — 0, 1, . . . , T. The investor has T choice vectors Xo = (^10, ^2o)'^, • • •, XT-1 = {xi^T-ii X2,T-i)^• The uncertainties are the unknown risky payoffs ?/(i) = (yn, . . . , yir)^• The info-gaps in these payoffs are described by 3^(a, y) in eq.(11.51) where the horizon of uncertainty a is unknown. The risk-free payoffs (?/2o, • • •, ?/2T) are known. Using the budget constraint in eq.(11.50), the discounted utility in eq.(11.52) is: T

U{x, y) = Y^ P^u{yitxi^t-i

+ y2tX2.t-i - pjxt)

(11.56)

t=o

xit can be either positive or negative. Let 0^ = 1 if Xu > 0 and 0^ = — 1 otherwise for t > 0. Define 0_i = 0 . The marginal utility is positive, so the lowest utility up to horizon of uncertainty a occurs when the risky payoffs yit are such that yuXi^t-i is minimal, for t = 1, . . . ,T. (Recall that ^lo is known.) The risky payoffs which minimize the utility at uncertainty a are yit — {l — a(t)t-i)yn. Thus the minimum in the definition of the robustness, eq.(11.54), is:

288

INFO-GAP DECISION THEORY

li{a,x,p)

=

min _ U{x,y)

(11.57)

y(i)ey{a,y) T

=

y^ 3^u[{l - a(t)t-i)yitXi^t-i t=0

+y2tX2,t-i -pjxt]

(11.58)

^

which defines Ct{a), the lowest anticipated consumption up to uncertainty a. The utihty aspiration u is feasible if the righthand side of eq. (11.58) is no less than u in the absence of uncertainty {a — 0). Because the marginal utility is positive and a 0 t - i ^ i t ^ i , t - i is also positive, the righthand side of eq. (11.58) decreases strictly monotonically as a increases. Hence, for any feasible aspiration, the robustness is the value of a at which the righthand side of eq.(11.58) equals u: u = u{ylx-i-plxo)

-Y^ph

(11.59)

(1 - S(X, ix)0t-l) 2/lt^l,t-l + y2tX2,t-l

'

T^

-

pjxt

'

ct{a)

If eq.(11.59) holds for a continuum of investments x, then its derivatives with respect to Xu and X2t also hold. Suppose there is a robust-satisficing investment which maximizes the robustness without constraint (we study the existence of such investments in section 11.5.5): dS. ^—- - 0 ,

i - 1 , 2; t = 0, . . . , T - 1

(11.60)

OXit

Differentiating eq.(11.59) with respect to x ^ (which we assume differs from zero) and X2t, for t = 0, . . . , T — 1, and using eq.(11.60) yields: du{ct)

—. Pit = act du{ct) —^ P2t = dct

d?i(Q+i)

_

P—^ (1 - a(/)0?/i,i+i ciQ-fi du{ct-^i) P—i 2/2,t+i dQ+i

nA a^\ (11.61)

(11.62)

where Q = Ct{a) is the argument of the utility function in eq.(11.59). These relations are the info-gap generalizations of the first-order conditions in the Lucas asset-pricing model [35, e q . ( l l ) , p.511]. T h e ordinary Lucas relations result when 3 = 0. Eq.(11.61) asserts t h a t the risky-asset price pu depends not only on the marginal rate of substitution as in the Lucas model, but also depends on the required robustness a. At low risk aversion the marginal utility is nearly

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constant, so the dominant term on the righthand of eq.( 11.61) is 1 — 30^. When (f)t = -\-l (e.g. the representative agent has positive risky holdings), this means that the risky-asset price tends to decrease as the robustness which the investor requires increases. On the other hand, for short sales of the risky asset (so ({)t = —1), the price tends to increase as the robustness which the investor requires increases. The basic trade-off theorem (section 3.1.7) asserts that robustness decreases as aspiration increases: a{x^u) decreases as u increases. This holds both for arbitrary investments x and for the robust-satisficing investments x{u) in eq.(11.55). Furthermore, the robustness vanishes at the greatest feasible aspiration: a{x^u) = 0 if it = U{x,y). (These results derive from the nested structure of info-gap models, e.g. y{a,y) in eq.(11.51), and the definition of the robustness, eq.(11.54).) This trade-off between robustness and utility-aspiration is illustrated schematically in fig. 11.9. Fig. 11.10 illustrates eq.( 11.61) for low risk aversion and 0^ = -f-l: the risky-asset price tends to decrease as the demanded robustness increases. Consequently, combining figs. 11.9 and 11.10, the risky-asset price rises as investor aspirations increase as shown in fig. 11.11. The solid dots in figs. 11.9-11.11 identify corresponding points. In fig. 11.9 the robustness vanishes at the maximal utility aspiration. In fig. 11.10 we see that zero robustness corresponds to maximal risky-asset price, which is the analog of the Lucas asset price. PIQ represents the price when the aspiration is maximal and the robustness is zero. In fig. 11.11 this maximal price corresponds to maximal utility aspiration. Sub-maximal utility-aspiration (equivalently: positive robustness-aspiration) will force a drop in the price of risky assets held by the representative agent {(pt — +1)-

Robustness a{x,u)

Aspiration

U{x,y)

Figure 11.9: Tradeoff of robustness against utility aspiration.

Robustness Figure 11.10: Price of risky asset vs. robustness, eq.(11.61), for (l)t = +1.

Aspiration

U{x,y)

Figure 11.11: Price of risky asset vs. utility aspiration for (/)t = -hi.

290

INFO-GAP DECISION THEORY

11.5.4

Equity Premium

Define the anticipated or 'best-estimate' rates of return to the risky asset: ^it = VitlVi.t-i-) for t = 1, . . . ,T. Likewise the known rates of return to the risk-free asset are r2t = y2tlP2,t-i- With these definitions we can write eqs.(11.61) and (11.62), for t = 0, . . . , T - 1, as: u\ct)

=

/3^'(Q+i)(l-S0,)n.t+i

(11.63)

u'{ct)

=

/5ix'(c,+i)r2,t+i

(11.64)

We now use these relations to show, in the three different ways, how the info-gap robust-satisficing decision paradigm provides insight into the equity premium puzzle. Robustness. Subtracting eq.(11.64) from eq. (11.63) results in the infogap generalization of a basic CAPM relation [35, eq.(4), p.507]: 0 = pu\ct+i)[(l

- S(/),)ri,,+i - r2,t+i],

t = 0, . . . , T - 1

(11.65)

This relation can be re-arranged (assuming I3u'[ct^i) > 0) to show that the premium for the risky asset is proportional to the required robustness: n,t+i - r2,t+i = 30tri,t+i

(11.66)

Investors who do not require robustness to uncertainty (3 = 0) also do not need a premium to attract them to have positive holdings on risky assets (0^ = -f-l). As the investor becomes more sensitive to Knightian uncertainty in the returns (as a increases), a greater premium is needed to induce investment in both risky and risk-free assets. Investors need not be terribly sensitive to Knightian uncertainty in order to explain the usual equity premium. For example, if Vi^t+i = 1-07 and r2,t+i = 1.01, then the robustness in eq.(11.66) which explains this 6% premium is S = 0.06/1.07 ^ 0.056. This is fairly low robustness compared to the variation of risky returns which is on the order of 15%. In fight of the monotonic trade-off between robustness and aspiration for utility, illustrated in fig. 11.9, this low robustness suggests that investors reduce their aspirations only slightly below the maximum. This is illustrated schematically by the open circle on the robustness curve of fig. 11.9. This small reduction in aspiration, in response to Knightian uncertainty in the returns, is responsible for the observed premium for the risky asset. Note that the only assumptions concerning the utility function u{c) are that it is continuous and that the marginal utility is positive. We have assumed nothing about the magnitude of Arrow-Pratt risk aversion. We have assumed time-separation of the discounted utility U{x^y). The time horizon T is arbitrarily large. Consumption growth. The equity premium can be understood in another manner from eqs.(11.63) and (11.64), with explicit reference to consumption growth. Define m = (3u'(ct^i)/u\ct). In stochastic pricing

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291

models the expectation of this quantity is sometimes called the stochastic discount factor [50]. Now eqs.(11.63) and (11.64) can be combined to yield the equity premium as: (n,t+i - r2,t+i)m =

^

(11.67)

1 - acpt

Consumption growth and equity premium imply robustness. If, as before, (/)t = + 1 , ri^i+i—r2,t+i = 0.06 and m = 1/1.01, then the robustness required by investors is again about 0.056. Risk aversion. Eq.(11.63) can be written as: 1 = {1 - a(t)t)mri,M

(11.68)

where as before m = /3i/'(Q+i)/i^'(ct). Both n.^+i and m depend on anticipations based on information available at time t = 0. At time 0 the investor could roughly estimate these future values as historical expectations. Specifically: Eo(mri,t+i) = Eo(m)Eo(ri,t+i) -f (To{m)ao{ri,t^i)poim,

ri,t-\-i)

(11.69)

where po{m, r^i,t+i) is the correlation coefficient between m and ri^t-\-i whose value is in the interval [—1, 1]. Eqs.(11.68) and (11.69) can be combined as: Eo(ri,t+i) -^2,t+i - T^ - I N f^ + ^oM^2,t+i (11.70) cro(ri,t+i) (1 - a(pt)cro{rid^i) where we have used eq.(11.64) to approximate Eo(m) as l/r2,t+i. Adopt the constant relative risk aversion utihty function u{c) — c^~'^/{I — 7). Thus m = P{ct-\-i/ct)~^. Now suppose that ln(ct+i/ct) is distributed normally with mean and standard deviation Eo[ln(ct+i/ct)] = 0.02 and ao[\n{ct^i/ct)] — 0.01. From this it results that cro(m) ^ O.OI7. Thus the upper limit of relation (11.70) is approximately: Eo(r,,^0-r..,i ^ :^^;^-^t+ 0.0l7r,,,, cro(ri,t+i) (1 - a0t)ao(ri,t+i)

(11.71)

Typical values are Eo(ri^^+i) = 1.07, cro(ri.t+i) = 0.15 and r2,t+i = 1.01. If S = 0 then the risk aversion coefficient required in relation (11.71) is 7 = 39.6. This value of 7 is much greater than usually estimated, which is the traditional equity premium puzzle [50, chap. 1]. On the other hand, for (^t — + 1 , if S = 0.055 then relation (11.71) implies 7 — 0.80 which is an entirely ordinary level of Arrow-Pratt risk aversion. The info-gap equity premium relations, eqs. (11.66), (11.67) and (11.71), each reveal a different aspect of the role of robust-satisficing in understanding the equity premium. Together they constitute the info-gap robustsatisficing version of the "empty fig. 4" in Mehra and Prescott's famous

292

INFO-GAP DECISION THEORY

paper [122]. Mehra and Prescott's figure showed that reasonable values of risk aversion and discount rate cannot be reconciled with observed equity premium, risk-free rate and consumption growth through a specific utilitymaximizing paradigm. Eqs.(11.66), (11.67) and (11.71) demonstrate that these quantities are coherent within an info-gap robust-satisficing framework.

11.5.5

Stationarity

Info-gap robust-satisficing stationarity at time t = 0 is the existence of a set of prices p and investments x and a utility aspiration u such that X is a robust-satisficing investment x{u) which maximizes the robustness, eq.(11.55), and the budget-constrained consumptions are non-negative for all payoffs at horizons of uncertainty a < a(x,iZ). In other words, at robustsatisficing stationarity the representative agent satisfices the discounted utility, maximizes the robustness to payoff-uncertainty, and has non-negative consumption within the budget if the realized uncertainty does not exceed the robustness. The existence of robust-satisficing stationarity was assumed, eq.(11.60), in deriving the asset price relations, eqs.(11.61) and (11.62). In this section we identify conditions under which this assumption holds. It is convenient to combine eqs.(11.54) and (11.57) to express the robustness of investment stream x, with utility aspiration u, as: S(x,u) = max{a : fi{a,x,p)>u}

(11.72)

We continue with the assumptions that u{c) is continuous and u\c) > 0 and (j)t-iyitXi^t-i > 0 for t > 0. Thus, from eq.(11.58), ii{a,x,p) is continuous in both x and a and strictly decreases as a increases. The robustness, a{x^u), of investments x with aspiration u is the greatest value of a satisfying ii{a,x,p) > u. Hence: li{a^x^p) = u implies

a{x^u) = a

(11.73)

Finally, an investment stream x which maximizes ii{a^x^p) also maximizes a(x,i^), where the value of the aspiration, u^ is /j,{a^x^p) and the corresponding maximal robustness is a. We explain this as follows. Eq.(11.73) implies that a plot of a vs. ii{a,x,p) is the same as a plot of a(x^u) vs. u. This is illustrated infig.11.12 (which is the same afig.11.9). Consider two different investment streams, x^^^ and x^'^\ for which Zt{a) in eq.(11.58) is non-negative for t > 0. From the monotonicity and continuity of /i(a, x,p) in a we conclude that if, for some horizon of uncertainty ai'.

fi{a,,x^'\p)>li{ai,x^^\p)

(11.74)

then: a{x^^\u)

> a{x^'^\u) = ai

for u = ij,{ai,x^'^\p)

(11.75)

C H A P T E R 11.

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ROBUST-SATISFICING BEHAVIOR

IJ,{a,x^

,p)

a{x,U)

fi{a,x^

/i(a,x,p)

Figure 11.12: Robustness evaluated from ii{a,x,p).

' ,p)

IJi{a,x,p)

Figure 11.13: Illustration of eqs.(11.74) and (11.75).

This is illustrated in fig. 11.13. Robust-satisficing investments, x{u)^ maximize the robustness, eq.(11.55). From relations (11.74) and (11.75) we find that all robust-satisficing investments are those investment streams which maximize /i(a,x,j9) on x. Finally we return to the concept of info-gap stationarity. From eq.(11.58) we recognize that ii{a,x,p) is the T-period discounted utility with risky payoffs (1 — a(f)t-i)yf and risk-free payoffs 2/2t- Ljungqvist and Sargent [113] discuss recursive methods for finding investment sequences x which maximize the discounted utility. Since maximizing /i(a,x,p) is equivalent to maximizing a(x,?l), we see that the robustness has a maximum, info-gap robust-satisficing stationarity exists, and eq.(11.60) can be satisfied if and only if this discounted utility can be maximized. In other words, robustsatisficing stationarity exists if and only if there is an investment stream X which maximizes this discounted utility. The utility aspiration at which the robustness is maximized is u = ii{a,x,p) and the maximal robustness is aix^u) — a.

11.5.6

Discussion

Info-gap robust-satisficing is a quantification of three concepts which have been discussed in economic literature: Knightian uncertainty, bounded rationality and robustness. Info-gap robust-satisficing explains the equity premium puzzle and suggests a method for modelling behavior under severe uncertainty. Knightian uncertainty. An info-gap model of uncertainty is a nonprobabilistic representation of uncertainty which is starkly minimalistic. Info-gap models provide no opportunity for insurance-like calculations. The theory of info-gap uncertainty provides one plausible quantitative model for Knight's concept of "true uncertainty" for which "there is no objective

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measure of the probability", as opposed to risk which is probabihstically measurable [103, pp.46, 120, 231-232]. Info-gap models, together with the robust-satisficing decision model, have been used to explain the home-bias phenomenon [31] and the Allais and EUsberg paradoxes (sections 11.1 and 11.2). Info-gap models have also been employed in financial risk assessment [26], modelling animal behavior under uncertainty [42], project management [33], biological conservation [145], and engineering analysis and design [25]. Further discussion of the relation between Knight's conception and infogap theory is found in section 13.5. Similarly, Shackle's "non-distributional uncertainty variable" bears some similarity to info-gap analysis [157, p.23]. Likewise, Kyburg recognized the possibility of a "decision theory that is based on some non-probabilistic measure of uncertainty." [107, p. 1094]. Info-gap models are not the only possible way to quantify Knightian uncertainty. On the contrary, Gilboa and Schmeidler [78], Epstein and Wang [72], Epstein and Miao [71] and others, achieve uninsurable uncertainty of a clearly Knightian type by replacing a single prior probability distribution with a set of distributions. These approaches are Knightian "true uncertainty" since the absence of a probability measure on the set of probability distributions makes the uncertainty uninsurable. Nonetheless, an info-gap model of uncertainty is a more extreme departure from the probabilistic tradition. In our formulation, preferences are generated by the robustness function without any distribution functions at all. Bounded rationality and robustness. The investor's ability to optimize future outcomes is severely limited. Past asset returns provide only a rough indication of future returns. The investor's access to information about and understanding of the relevant social and economic forces is far too limited to enable reliable assessment of maximal future behavior. Furthermore, the great variability of the returns motivates the choice of a strategy which balances aspiration for utility against aspiration for immunity to uncertainty. The info-gap robust-satisficing decision strategy does precisely that. Hansen, Sargent and Tallarini [87, p.873], "show how a preference for robustness lies concealed within the quantity implications of the permanent income model". That is, a desire for robustness against uncertainty influences the choices of economic agents. This insight can be viewed as a starting point of the current analysis, which shows that desire for robustness is as important as desire for material utility. In that spirit, the theorem which underlies this analysis establishes an irrevocable trade-off between robustness and reward. This trade-off motivates the robust-satisficing strategy, very deliberately violating the classical axiom of economic rationality. The investor does not attempt to maximize utility, or even a risk-adjusted utility. In the info-gap decision model, utility is satisficed; what is maximized is robustness to uncertainty. The "rationality" of this strategy is expressed in the trade-off of the aspiration for utility against the aspiration for robustness to uncertainty (illustrated in fig. 11.9). This trade-off is a general

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theorem which asserts that maximal utihty is invariably accompanied by zero robustness. Utihty-maximization is entirely unreliable. Equity premium. Kocherlakota concludes his discussion of the equity premium puzzle with the comment: The universality of the equity premium tells us that, like money, the equity premium must emerge from some primitive and elementary features of asset exchange that are probably best captured through extremely stark models. With this in mind, we cannot hope to find a resolution to the equity premium puzzle by continuing in our current mode of patching the standard models of asset exchange with transactions costs here and risk aversion there. Instead, we must seek to identify what fundamental features of goods and asset markets lead to large risk adjusted price differences between stocks and bonds. [106, p.67] The info-gap robust-satisficing decision model offers the possibility of new insight. Two ideas are prominent. First, risk aversion is a multi-faceted phenomenon which is not captured entirely by utility-function curvature. Especially in situations of great Knightian uncertainty, where betting is not a particularly useful concept, risk aversion is expressed in part by the willingness to forego utility in exchange for robustness-to-failure. Second, the info-gap robust-satisficing decision model demonstrates that the classical axiom of utility maximization can be relaxed without losing touch with economic intuition and data. The fundamental psychological premise — that utility is desirable — does not imply that maximal utility is most desirable. In this section we have shown that realistic and sensible economic reasoning can be based on, and modelled by, the concept of satisficing the utility and maximizing the robustness with non-probabilistic info-gap models of uncertainty.

Chapter 12

Retrospective Essay: Risk Assessment in Project Management The uncertainties which persist as causes of profit are those which are uninsurable because there is no objective measure of the probability of gain or loss. This is true especially of the prediction of demand. It not only cannot be foreseen accurately, but there is no basis for saying that the probability of its being of one sort rather than another is of a certain value—as we can compute the chance that a man will live to a certain age. Situations in regard to which business judgment must be exercised do not repeat themselves with sufRcient conformity to type to make possible a computation of probability. Frank H. Knight [104, p. 120] The best-laid schemes o' mice an' men Gang aft a-gley An' lea'e us nought but grief an' pain For promised joy. Still thou art blest, compared wi' me! The present only toucheth thee: But och! I backward cast my e'e, On prospects drear! An' forward, though I canna see, I guess an' fear! Robert Burns, To a Mouse [41]

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In this chapter we look back over some of the main ideas and methods developed in this book: information-gap uncertainty, robustness, opportuneness, preference reversal, risk, information, and value judgments. Having acquired, in our study up to now, a familiarity with the quantitative aspects of info-gap decisions, the discussion in this chapter is entirely qualitative: not a single equation will appear. The discussion is focused on managing uncertainty in a multi-task project. However, the considerations are relevant to generic strategic decisions under severe uncertainty.

12.1

Info-Gap Uncertainty: W h a t Is It?

Even if we cannot always compute probabilities it is still sometimes useful to think in terms of frequent or rare outcomes. Personal and professional experience are powerful engines for calibrating one's degree of belief in the likelihood of various contingencies. However, in many planning decisions one cannot even identify the dominant contingencies, let alone gauge their likelihood. In such situations, all too prevalent in our turbulent world, how are we to arrange our thinking about uncertainty? In the face of severe lack of information, uncertainty is the gap between what is known and what must be known for a responsible decision. The phenomenon of severe uncertainty lies in that tantalizing gap between what we do know and what we could know.^ An info-gap is a lack of data or an error in matters of fact, but it is also a lack of understanding or a deficiency in models and theories. Info-gaps can either increase or decrease with the accumulation of new facts and understanding. This is unlike most common conceptions of uncertainty, where new data dispell uncertainty or at least leave it unmodified. In contrast, info-gap uncertainty relates to human knowledge and understanding; new insight may reveal greater uncertainty than was previously appreciated. Surprises may change one's perspective; revolutionary innovations or discoveries may force fundamental revision of thought. We will review the implications of info-gap uncertainty for assessing risk in managing multi-task projects. The concept of robustness is central to the info-gap approach, and we will express it as the greatest info-gap consistent with satisfactory performance. Robustness-to-uncertainty is desirable, and the decision maker must make a qualitative value judgment of how much robustness is needed. Next we will discuss the concept of robustness-touncertainty as an assessment of risk. We will show how to characterize gambling-like behavior in the context of info-gap uncertainty. We will use info-gap robustness to explore the value of gathering information or restructuring the project. We will also discuss the role of optimization in strategic planning under severe uncertainty. Uncertainty is not only pernicious; it also entails the possibility of de^Info-gap uncertainty is first discussed in chapter 2.

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sirable but unanticipated windfall reward. How are risks and opportunities related? Are they always antagonistic and mutually exclusive? Finally, we will touch on the role of quantitative decision support systems in assessing risk and exploiting uncertainty.

12.2

Info-Gap Uncertainties in Project Management

Information-gap uncertainties — disparities between what the manager actually knows and what really needs to be known for assured successful project completion — are widely acknowledged but not always explicitly conceptualized and included in decision-making. Let us consider two examples. Info-gap uncertainty in the demand curve. In the quote at the beginning of this chapter Frank H. Knight identifies the prediction of demand as the archetypal source of economic uncertainty. One may know the demand from last season, or last week, but, as Neils Bohr was fond of saying, prediction is always difficult, especially of the future [124, p.320]. Increasing info-gap uncertainty 1 Representative J Upper envelopes

Price

Last year's demand curve 1 Representative I Lower envelopes

Quantity Demanded Fi gure 12.11 Demand curve (middle) and two representative uncertainty envelopes. Despite the futility of precise prediction, and the tremendous difficulty of probabilistic modelling, one still has some conception of how the uncertainty is structured. To perceive an info-gap uncertainty means to capture its basic pattern of clustering. If last year's demand curve falls off in such-and-such a manner with increasing price, one cannot expect the same behavior next year. Indeed the differences may be great. But experience may lead one to

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expect the price-vs.-quantity curve to decrease steadily like the thick line in fig. 12.1, not to waver up and down. Furthermore, one may anticipate larger fluctuations in demand at low price and high demand, while lower fluctuations may tend to prevail at high price and low demand. That is, the general contours of an envelope of demand curves may be accessible, as illustrated by representative envelopes in flg. 12.1, even though the size of the demand-curve uncertainty envelope is unknown. The info-gap uncertainty increases with the size of the envelope, as indicated in the flgure. This is much weaker information than knowing the likelihood of different specific demand curves. Nonetheless, the envelopes of uncertainty provide a tool with which to model and manage the uncertainty, without talking about probabilities at all. An info-gap is a disparity between what we do know and what we need to know. In the demand-curve example we know basically three things: (1) last year's curve, (2) demand curves decrease monotonically, and (3) fluctuations tend to be larger at large demand. What we don't know is the precise curve which will prevail in the future, nor the probabilities of particular realizations. This info-gap is quite substantial, but a little information can go a long way, especially if it is not corrupted by unwarranted and unveriflable assumptions. An info-gap model of uncertainty encodes the available information, and expresses the information gap around this information, with a family of nested sets of possible events. In the case of demand curves one might use a slope-bound info-gap model to represent the fragmentary information. Info-gap uncertainties in task durations and expenditures. Another common focus of uncertainty in project planning and management is the duration and cost of the individual tasks. The planner knows the projected budget and anticipated duration for each task. There also usually is some qualitative information about how dependable these projections are. For a routine procedure, one which the flrm has implemented repeatedly in the past, the uncertainties may be small. On the other hand, a task involving emerging technology or untried methods may entail much greater uncertainties. In any case, there is a gap between the information available to the planner and the information needed for fully confident decision-making. The info-gap is usually substantial, and the data rarely enable one to verify the choice of a probability distribution of time and cost over-runs. However, an info-gap perception captures the gist, without introducing unwarranted extrapolations. Fig. 12.2 portrays an info-gap picture of time and expenditure uncertainties. Each task has a known nominal duration and nominal cost. However, the actual values may vary, and some may be expected to vary more than others. The horizon of uncertainty is unknown, but approximate assessments of the relative ranges of variation are within the grasp of the decision maker. Fig. 12.2 shows the decision maker's info-gap conceptualization of the duration- and expenditure-uncertainties. For example, we

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Unknown horizon iof uncertainty

Duration

Nominal duration

1

:

+1

' T

2 Task

i k

Expenditure

^'

Unknown horizon of i uncertainty

^r

Nominal cost ^

1

2 Task

Figure 12.2: Info-gap uncertainties in duration and expenditure of tasks in a multi-activity project. see from fig. 12.2 that task 3 has the smahest relative variation in duration (upper frame) and the greatest relative variation in expenditure (lower frame). These perceptions can be encoded, non-probabilistically, in an infogap model. Origins of info-gap uncertainties. Info-gap uncertainties are intimately linked with two other managerial facts of life: high project speed and lack of managerial attention. Project speed is nearly always a competitive requirement. In addition, managers often do not have adequate time for project planning. Empirical studies portray the manager's life as leaving little time for reflection and analysis amid the pressure of short-term, interrupted, and sometimes chaotic activities. Time pressures are further amplified by telefaxes, cellular phones, and electronic mail systems. In short, uncertainty, speed, and scarcity of attention are very much interrelated. The need for speed reduces the manager's ability to give careful and in-depth attention, the lack of attention reduces knowledge and familiarity, and that is precisely info-gap uncertainty. The adverse impacts of high project speed and diffuse managerial attention are often exacerbated by structurally inherent uncertainties. In the

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construction industry, Laufer and Tucker [111] concluded that uncertainty is not an exceptional state in the otherwise predictable process of construction work. In fact it is a permanent feature in the realm of construction, resulting from conditions prevailing at the construction site and its environment. Recent studies have shown that uncertainty originates at high levels of the project hierarchy and is rooted in early stages of the project life. Laufer [109] and Howell et al [94] found that many construction projects start at a high level of uncertainty due to the incompleteness and instability of project objectives formulated by the project owner.

12.3

Robustness: Greatest Tolerable Info-Gap

After the project manager conceptualizes the info-gap uncertainties, the critical question becomes: how robust is the current plan? Are the schedule of activities, the contractual commitments of suppliers and subcontractors, the array of financial and manpower reserves, and the other related facets of the project immune or vulnerable to uncertainties? What is the greatest level of info-gap uncertainty which the project can tolerate without entailing the possibility of failure? The robustness^ to info-gap uncertainty is an expression of the overall degree of resistance of the project to the adverse contingencies which the ambient uncertainty holds in store. Robustness analysis identifies critical task-paths. In particular, the planner can distinguish between nominal

critical paths and uncertainty critical paths, as we now explain. ^\^^^^^

M—JiW-^0—M ^igj

12

13

14

J16

^|iQ^-^ii

15

Figure 12.3: A 16-task project schedule. Complex multi-task projects are rarely arranged in simple serial order. More often the flow chart of task implementation diverges and converges •^The info-gap conception of robustness is first introduced in section 3.1.

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in a complex pattern, for which fig. 12.3 is the simple schematic 16-task example studied previously in section 3.2.6 on p.64. All 16 tasks must be completed; some before others, some in parallel. Some tasks, like task 3, sprout out of other finished and partially finished tasks (tasks 2 and 6). The analysis of the flow chart in fig. 12.3 shows that the 16 tasks are arranged in five distinct but partly overlapping task paths. For instance, the task sequence: 1^2^3-^4^16 is one task path, and another, partially overlapping one, is: l->5->6-^3-^4^16 Critical paths are defined with respect to the various constraints imposed on the project, in particular time and budget limitations. The nominal critical path with respect to time constraints is the path whose nominal, anticipated, duration is greatest. This path nominally controls the timeto-completion of the project; nominally, but not necessarily in practice. Nominal critical paths, as important as they are in evaluating and managing a complex project, may fade in importance when considering the adverse impact of uncertainty. It can and frequently does happen that a path whose nominal duration is short (and hence not nominally critical) may be very vulnerable to uncertainty. Such a path can turn out in practice to control the time-to-completion by extending beyond the duration of the nominal critical path due to adverse occurrences. The uncertainty critical path may be totally different from the nominal critical path. In planning for uncertain surprises one must carefully distinguish between nominal-critical and uncertainty-critical paths. The latter are identified by locating particular vulnerability to uncertainty, and by understanding the propagation and multiplication of this vulnerability due to the structure of the project. Quantitative decision-support tools can be useful here, as we will indicate later.

12.4

Value Judgments: How Robust Is Robust Enough?

Up to now we have identified the following stages in the info-gap management of uncertainty. • Conceptualize the info-gap uncertainties. • Evaluate the project robustness to the info-gap uncertainties. • Identify the uncertainty-critical paths. At this point, the project manager must make a value judgmenti"^ • Decide how^ much robustness is needed. ^Value judgments such as this are studied in chapter 4.

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Robustness to uncertainty is desirable since it implies successful completion despite substantial unexpected adversity. But there are two totally different types of answers to the question: "How much robustness is needed?". The first, which we address now, considers the need for robustness as a protection against adverse uncertainty. The second approach, which we defer to later, recognizes that uncertainty may entail favorable surprises, and that robustness may imply lethargy or rigidity which prevent the exploitation of opportunity. Prior experience is sometimes useful for determining whether a given project plan is adequately robust. We have already described how the manager conceptualizes info-gap uncertainties. We have shown how the relative intensities of the diff'erent foci of uncertainty are judged, as in figs. 12.1 and 12.2. With this in mind, the planner can evaluate the robustness in comparison with similar projects and project-environments. For example, Laufer [110, p.65] discusses an example of "on-line learning by doing" for reducing uncertainty. In an attempt to increase speed and reduce vulnerability to uncertainty, the managers of a production project decided to "blitz-prototype" the first production line and then rapidly apply the experience to the other production lines as well. Laufer compares this to clipping a few snips from a hedge and then stepping back to see the effect and correct the mistakes. This sort of adaptive implementation is valuable for garnering on-line understanding of the project environment as a means of 'robustifying' the implementation. Experience can be a useful guide to the management of uncertainty. But relevant experience is not always available, nor is it even always a reliable guide for the future. Another approach to deciding how much robustness is needed is to calibrate robustness in terms of some additional quantity whose meaning is clear in terms of quality or value. More specifically, when the manager must choose between two options, one of which is more robust than the other, it must be decided if the increment in robustness is 'substantial' or 'marginal'. Rather than comparing against past experience with a similar situation, we relate the increment in robustness to an increment in a different parameter whose variations are qualitatively meaningful, thereby "calibrating" the increment in robustness. Consider the following example. A production manager must assure the manufacture of 10,000 units per day, and the current plan calls for installing 50 machines. Since machines can break down, maintenance teams can go on strike, spare parts can become temporarily unavailable, and so on, the immunity to uncertainty is enhanced by installing more machines, say 80. By considering the production process, the maintenance facilities, the spareparts supply network, and the ability to warehouse extra production, the manager has some sense of how much the robustness is increased by using 80 rather than 50 machines. But having a sense of augmented robustness is not the same as making the qualitative value judgment that the increment is substantial rather than marginal. One looks for some collateral parameter

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Robustness

(2)

(2')1

Substantial Substantial 10,000

5,000

50

70

80

Units/Day

Number of Machines

(50 Machines)

(10,000 units/day)

Figure 12.4: Calibrating the robustness. which also enhances robustness and which has clear qualitative meaning in linguistic terms such as 'large' or 'small'. For instance, based on understanding the production system and its relation to the supply and maintenance networks, the manager confidently makes the judgment that, with 50 machines, a production requirement of only 5,000 units per day is substantially less demanding and more robust to uncertainty than 10,000 units per day. This is indicated in the left-hand frame of fig. 12.4, which we summarize as follows: (1) Producing 5,000 units/day is substantially less demanding than producing 10,000 units/day. (2) The correlative change in robustness is a substantial increase. Now back to the number of machines. Suppose that increasing from 50 to 80 machines entails an increment in robustness which is similar to the increment in robustness resulting from reducing the output requirement from 10,000 to 5,000, as shown in the right-hand frame of fig. 12.4. We then conclude that 80 machines is indeed substantially better, significantly more robust, than 50 machines. That is: (2^) The increment in robustness is the same as (2) and hence substantial. Hence: (3) Using 80 machines rather than 50 is a substantial improvement. The inference from (1) and (2) to (2') and (3) is not a rigorous mathematical deduction. It is a value judgment based on analogy which becomes plausible if the correlated quantities — robustness, production rate, and number of machines — are part of a coherent and integrated whole. Do 80 machines provide enough robustness? If the manager felt confident with 50 machines and a 5,000 unit/day production requirement, then one

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should feel confident with 80 machines and a 10,000 u n i t / d a y requirement. Alternatively, since the curve in the right-hand frame of fig. 12.4 begins to flatten out around 70 machines, the manager may feel t h a t even 70 machines is adequate; t h a t the marginal utility — in terms of robustness — is small beyond t h a t level. And in any case, due to the strongly diminishing marginal robustness, only a vast increase beyond 80 machines would bring substantial improvement in robustness.

12.5

Risk and the Robustness-vs.-Reward Trade-off V

Immunity to Jncertainty

\ \

Robustness versus \ Reward

Reward Figure 12.5: Trade-off of robustness against reward. One of the central decisions which a manager must make in dealing with uncertainty is: where to position the firm in the inexorable trade-off between robustness and reward.^ Fig. 12.5 illustrates the contention, which can be proven mathematically, t h a t as the demand for reward increases, the immunity to uncertainty must fall off. The planner must decide where on this robustness-vs.-reward curve to operate. Or, if one is dissatisfied with any of the options entailed in the trade-off, then structural changes in the project must be sought which shift the position of the curve and improve the immunity without relinquishing reward. In subsequent sections we will discuss three methods for enhancing robustness to uncertainty: • Gathering information. • Restructuring the project. • Monitoring in real time. Before this, however, we will examine the robustness-vs.-reward trade-off in fig. 12.5 as it bears on the concept of risk in the info-gap treatment of uncertainty. The dictionary tells us t h a t risk is a "possibility of loss or injury", a "peril", a "dangerous element or factor" [172]. W i t h this intuitive back"^This trade-off is first discussed in section 3.1.7.

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ground we see that risk can be conceptualized and evaluated in various different ways. Fig. 12.6 shows the robustness-vs-reward curves for two different decision makers, or for two different project plans contemplated by the same manager. Robustness curves such as these characterize the risk-taking propensity of the decision maker (or the riskiness of the project): courageous or cautious depending on whether great or small demands are (or can be) made on the environment. The shape and position of the robustness-vs.-reward curve expresses one aspect of the decision maker's sensitivity to risk, as we will now explain.^

L

immune

Immunity Q,^ ' , robustness to premium \ Uncertainty ^i (Robustness) ^

r"**^^^

^\^

\i

\^ reward premium

vulnerable

•'

' ^2

modest

Reward

demanding

Figure 12.6: Robustness versus demanded reward for two different decision makers. Illustrating robust risk sensitivity. The reward axis in fig. 12.6 is labelled 'modest' near the origin and 'demanding' away from the origin, and the immunity-to-uncertainty axis is labelled 'vulnerable' and 'immune' for small and large values of robustness, respectively. Consider two decision makers with different project plans and different information and uncertainties. Their robustness functions are shown in the figure: decision maker 2 is relatively immune to uncertainty even when great demands are made on the environment, while decision maker 1 is relatively vulnerable to uncertainty even when modest reward is demanded. Decision maker 2 will look speculative, audacious or daring in the eyes of decision maker 1, while decision maker 1 will appear conservative, cautious, or even timid in decision maker 2's eyes. Consider for instance the arrows rising from the point labelled r on the reward axis. For this value of demanded reward the upper decision maker is relatively immune to uncertainty, while the lower decision maker is relatively vulnerable. In other words, since 0^2 is greater than a i . decision maker 2 can tolerate more uncertainty than decision maker 1. Decision-maker 2 enjoys a positive ^The info-gap approach to gambling and risk-sensitivity is discussed in chapter 6.

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robustness premium with respect to decision maker 1. Similarly, consider the arrows stemming from the robustness a on the immunity axis of fig. 12.6. At this level of uncertainty, decision maker 1 is able to demand only a low guaranteed return, r i , while decision maker 2 can demand greater return r2- This implies that, under particular conditions, decision maker 1 is willing to make a smaller commitment of resources than decision maker 2. Again, decision maker 1 is seen to be more timid in the face of uncertainty than decision maker 2. Decision-maker 2 has a positive reward premium as compared to decision maker 1.

12.6

Improving Robustness by Gathering Information

If the manager is dissatisfied with the perceived level of risk, one possible response is to reduce uncertainty by gathering information. Even if information grew on trees, its harvesting would require the expenditure of resources. Consequently the central question is: What foci of uncertainty should be explored? Or equivalently: What information should be acquired? The basic handle on these queries is: What is the value of information? In our discussion of robustness in section 12.3 we stressed the importance of identifying uncertainty-critical paths, as distinct from nominal-critical paths. Uncertainty-critical paths are task sequences whose vulnerability to adverse uncertainty may lead to large deviations from ordinary and anticipated performance, causing violation of project objectives under conditions which, nominally, would not result in failure. Clearly, the important foci of uncertainty are the uncertainty-critical paths. The aim should be to gather information so as to equalize the vulnerability of all paths by reducing the vulnerability of the uncertainty-critical paths. We can crystallize our thinking by concentrating on the value of information. In particular, we will discuss two types of values: a demand value and a robustness premium of information. As we discussed earlier and illustrated in figs. 12.5 and 12.6, the robustness-vs.-reward curve decreases monotonically: this is the inexorable trade-off between immunity-to-uncertainty and demanded reward. Furthermore, information is valuable in the sense that it enhances the robustness to uncertainty. Consequently, in fig. 12.7 we show two robustness curves where the upper curve corresponds to a situation of greater information. We have "reached" the more comfortable and protected situation in the upper curve after investing resources in the acquisition of information.^ The question is: what is the value of this increment of information? The demand value of the information needed to move from the lower ^Chapter 7 discusses the "informativeness" of an info-gap model of uncertainty and characterizes the type of information which entails an increase in robustness. Info-gap learning is studied in chapter 8.

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Immunity to Uncertainty

Robustness premium More information Less information Reward

— •

Demand value Figure 12.7: Two robustness curves: before acquiring information (lower) and after (upper).

to the upper robustness curve is the increment in reward which can be demanded after acquiring the new information. The horizontal arrows in fig. 12.7 show the increment in reward which the manager can now demand as a result of enhancing the information. This is a "demand value" in the economic sense that the manager would be willing to pay any quantity up to, but not more than, this increment in reward in exchange for the information. That is, the demand value of information is expressed explicitly in the units of the reward by which the project itself is evaluated. We note that the demand value depends on the level of ambient uncertainty: where on the robustness curve the project is operating. Information has no absolute value; its value is relative to the environment. In particular, we see from this pair of curves that if the project were operating at a low level of demanded reward and at a high level of robustness, then information is of relatively lower value. The robustness premium of information expresses the increment of immunity resulting from the added information, and is shown in fig. 12.7 by the vertical arrows. The robustness premium of information speaks directly to the issue of immunity, but it lacks the economic overtone of the demand value. The robustness premium, like the demand value, depends on the operating point at which the project is functioning. Both information values, demand value and robustness premium, are concepts by which the manager chooses the foci of uncertainty upon which to concentrate the information-gathering effort.

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Improving Robustness by Restructuring

Changing the structure of the flow-chart in which the tasks of a project are implemented is a standard and obvious method for quickening the completion of the project. If everything else remains the same, then it is quicker to do two things in parallel than in sequence. The point which we will discuss in this section is that one can sometimes also enhance the robustness to uncertainty by restructuring the flow chart of task execution. We will make three points. First, restructuring for robustness enhancement is done by isolating highly uncertain tasks. Second, restructuring can augment the uncertainty of a downstream task, sometimes inadvertently. Third, restructuring for speed-up will tend to robustify the project, but not necessarily. Laufer identifies and discusses nine principles for managing uncertainty and enhancing speed in project management. One principle is to isolate highly uncertain tasks, or at least to loosen their connections to the rest of the project [110, chapter 3]. We will discuss this idea in the context of info-gap management of uncertainty by project restructuring. Refer again to the schematic flow chart of task sequences in fig. 12.3 on p.302. The vertical arrow rising out of the middle of task 6 means that task 3 can begin when task 6 is partly complete and task 2 is finished. For example, suppose the project is the design and production of a modular electronic unit. Task 2 is the specification of the space into which the unit must fit, task 3 is the design of the outer casing of the unit, and task 6 is the design of the individual electronic boards in the unit. Task 3 depends on the number and size of the boards, but not on their detailed electronic specifications, which allows task 3 to begin before task 6 is done. In fact, design stage 6 is highly uncertain because the board design is a complex process involving the development and testing of new circuits. Thus it is desirable to start task 3 as soon as possible, so as to avoid the time delays inherent in the design details of task 6. Furthermore, task 4, which is the parts-purchasing stage, is a long process due to supplier delays. In fact, the task-path 1-^5-^6-^3-^4—> 16 is uncertainty-critical since tasks 6 and 4 are both very vulnerable to uncertain contingencies. We achieve partial isolation of the uncertain task 6 by allowing the earliest possible start on task 3. However, task 4, following task 3, is also dependent on results of the electronic design in task 6. By isolating task 3 from task 6 we have inadvertently increased the uncertainty of task 4 further downstream, since purchase orders made from incomplete electronic designs may be erroneous. The example we have just discussed illustrates the potentially advantageous or dangerous interactions between restructuring for speed-up and restructuring for enhanced robustness. Reducing path durations introduces added slack time to the project, thereby enhancing robustness, if all else remains the same. But all else need not remain the same. New uncer-

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tainties may be introduced into downstream tasks. Also, running parallel tasks makes the project more complex, increases the need for communication between task teams, and makes the project possibly more vulnerable to local instabilities resulting from faulty or delayed task implementation. All these considerations make it difficult and delicate to decide whether and how to restructure. Quantitative decision-support tools can assist in project restructuring, though no computer program, however sophisticated, is a panacea. This example also illustrates the importance of monitoring the project development in real time. The uncertainty in task 4 — parts purchasing — derives in large measure from the unknown electronic design in task 6. This is a specific case of the general rule that downstream uncertainties are contingent upon upstream results. Early in the project, the large uncertainty of task 4 contributes substantially to making the paths in which it participates uncertainty-critical. But the uncertainty in task 4 is greatly reduced as soon as the electronic design is completed. This changes the entire picture; task 4 is then much clearer and more certain, and paths running through it are no longer uncertainty-critical. Now managerial attention and project resources can be directed in other directions. New options and imperatives for restructuring the project and for gathering information open up.

12.8

W h a t Should We Optimize?

Project managers and owners naturally desire high performance, but we have seen that the robustness to the uncertainty surrounding a project decreases as the aspiration for performance increases. Robustness trades-off against performance. So the question arises: what should we optimize? While high performance is better than low performance, reliable attainment of specified goals is also valuable. The attempt to maximize performance may be unrealistic and self-defeating. Info-gap theory can show the way to sensible preferences between project options. Consider project 1, for which our best models and understanding indicate that, if all goes according to plan, the reward will be r i . That is, ri is our best-model estimation of the performance of this project. However, severe uncertainties surround at least parts of the project. We already know that the robustness to these uncertainties increases as we decrease our aspirations for reward. Furthermore, a basic theorem of info-gap theory asserts that best-model aspirations have absolutely no robustness to uncertainty.^ The robustness curve for this project is shown schematically in fig. 12.8. The robustness of reward ri is zero, and only lower aspirations such as r[ have positive immunity to adverse surprises. Adopting project 1 with the anticipation of performance as good as "These ideas are discussed further in section 3.1.8.

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Robustness

low

Project performance Figure 12.8: Robustness vs. performance for project 1.

Robustness

low Project performance Figure 12.9: Robustness vs. performance for two projects.

r i would be risky and unrealistic since this outcome has no tolerance to uncertainty. T h e manager can "migrate up" the robustness curve in search of lower b u t adequate reward such as r[ for which the robustness, S i , is sufficiently large to induce confidence in success. Now consider a different project,^ 2, for which the best estimate of the reward is r2. If r2 is less t h a n r i , then the first inchnation would be to prefer plan 1 over plan 2. However, neither plan can reliably provide reward as large as its best-estimate, r i or r2. We must explore the robustness curves of these plans. Fig. 12.9 shows a typical situation: the robustness curves for the two projects cross. Both robustness curves hit the performance-axis at zero robustness, since best-estimated outcomes have no immunity to uncertainty. P l a n 2 hits the axis to t h e left of plan 1 since plan 2 has lower best-estimated outcome. However, the slope of the robustness curve of plan 2 is greater t h a n the slope of the robustness of plan 1, so these curves cross at some level of reward. If reward r[ was adequate to motivate plan 1, then now plan 2 is even more desirable because plan 2 is more reliable t h a n plan 1 in achieving outcome r[. By considering the full robustness curves, rather t h a n just the zero-robustness endpoints, our preference between the projects has changed. Based on the nominal best-estimated performance we preferred plan 1 over plan 2, but considering the trade-off between robustness and performance, the preference has been reversed. So, what should we optimize? The robustness curve is a Pareto efficient frontier of possible outcomes and reliabilities. Any increase in aspiration for performance entails a decrease in robustness to uncertainty. W h a t the manager optimizes is the combined judgment of outcome and confidence in attaining t h a t outcome in the turbulent uncertainty of actual implementation. Contrast this with the strategy of direct optimization: first formulate ^Or a different implementation of the same project.

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the best models of the alternatives, and then choose the alternative for which the best-estimated outcome is greatest. Direct optimization is not a formula for success, since it ignores the dimension of uncertainty in those estimates.^ Even if the estimates themselves entail probabilities, the info-gaps in these probabilities have been ignored.

12.9

The Other Face of Uncertainty: Opportuneness

Up to now we have discussed the adverse and threatening character of uncertainty, and we have concentrated on evaluating and enhancing the robustness of the project. We have stressed the following elements in the assessment and management of info-gap uncertainties: • Conceptualize the info-gaps. • Evaluate the project robustness to the info-gaps. • Identify the uncertainty-critical paths. • Judge, in qualitative terms, how much robustness is needed. • Assess the info-gap risk as a trade-off between robustness-to-uncertainty and reward. • Augment the project robustness by: o Gathering valuable information. o Restructuring the project plan. o Monitoring the project in real time. Robustness to uncertainty is important when we concentrate on the pernicious possibilities entailed by unknown variation. However, variations can be propitious and surprises can be beneficent. The dynamic flexibility which is so important for survival, as Rosenhead [147] has stressed, thrives on emergent opportunities. The opportuneness of a project is a measure of the possibility of unanticipated windfall reward.^^ Unanticipated but favorable opportunities arise only in an environment of uncertainty. If everything is known ahead of time, then there are no surprises, for better or for worse. There is a fundamental and irrevocable connection between the magnitude of the ambient uncertainty and the degree of windfall reward which can be gained, as shown in fig. 12.10. This relation is precisely the reverse of the trade-off" between robustness and reward shown in fig. 12.5. In fig. 12.5, robustness to the pernicious threats of uncertainty increases as the demanded critical reward is reduced. In fig. 12.10, the ambient uncertainty must rise in order to enable the attainment of large windfall gain. A project is highly opportune if wonderful windfall is possible even at low ambient uncertainty. Hence opportuneness and uncertainty increase in opposite directions on the vertical axis of fig. 12.10. ^The strategies of direct optimization and robust-satisficing are compared in section 11.4. ^°Opportuneness is introduced in section 3.1.

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T Uncertainty

Opportuneness

Opportuneness versus Windfall

i Windfall Reward F i g u r e 12.10: O p p o r t u n e n e s s a n d uncertainty.

Fig. 12.10 can be viewed as a trade-off. The figure shows that large ambient uncertainty must be accepted in order to facilitate great windfall gains. In other words, certainty must be relinquished in exchange for the possibility of favorable surprises. Figs. 12.5 and 12.10 summarize the two basic trade-off's which are presented by an uncertain environment: robustness vs. required reward, and certainty vs. windfall gain. We have explored a number of methods for enhancing robustness to the pernicious impact of uncertainty. The question now arises: can windfall opportunities be exploited or even enhanced at the same time that one protects against the pernicious potential of uncertainty? It is perhaps not surprising that robustification can eliminate favorable impacts from fluctuations. In other words, actions which protect the project against the vagaries of environmental uncertainty may tend to diminish the potential for windfall. What is less obvious is that this is not universally the case. It can occur that, in choosing a more robust over a less robust alternative, the result is to enhance both the immunity to failure and the opportuneness of the project for unanticipated windfall.^^ Robustification can make the project less responsive to all stimuli or only to pernicious stimuli. In the former case, robustness and opportuneness are antagonistic, while in the latter case they are mutually supportive. The distinction between the two faces of uncertainty — pernicious and propitious — is critical for enhancing immunity to failure while not making the project lethargic and insensitive to emergent possibilities. Uncertainty, in its broadest sense, is neither good nor bad; rather, it is the management of uncertainty which can be either successful or catastrophic. One of Laufer's conclusions is that "You are not usually expected to eliminate risk. You are expected to take the right risk." Of course, one's attitude to adverse and favorable uncertainties is not symmetric: we are (usually) risk-adverse and, as Laufer continues: "when the negative impact of a risk is intolerable, you have no choice but to eliminate it." [110, p.78] ^ ^ Antagonism and sympathy between robustness and opportuneness are discussed in chapter 5.

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Operational or pragmatic judgments depend on judgments of degrees as well as kinds of uncertainty: favorable or adverse. When we lack the sixth-sense for uncertainty we become disoriented and our decisions are made by rote rule or, worse, become arbitrary. The skillful fighter pilot is distinguished not only by being uninhibited by danger and uncertainty, but by betting correctly: by judging uncertainties successfully. Like night-vision or color-vision, the manager must have 'uncertainty-vision' with which the project is guided away from pernicious uncertainties and towards propitious ones.

12.10

Quantitative Decision Support Systems

There is no replacement for intuitive and contemplative judgment by an experienced and informed manager. The human mind is unsurpassed by machines in its ability for insightful understanding and creative problemsolving. Nonetheless, quantitative decision-support tools can be useful in managing uncertainty. In particular, the conceptualization of info-gap uncertainties can be quantified, and robustness to these uncertainties can be assessed numerically. This in turn is the basis for identifying uncertaintycritical task paths. Finally, quantitative and computerizable algorithms enable the decision maker to quickly explore a wide range of alternatives for enhancing the project robustness. It is the manager's responsibility to know what quantitative tools can be applied. Perhaps the most important message is that the info-gap conception of uncertainty, because of its sparsity and simplicity, is quantifiable when other approaches are not.

Chapter 13

Implications of Info-Gap Uncertainty Meaning is wider in scope as well as more precious in value than truth, and philosophy is occupied with meaning rather than with truth. Making such a statement is dangerous; it is easily misconceived to signify that truth is of no great importance under any circumstances; while the fact is that truth is so infinitely important when it is important at all, namely, in records of events and descriptions of existences, that we extend its claims to regions where it has no jurisdiction. But even as respects truths, meaning is the wider category; truths are but one class of meanings, namely, those in which a claim to verifiability by their consequences is an intrinsic part of their meaning. Beyond this island of meanings which in their own nature are true or false lies the ocean of meanings to which truth and falsity are irrelevant. We do not inquire whether Greek civilization was true or false, but we are immensely concerned to penetrate its meaning. John Dewey, [59, p.247]

In this chapter we consider a number of impUcations of uncertainty in general and of info-gap uncertainty in particular. Section 13.1 discusses the role of uncertainty modelling in the holistic study and control of complex systems. We explore the tension between predicting future behavior and the inability to know future innovations. Holistic modelhng, it will be suggested, should entail the symbiosis of system models and uncertainty models. In section 13.2 we study the question: why are words sometimes so difficult to define precisely? In response, we explore the relation between info-gap uncertainty and linguistic ambiguity. We build on this to develop 317

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an understanding of the relation between knowledge, which we express with words, and reality, both of which entail uncertainty. In section 13.3 we consider the implications of uncertainty modelling for a question which is perennial for both epistemologists and practical decision makers: is a judgment of warrant for a proposition relative or absolute? More specifically, because strongly warranted propositions are relevant to the choice of an action, we ask whether or not one should expect universal agreement about an inference, from uncertain evidence, regarding the degree of warrant for a proposition. The answer draws on the central role played by the uncertainty model in such an inference, and on the fact that the choice of an uncertainty model is prior to, and distinct from, the inference. In section 13.4 we examine the related question: why is an info-gap inference credible? We examine the relation between warrant for a proposition and credence for a decision methodology. We suggest that info-gap inferences are motivated by a belief in the continuity between cause and effect. This somewhat metaphysical investigation bears practical fruit as well, leading to the formulation of an info-gap opportuneness inference which, it turns out, is not entirely foreign to the discussion of inference. Finally, in section 13.5, we explore the bearing of info-gap uncertainty on Frank Knight's famous distinction between risk and uncertainty, which he understood as a distinction between probabilistic uncertainty and "unmeasurable" uncertainty. The info-gap concept helps bridge the gap between these two facets of the unknown. This has both practical consequences, as well as a range of methodological and philosophical ones.

13.1

Holism and Uncertainty

Generality — abstraction from distracting mundane detail — is the Archimedean lever of constructive thought. As Whitehead put it, "It is the large generalisation, limited by a happy particularity, which is the fruitful conception." [175, p.35]. As one searches for generalizations one recognizes that the difference between a large generalization and a theory of cosmic scope is only a matter of degree. It is no idle conceit which drives theorybuilders to stretch the domain of their thought to the farthest limits. If we can understand the world at all it is because of the myriad interconnections of its parts. From belief in the power of generalization and in the unity of the causal world it is a small step to the aspiration for a grand theory of everything. Holistic generalization is a vitalizing complement to the analytical specialization of modern science, technology, medicine, social planning, and every other field which struggles for a comprehensive understanding and mastery of complex systems. Evidence of the practical importance of the broad view is found, for instance, in the prevalence of teamwork which has become characteristic of innovative effort in these and other domains. Indeed, creativity itself — breaking away from common modes of thought —

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is sometimes the fruit of cross-fertilization between distant fields. However, uncertainty is the Achilles heel of any holistic approach. This is particularly true in dealing with people. The willful behavior of human beings is strongly influenced by what they know. The inability to predict today what will be discovered tomorrow is a major impediment to comprehensive modelling or management of individual or collective behavior. The impossibility of current knowledge of future discovery is a cornerstone of much of Karl Popper's thought on history and determinism.^ In arguing for indeterminism in human affairs and the growth of knowledge Popper wrote: Knowledge may conquer new problems. But in doing so, it will create new problems which it cannot solve; at least not at once. For it cannot foreknow its own future conquests. [137, p. 109] Popper has many arguments swirling around this idea of not knowing today what we will know tomorrow. It is the heart of his argument for indeterminism in the growth of knowledge, and it is the heart of his argument against Marxism and all other forms of predictive history as the basis for comprehensive centralized social planning and control. Popper's no-foreknowledge argument led him quite early to a strong stand against holism: I personally believe that quite a good case may be made for the view that the holistic way of thinking (whether about 'society' or about 'nature'), so far from representing a high level or late stage in the development of thought, is characteristic of a prescientific stage. [135, p.76] Holistic thinking — today called 'integration', 'synthesis', sometimes even 'relevance' when one holistically brings diverse things together — is very much the rage in many areas of education and practice. Popper howls at holism because Utopian theories are holistic, and in their place he advocates what he calls piecemeal social engineering: The characteristic approach of the piecemeal engineer is this. Even though he may perhaps cherish some ideas which concern society 'as a whole' — its general welfare, perhaps — he does not believe in the method of re-designing it as a whole. Whatever his ends, he tries to achieve them by small adjustments and re-adjustments which can be continually improved upon. [135, p.66] There is much merit in Popper's argument, especially as it strikes at the root of all totalitarian societies from Plato's republic to modern communism ^This idea was developed separately and in different ways by Shackle [157, pp.3-4, 156, 239, 401-402]. Further discussion of Shackle-Popper indeterminism appears in [27].

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[136]. The delicate point, which Popper is well aware of, is that a broad theoretical grasp is exceedingly valuable when it is well founded and not naively Utopian or otherwise pernicious. The difficult judgment to make is how broad, as against how piecemeal, a policy or program or strategy should be. Theories need to strive for holism. The great secret of scientific success, as of mathematical analysis, is that the general, the abstract, the all-inclusive, is often simpler and more powerful than what is particular and case-specific. "Nevertheless," as Isaiah Berlin wrote, the sense of reality or of history which enables us to detect the relationships of actual things and persons is acquaintance with particulars, while all theory deals with attributes and idealised entities — with the general. [34, p.35] So it would seem that, to reconcile Popper's piecemeal technology with the nature and needs of scientific analysis, one needs to establish a strict border between theory and practice. On the one side, theories aim at generality, since only in that way can they hope to capture the patterns and laws of the subject. On the other side, programs and policies must shun theoretical generality and adhere to very world-bound practical experience. Nonetheless, the border must be permeable in both directions. On the theoretical side, learning from error is the core of the scientific method, and error occurs only by experience. On the policy side, theory can assist in identifying what is practical since theory grasps the big picture: the laws which govern what is feasible. Recognizing that uncertainty is the Achilles heel of holism, as expressed by Shackle-Popper indeterminism, we can say something more about the relation between theory and practice, between the holistic view and the practical constraints of evolving knowledge. Models are of two sorts: models of systems and models of uncertainties. System models are developed by scientists, engineers and analysts of all breeds. Quite frequently, models of uncertainty are developed independently by specialists — statisticians for instance — and borrowed by system-modellers as needed. One deficiency of this division of labor is that the uncertainty models may not thoroughly match the uncertainties inherent in the system being modelled. A second problem is that the system modeller, in striving for theoretical generality, tends to overlook uncertainty as an integral part of the greater phenomenon. A theory is complete, it will be claimed by the enthusiastic system modeller, when uncertainties have been removed and predictive power is complete. This overly ambitious or naive holism can be avoided by incorporating uncertainty modelling into the study of systems themselves. The system modeller can be holistic by filling out the system model with an uncertainty model. By integrating an understanding of how uncertainty wanes but also waxes with time and experience, the system modeller fixes a handle on that very aspect — the unknown — which confounds the ability to master-mind

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the future. In short, an uncertainty model is the hoUstic complement of a system model. A decision theory, such as info-gap theory, provides a framework for integrating system and uncertainty models. Models of uncertainty have a further role to play in watchdogging theoretical prediction. Again Popper: "[I]t is one of the most characteristic tasks of any technology to point out what cannot be achieved^'^ [135, p.61]. In our development of info-gap decision theory we have stressed that the analysis of uncertainty is as much a guard against inept expectations, as it is a doorway to decision. A holistic theory with one foot firmly braced on a system model and the other on a model of uncertainty, will serve the decision maker both in making choices and in abstaining from precipitious action. This is a well-tempered and balanced sort of holism.

13.2

Language, Meaning and Uncertainty

Why is it sometimes so difficult to tell each other what we mean, especially on subjects of great importance? Statements such as "We demand justice" or "I love you" are colossally difficult to define. Even the seemingly innocuous proposition "Today is the 6th of November" is problematical if we recall that that date is the anniversary of the Russian October Revolution. The "growing inadequacy of words, and especially certain key words which have always functioned in our moral and political discourse" has been held responsible by Susanne Langer for keeping us "in a state of nervous tension verging on hysteria". [108, p.142] Words can be ambiguous; meanings can be unclear. So much is certain. But since we use language for so many critical tasks, pre-eminently for expressing facts, knowledge, understanding and truths, one worries that linguistic uncertainties may undermine the validity of such enterprises. If we cannot be sure of the meanings of words, how can we confidently use words to represent knowledge? And when we represent knowledge with words, what relation to reality can such knowledge have? In this section we will explore an info-gap approach to reconciling linguistic uncertainty with the process of obtaining and representing knowledge. The concept of "representation" will occur in two senses, each with attendant uncertainty. In one sense, language is ambiguous in representing what we know. In the second sense, what we know is an uncertain representation of what is. We will make some sense of these uncertainties, and gain some confidence in our ability to manage them, through an info-gap conception of both natural or real-world uncertainty and of our own linguistic uncertainty. To be more specific, the argument contains two assertions which are summarized now and expounded later. First, nature itself is uncertain, though not necessarily probabilistic as in many interpretations of quantum mechanics. That is, there are inherent and unavoidable uncertainties in the ^Italics in the original.

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real world which have nothing to do with the frailties of human knowledge. We will refer to this as ontological uncertainty. Second, our language, by its very essence, entails uncertainty in meanings. That is, language is uncertain not only when we use it poorly, but also when we exploit it to the full extent of its potential. We will refer to this as epistemic uncertainty. Ontological and epistemic uncertainties are sometimes referred to as objective and subjective uncertainties, respectively Many people are reluctant, myself included, to assert that any proposition is absolutely true, irrevocable, undeniable and resistant to any possible modification. At the same time, such people as myself also act through convictions based on what we take to be truths of various sorts. John Locke said it with sweet and inimitable 17th century charm: If we will disbelieve everthing, because we cannot certainly know all things; we shall do muchwhat as wisely as he, who would not use his legs, but sit still and perish, because he had no wings to fly [114, p.57, Li.5.] One way to justify this pragmatic acceptance of truths is to recognize that there can legitimately, realistically, be uncertain truths. Not that one is unsure of the truth of some proposition, but that the proposition, the property of the world itself, is uncertain. Because of the ontological claim, this is not an oxymoron. Combining the ontological and the epistemic claims, we are assured that uncertainty arises on both sides of the great divide: on one side, nature is inherently uncertain; on the other side, our apprehensions of sensations, our references, and our ascriptions of meaning are inherently uncertain. Once one accepts a concept of uncertain truth, in this two-sided sense, then the sharp edge to the denial of apodictic truth is blunted, as we will see. The task before us is to explain a conception of knowledge as accurate representation of reality. That is, we wish to understand in what sense there is a correspondence between human knowledge and external reality which motivates and vindicates the search for truth-indicative justifications of belief. To do this we will employ the concepts of ontological and epistemic uncertainty and the following two ideas: 1. Skepticism about the possibility of apodictic propositions. 2. A holistic conception of linguistic meaning: words attain meaning only in a broad linguistic and experiential context. In the simplest sense, for instance, this is the claim that one cannot ask for the meaning of the noun 'mist' without specifying whether one asks in English or in German. We begin with a rough metaphor for "accurate representation" of reality by language.

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The accurate representation of a dot is something dot-hke. The accurate representation of a smudge is something smudge-Uke. Carrying on with this dot/smudge image: What ontological uncertainty means is that what we can know about is not a crisp dot but a more or less foggy smudge. Common examples of smudgy reality derive plentifully from Heisenberg's uncertainty principle. There are also non-quantum examples of natural uncertainties such as chaotic dynamics [67] and computationally indeterminate systems [130]. What epistemic uncertainty means is that what we can know and express verbally is likewise more or less smudgy and uncertain. Common examples of smudgy knowledge range from sloppy thinking up to rigorous but fragmentary scientific theories. Something like the smudge-metaphor for epistemic uncertainty surfaces in Quine's discussion of the "penumbra of vagueness" surrounding what he calls the "stimulus meaning" of words [143, p.41]. Quine wrote that the "[s]timulations eliciting a verbal response . . . will not be a mere point in quality space." Instead, they will "sprawl freely" [143, p.85]. These are very info-gap-like ideas, as we will see. We now have a clear path to understanding knowledge as accurate but fallible representation of reality. Epistemic uncertainty is an accurate representation of ontological uncertainty; both are smudgy, from which derives the fidelity of the one to the other. But note also that a smudge is a collection of dots, which implies a limitation on the accuracy of the representation. An epistemic smudge is mapped to an ontological smudge, but not dot-by-dot; we do not know which dot in the epistemic smudge corresponds to which dot in the ontological smudge. This implies that there are no apodictic assertions about reality, but it also entails a meaningful though fallible correspondence between words and reality. We must now show how this is compatible with the idea of linguistic holism. Uncertain knowledge — epistemic uncertainty — is linguistically holistic in a Quine-like sense. Quine insists that we cannot ask for the meaning of an isolated word or sentence; meanings are developed holistically, in the broader context of usage and experience [142, p.43; 143]. Words have meaning not in isolation but rather in relation to other words, to sentences, to a language as a whole, and to the personal and collective experience which has generated the language. Epistemic uncertainty as construed from an infogap perspective is also holistic, in several senses which together support Quine's linguistic holism, as we now will see. In using the term 'uncertainty' we are thinking of the info-gap concepts of clustering and nesting.'^ As we know, an info-gap model of uncertainty is an unbounded family of nested sets. The elements of these sets are possible realizations. These realizations may be humble events, like the interest rate tomorrow morning, or they may be grand propositions about the world in general. "^It may be possible to develop an analogous explication based on probability models, but that will be left to the probabilists.

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An info-gap model entails two levels of uncertainty. The first level is that we do not know what event will occur or which proposition is true, though we can pick out an infinity of feasible candidates. The second level of info-gap uncertainty is that we do not know the horizon of uncertainty: how far afield feasible candidates may lie. What this means is that any event (or any proposition) is feasible, at some horizon of uncertainty, under any given uncertainty model. From this we see that an info-gap model of uncertainty — an unbounded family of nested sets — contains among its sets every element of the universe of discourse. An info-gap model is not a limited local point-like phenomenon, but is defined in terms of the entire space. Since an info-gap model pervades the entire space, the model changes in character (meaning, signification, ostension, etc.) if any province of the space of possibilities is altered. We have completely specified an info-gap model only after specifying, at least implicitly, the entire universe of possible events. This corresponds to Quine's conception of the holism of meaning, and it derives directly from the character of info-gap uncertainty. Uncertainty is holistic in a way which, evidently, certainty is not. There is an additional sense in which holism arises in the info-gap representation of linguistic uncertainty. We understand that, since an info-gap model is an unbounded family of nested sets of events, any event belongs to any info-gap model. But if this is so, what distinguishes diff'erent info-gap models? Distinct info-gap models are distinguished by their relative position in the space of events. Now relative position is a potent concept (as first Galileo and then Leibniz and later Mach and Einstein realized in connection with physical space). We cannot ask: what is the position of this point (or this info-gap model) without reference to all other points (or models) in the universe of discourse. To ask for the position of a single spatial point is meaningless. Leibniz' great insight into what became general relativity was: space is the potential for relation between objects [151, p.249]. Space itself has no local meaning; space has only global, holistic, universal meaning. In the same way, by analogy, info-gap models of uncertainty are defined relative to each other in the space of possible events. Info-gap models, as relative entities, are endowed with a space-like holism. To summarize, we have seen that epistemic and ontological uncertainties imply an accurate but fallible representation of reality in the sense that what is known, and what is known about, are both uncertain in a smudge-like info-gap sense. Also, this account of knowledge is holistic because of the unbounded nested structure and the space-like relativism of info-gap models of uncertainty. So what about truth? Why should one expect warranted knowledge to be truth-indicative? We have no clear line of sight on reality. As Rorty has asserted, we have no privileged representations which are "automatically and intrinsically accurate" [146, p. 170]. So much is true, but now I part from Rorty.

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We do have "accurate representations" in the smudge-for-a-smudge sense, though not dot-for-dot. Epistemic uncertainty takes its bearings from ontological uncertainty. This seems to me to be the basis for a falhbihst truth-indicativeness of warranted knowledge based on evidence.^ It also points to the great difficulty of verifying knowledge: learning is a process of updating an uncertainty model. This, as we have seen in chapter 8, is a fundamentally difficult decision problem. Furthermore, we obtain truthindicativeness of knowledge even if there are no apodictic propositions, or at least none of any consequence, though the indications of truth are always accompanied by uncertainty. Finally, and perhaps most uniquely to this uncertainty-based approach, one obtains a concept of uncertain truth as well as the more conventional concept of uncertain knowledge.

13,3

Warrant and Uncertainty

Our involvement with decision-making with uncertain evidence has been motivated primarily by the practical need to choose a course of action. Nonetheless, the decisions under uncertainty which have motivated our study of info-gap decision theory are all special cases of the rather theoretical problem of inferring the degree of warrant for a proposition from evidence which is uncertain. So much hangs, both in theory and in practice, on the status of truth certificates of this sort that tremendous effort has been invested in exploring the validity of rational inference. Everyone agrees that one can err by being misled by faulty evidence. The tough question is, can one establish some universal and incontrovertible method for assessing the degree of warrant of a proposition founded on uncertain evidence? Should this method lead all rational people to essentially the same assessment of degree of confirmation? The short answer to the 'tough question' mentioned above is: emphatically no, as will be explained in this section. The argument will proceed as follows. 1. Inference from uncertain evidence depends upon a model which quantifies degrees of uncertainty, as well as upon a logical framework established by axioms, definitions, rules of inference and common notions of various sorts. This apparatus — the uncertainty model and the logical framework — is the baggage of a decision theory. These entities are themselves not subject to rational analysis within that framework, though they may of course be analysed in a larger framework.^ 2. There are many different types of uncertainty models. The choice of an uncertainty model is prior to the inference and independent of it. ^We will encounter a variation on this argument when we discuss credence and truthindicativeness of info-gap decisions on p.336. ^This is what makes the updating of an uncertainty model such a delicate matter, as we saw in section 8.2.6.

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and sometimes controversial and exceedingly difficult to justify. The selection of an uncertainty model can sometimes be anything but selfevident.^ 3. Different logical frameworks and models of uncertainty can assign different degrees of warrant for one and the same proposition. 4. There are at least two types of warrant for a proposition: holistic integration of the proposition and the evidence into the contextual understanding of the problem, and piecemeal confirmation by severe tests based on an uncertainty model. While these two approaches may be combined, they highlight different aspects of the process of confirmation and again indicate a type of relativism of warrant. 5. We will discuss an example of the great differences in an inference which can innocuously arise from choosing different uncertainty models. I am not saying that inferences of truth or falsity are relative to culture, or psychology, or gender, or historical moment in some cynical or deterministic sense. I am however saying that the warrant for a proposition is relative to some conceptual starting points — especially the choice of a model of uncertainty — and this relativism is real and substantial.^ The implication of info-gap decision theory, in contrasting it against the army of statistical decision theories, is that distinct conceptualizations of uncertainty generate distinct methods and results of decision from uncertain evidence.^ ^There can be alternative logical frameworks as well, though we won't pursue this point here, except to note that many thinkers have pointed out that the logical framework of a subject can be quite controversial. Savage said it well: It is often argued academically that no science can be more secure than its foundations, and that, if there is controversy about the foundations, there must be even greater controversy about the higher parts of the science. As a matter of fact, the foundations are the most controversial parts of many, if not all, sciences. [154, p.l] ^The term 'relativism' has come to bear a controversial connotation in recent decades, through no fault of the word itself. I will continue to use the term since lexically it fills the bill. For those who prefer a different epithet, Haack's description of perspectivalism [80, p.208] is rather close to what I have in mind. See also Haack's essay "Reflections on relativism: From momentous tautology to seductive contradiction" in [82]. ^ A similar idea is expressed by Savage in discussing the personalistic view of probable inference: The criteria incorporated in the personalistic view do not guarantee agreement on all questions among all honest and freely communciating people, even in principle. That incompleteness, if one will call it such, does not distress me, for I think that at least some of the disagreement we see around us is due neither to dishonesty, to errors in reasoning, nor to friction in communication, though the harmful effects of the latter are almost incapable of exaggeration. [154, pp.67-68]

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1. Inference with uncertain evidence depends upon an uncertainty model. A judgment such as "beyond a reasonable doubt" requires assessment of a degree of uncertainty. Whether the subject is legal or technological or scientific, any inference from uncertain evidence requires some quantification of the uncertainty. This idea underlies all methods of statistical inference, the idea of (statistical) level of significance, the idea of a robust severe test of truth discussed in section 8.2.5, the theory of info-gap decision, and so on. This attitude — that judgments depend on assessments of uncertainty — was expressed quite clearly by Keynes, who asserts: Part of our knowledge we obtain direct; and part by argument. The Theory of Probability is concerned with that part which we obtain by argument, and it treats of the different degrees in which the results so obtained are conclusive or inconclusive. . . . The method of this treatise has been to regard subjective probability as fundamental and to treat all other relevant conceptions as derivative from this. [99, pp.3, 281-282] Even more explicitly, among Carnap's "basic conceptions" is the contention that all inductive reasoning, in the wide sense of nondeductive or nondemonstrative reasoning, is reasoning in terms of probability. [43, p.v] It would be a mistake to view these brief almost slogan-styled statements as comprehensive reflections of these men's positions. However, they represent the attitude, to which I subscribe, that inference with uncertain evidence requires a quantification of the uncertainties. The point is not that inference with uncertainty may be inconclusive because of the uncertainties (though this is obviously true). Rather, the point is that inference is contingent upon the choice of an uncertainty model, just as deductions are contingent upon the axiomatic framework within which they are made. Even if one can ascribe an objective rationality to the logical process of reaching an inference with uncertain evidence, relativism enters through the choice of an uncertainty model upon which this process depends. What is crucial is that different choices of an uncertainty model can, and often do, lead to different inferences: different degrees of warrant for one and the same assertion. We will consider a seismic example shortly 2. Choosing an uncertainty model. Everyone recognizes the diversity of uncertainty models. The statisticians have warehouses full of them. Where Keynes, Carnap and nearly all modern philosophers have erred, however, is in believing that probability is the only means of handling uncertainty^ The choice of an uncertainty model is much more difficult and ^Popper [137], Shackle [157] and Kyburg [107] are notable exceptions.

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controversial than that. The strict probabihst's attitude fails to account for inference procedures which grapple with uncertainty and yet are utterly non-probabilistic. These procedures are important in engineering, strategic planning, economics, management and elsewhere, as has been extensively illustrated in earlier chapters. The focus of this book has been to use nonprobabilistic info-gap theory of uncertainty to draw inferences and make decisions in ways and with results which may differ from classical results. Indeed, Kyburg recognized that it might be the case that some novel procedure could be used in a decision theory that is based on some non-probabilistic measure of uncertainty. [107, p.1094] The rigors of practical necessity have in fact led to the emergence of new methods for quantifying and exploiting uncertain information. By far the most well known is possibility theory based on fuzzy logic, which is extensively used in electro-mechanical control and decision systems and elsewhere. Possibility theory, while axiomatically distinct from probability, is nonetheless similar to the extent that it quantifies uncertainties with non-negative real functions (membership functions rather than probability densities). Info-gap theory deals with models of uncertainty which employ geometrical rather than measure-theoretic tools, and which are utterly different from both probability and possibility models. None of the traditional interpretations of probability — Laplace's enumeration of basic events, subjective preferences, frequency of recurrence — provide an explanation of info-gap models. In a more modern context, it is yet to be resolved whether "generalized information theory" is sufficiently general to encompass info-gap decision theory [101, p.37]. And what is more important, the axiomatic basis of info-gap models is completely different from Kolmogorov's axiomatization of probability [20]. 3. Warrant and uncertainty. I can agree with Carnap's assertion, quoted on p.327, after one modification: All inductive reasoning, in the wide sense of nondeductive or nondemonstrative reasoning, is reasoning in terms of uncertainty. Now here is the central point in connection with the issue of relativism. Once we establish an axiomatically distinct framework for uncertainty, we arrive at new possibilities for inference and decision with uncertain evidence. The same evidence may lead different decision makers to different decisions (about warrant for a proposition, for instance), if they assess uncertainty differently. This comes to a dramatic boiling point when one wishes to decide on the validity of an uncertainty model itself, as we saw in chapter 8. To avoid circular reasoning, an uncertainty model cannot be used to establish its own validity by severe testing, since a severe test depends upon an uncertainty model. One must go outside or above by introducing some new criterion, or some more general uncertainty model.

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Much of the battle between schools of decision theorists is over the concept of uncertainty, its quantification and verification. In much the same way that early advocates of probability methods struggled against estabHshed patterns of scholastic deduction [85], diff'erent schools of inference with uncertainty vie for intuitive acceptance by decision makers. Inferences, I grant, can be objective and rational with respect to an axiomatic framework and a model of uncertainty. However, the justification of the conclusions is contingent upon the choice, prior to reasoning, of this conceptual framework. 4. Two types of warrant: holistic and piecemeal. We now come to the question: what constitutes warrant for a proposition? In an engrossing essay on inference Susan Haack explains that how warranted an empirical claim is depends on how well it is supported by experiential evidence and background beliefs, how reasonable those background beliefs are, independent of the belief in question, and how much of the relevant evidence the evidence includes. . . . Briefly and very roughly, how well evidence supports a claim depends on how well the claim is explanatorily integrated with the evidence. [83, p. 11] What is described here — which Haack refers to as "foundherentism" [80] — bears some resemblance to Piaget's concept of the growth of a cognitive structure in the process of learning. In a like manner Quine asserts that "[t]he unit of empirical significance is the whole of science." [144, p.42] An overall picture emerges in which the claim and the evidence each play an integral part. This is a model of the holistic growth of understanding, the development of a coherent picture of the world. A related idea is expressed in a rather diff'erent philosophical milieu by Paul M. Churchland, who asserts that "global excellence of theory is the ultimate measure of truth and ontology at all levels of cognition, even at the observational level." [48, p.35] In section 13.1 we recognized the limitations of a holistic approach to the growth of knowledge, and stressed the need for supplementing it with tentative piecemeal development. While the aim of theoretical or conceptual inquiry may legitimately be the nurturing of a grand theory of cosmic scope, the process of attaining such a goal must entail step by step severe tests of the theory. But the distinction is not only between goals and means. Foundherentism (which is holistic) and severe testing (which is piecemeal) are both methods of warrant. These methods are different and they may or may not return the same verdict. How one weighs these two types of warrant in reaching a final decision on the degree of confirmation is a complex judgment which itself is subject to considerable variability. Furthermore, these methods of confirmation stress different aspects of the proposition being investigated. Explanatory integration stresses the contextual significance of

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the proposition, while a severe test focuses more on the structural or formal relation between the proposition and the evidence.^^ So is there a universal methodology, which all rational beings can be expected to accept, for actually evaluating the degree of warrant of a proposition based on uncertain evidence? It would seem not, since the choice of a logical framework of reasoning and the choice of a model of uncertainty are prior to the judgment of warrant and are clearly not unique. Furthermore, warrant can and should involve holistic considerations of the explanatory coherence of the global picture which encompasses the proposition and the evidence, as well as the results of highly specific step by step testing. The problem of combining these conceptually distinct aspects of the confirmation process is not likely to find a single resolution on rational grounds alone. 5. Example: Seismic safety. A major challenge in civil engineering design for seismically active regions is the prevention of life-threatening structural damage resulting from earthquakes. Also important is the reduction of seismically-induced damage to equipment and secondary systems (power systems, communications equipment, etc.). The design of seismically safe structures is complicated by the wide temporal and spatial variability of ground motion during an earthquake and its complex and imperfectly understood interaction with structures. Consequently, design decisions depend intimately upon how the uncertainties are modelled. We will briefly illustrate that the choice of a model of uncertainty is anything but unique, and that different choices can entail dramatically different design decisions. These choices can be expressed in terms of inferences about the truth or falsity of a statement. An assertion such as 'This building design is seismically safe' will be assigned different truth-values depending on which uncertainty model is used. The consequences of such an inference can be substantial. Probabilistic models have been used in recent decades to represent the uncertainty of vibrating structures [68, 162] and seismic ground motion [163, 178]. The concern about these models arises from the fact that a stochastic model represents typical events much more reliably than rare events, especially when the model is based on limited information. In discussing fatigue failure of offshore structures, Hartt and Lin comment that it is the "extreme, infrequent stress excursions which may be important either with regard to direct damage or to subsequent interaction effects." [89, p.91]. Rare events in probabilistic models are described by the tails of the distribution, while probability distributions are usually specified in terms of mean and mean-variation parameters. This makes probabilistic models risky de^The distinction between explanatory integration and piecemeal severe testing by infogap inference is not quite as clear cut as it may seem. In section 13.4.1 we will show that in fact Haack's theory of foundherentism can itself be integrated into a robust severe test or, alternatively, a robust severe test can serve as an element in Haack's analysis. However, for the present discussion it is sufficient to note that a severe test can be much less global than Haack's theory of warrant, and that the overall emphases of these approaches can be quite different.

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sign tools, since it is rare events, the catastrophic ones, which must underhe the rehable design. It is immediately evident that probabilistic models which differ from each other only in minor variations of their means and variances may result in quite different anticipations of the frequency or severity of rare events. Such models are quite difficult to distinguish empirically, each being essentially consistent with available data. However, these different uncertainty models result in very different design decisions. The variety of relevant uncertainty models is, as we know from our study of info-gap models, much greater than the collection of probability models. Furthermore, info-gap models are eminently suitable for modelling fragmentary information about seismic uncertainty. Global features of seismic events, such as total or instantaneous energy, can be used to define unbounded families of sets of seismic events. It is fairly straightforward to formulate an info-gap model as the family of minimal sets of all seismic events consistent with available fragmentary data constraining the seismic occurrence. One can then optimize the robustness and satisfice the design with respect to these sets of conceivable events. In this way information about earthquake uncertainty is incorporated in the structural design decisions without postulating stochastic properties of seismic phenomena [14, 63, 64, 159, 165]. Furthermore, one can formulate an info-gap model for the uncertainty of seismic loads in such a way as to be homologous to a probabilistic model based on the same information. How this stochastic homology is implemented is a technical matter and not entirely unambiguous. However, experience has shown that traditional deterministic and probabilistic predictions of extremal response tend to be substantially lower than those predicted by stochastically homologous info-gap models [28, pp.331-332]. This suggests that, while these design-base uncertainty models are equivalent in the sense that the choice of a model is underdetermined by the available data, these models can lead to very different designs, one being substantially more conservative than the other. Referring to the 'tough question' on p.325 which motivated this section, we see that rational people may quite honestly come to opposite conclusions, from the same evidence, concerning the truth of a proposition such as "This building design is seismically safe".

13.4

Credence for Info-Gap Inference

In the previous section we explored the idea that the justification or warrant for a proposition is relative to the choice of an uncertainty model (among other things) when the proposition is tested with uncertain evidence.^^ In ^^To avoid confusion it should be pointed out that, when using evidence for an inference about a proposition, the evidence can be uncertain in either or both of two senses. The evidence may be inconclusive in regard to the proposition. Or, the evidence may itself be uncertain, meaning that the evidence in hand — measurements etc. — could have come

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this section we will explore the question: why should one be inclined to accept the structure of an info-gap inference based on a robust severe test? Why is an info-gap decision credible? We will first re-iterate, in a somewhat different manner than before, what we mean by info-gap inference based on a robust severe test. We will then discuss the distinction, which is in part a matter of degree, between the warrant implied by a decision procedure, and the credibility of that procedure as a whole. We will then consider the credibility of info-gap decisions. Finally we consider an info-gap inference based on the opportuneness function rather than on the robustness function.

13.4.1

Info-Gap Inference and Robust Severe Tests

In skeletal form, an info-gap inference embraces the following four ideas [20]: 1. To make an info-gap inference about the justification of (or warrant for) an hypothesis HQ means to accept or reject HQ, usually against a competing hypothesis Hi, by subjecting HQ to a robust severe test based on info-gap models of uncertainty.^^ 2. The implementation of a robust severe test with info-gap models requires evaluating a robustness, a. As in eqs.(3.1), (3.4) and (3.10), robustness is the maximum horizon of uncertainty a for which a collection o^conditions holds for all realizations up to uncertainty a in an info-gap model U{a,u), a > 0. 3. The hypothesis HQ passes the robust severe test if the robustness is "large". More precisely, the qualitative meaning of the value of S, whether it is "stupendously large", "moderately small", etc., indicates a degree of warrant (or rejection) of the hypothesis. 4. The calibration of a in qualitative terms ("large", "small", "good enough", etc.) is a value judgment which can be made in various ways, as discussed in chapter 4 and elsewhere throughout the book. It is necessary to amplify the sense in which the robustness a serves as a measure of the warrant of an hypothesis. In eq.(3.1) on p.38 we introduced the robustness as the greatest level of info-gap uncertainty at which specified minimal requirements are satisfied. This expresses, in embryonic form, the basic idea of satisficing on an info-gap family of nested sets of uncertain realizations. This robustness can be interpreted as a measure of warrant for an hypothesis HQ based on evidence ^ * in the following way. We understand "evidence" in a very broad sense: measurements, models, understanding of the relevant processes, and so forth. The evidence £"*, however, is uncertain: out differently. It is the latter sense which is implied by an info-gap model of uncertainty. This evidential uncertainty, of course, entails inferential uncertainty as well. ^^As should be clear by now, and as will be re-stated in item 3 below, acceptance or rejection is a matter of degree. One may in fact be unable to decide conclusively.

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the measurements could have come out differently, the models could be erroneous, our understanding of the factors governing the processes could be faulty or incomplete, etc. These uncertainties are encoded in an info-gap model U{a,u) (or possibly in more than one model, as in section 3.4). We can now state the robustness of the assertion "£^* implies i/o" in a manner quite similar to eq.(3.1): a — maxja : E adequately supports HQ for all evidence E G U{a, u)} (13.1) a is the greatest deviation of possible evidence, E^ from the specific evidence in hand, £^*, which still adequately supports the hypothesis we wish to test. The "minimal requirement" in eq.(3.1) appears in eq.(13.1) as the requirement that the evidence provide "adequate support" for the hypothesis. The demand for no more than adequate support of the hypothesis is a satisficing rather than an optimizing requirement. What constitutes "adequate support" is a difficult epistemological question. From the point of view of an info-gap inference we are free to supply alternative criteria, understanding that different criteria may have different levels of credence. One very comprehensive and convincing criterion for adequacy of evidence is Haack's "foundherentist" theory of warrant. As we saw on p.329, Haack advocates three-pronged explanatory integration: "[djegree of supportiveness", "degree of independent security" and "dimension of comprehensiveness" [80, p.87]. According to Haack's analysis, experiential evidence and background beliefs support the hypothesis to the extent that they are "favorable" to the hypothesis, though the idea is complicated by the fact that evidence, background beliefs and hypothesis may legitimately be mutually dependent [80, pp.82, 84]. Consequently, the second requirement is that the reasons for accepting the evidence which supports the hypothesis must be "secure", independent of belief in the hypothesis which is tested, recognizing that security is also a matter of degree. Finally, the adequacy of the support which is provided by the evidence depends on how comprehensive it is. So, "adequate support" in eq.(13.1) can be interpreted in terms of Haack's three-pronged criterion of degree of warrant by explanatory integration. However, we are also free to adopt other perhaps weaker senses of evidential adequacy. For instance, the info-gap inference tested by the robustness in eq.(13.1) could be part of the first of Haack's requirements by expressing a degree to which the evidence is favorable to the hypothesis. It is important to point out that "adequate support" of the evidence E for the hypothesis HQ rarely means that E deductively implies HQ. In fact, the evidence may contain conflicting reports, some which support and some which refute the hypothesis. Deduction is usually much too strong a requirement for warrant under uncertainty. What the robustness in eq.(13.1) is expressing is the greatest deviation of possible evidence E from the actual evidence ^*, within which E still adequately supports HQ by some criterion of justification. In addition, the judgment of adequacy places the decision

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maker is a central position, very much as in Haack's analysis [80, p. 102]. The human analyst must both formulate the criterion of adequacy and, more importantly, evaluate the qualitative import of a in linguistic terms.

13.4.2

Warrant and Credence

We have re-iterated and discussed info-gap inference based on robust severe tests as a method for warranting an hypothesis. Before exploring the credibility of this decision procedure it is important to clarify the distinction between warrant and credence. Briefly, warrant is a judgment of evidence for a proposition, while credence is the belief in the truth of that proposition based on the credibility of the procedure by which warrant is sought. We will expand on four aspects of this distinction: (1) Warrant or justification for a proposition is diff'erent from credence for or belief in the proposition. (2) There is no evidential justification for credence. (3) The analysis of the basis of the credibility of a procedure for warrant is essential for responsible decision-making. (4) Credence can be an indication of truth. 1. Warrant is not belief. Evidential justification or warrant for a proposition or theory is different from belief in the proposition or theory. This is not controversial, at least among empiricists and pragmatists of various breeds. For instance van Fraassen distinguishes between "judgment of evidence" and "judgment of opinion or credence" [74, p.277]. Haack distinguishes between a proposition being more or less warranted and more or less firmly held [80, pp.90, 104; 81, p.l08; 84, pp.73-74]. In much practical work on decisions, such as has occupied us in most of this book, a judgment of evidence is a technical assessment. This may for instance be an info-gap analysis or a statistical test of a binary hypothesis based on a level of significance. In the latter case the judgment to be made is: if the level of significance is small then reject the null hypothesis. On the other hand, a judgment of opinion or credence associated with a statistical hypothesis test is based on what engineers unabashedly call the "philosophy" of the test: its conceptual content, structure and character. For instance, a binary test of significance depends only on a probability conditioned on the null hypothesis; the alternative hypothesis influences the level of significance only indirectly, for instance by determining whether the test is one-tailed or two-tailed. These attributes of the test may or may not elicit confidence. The credence one gives to the test, regardless of the value of the level of significance, depends on how kindly one takes to the philosophy of the test. Thus one may have great confidence in a test and believe its recommendation even if the technical warrant is only modest or, vice versa, one may disbelieve a test even if the technical warrant (e.g. level of significance) takes an extreme value. One of the confusing things about the distinction between warrant and credence is that both of them are analog quantities: they come in a continuum of values. There are degrees of warrant and degrees of credence. A

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degree of warrant, for instance a level of statistical significance or a value of info-gap robustness, induces something which may appear quite similar to a degree of credence. This is especially true for Bayesians for whom probabilities are in any case degrees of belief. However, the distinction between (degrees of) warrant and (degrees of) credence is a distinction between different stages or aspects of the decision process: the former is the technical justification of the proposition based on evidence, while the latter is the individual non-evidential evaluation of the credibility of the decision methodology itself. 2. N o evidential justification of belief. This distinction between warrant and credence as belonging to different facets of the decision process highlights the assertion that it makes no sense to seek a justification, in an evidential sense, for a judgment of credence. Consider, for example, our earlier discussion of statistical significance tests. The credence one gives to a particular test, based on understanding its structural features, is not a judgment of evidence (at least not of the evidence with which a level of significance is evaluated), but some sort of an intuitive semantic judgment, what physicists like to call "physical reasoning". These intuitive linguistic judgments are exceedingly important in practical decision making. However, they are not evidential inferences, and they are not at all like a determination of warrant. 3. Analysis of belief is essential. Justification of method, by which we establish the credibility of a decision procedure, is not an empirical matter, not an issue to be resolved by a measure of empirical adequacy, but it is nonetheless important to justify one's method in some fashion. This is an exceedingly thorny problem, and I am skeptical of any conclusive resolution. However, the analysis of credibility is important for the intellectual integrity of the decision procedure, and it is conducive to the discovery of new paradigms of decision. Let us briefly explore the latter idea. To ask why a logical pattern like the syllogism or the level-of-significance test is convincing to the human mind is somewhat like asking why the musical octave is pleasing to the human ear. Justification of method is in part a matter of psychology, and here I am sympathetic to Piaget's early program of "genetic epistemology" [79]: by studying patterns of human thought we can discern patterns of logical method. The implication however is not that preferred patterns of thought are necessarily species specific. Furthermore, even if some thought schemes are explained psychologically or phylogenetically, we need not channel all our analytical efforts into the program of materialist neuro-philosophers like the Churchlands who would take up the electrode probe and look for the neurophysiological basis of human preferences for particular cognitive paradigms. To continue the musical metaphor: the study of the theory of scales and harmony proved exceedingly fruitful in revealing new possibilities for musical invention. Likewise, the analytical philosophy of decision algorithms is a good starting point for searching out new structures for decisions, and

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for stretching the human mind to accept new modes of reasoning. The hallmark of a good scientific question is its proximity to a good philosophical question, and that is true of much research on the theory of decision-making with uncertainty. By exploring the philosophical basis of the credibility of a specific decision algorithm, and of inference in general, we are invited to keep our eyes open for new paradigms of decision which had not previously surfaced. Plasticity is the essence of the human mind, and if some logical patterns acquire their credibility through evolutionary processes, so be it; we are however not constrained by physiology nearly so much as we are constrained by the limitations of inventiveness. Philosophy, at least for the non-philosopher, is an Archimedean lever by which we can leverage ourselves out of some of our preconceptions.^*^ After exploring the credibility of infogap decision, to which we soon turn, we examine in section 13.4.4 a rathej unconventional inference scheme based on the opportuneness function /? rather than on the robustness function a. 4. Credence and the indication of truth. The credence of a decision procedure is sometimes construed to refer to the truth-indicativeness of the procedure: giving credence to a procedure means believing that its decisions are true reflections of reality. We have not explicitly taken this approach. Rather, we have discussed credence in terms of the credibility of the underlying conceptual structure and character of the decision procedure from the perspective of the decision maker. (For instance, in section 13.4.3 we will suggest that an info-gap inference is credible if one has a prior belief in the continuity of cause and eff"ect.) However, there is an element of overlap between credence as harmony with the decision maker's other beliefs and credence as truth-indicativeness. This becomes evident if one is willing, as I am, to adopt a classical pragmatist understanding of truth. In the epigraph to this chapter we read Dewey's definition that "truths are . . . those [propositions] in which a claim to verifiability by their consequences is an intrinsic part of their meaning." In a similar vein Dewey writes elsewhere that "truth means fulfillment of the consequences to which an idea or proposition refers." [57, p. 123]. People have many and diverse beliefs, not all of which are in any direct sense verifiable from their consequences (even ignoring the methodological difficulties of verifiability). However, let us suppose that one is willing, as again I am, to accept that one's galaxy of beliefs is to some significant extent pulled into shape by the powerful gravitational forces of reality. If so, then har^This is surely what Dewey (no doubt a philosopher) had in mind when he wrote: As long as we worship science and are afraid of philosophy we shall have no great science; we shall have a lagging and halting continuation of what is thought and said elsewhere. As far as any plea is implicit in what has been said, it is, then, a plea for the casting off of that intellectual timidity which hampers the wings of imagination, a plea for speculative audacity, for more faith in ideas sloughing off a cowardly reliance upon those partial ideas to which we are wont to give the name of facts. [59, pp.254-255]

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mony between a decision paradigm and one's beliefs at large is indication of compatibility between that paradigm's decisions and the reality which has honed the beliefs in question. Credence, then, becomes truth-indicative.

13.4.3

Credibility of Info-Gap Inference

We have re-iterated the structure of info-gap inference based on a robust severe test of uncertain evidence in section 13.4.1, and we have discussed the distinction between evidential warrant for a decision and non-evidential credence for the decision procedure itself in section 13.4.2. We are now in a position to examine the credibility of info-gap inference [20]. One justification for accepting an info-gap inference is a prior belief in the continuity of cause and effect: very diff'erent effects come from very different causes, while small changes in the cause produce small changes in the effects. This is a pebble-splash idea: if you see a large ripple, a pebble must have hit the water near by. The evidence is the size of the ripple, and the robustness calibrates how convincingly "large" the ripple is. A robust severe test accepts an hypothesis (the 'cause') if it is insensitive to large variation of the evidence (the 'effect'). (If the robustness is large then the ripple could have been much different in size and would still have come from a nearby pebble-strike.) A robust severe test calibrates the 'distance' of evidence: even evidence quite different from the evidence in hand would arise from the same hypothesis (if it has passed a robust severe test), so that only evidence far beyond the pale of experience would arise from the competing hypothesis. If the competing hypothesis were true, the evidence would have differed greatly from what was observed. The cause-effect continuity is very characteristic of info-gap models of uncertainty which entail a nested view of the universe of possibilities. A belief in the continuity of cause and effect is a natural entree to accepting the credibility of a robust severe test based on info-gap uncertain evidence. Info-gap inference is rooted in interpreting uncertainty in terms of the geometrical ideas of clustering and information gap. An intuition which is somewhat analogous to the cause-effect continuity can be enlisted to support the credibility of probabilistic severe tests such as a statistical significance test of a binary hypothesis. Probabilistic inference is credible if one holds a prior belief in the continuity of cause and frequency of effect: events which are rare with respect to one cause can be caused frequently by distant causes, while small changes in the cause produce small changes in the likelihoods. This again is a pebble-splash idea: if you see a large ripple, the pebble must have hit the water near by. Now, however, the calibration of what is a "large" ripple is done in terms of probability. Neither of these continuity justifications can be viewed as incontrovertible. One could imagine discontinuities between cause and effect, in which greatly disparate causes mimic each other's effects. Similarly, one can imagine discontinuities in conditional probabilities, so that large changes in the

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cause (the conditioning variable) produce small changes in the probability distribution. The result would be that what is frequent with one cause becomes frequent with a greatly different cause as well. Nonetheless, both of these continuity intuitions may instill confidence. The decision maker's past experience and current judgment are critical in establishing (or dispelling) confidence. Furthermore, one can hold to one, or the other, or both beliefs simultaneously without logical inconsistency. There may well be additional ways of decomposing info-gap inference to more fundamental belief structures. Such investigation is worthwhile for two reasons, even though no such decomposition can lead to irrefutable and universal credibility. First, by unearthing the fundaments of plausibility we either strengthen or weaken our conviction and come to understand when, and with what sorts of evidence, a particular procedure is suitable. Second, by exploring the reasons for accepting a particular procedure we can uncover motivations for alterative procedures as yet unexplored. In section 13.4.4 we will briefly consider an example of this latter development. We noted at the beginning of section 13.4.2 that warrant is the degree of evidential support for a proposition, while credence is the belief in that proposition based on the credibility of the procedure by which warrant is sought. Credibility of an inference procedure could be based on an epistemological evaluation of the procedure. For instance, Haack's three-fold theory of explanatory integration provides criteria for the strength of a warrant. One may legitimately evaluate the credibility of a given inference procedure by evaluating the procedure against Haack's criteria: the procedure is credible if and only if it meets the criteria of strong warrant. However, this is not what we have done, and deliberately so. We have not asked: does an info-gap inference evaluate supportiveness, independence and comprehensiveness of the evidence. An info-gap inference could do these things, depending how one implements the "adequate support" clause in eq.(13.1) on p.333, but it doesn't have to. Nor should the credibility of the procedure necessarily reflect whether or not it does. Requiring an inference procedure to meet the full panoply of epistemological criteria is extremely severe. One may sometimes wish to do so, but not always. The most obvious situation in which one may forgo the severity of a full-scale epistemological evaluation is when no known procedure meets the test but nonetheless some decision must be made. This is the daily bread of decision makers in many strategic situations fraught with severe uncertainty.^'^

^"^To the philosophical community I would suggest that the classical project of establishing universal criteria for warrant and credence should be broadened. Classically one asked: what procedures of inference are (possibly in some qualified sense) always valid? It is no less important to inquire: what procedures can sometimes be valid? Or, if one still prefers questions stated with absolutes: what procedures can never be valid? Decision analysts are often interested in those procedures falling between the extremes of universal validity and universal uselessness.

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Info-Gap Inference with the Opportuneness Function

We have explained that the continuity of cause and effect motivates warranting a proposition by satisficing the evidential support while maximizing the immunity to uncertainty. As seen in eq.(13.1) on p.333, this is precisely what is done in an info-gap inference with a robust severe test. But the cause-effect continuity is a property of info-gap models in general and can motivate the plausibility of other inference schemes as well. In particular, we will now briefly examine the credibility of an inference scheme based on the opportuneness function rather than the robustness function. The bare-bones of what might be cahed an info-gap opportuneness inference is based on the following four ideas, which are parallel to the foursome enumerated at the beginning of section 13.4.1: 1. To make an info-gap opportuneness inference about the justification of (or warrant for) an hypothesis HQ means to accept or reject HQ^ usually against a competing hypothesis Hi, by subjecting HQ to a severe test of the info-gap opportuneness based on info-gap models of uncertainty. 2. The implementation of a severe test of opportuneness with info-gap models requires evaluating an opportuneness, /3. As in eqs.(3.2), (3.5) and (3.11), opportuneness is the minimum horizon of uncertainty a for which a collection of conditions holds for some realizations at all levels of uncertainty down to a in an info-gap model U{a, u), a > 0. 3. The hypothesi^s HQ passes the severe test of opportuneness if the opportuneness (3 is "small". More precisely, the qualitative meaning of the value of /5, whether it is "extraordinarily small", "moderately large", etc., indicates a degree of warrant (or rejection) of the hypothesis. 4. The calibration of /? in qualitative terms ("large", "small", "good enough", etc.) is a value judgment which can be made in various ways in analogy to the qualitative calibration of the robustness function. Eq.(3.2) on p.38 expresses the kernel of the concept of windfalhng, as opposed to satisficing. The opportuneness function /? is the lowest level of info-gap uncertainty at which sweeping success is possible, though not necessarily guaranteed. (3 can be used as a decision function in an info-gap opportuneness inference regarding an hypothesis HQ based on evidence ^*. In analogy to eq.(13.1), the uncertainty in the evidence £"* is encoded in an info-gap model U{a,u), and the opportuneness is the least level of uncertainty at which some body of evidence E provides overwhelming support for the hypothesis: (3 — min{a : E overwhelmingly supports H^ for some evidence E G Z//(a, S)} (13.2)

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/3 is the smallest deviation of possible evidence E from the actual evidence E* which entails overwhelming support for HQ. Speaking "opportunistically", (3 tells us how much the data must be fudged in order to obtain prodigious corroboration of the hypothesis. As usual, "big is bad" for /3 and the opportuneness inference is strong if /? is small. If the evidence in hand E* is close to strongly windfalling evidence E, and if one believes in the continuity of cause and effect, then the warrant for HQ is strong and credible. This is again a pebble-splash idea, and the size of the ripple is calibrated by the opportuneness function. We know, from chapter 5 and elsewhere, that the robustness and opportuneness entailed in a decision can be either sympathetic or antagonistic. That is, the opportuneness for windfall reward may either increase or decrease as we enhance the robustness against the pernicious potential of uncertainty. An analogous situation of course arises in robust and opportune inferences: the strengths of their warrants for the hypothesis being tested may be either conflicting or mutually supporting, and these degrees of warrant may either vary together or oppositely as the details of either inference are varied. It is interesting to note that Haack uses an argument in support of her theory of justification which combines, in mutual support, elements which are reminiscent of both robust satisficing and opportune windfalling, though of course without reference to info-gap uncertainty. She writes, in summarizing one of several arguments:

So the more justified A is in believing that p, the closer his evidence is to all the relevant evidence, and . . . the less room his evidence leaves for rivals to p. (Italics in the original) [80, p.217]

The first clause in italics is reminiscent of opportuneness inference, while the second clause suggests robust info-gap inference. What is claimed in the first clause in italics can be interpreted approximately as follows. The evidence in hand is E*, while all relevant evidence is E in eq.(13.2); E would be overwhelmingly supportive if we had it rather than just E"*. The first clause asserts that close proximity of E* to E strongly justifies A's belief in proposition p. This is quite like a windfalling info-gap opportuneness inference which is strong if (3 is small. The metaphor in the second clause, "leaving less room" for rival propositions, is suggestive of the robust infogap inference in eq.(13.1): large a means that vast change in the evidence is needed to adequately support the rival hypothesis; there is "no room" for the alternative hypothesis. A robust satisficing inference is strong if a is large.

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Risk and Uncertainty

Frank H. Knight stressed the distinction between probabihsticahy measurable uncertainty, which he called risk, and unmeasurable uncertainty, which he referred to as " 'true' uncertainty" [103, p.20]. It is the latter, rather than risk, which Knight saw as underlying the greatest challenges to economic agents and analysts: Business decisions . . . deal with situations which are far too unique, generally speaking, for any sort of statistical tabulation to have any value for guidance. The conception of an objectively measurable probability or chance is simply inapplicable. . . . It is this true uncertainty [Knight's italics] which by preventing the theoretically perfect outworking of the tendencies of competition gives the characteristic form of 'enterprise' to economic organization as a whole and accounts for the peculiar income of the entrepreneur. [103, pp.231-232] The distinction between probabilistic risk and generic uncertainty has been widely accepted by economists. Samuelson notes that: Thinkers have naturally questioned whether the phenomena of uncertainty [original italics] can be usefully handled by the quasimathematical notions of 'probability.' Certain subsets of uncertainty — those dealing with risks, gambling, insurance, repetitive inventory and quality control of production, and even with tactics of repeated investing — are thought to lend themselves better to useful employment of probability procedures. [152, pp.503-504] Arrow follows Knight and accepts in a "fundamental sense" that the "seemingly mechanical nature of the probability calculus . . . [leads to its] failure to reflect the tentative, creative nature of the human mind in the face of the unknown." Arrow suggests, however, that this "seems to lead only to the conclusion that no theory can be formulated for this case." [5, p.l9]. Since virtually all treatments of economic uncertainty have been probabilistic, it has been necessary to provide an explanation for why these treatments can be expected to work at all. The Bayesian approach, based on personal subjective probabilities, has usually underwritten these attempts "by reducing all uncertainty to risk through the use of beliefs expressible as probabilities." [120, p.207] It is evidently the informational demands of statistical models which signaled to Knight the limitation of their usefulness in dealing with the severe lack of information accompanying the decisions which concerned him. Info-gap models, in contrast, depend on less information and entail lower predictive power than probabilistic models of uncertainty. Knight employed

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a frequentist interpretation of probability, as evidenced in the quotation on p.297. Info-gap models neither require nor imply recurrence frequencies. Typically, an info-gap model is formulated minimalistically in response to the question: what is known. An info-gap model is a family of sets. Each set in the family is the smallest collection of realizations which is consistent with the available information, up to an horizon of uncertainty specified by the value of the uncertainty parameter for that set in the info-gap family. The degree of extrapolation from finite data to the uncountable infinity of an info-gap family of sets is typically far less than the corresponding extrapolation entailed in adopting a probability density. In this sense infogap models are partial reahzations of Knight's 'true uncertainty'. The import of info-gap decision theory is that we need neither accept Arrow's pessimistic suggestion that Knight's "true uncertainty" cannot be quantified, nor rely exclusively on Bayesian methods and subjective probabilities. Rather, our response to Knight's concern about the diversity of uncertain phenomena must be to diversify our quantifications of uncertainty and the consequent decision theories which they engender. Each theory will refiect different features of the boundless dimensions of human doubt and ignorance. In this sense, then, info-gap models of uncertainty do not replace but rather supplement probability models even when the latter are implement able. Furthermore, hybrids of these two (as well as other) classes of uncertainty models are to be sought. Some moves in this direction were made in chapter 10 and elsewhere in this book. The emergence of info-gap decision theory as a viable alternative to probabilistic methods helps to reconcile Knight's dichotomy between risk and uncertainty. But more than that, while info-gap models of severe lack of information serve to quantify Knight's 'unmeasurable uncertainty', they also provide new insight into risk, gambling, and the entire pantheon of classical probabilistic explananda. We realize the full potential of the new theory when we see that it provides new ways of thinking about old problems. Conversely, the greatest difficulty in assimilating the new methodology of info-gap decision theory is presented by classical problem-formulations which are incompatible with the new approach. This incompatibility results primarily from the fact that classical methods capture only part of the phenomena of uncertainty: probabilistic theories generate probabilistic questions, while uncertainty is not exclusively probabilistic. I have quoted above only a few of the authors who have pointed to situations in which probability is not fully suited to the tasks it faces. Yet these criticisms have yielded relatively little in the way of fundamental change in method. Almost all of the responses to such criticism have sought modifications of probabilistic models, since the problems to be solved have themselves been defined probabilistically. In addressing an analogous situation in the field of mind/brain research, Patricia Churchland writes: I suspect that any grand-scale criticism of a deeply entrenched, broadly encompassing theory will seem to have a self-refuting

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flavor so long as no replacement theory is available. The reason is this: the available theory specifies not only what counts as an explanation but also the explananda themselves. [46, p.398] Info-gap decision theory clearly presents a 'replacement theory' with which we can more fully understand the relation between classical theories of uncertainty and uncertain phenomena themselves. About uncertainty — chance, doubt, contingency, ignorance — there is much yet to be learned. The classical interpretations of probability, such as recurrence frequency or degree of belief, do not entail all there is to be understood. For instance, how are we to understand the emergence of uncertainty, even of highly structured probabilistic patterns, in completely deterministic systems like chaotic mechanical oscillators? What is the origin of this uncertainty and what is the full gamut of possible conceptualizations? In discussing the implications of chaotic dynamics, Ekeland observes: Randomness appears because the available information, though accurate, is incomplete. Part of the information is withheld from us. . . . So, if determinism means that the past determines the future, it can only be a property of reality as a whole, of the total cosmos. As soon as one isolates, from this global reality, a sequence of observations to be described and analyzed, one runs the risk of finding only randomness in that particular projection of the deterministic whole. . . . These subsystems may exhibit randomness, even though the total system is deterministic. . . . Like the queen of England, determinism reigns but does not govern. Its power nominally extends over vast territories, where local rulers are in fact independent, and even turn against it. [67, pp.62-63] On this view, probabilistic regularities are an artefact of the finite circumstances of the observer, an empirical device needed to characterize a partial picture of a broader context. Like the prisoner in Plato's cave [132, bk. VII], we see only a projection of the world, and this distortion leads us, sometimes, to probabilistic models. But this epistemic explanation of the origin of probabilistic patterns in chaotic dynamics does not explain why these projections of a fuller reality have this rather than some other structure. Need we view a fully deterministic system as the limit of the probabilistic regularities of its projections, or is probability instead an expression of particular cognitive and perceptual preferences? Would a different, possibly non-human, mind see other very different patterns in these projections? The relation between order and chaos is complicated by the fact that we cannot see either of them as being the universal causal precursor of the other. In some situations we find order growing out of chaos, as in self-organizing systems so most admired by biologists [140]. In other domains, like chaotic dynamics, the reverse is the case and chaos rises out of order.

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Expressing an idea somewhat similar to Ekeland's, John Dewey seems to say that uncertainty is a consequence of human experience. Dewey envisions that each person, in formulating and implementing actions, breaks off part of a larger reality and ignores (or is ignorant of) all the rest. Chance, says Dewey, is the result of this partial assimilation of a wider reality [55, pp.97-98]. But more than that, Dewey sees doubt as an inherent part of conscious awareness. One does not choose to doubt; doubt is an unavoidable attribute of the human circumstance [56, p. 129]. That chunk of the wider world which one experiences and knows, that projection of the greater reality which comes one's way, is insufficient to characterize the entirety which must remain unknown. Uncertainty, in this view of things, is the conception of alternative possible worlds between which we cannot distinguish. Uncertainty is that consequence of our interaction with the world which limits our ability to completely identify reality and constrains us to do no better than to identify sets of possible worlds. This is very much an info-gap conception of uncertainty, and it is a much weaker and less binding conception than probabilistic interpretations of uncertainty. The concept of alternative possible worlds is at first sight reminiscent of the frequentist conception of probability. In von Mises' explanation, for instance, the probability of an event is the limiting value of the ratio of occurrence of that event to the occurrence of all other events, in all subsequences in an ensemble of possible sequences [123]. Alternative possible worlds, in a frequentist interpretation, are alternative sequences in the uncountably infinite space of possibilities. This frequentist conception of possibility is strongly distinguished from necessity, since possible sequences by no means necessarily occur, or they may occur with vanishing limiting frequency. The ensemble of possibilities is spacious enough to include widely varying alternatives. However, the idea of alternative possible worlds is a hoary one, and Leibniz' conception will illuminate the very great difference between the frequentist conception of an ensemble of possible sequences, and the infogap conception of possible worlds. [S]ince there is an infinity of possible worlds, there is also an infinity of possible laws, some proper to one world, others proper to another, and each possible individual of a world includes the laws of its world in its notion. [112. p.71.] What is entailed in Leibniz' conception of alternative possible worlds is that each world is governed by different laws, that there is no coherent unity between different worlds, each of which is, so to say, a world unto itself. Leibniz, the master metaphysician, did not imagine that some met a-world subsumes all these distinct possible worlds; he did not intend that some meta-laws govern a super-ensemble of possible worlds (though of course one and the same God and Creator enables all possible worlds). This is quite

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different from the frequentist conception of possible sequences in an ensemble. The whole idea of probability is that there is some underlying pattern shared by all the sequences. The axioms of probability impose structure and coherence on the ensemble as a whole. For instance, the additivity axiom states that the probability of the union of disjoint events equals the sum of their individual probabilities. This imposes long-range order on the ensemble, since the complement of a rare event must be frequent, and vice versa. Probability is a highly structured model of uncertainty, entailing long-range order in the ensemble of possibilities.^^ The probabilistic conception of an ensemble is quite different from the info-gap conception which, much like Leibniz' conception of nomologically distinct worlds, implies little structural coherence among the elements of an info-gap model.^^ About uncertainty there is indeed much left to be explored, and Knight was correct to point out that measurable probabilistic uncertainty is only part of the story. On an immediately practical level, the thrust of info-gap decision theory is that Knight's unmeasurable uncertainty is more accessible, less unmeasurable, than before. On a more philosophical level, what we can learn from the discussion of alternative theories of uncertainty bears on the primary methodological problem in the pursuit of knowledge: how to formulate our questions about the unknown? In one sense, this methodological problem touches the polemic between empiricists and realists. Having broadened our conceptions of uncertainty we can now frame new questions about uncertainty as a real natural phenomenon. For example, so long as probability was identified as synonymous with uncertainty, asking if nature is probabilistic (as in some interpretations of quantum mechanics) was tantamount to asking if nature is uncertain. However, in light of a plurality of uncertainty models, one can be a non-probabilistic nature-is-uncertain realist. If probability looked like an implausible candidate for the true fabric of nature, as Einstein felt ("God does not play dice"), one can now possibly advocate the reality of uncertainty without endorsing probabilistic laws of nature. This may be pertinent to the development of new understandings of specific quantum uncertainties, such as indeterminism in the polarization of light quanta in tourmaline crystals [61, pp.4-7] which seems to challenge the principal of causality. In a related manner, the methodological problem — how to frame questions about the unknown — relates to what Haack refers to as the project of ratification: "what is the connection between a belief's being well-supported by good evidence, and the hkelihood that it is true?" [80, p.l] The word 'likelihood' has an almost exclusively probabilistic lexical definition, so its ^^Bayesian subjective probability does not involve a frequentist interpretation of an ensemble of sequences, but it is nonetheless axiomatically identical to the probability theory to which the frequentist interpretation applies. In other words, Bayesian probability entails the same long-range order and substantial structuring of events as frequentist probability. ^^See [20] for a discussion of the non-representability of an info-gap model in terms of a probability model.

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use nearly commits one (perhaps inadvertently) to a probabilistic program. We know that severe tests of warrant depend on the use of some model of uncertainty, but not necessarily on a probability model. Info-gap decision theory is, in essence, the study of severe tests which are based on infogap models of uncertainty. The project of ratification can, and should, be pursued with a more general view than a purely probabilistic one. In yet another sense, the methodological problem is one of holism versus plurahsm. Must we look for a coherent synthesis of probabilistic and info-gap conceptions of uncertainty, or can we, as Henry Adams presciently suggested, learn to live and even benefit from a plurality of conflicting conceptions: Images are not arguments, rarely even lead to proof, but the mind craves them, and, of late more than ever, the keenest experimenters find twenty images better than one, especially if contradictory; since the human mind has already learned to deal in contradictions. [1, p.489] Contradictions within a logical framework are not acceptable. However, logical inconsistency between distinct frameworks, or between systems based on conceptually disparate models of uncertainty, cannot incur censure until some larger framework embraces the distinct systems. Adams suggests that we need not necessarily insist on corralling all of our logical systems within a single fence. The arguments supporting Popper's conception of piecemeal engineering on p.319 lend credence to this suggestion. The diversity of axiomatically disparate uncertainty models, and the necessity of using an uncertainty model in drawing inferences from uncertain evidence, also indicate the wisdom of Adams' suggestion. Ephraim Lessing would probably have agreed. For Lessing, the dialectic of dispute is essential for any mental or moral progress: I am not duty-bound to resolve the difficulties I create. May my ideas always be somewhat disjunct, or even appear to contradict one another, if only they are ideas in which readers will find material that stirs them to think for themselves. (Quoted in [3, p.8])

References 1. Adams, Henry, 1918, The Education of Henry Adams. edition, 1973.

Houghton Mifflin

2. Arbocz, J. and J.G. Williams, 1977, Imperfection surveys on a 10 ft. diameter shell structure, Amer. Inst, of Aeronautics and Astronautics Journal, 15: 949-956. 3. Arendt, Hanna, 1960, On humanity in dark times: Thoughts about Lessing. In: Men in Dark Times. Harcourt Brace and Jovanovich. 4. Arrow, Kenneth J., 1971, Essays in the Theory of Risk Bearing, Markham Publishers, Chicago. 5. Arrow, Kenneth J., 1984, Individual Choice Under Certainty and Uncertainty: The Collected Papers of Kenneth J. Arrow, Vol. 3, Belknap Press, Cambridge, Massachusetts. 6. Baker, Elliot, 1976, KlynVs Law, Ballantine Books, New York. 7. Barberis, Nicholas, and Richard H. Thaler, 2005, A survey of behavioral finance, in [167], pp. 1-75. 8. Bardossy, A., and L. Duckstein, 1995, Fuzzy Rule-Based Modeling with Applications to Geophysical, Biological and Engineering Systems. Boca Raton, Florida: CRC Press. 9. Bardsen, Gunnar, 0yvind Eitrheim, Eilev S. Jansen, and Ragnar Nymoen, 2005, The Econometrics of Macroeconomic Modelling, Oxford University Press. 10. Bell, David E., Howard Raiffa and Amos Tversky, eds., 1988, Decision Making: Descriptive, Normative, and Prescriptive Interactions, Cambridge University Press. 11. Ben-Haim, Yakov, 1985, The Assay of Spatially Random Material. Kluwer Academic Publishers, Dordrecht, The Netherlands. 12. Ben-Haim, Yakov, 1994, A non-probabilistic concept of reliability. Structural Safety, 14: 227-245. 13. Ben-Haim, Yakov, 1994, Fatigue lifetime with load uncertainty represented by convex model, ASCE J. Engineering Mechanics, 120: 445-462. 14. Ben-Haim, Yakov, 1996, Robust Reliability in the Mechanical Sciences, Berlin, Springer-Verlag.

347

348

REFERENCES

15. Ben-Haim, Yakov, 1998, Reliability of vibrating structures with uncertain inputs. Shock and Vibration Digest, 30: 106-113. 16. Ben-Haim, Yakov, 1998, Sequential tests based on convex models of uncertainty. Mechanical Systems and Signal Processing, 12: 427-448. 17. Ben-Haim, Yakov, 1999, Design certification with information-gap uncertainty. Structural Safety, 21: 269-289. 18. Ben-Haim, Yakov, 1999, Decisions with severe uncertainty and non-transitive ordinal preferences, Technion Mechanical Engineering Report TME 466. 19. Ben-Haim, Yakov, 1999, Info-Gap orders and the value of information, Technion Mechanical Engineering Report TME 465. 20. Ben-Haim, Yakov, 1999, Set-models of information-gap uncertainty: Axioms and an inference scheme. Journal of the Franklin Institute, 336: 10931117. 21. Ben-Haim, Yakov, 2000, Robust rationality and decisions under severe uncertainty. Journal of the Franklin Institute, 337: 171-199. 22. Ben-Haim, Yakov, 2000, Robustness of model-based fault diagnosis: Decisions with information-gap models of uncertainty. International Journal of Systems Science, 31: 1511-1518. 23. Ben-Haim, Yakov, 2001, Info-gap value of information in model up-dating. Mechanical Systems and Signal Processing, 15: 457-474. 24. Ben-Haim, Yakov, 2004, Uncertainty, probability and information-gaps, Reliability Engineering and System Safety, 85: 249-266. 25. Ben-Haim, Yakov, 2005, Info-gap Decision Theory For Engineering Design. Or: Why 'Good' is Preferable to 'Best', appearing as chapter 11 in Engineering Design Reliability Handbook, Edited by Efstratios Nikolaidis, Dan M.Ghiocel and Surendra Singhal, CRC Press, Boca Raton. 26. Ben-Haim, Yakov, 2005, Value at risk with info-gap uncertainty. Journal of Risk Finance, vol. 6, # 5 , pp.388-403. 27. Ben-Haim, Yakov, 2006, Peirce, Haack and Info-gaps, to appear in Susan Haack: The Philosopher Responds to Critics, edited by Cornelis de Waal, Prometheus Books. 28. Ben-Haim, Yakov, Genda Chen and T.T. Soong, 1996, Maximum Structural Response Using Convex Models, ASCE J. Engineering Mechanics, 122: 325333. 29. Ben-Haim, Yakov, Scott Cogan and Laetitia Sanseigne, 1998, Usability of Mathematical Models in Mechanical Decision Processes, Mechanical Systems and Signal Processing. 12: 121-134. 30. Ben-Haim, Yakov and Keith W. Hipel, 2002, The graph model for conflict resolution with information-gap uncertainty in preferences. Applied Mathematics and Computation, 126: 319-340. 31. Ben-Haim, Yakov and Karsten Jeske, 2003, Home-bias in financial markets: Robust satisficing with info-gaps, Federal Reserve Bank of Atlanta, Working Paper Series, 2003-35, Dec. 2003. SSRN abstract and full paper at: http://ssrn.com/abstract=487585.

REFERENCES

349

32. Ben-Haim, Yakov and Isaac ElishakofF, 1990, Convex Models of Uncertainty in Applied Mechanics, Amsterdam, Elsevier. 33. Ben-Haim, Yakov and Alexander Laufer, 1998, Robust reliability of projects with activity-duration uncertainty, ASCE Journal of Construction Engineering and Management. 124: 125-132. 34. Berlin, Isaiah, 1997, The Sense of Reality: Studies in Ideas and Their History. Ed. by Henry Hardy. Farrar, Straus and Giroux. 35. Blanchard, Olivier Jean and Stanley Fischer, 1989, Lectures on Macroeconomics, MIT Press, Cambridge, MA. 36. Blinder, Alan S., 1998, Central Banking in Theory and Practice, Lionel Robbins Lecture, MIT Press, Cambridge. 37. Bonen, Zeev, 1969, On the planning of development projects, 3rd Annual Israel Conference on Operations Research, Tel Aviv, July 1969. 38. Bossaerts, Peter, 2002, The Paradox of Asset Pricing, Princeton University Press, Princeton. 39. Brainard, William C , 1967, Uncertainty and the effectiveness of policy. The American Economic Review, vol. 57, # 2 , pp.411-425. 40. Brauer, Fred and Carlos Castillo-Chavez, 2001, Mathematical Models in Population Biology and Epidemiology, Springer, New York. 41. Burns, Robert, 1786, To a Mouse, 7th stanza. 42. Carmel, Yohay and Yakov Ben-Haim, 2005, Info-gap robust-satisficing model of foraging behavior: Do foragers optimize or satisfice? The American NaturaUst, 166: 633-641. 43. Carnap, Rudolf, 1962, Logical Foundations of Probability, 2nd ed.. University of Chicago Press. 44. Carroll, Lewis, 1895, Pillow Problems, Re-issued by Dover Press, New York, 1958, riddle 72. 45. Choi, Y.B., 1993, Paradigms and Conventions: Uncertainty, Decision Making, and Entrepreneur ship. University of Michigan Press, Ann Arbor. 46. Churchland, Patricia Smith, 1986, Neurophilosophy: ence of the Mind-Brain, MIT Press.

Toward a Unified Sci-

47. Churchland, Paul M., 1979, Scientific Realism and the Plasticity of Mind, Cambridge University Press. 48. Churchland, Paul M., 1985, The ontological status of observables: In praise of the superempirical virtues, in [49], pp.35-47. 49. Churchland, Paul M. and CliflFord A. Hooker, eds., 1985, Images of Science: Essays on Realism and Empiricism, with a reply from Bas C. van Fraasen, University of Chicago Press. 50. Cochrane, John H., 2001, Asset Pricing, Princeton University Press, Princeton. 51. Conlisk, John, 1996, Why bounded rationality? Journal of Economic Literature, vol. XXXIV, pp. 669-700. 52. Crocker, Malcolm J., ed., 1998, Handbook of Acoustics, John Wiley.

350

REFERENCES

53. Daley, D.J. and J. Gani, 1999, Epidemic Modelling: An Introduction, Cambridge University Press. 54. Darwin, Charles, 1872, The Origin of Species, Harvard Classics, 1937 rendition of the 6th ed. 55. Dewey, John, 1893, The superstition of necessity. The Monist, III: 362-379. Reprinted in [91], pp.91-100. 56. Dewey, John, 1908, Does reality possess practical character? First published in Essays, Philosophical and Psychological, in Honor of William James. Reprinted in [90], pp. 121-133. 57. Dewey, John, 1911, The problem of truth. Old Penn, Weekly Review of the University of Pennsylvania. Reprinted in [91], pp. 101-130. 58. Dewey, John, 1924, Logical method and law. Philosophical Review, 33: 560-572. Reprinted in [90], pp.355-362. 59. Dewey, John, 1931, Philosophy and Civilization, appearing in Intelligence in the Modern World: John Dewey's Philosophy, Joseph Ratner, ed., 1939, Modern Library. 60. Diekmann, O. and J.A.P. Heesterbeek, 2000, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley & Sons, 61. Dirac, P.A.M., 1958, The Principles of Quantum Mechanics, 4th ed., Oxford University Press. 62. Drake, S., 1980, Galileo, Oxford University Press. 63. Drenick, Rudolf F., 1968, Functional analysis of effects of earthquakes, 2nd Joint United States-Japan Seminar on Applied Stochastics, Washington, D.C., Sept. 19-24, 1968. 64. Drenick, Rudolf F., 1970, Model-free design of aseismic structures. Journal of the Engineering Mechanics Division, Proceedings of the ASCE, 96: 483493. 65. Dubois, Didier and Henri Prade, 1988, Possibility Theory: An Approach to Computerized Processing of Uncertainty, New York, Plenum. 66. Einstein, Albert, 1950, Out Of My Later Years, Thames and Hudson Pub., London. 67. Ekeland, Ivar, 1988, Mathematics and the Unexpected, University of Chicago Press. 68. Elishakoff, Isaac, 1983, Probabilistic Methods in the Theory of Structures, Wiley, New York. 69. Elishakoff, Isaac and Pierluigi Colombi, 1993, Combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters. Computer Methods in Applied Mechanics and Engineering, 104: 187-209. 70. Ellsberg, D., 1961, Risk, ambiguity and the Savage axioms. Quarterly Journal of Economics, 75: 643-669.

REFERENCES

351

71. Epstein, Larry G. and Jianjun Miao, 2003, A two-person dynamic equilibrium under ambiguity, Journal of Economic Dynamics and Control, 27: 1253-1288. 72. Epstein, Larry G. and Tan Wang, 1994, Intertemporal asset pricing under Knightian uncertainty, Econometrica, 62: 283-322. 73. Fabrycky, W.J., G.J. Thuesen and D. Verma, 1998, Economic Analysis, 3rd ed., Prentice-Hall.

Decision

74. Fraassen, Bas C. van, 1985, Bas C. van Fraassen replies to his critics, in [49], pp.243-308. 75. Fuller, J.D. and Y. Gerchak, 1989, Risk aversion and plant size: theory and application to tar-sand oil plants, Canadian J. of Economics, 22: 164-173. 76. Galbraith, J., 1973, Designing Complex Organizations, Addison-Wesley Publ. Co. 77. Galileo Galilei, 1638, Dialogues Concerning Two New Sciences, Trans, by H. Crew and A. de Salvio, 1914. Re-issued by Dover Publ., 1954. 78. Gilboa, L and D. Schmeidler, 1989, Maxmin expected utility with nonunique prior. Journal of Mathematical Economics, 18: 141-153. 79. Ginsburg, Herbert and Sylvia Opper, 1969, PiageVs Theory of Intellectual Development: An Introduction, Prentice-Hall. 80. Haack, Susan, 1993, Evidence and Inquiry: Towards Reconstruction in Epistemology. Blackwell. 81. Haack, Susan, 1996, Science as social? Yes and no, reprinted in [82], pp. 104122. 82. Haack, Susan, 1998, Manifesto of a Passionate Moderate: Essays, University of Chicago Press.

Unfashionable

83. Haack, Susan, 2001, An Epistemologist in the Bramble-Bush: At the Supreme Court with Mr. Joiner. Journal of Health Politics, Policy and Law, 26: 217-248. 84. Haack, Susan, 2003, Defending Science — Within Reason: Between Scientism and Cynicism, Prometheus Books. 85. Hacking, Ian, 1975, The Emergence of Probability: A Philosophical Study of Early Ideas About Probability, Induction and Statistical Inference, Cambridge University Press. 86. Hall, J., 1653, Paradoxes. 'uncertainty' lb.

Quoted in Oxford English Dictionary [126] at

87. Hansen, L.P., Thomas J. Sargent and T.Tallarini, 1999, Robust permanent income and pricing. Review of Economic Studies, 66, 873-907. 88. Hardy, G.H., J.E. Littlewood and G. Polya, 1934, Inequalities, Cambridge University Press, London. 89. Hartt, W.H. and N.K. Lin, 1986, A proposed stress history for fatigue testing applicable to offshore structures. International Journal of Fatigue, 8: 91-93. 90. Hickman, Larry A. and Thomas M. Alexander, eds., 1998, The Essential Dewey, vol. 1: Pragmatism, Education, Democracy, Indiana University Press.

352

REFERENCES

91. Hickman, Larry A. and Thomas M. Alexander, eds., 1998, The Essential Dewey, vol. 2: Ethics, Logic, Psychology, Indiana University Press. 92. Hipel, Keith W. and Yakov Ben-Haim, 1999, Decision making in an uncertain world: Information-gap modelling in water resources management, IEEE Trans., Systems, Man and Cybernetics, Part C: Applications and Reviews, 29: 506-517. 93. Hilton, Ronald W., 1981, The determinants of information value: Synthesizing some general results. Management Science, 27: 57-64. 94. Howell, G.A., A. Laufer and G. BaUard, 1993, Uncertainty and project objectives. Project Appraisal, 8(1): 37-43. 95. James, William, 1884, Unitarian Review, XXII, 205. Quoted in Oxford English Dictionary [126] at 'un-prefix' 7b. 96. Kanno, Y. and Izuru Takewaki, Robustness analysis of trusses with separable load and structural uncertainties. International J. Solids and Structures, to appear. 97. Kaufman, Martin, 1971, Homeopathy in America: Medical Heresy, Johns Hopkins Press, Baltimore.

The Rise and Fall of a

98. Keeney, Ralph L. and Howard Raiffa, 1993, Decisions with Multiple Objectives: Preferences and Value Tradeoffs, Cambridge University Press, New York. 99. Keynes, John Maynard, 1929, A Treatise on Probability, Macmillan and Co., London. 100. Kirkpatrick, S.W. and B.S. Holmes, 1989, Effect of initial imperfections on dynamic buckling of shells, ASCE Journal of Engineering Mechanics, 115: 1075-1093. 101. Klir, George J., 2004, Generalized information theory: aims, results and open problems. Reliability Engineering and System Safety, 85: 21-38. 102. Klir, George J. and Tina A. Folger, 1988, Fuzzy Sets, Uncertainty, and Information, Englewood Cliffs, NJ, Prentice Hall. 103. Knight, Frank H., 1921, Risk, Uncertainty and Profit. Houghton Mifflin Co. Re-issued by University of Chicago Press, 1971. 104. Knight, Frank H., 1951, The Economic Organization, Harper Torchbooks. 105. Kohelet, or Ecclesiastes. 106. Kocherlakota, Narayana R., 1996, The equity premium: It's still a puzzle. Journal of Economic Literature, 34: 42-71. 107. Kyburg, Henry E. jr., 1990, Getting fancy with probability, Synthese, 90: 189-203. 108. Langer, Susanne K., 1962, Philosophical Sketches. Mentor Book. 109. Laufer, Alexander, 1991, Project planning: Timing issues and path of progress. Project Management Journal, vol. 22(2), 39-45. 110. Laufer, Alexander, 1997, Simultaneous Management: Managing Projects in a Dynamic Environment, AMACOM, American Management Association, New York.

REFERENCES

353

111. Laufer, Alexander and R.L. Tucker, 1988, Competence and timing dilemma in construction planning, Construction Management and Economics, 6: 339-355. 112. Leibniz, Gottfried Wilhelm, 1686, From the letters to Arnauld. In Philosophical Essays, Roger Ariew and Daniel Garber, editors and translators. Hackett Publishing Co., 1989. 113. Ljungqvist, Lars and Thomas J. Sargent, 2000, Recursive Theory, MIT Press, Cambridge, USA.

Macroeconomic

114. Locke, John, 1706, An Essay Concerning Human Understanding. tion. Roger Woolhouse, editor. Penquin Books, 1997.

5th edi-

115. Lyu, Michael R., 1996, Handbook of Software Reliability Engineering, IEEE Computer Society Press, McGraw-Hill. 116. Mack, R.P., 1971, Planning and Uncertainty: Decision Making in Business and Government Administration, Wiley, New York. 117. March, James G., 1978, Bounded rationality, ambiguity, and the engineering of choice. The Bell Journal of Economics, 9: 587-608. Also appearing in [10], pp.33-57. 118. March, James G., 1994, A Primer on Decision Making: How Decisions Happen, The Free Press, A Division of Macmillan, Inc., New York. 119. Margenau, H. and G.M. Murphy, 1964, The Mathematics Chemistry, vol. 2, Van Nostrand-Reinhold Co.

of Physics and

120. Mas-Colell, A., M.D. Whinston and J.R. Green, 1995, Microeconomic Theory, Oxford University Press. 121. Mayo, Deborah G., 1996, Error and the Growth of Experimental University of Chicago Press.

Knowledge,

122. Mehra, Rajnish and Edward C.Prescott, 1985, The equity premium: A puzzle. Journal of Monetary Economics, 15: 145-161. 123. Mises, R. von, 1957, Probability, Statistics and Truth. London, George Allen and Unwin. Re-issued by Dover Publications, 1981. 124. Moore, Walter, 1989, Schrodinger: Life and Thought, Cambridge University Press. 125. Neumann, John von and Oskar Morgenstern, 1953, Theory of Games and Economic Behavior, 3rd ed., Princeton University Press. 126. Oxford English Dictionary, 1999, 2nd edition on CD-ROM, Version 2.0. 127. Peirce, Charles Sanders, 1878, The Doctrine of Chances, Popular Science Monthly. Appearing in [129], pp.61-81. 128. Peirce, Charles Sanders, 1891, The architecture of theories. The Appearing in [129], pp.157-178.

Monist.

129. Peirce, Charles Sanders, 1998, Chance, Love, and Logic: Philosophical Essays, ed. by M.R. Cohen, University of Nebraska Press. 130. Penrose, Roger, 1994, Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press.

354

REFERENCES

131. Piaget, Jean, 1967, Six Psychological Studies, Translated by Anita Tenzer. Random House. 132. Plato, Republic, in The Collected Dialogues of Plato, including the Letters. Edith Hamilton and Huntington Cairns, editors, Princeton University Press, 1961. 133. Pious, Scott, 1993, The Psychology of Judgment McGraw-Hill, New York.

and Decision

Making,

134. Polya, G., 1954, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning. Princeton University Press, Princeton, N.J. 135. Popper, Karl R., 1957, The Poverty of Historicism, edition 1964.

Harper Torchbooks

136. Popper, Karl R., The Open Society and Its Enemies. Vol.1: The Spell of Plato, 1966. Vol.2: The High Tide of Prophecy: Hegel, Marx and the Aftermath, 1962. 137. Popper, Karl R., 1982, The Open Universe: An Argument for ism. Rout ledge.

Indetermin-

138. Porter, T.M., 1986, The Rise of Statistical Thinking: 1820-1900, Princeton University Press. 139. Pratt, J., 1964, Risk aversion in the small and in the large, 32: 122-136.

Econometrica,

140. Prigogine, Ilya and Isabelle Stengers, 1984, Order Out of Chaos: Man's New Dialogue With Nature. Bantam Books. 141. Prior, Matthew, 1718, Solomon, III, 290. Quoted in Oxford English Dictionary [126] at 'uncertain' 6a. 142. Quine, Williard V., 1951, Two dogmas of empiricism, appearing in [144], pp.20-46. 143. Quine, WiUiard V., 1960, Word and Object, MIT Press. 144. Quine, Williard V., 1980, From a Logical Point of View: Nine LogicoPhilosophical Essays. 2nd ed., Harvard University Press. 145. Regan, Helen M., Yakov Ben-Haim, Bill Langford, Will G. Wilson, Per Lundberg, Sandy J. Andelman, Mark A. Burgman, 2005, Robust decision making under severe uncertainty for conservation management, Ecological Applications, vol. 15(4): 1471-1477. 146. Rorty, Richard, 1979, Philosophy and the Mirror of Nature, Princeton University Press. 147. Rosenhead, Jonathan, ed., 1986, Rational Analysis For a Problematic World: Problem Structuring Methods For Complexity, Uncertainty and Conflict, Wiley. 148. Rota, Gian-Carlo, 1992, Discrete Thoughts, in Discrete Thoughts: Essays on Mathematics, Science and Philosophy, by M. Kac, G.-C.Rota and J.T. Schwartz, Birkhauser. 149. Rousseau, Jean-Jacques, 1762, Emile, or Education. Quoted in The Living Thoughts of Rousseau, Romain Rolland, ed., Cassell, London.

REFERENCES

355

150. Rubinstein, Ariel, 1998, Modeling Bounded Rationality, MIT Press. 151. Russell, Bertrand, 1937, A Critical Exposition of the Philosophy of Leibniz, 2nd ed. George Allen & Unwin, London. 152. Samuelson, Paul A., 1983, Foundations of Economic Analysis. edition. Harvard University Press.

Enlarged

153. Savage, Leonard J., 1962, Subjective Probability and statistical practice. In: G.A. Barnard and D.R. Cox, eds.. The Foundations of Statistical Inference: A Discussion. London, Methuen. 154. Savage, Leonard J., 1972, The Foundations of Statistics. tion. Dover publications.

2nd revised edi-

155. Schwartz, Barry, 2004, The tyranny of choice. Scientific American, vol. 290, # 4 , pp70-75. 156. Schweppe, Fred C , 1973, Uncertain Dynamic Systems, Prentice-Hall, Englewood Cliffs. 157. Shackle, G.L.S., 1972, Epistemics and Economics: A Critique of Economic Doctrines, Transaction Publishers, 1992, originally published by Cambridge University Press. 158. Shafer, Glenn, 1988, Savage revisited, in [10], pp. 193-234. 159. Shinozuka, M., 1970, Maximum structural response to seismic excitations. Journal of the Engineering Mechanics Division, Proceedings of the ASCE, 96: 729-738. 160. Simon, Herbert A., 1982, Reply: Surrogates for uncertain decision problems, appearing in H.A. Simon, Models of Bounded Rationality, vol.1, MIT Press,pp.235-247. 161. Simon, Herbert A., 1983, Reason in Human Affairs, Stanford University Press, Stanford. 162. Soong, T.T., 1981, Probabilistic Modelling in Science and Engineering, Wiley, New York. 163. Soong, T.T., 1990, Active Structural Control: Theory and Practice, Wiley, New York. 164. Stark, H. and J.W. Woods, 1994, Probability, Random Processes and Estimation Theory for Engineers, Prentice Hall. 165. Takewaki, Izuru and Yakov Ben-Haim, 2005, Info-gap robust design with load and model uncertainties. Journal of Sound and Vibration, 288: 551570. 166. Thaler, Richard H., 1992, The Winner^s Curse: Paradoxes and Anomalies of Economic Life, Princeton University Press. 167. Thaler, Richard H., ed., 2005, Advances in Behavioral Finance, Volume II, Princeton University Press. 168. Thompson, J.M.T., 1982, Instabilities and Catastrophes in Science and Engineering, John Wiley. 169. Thompson, J.M.T. and H.B. Stewart, 1986, Nonlinear Dynamics and Chaos: Geometric Methods for Engineers and Scientists, John Wiley.

356

REFERENCES

170. Treatment Guidelines, Drugs for lipids, The Medical Letter, Vol. 3, issue 31, March 2005. 171. Wald, Abraham, 1947, Sequential Analysis. tions, 1973.

Re-issued by Dover Publica-

172. Webster^s Ninth New Collegiate Dictionary, 1985, Merriam-Webster. 173. Weekley, Ernest, 1921, An Etymological Dictionary of Modern English. Reissued by Dover Pubhcations, 1967. 174. Wheeler, John A., quoted in Scientific American, Dec. 1992, p.20. 175. Whitehead, Alfred North, 1925, Science and the Modern World, Lowell Lectures, 1925. A Mentor Book, 1960. 176. Wiener, Norbert, 1956, I Am A Mathematician: MIT Press.

The Later Life of a Prodigy.

177. Wilson, Edward O., 1998, Consilience: The Unity of Knowledge. Random House, New York. 178. Yao, J.T.P., 1985, Safety and Reliability of Existing Structures, Boston.

Pitman,

179. Zadeh, L.A., 1972, A fuzzy set-theoretic interpretation of linguistic hedges. Journal of Cybernetics, vol.2, pp. 4-34. 180. Zadeh, L.A., 1973, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man and Cybernetics, vol.3, pp. 28-44.

Author Index Adams, Henry 207, 346, 347

Brauer, Fred 83, 349

Alexander, Thomas M. 336, 344, 351

Burgman, Mark A. 256, 294, 354

Andelman, Sandy J. 256, 294, 354

Burns, Robert 297, 349 Carmel, Yohay 280, 349

Arbocz, J. 25, 347

Carnap, Rudolf 327, 349

Arendt, Hanna 346, 347

Carroll, Lewis 35, 349

Arrow, Kenneth J. 200, 277, 341, 347

Castillo-Chavez, Carlos 83, 349 Chen, Genda 331, 348

Baker, Elhot 1, 347 Ballard, G. 302, 352

Choi, Y.B. 15, 349

Barberis, Nicholas 125, 347

Churchland, Patricia Smith 342, 349

Bardossy, A. 14, 347 Bardsen, Gunnar 91, 347

Churchland, Paul M. 231, 329, 334, 349

Bell, David E. 347

Cochrane, John H. 285, 291, 349

Ben-Haim, Yakov 12, 15-20, 30, 31, 43, 46, 54, 60, 65, 116, 133, 173, 176, 188, 216, 250, 253, 256, 263, 280, 294, 328, 331, 332, 337, 345, 347-349, 352, 354, 355

Cogan, Scott 43, 348 Conlisk, John 279, 349 Crocker, Malcolm J. 222, 349 Dacso, Cliff 86 Daley, D.J. 83, 350

Berhn, Isaiah 320, 349 Blanchard, Olivier Jean 288, 290, 349 Blinder, Alan S. 75, 78, 349

Darwin, Charles 279, 350 Dewey, John 125, 126, 317, 336, 344, 350 Diekman, O. 83, 350

Boltzmann, Ludwig x Bonen, Zeev 15, 349

Dirac, P.A.M. 345, 350

Bossaerts, Peter 285, 349

Drake, S. 10, 350

Brainard, Wilham C. 75, 76, 78, 86, 349

Drenick, Rudolf F. 15, 331, 350

357

Dubois, Didier 14, 350

358

AUTHOR INDEX

Duckstein, L. 14, 347

Howell, G.A. 302, 352

Einstein, Albert 115, 350

James, William 37, 352

Eitrheim, 0yvind 91, 347

Jansen, Eilev S. 91, 347

Ekeland, Ivar 323, 343, 350

Jeske, Karsten 294, 348

Elishakoff, Isaac 15, 17-19, 249, 330, 349, 350

Kanno, Y. 54, 352

Ellsberg, D. 268, 350

Keeney, Ralph L. 43, 352

Epstein, Larry G. 294, 351

Keynes, John Maynard 13, 327, 352

Fabrycky, W.J. 71, 351

Kirkpatrick, S.W. 25, 352

Fischer, Stanley 288, 290, 349

Klir, George J. 14, 352

Folger, Tina A. 14, 352

Knight, Frank H. 15, 149, 294, 297, 341, 352

Fraassen, Bas C. van 334, 349, 351 Fuller, J.D. 156, 201, 351 Galbraith, J. 15, 351 Galileo, Galilei 9, 50, 351 Gani, J. 83, 350 Gerchak, Yigal 156, 201, 351 Gilboa, I. 294, 351 Ginsburg, Herbert 208, 335, 351 Green, J.R. 125, 268, 272, 341, 353 Greenspan, Alan 267

Kaufman, Martin 129, 352

Kocherlakota, Narayana R. 285, 295, 352 Kyburg, Henry E. jr. 294, 327, 328, 352 Langer, Susanne K. 321, 352 Langford, Bih 256, 294, 354 Laufer, Alexander 15, 65, 294, 302, 304, 310, 314, 349, 352, 353 Leibniz, Gottfried Wilhelm 22, 344, 353

Haack, Susan 326, 329, 333, 334, 340, 345, 351

Lessing, Ephraim 346

Hacking, Ian 11, 12, 329, 351

Littlewood, J.E. 24, 351

Hall, J. 37, 351

Ljungqvist, Lars 293, 353

Hansen, L.P. 294, 351

Locke, John 118, 322, 353

Hardy, G.H. 24, 351

Lundberg, Per 256, 294, 354

Hartt, W.H. 330, 351

Lyu, Michael R. 137, 353

Heesterbeek, J.A.P. 83, 350

Mack, R.P. 15, 353

Hickman, Larry A. 336, 344, 351

March, James G. 15, 125, 267, 353

Hilton, Ronald W. 186, 352

Margenau, H. 28, 353

Hipel, Keith W. 16, 30, 250, 263, 352

Mas-Colell, A. 125, 268, 272, 341, 353

Holmes, B.S. 25, 352

Mayo, Deborah G. 216, 353

Hooker, Clifford A. 334, 349

Mehra, Rajnish 285, 292, 353

Lin, N.K. 330, 351

AUTHOR INDEX

359

Miao, Jianjun 294, 351

Sanseigne, Laetitia 43, 348

Mises, R. von 12-14, 344, 353 Moore, Walter 299, 353

Sargent, Thomas J. 293, 294, 351, 353

Morgenstern, Oskar 272, 353

Savage, Leonard J. 13, 326, 355

Murphy, G.M. 28, 353

Schmeidler, D. 294, 351

Namath, Joe 249

Schwartz, Barry 279, 355

Neumann, John von 272, 353

Schweppe, Fred C. 15, 21, 355

Nymoen, Ragnar 91, 347

Shackle, G.L.S. 15, 294, 319, 327, 355

Opper, Sylvia 208, 335, 351 Peirce, Charles Sanders 9, 19, 353 Penrose, Roger 323, 353

Shafer, Glenn 185, 355 Shinozuka, M. 331, 355

Piaget, Jean 208, 354

Simon, Herbert A. 39, 76, 96, 279, 355

Plato 343, 354

Solomon 249

Pious, Scott 267, 354

Soong, T.T. 330, 331, 348, 355

Polya, G. 24, 118, 351, 354

Stark, H. 12, 355

Popper, Karl R. 319-321, 327, 346, 354

Stengers, Isabelle 343, 354

Porter, T.M. 10, 354 Prade, Henri 14, 350 Pratt, J. 277, 354

Steward, H.B. 28, 355 Takewaki, Izuru 54, 331, 352, 355 Tallarini, T. 294, 351

Prescott, Edward C. 285, 292, 353

Thaler, Richard H. 125, 267, 347, 355

Prigogine, Ilya 343, 354

Thompson, J.M.T. 28, 355

Prior, Matthew 37, 354

Thuesen, G.J. 71, 351

Quine, Willard V. 118, 323, 329, 354

Tucker, R.L. 302, 353

RaifFa, Howard 43, 347, 352 Regan, Helen M. 256, 294, 354 Rolland, Romain 208, 354 Rorty, Richard 324, 354

Tversky, Amos 347 Verma, D. 71, 351 Wald, Abraham 60, 356 Wang, Tan 294, 351 Weekley, Ernest 149, 356

Rosenhead, Jonathan 313, 354

Wheeler, John A. 185, 356

Rota, Gian-Carlo 9, 354 Rousseau, Jean-Jacques 208, 354

Whinston, M.D. 125, 268, 272, 341, 353

Rubinstein, Ariel 125, 355

Whitehead, Alfred North 318, 356

Russeh, Bertrand 22, 324, 355

Wiener, Norbert 207, 356

Samuelson, Paul A. 341, 355

Williams, J.G. 25, 347

360

AUTHOR INDEX

Wilson, Edward O. 149, 185, 356 Wilson, Will G. 256, 294, 354 Woods, J.W. 12, 355 Yao, J.T.R 330, 356 Zadeh, L.A. 14, 356

Subject Index Abstraction, power of, 318

Brainard's dictum, 78, 86

Acoustic controller, 223 Acoustic pressure, uncertain spatial variation, 223 Additivity axiom of expected utility theory, violation of, 272

Calibration by consequence severity, 121, 123, 127, 304 by information gap, 56, 63 by normalization, 56, 94, 116, 118 by prior experience, 56, 118, 304

Allais 'paradox', 6, 272 info-gap explanation, 272 Analogical reasoning, 63, 117, 121, 304, 305 Antagonistic immunity functions, 4, 48, 74, 98, 125, 131, 133, 136, 145, 161, 163, 169, 170, 179, 181, 299, 314, 340 new approach, 146 Antenna design, 180 Apodictic proposition, skepticism about, 322 Arrow-Pratt risk aversion, 291

Calibration of immunity functions, 4 of robustness function, 116 Cantilever, design of, 50, 190, 248, 265 Capital asset pricing models, 285 Cause and effect, continuity of, 337, 339 Cause and frequency, continuity of, 337 Center point, updating, 228

Assay of material, 80

Center point of info-gap model, 31, 187

Asset-pricing model, info-gap, 288

Cholesterol control, 86

Lucas, 288

Classifier, ambiguity of, 210, 211

Asymptotic robustness, 210, 211, 214 Axioms of info-gap uncertainty, 2, 31, 328

info-gap supervision of, 209 linear, 229 robust optimal, 212, 215 robustness of, 209

Bio-terror preparedness, 83

Clustering and info-gap uncertainty, 2. 17-19. 31. 299, 323, 337

Bounded rationality, 294 361

362

SUBJECT INDEX

Clustering of events, 2

Damage, increment of, 251

Cognitive structures, modification in learning, 208

Deciding and learning, 207

Coherence and preference, 235

Decision analyst, x

Coherence function, 232, 233, 236, 239, 243

Decision maker, executive, xi

Coherence function and robustness, 233 Coherence of info-gap models, 6, 231, 233, 235, 238-240, 243 Commitment premium, 156, 159 Comparing two decision strategies, 181 Consensus and uncertainty, 6, 231 Conservation biology, 256 Construction time, uncertain, 156, 201

Decision algorithm, generic, 41

Decision models, components of, 44, 65 Decision vector, 38, 45 Decisions, sequential, 60 Demand curve, uncertainty of, 299 Demand value of information, 189, 190, 193, 200, 202-204, 308 positive or negative, 194, 196 Doubt and consciousness, 344 Drug selection, 86 Dynamical system, hnear, 250, 254, 263

Content of learning, 207 Continuity of cause and effect, 337, 339

Ellipsoids, properties of, 33

Continuity of cause and frequency, 337

Empirical robustness function, 217, 220, 224, 225, 227

Contraction axiom, 31

Empiricism and realism, 345

Convex model, 17, 34

Employee productivity, 127

Convexity, definition, 35

Endangered species, 256

Convexity and info-gap models, 17

Envelope function, 22

Convexity and uncertainty, 19

Environmental monitoring, 80

Credence and warrant, 334

Environmental protection, 121, 246

Credence for info-gap inference, 331, 337

Epidemiological models, uncertainty of, 83

Critical path, nominal, 69, 171, 302

Epistemic uncertainty, 322-324

uncertainty, 70, 171, 197, 302, 308

Ellsberg 'paradox', 6, 268

Equity premium puzzle, 285 Error and learning, 208, 227

Critical reward, 39

Estimating a probability density,

Crossing of robustness curves, 49, 73, 77, 83, 85, 91, 95, 160, 195, 259, 262, 271, 275, 279, 283, 312

88 Evolution, biological, 279 Expansion axiom, 31

SUBJECT INDEX

Expected dis-utility, 87 Expected utility risk aversion, infogap explanation, 277 Expected utility theory, 2, 256, 272 Explanatory integration and warrant, 329, 330, 333, 338 Financial investment, 70, 204, 248 Foundherentism, 329, 333 Fourier expansion, 33 Function of learning, 207 Fuzzy logic, 14

363

Increment of information, behaviorally significant, 235 Indeterminism, Shackle-Popper, 319, 320 Inference, analogical, 119, 305 based on uncertainty model, 325, 327, 328 info-gap, 332 plausible, 63, 119, 305 robust, 332, 340 Info-gap, 2, 15, 16, 18, 22, 298, 300 Info-gap inference, 332

Gambling, 4, 47, 93, 102, 150, 152, 163, 199, 200, 203, 278, 298, 342

Info-gap inference, credence for, 331, 337

Gap function, 212, 220, 224, 227

Info-gap model, combined, 26

Generality, power of, 318

convexity and, 17

Harmonic oscillator, 133, 265

cumulative energy bound, 21

Heart disease, 86

discrete, 30

Holism, linguistic, 322, 323

ellipsoid bound, 21, 26, 34, 60, 71, 180, 210, 220, 229, 276

Holism and uncertainty, 318 Holism of info-gap model, 324

energy bound, 20, 32, 34, 51, 251

Holism of meaning, 322

envelope bound, 22, 25, 26, 94, 134, 167, 186, 201, 204, 228, 236

Holistic generalization, 318, 319

evidence against, 6, 208, 215

Home-bias paradox, 280

family of nested sets, 17

Horizon of uncertainty, 17, 21, 26, 31, 43, 186, 324, 342

formulation of, 16, 20, 32

Hybrid uncertainties, 6, 26, 56, 250, 342 Immunity functions, 3, 38, 39, 43 antagonism and sympathy, 4, 48, 74, 98, 125, 130, 132, 146, 163, 169, 179, 181, 299, 314, 340 calibration of, 4 components of, 44, 65

Fourier bound, 25 Fourier elhpsoid bound, 34, 157, 160, 187, 191, 205, 238, 240 generalized envelope bound, 23, 28, 29 holism of, 324 informativeness of, 5, 63, 186, 191 instantaneous energy bound, 21

general, 99

interval bound, 35, 67, 139, 241, 242, 248

reward coherence of, 48, 130

interval uncertainty, 179, 195

364

SUBJECT INDEX

Minkowski norm, 24

Information equivalence, 96

non-convex, 27, 28

Information surrogates, 96

slope bound, 25, 34, 78 two levels of uncertainty, 17

Informativeness of info-gap model, 63, 96, 97, 173, 186, 191

unbounded, 210

Intelligence, Piaget's theory, 208

uniform bound, 17, 32, 34, 93, 157, 181, 187, 228, 235

Inverse of robustness function, 81, 90, 260, 282

updating, 6, 215, 216, 218, 222, 224, 225, 227, 228, 328

Investment, robust satisficing, 287

Info-gap model of linguistic uncertainty, 324 of uncertain probability distribution, 89, 253 Info-gap models, coherence of, 231, 233, 235, 238-240, 243 origin of, 3, 15 simple and complex, 194 Info-gap opportuneness inference, 339, 340 Info-gap robust inference, 332, 340 Info-gap uncertainty, axioms, 31 implications, 317 spatial analogy, 22, 324 Information, demand value of, 5, 189, 190, 193, 194, 200, 202-204^ 308 fair price for, 188 robustness premium of, 189, 190

Investment decision, 179, 204 Investment portfolio, 70, 248 Investment riskiness, 179 Knightian uncertainty, 76, 285, 293, 318, 341, 342, 345 Knowledge, accurate representation of reality, 322, 324 Knowledge, pursuit of, 345 Language and uncertainty, 321 Learning, 6 perspectives on, 207 Learning and deciding, 207 Linear system with uncertain coefficients, 28 Linguistic ambiguity, 14 Linguistic holism, 322, 323 Linguistic uncertainty, info-gap model of, 324

value of, 5, 308

Lottery and info-gap uncertainty, 3, 150

windfall cost of, 197, 198, 200, 203

Lucas asset-pricing model, 288

Information and robustness, 96

Manufacturing process control, 58

Information and uncertainty, 1

Maximum likelihood decision rule, 212, 227

Information and windfall opportuneness, 198

Maximum likelihood estimate, 88

Information enhances robustness, 188, Meaning, holistic, 322 202 Meaning and uncertainty, 321

SUBJECT INDEX

365

Metatarsal pathology, 219

Piecemeal engineering, 319, 346

Models, two types, 320

Plantar pressure, 219

Monetary policy, 75

Plausible inference, 63, 119, 305

Multi-criterion reward, 43, 59

Poisson distribution, 250

Multi-tasking computer jobs, 137

Portfolio investment, 70

Nearest neighbor decision rule, 61, 212

Possible worlds and uncertainty, 344

Nesting and informativeness of infogap models, 186 Nesting axiom, 31, 232, 244 Nesting of uncertainty sets, 17, 21, 31, 97, 300, 323, 337 Noise, acoustic, 222 Nominal critical path, 69, 171, 302 Objective uncertainty, 322 Ontological uncertainty, 322-324 Opportune windfalling, 46, 102, 164 Opportuneness and robustness, 3 Opportuneness function, 38, 39, 43, 73, 101, 131, 161, 197, 313, 339 big is bad, 38, 46 generic, 101 monotonicity, 46 preferences induced by, 46 Opportuneness function and infogap inference, 339, 340 Opportuneness inference, 339, 340 Opportuneness premium, 151, 162, 169, 170 Opportunity and windfall costs, 200 Optimization, direct, 312

Pragmatic truth, 322 Preference and rationality, 124 Preference and uncertainty, 4, 231 Preference preservation, 231, 232, 235, 236, 243 Preference reversal, 49, 73, 77, 83, 85, 91, 95, 160, 195, 259, 262, 271, 275, 279, 283, 312 Preferences, induced by immunity functions, 45, 100, 102, 124, 126, 166 induced by probability of immunity, 264 non-transitivity, 166 Principal-agent contract bidding, 241 Probability, uncertain perception of, 268, 269, 273, 278 Probability and info-gap uncertainty, comparison, 18 Probability and uncertainty, 5, 298 Probability density, estimation of, 88 Probability distribution, info-gap uncertainty of, 89, 253, 256, 259, 269, 273 Probability of failure, 254 upper bound, 263

Participation matrix, 66

Probability of success, 279, 283, 284

Path robustness, 68

Process control, 58

Pendulum-like systems, 27

Production with uncertain costs, 92, 166

Perspectivalism, 326

366

SUBJECT INDEX

Profit maximization, 92 Project management, 64, 137, 171, 181 gathering information, 196, 308

Risk neutral line, 164, 166, 168, 182 Risk neutrality, 151, 163, 171 Risk sensitivity, 150

restructuring to enhance robustness, 310

with one robustness curve, 152, 169, 171, 307

Pursuit of knowledge, methodology, 345

with robustness and opportuneness, 161, 170

Queueing problems, 259

with two robustness curves, 154, 170, 171, 307

Random arrival time, 259 Randomness, origins of, 343 Rationality and preference, 101, 124 Realism and empiricism, 345 Relativism, 326, 328 Reliability, structural, 54 Reward algorithm 7^*, 101 Reward algorithm 7^*, 99 Reward coherent action, 132, 135, 155 Reward function, 39

Risk taking, 4, 47, 93, 102, 149, 150, 152, 163, 199, 200, 203, 298, 342 Riskiness of investment, 73, 179 Robust inference, 332, 340 Robust optimal classifier, 212, 215 Robust rationahty, 92, 97, 125, 126 Robust satisficing, 6, 45, 60, 71, 78, 91, 93, 97, 100, 130, 132, 134, 146, 152, 154, 155, 158-161, 164, 168, 169, 176, 178, 199, 201, 252, 257, 262, 263, 267, 271, 274, 279, 281, 283, 331, 340

Reward premium, 151, 155, 159, 172, 185

Robust satisficing investment, 287

Rhinoceros, Sumatran, 256

Robust severe test, 215, 216, 218, 225, 327, 328, 332, 337, 339, 346

Risk, etymology, 149 info-gap concept, 150 lexical meaning, 306 Risk and info-gap uncertainty, 150, 341, 342, 345 Risk and uncertainty, 341, 342, 345 Risk aversion, 6, 291 Arrow-Pratt, 291 info-gap, 295 Risk aversion and proclivity, 149, 151, 152, 154, 156, 158, 161-163, 169, 170

Robustness, asymptotic, 210 enhanced by information, 188, 202 Robustness and opportuneness, 3 Robustness and warrant, 332, 333, 345 Robustness as measure of validity, 42, 62, 332, 333 Robustness curves, crossing, 49, 73, 77, 83, 85, 91, 95, 160, 195, 259, 262, 271, 275, 279, 283, 312 Robustness function, 38-43, 100, 302

SUBJECT INDEX

367

bigger is better, 38, 45, 115, 124, 218

Severe test, robust, 215, 216, 218, 225, 327, 328, 332, 337, 339, 346

calibration by consequence severity, 121, 123, 127, 304

Severe uncertainty, xi, 1, 5, 11, 16, 92, 149, 298, 338, 342

calibration by information gap, 56, 63

Shackle-Popper indeterminism, 319, 320

calibration by normalization, 56, 94, 116, 118

Shape matrix, updating, 229

calibration by prior experience, 56, 118, 304

Space, analogy to info-gap uncertainty, 22

empirical, 217, 220, 224, 225, 227

Standardization class, 284

generic, 40, 269, 274

Statistics, emergence of, 10

inverse of, 81, 90, 260, 282

Stone grinder's theorem, 20

monotonicity, 46

Structural reliability, 54

Poisson randomized, 252

Structure of learning, 207

preferences induced by, 45

Subjective uncertainty, 322

significant increment, 118

Sympathetic immunity functions, 4, 48, 74, 98, 125, 131, 133, 144, 145, 161, 163, 170, 179, 181, 299, 314, 340

Robustness premium, 151, 153, 154, 156, 158, 161, 164, 169, 170, 172, 174, 175, 178, 188, 196, 308, 309 Robustness premium and resource commitment, 176

Software architecture, 137

new approach, 146

Robustness premium increases with information, 173, 175

Task path participation matrix, 66

Robustness premium of information, 189, 190

Tracking problem, 78

Robustness threshold, 55 Satisficing, 38, 73, 74, 100, 102, 200, 272, 332, 339, 340

Threshold robustness, 55 Trade-off, certainty vs. windfall reward, 47, 74, 131, 136, 144, 167, 198, 201, 203, 313

Seismic safety, 330

robustness vs. critical reward, 46, 72, 123, 130, 136, 144, 152, 167, 171, 193, 202, 210, 252, 306, 308, 312-314

Sequential decisions, 60

uncertainty vs. reward, 175

Search and evasion, 78, 205

Serial network, robustness of, 68

Translation axiom, 31

Set inclusion and informativeness of info-gap models, 186

Truth, pragmatic, 322 uncertain, 322

Severe test of opportuneness, 339 Severe test of truth, 216

Unbounded info-gap model, 210

368

SUBJECT INDEX

Uncertain density, Keynes' example, 13

Updating info-gap models, 215, 216, 218, 222, 224, 225, 227, 228

Uncertain truth, 322

Utility, uncertain perception of, 276, 278

Uncertainties, hybrid, 6, 26, 56, 250, 342 Uncertainty, info-gap, 15, 16, 18, 22, 298, 300

Value judgments, 4, 115, 124, 126, 253, 298, 303, 304, 332, 339

hmitation to knowledge, 319

Value of information, 189, 190, 193, 194, 196, 200, 202-204, 308

non-probabilistic models, 12

Vibration enhancement, 133

objective, 322

Vibration suppression, 50

ontological, 322-324 perceptions of, 10, 341, 342 severe, xi, 1, 5, 11, 16, 92, 298, 338, 342 subjective, 322 two faces of, 3, 313 Uncertainty and consensus, 231

Warrant, holistic, 326, 329 method for assessing, 325 piecemeal, 326, 329 Warrant and credence, 334 Warrant and explanatory integration, 329, 330, 333, 338

Uncertainty and convexity, 19

Warrant and robustness, 332, 333, 345

Uncertainty and holism, 318

Warrant and uncertainty, 325, 328

Uncertainty and language, 321

Warranted knowledge, truth indicative, 324, 325, 345

Uncertainty and meaning, 321 Uncertainty and possible worlds, 344 Uncertainty and preference, 231 Uncertainty and probability, 298

Windfall and opportunity costs, 200 Windfall cost of information, 197, 198, 200, 203

Uncertainty and risk, 341, 342, 345

Windfall opportuneness and information, 198

Uncertainty critical path, 70, 171, 302, 308

Windfall reward, 39

Uncertainty envelope, updating, 228 Uncertainty model, complement to system model, 320 Uncertainty models, diversity of, 327, 346 Uncertainty parameter, 17, 31 probabihstic, 262 Uncertainty weight, 68, 139, 143, 171, 196, 201, 248

Windfalling, 39, 73, 74, 102, 198, 200. 339