A Theory of Case-Based Decisions -...

[20:11 2001/5/8 G:/TeX/Keyword/Gilboa/gilboa.tex] Ref: 3751 GILBOA: A Theory of Case-Based Decisions Frontmatter Page: iii 1–199

A Theory ofCase-Based Decisions

ITZHAK GILBOA

and

DAVID SCHMEIDLER

[20:11 2001/5/8 G:/TeX/Keyword/Gilboa/gilboa.tex] Ref: 3751 GILBOA: A Theory of Case-Based Decisions Frontmatter Page: iv 1–199

published by the press syndicate ofthe university of cambridgeThe Pitt Building, Trumpington Street, CambridgeUnited Kingdom

cambridge university pressThe Edinburgh Building, Cambridge cb2 2ru, UK40 West 20th Street, New York, ny 10011-4211, USA10 Stamford Road, Oakleigh, vic 3166, AustraliaRuiz de Alarcón 13, 28014 Madrid, SpainDock House, The Waterfront, Cape Town 8001, South Africa

http://www.cambridge.org

© Itzhak Gilboa and David Schmeidler

This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 2001

Printed in the United Kingdom at the University PressCambridge

Typeface 9.75/12pt Trump Medieval System LATEX 2ε [kw]

A catalogue record for this book is available fromthe British Library

ISBN 0 521 80234 2 hardbackISBN 0 521 00311 3 paperback

[20:11 2001/5/8 G:/TeX/Keyword/Gilboa/gilboa.tex] Ref: 3751 GILBOA: A Theory of Case-Based Decisions Frontmatter Page: vii 1–199

CONTENT S

Acknowledgments x

1 Prologue 11 The scope of this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Meta-theoretical vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Theories and conceptual frameworks . . . . . . . . . . . . . . . . . . . . . . . 42.2 Descriptive and normative theories . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Behaviorist, behavioral, and cognitive theories . . . . . . . . . . . . 162.5 Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 Deviations from rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7 Subjective and objective terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Meta-theoretical prejudices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Preliminary remark on the philosophy of science . . . . . . . . . 223.2 Utility and expected utility “theories” as concep-

tual frameworks and as theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 On the validity of purely behavioral economic theory . . . . 253.4 What does all this have to do with CBDT? . . . . . . . . . . . . . . . . . 27

2 Decision rules 294 Elementary formula and interpretations . . . . . . . . . . . . . . . . . . . 29

4.1 Motivating examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Aspirations and satisficing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Comparison with EUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5 Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Variations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1 Average similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Act similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Case similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

vii

[20:11 2001/5/8 G:/TeX/Keyword/Gilboa/gilboa.tex] Ref: 3751 GILBOA: A Theory of Case-Based Decisions Frontmatter Page: viii 1–199

Contents

6 CBDT as a behaviorist theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.1 W-maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Cognitive specification: EUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.3 Cognitive specification: CBDT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.4 Comparing the cognitive specifications . . . . . . . . . . . . . . . . . . . . 57

7 Case-based prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Axiomatic derivation 628 Highlights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 Model and result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

9.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659.2 Basic result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679.3 Learning new cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689.4 Equivalent cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699.5 U-maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

10 Discussion of the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Conceptual foundations 9112 CBDT and expected utility theory. . . . . . . . . . . . . . . . . . . . . . . . . . 91

12.1 Reduction of theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9112.2 Hypothetical reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9312.3 Observability of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9512.4 The primacy of similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9612.5 Bounded rationality? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

13 CBDT and rule-based systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9813.1 What can be known? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9813.2 Deriving case-based decision theory . . . . . . . . . . . . . . . . . . . . . . . . 10013.3 Implicit knowledge of rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10513.4 Two roles of rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5 Planning 10914 Representation and evaluation of plans . . . . . . . . . . . . . . . . . . . . 109

14.1 Dissection, selection, and recombination . . . . . . . . . . . . . . . . . . 10914.2 Representing uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11214.3 Plan evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11414.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

15 Axiomatic derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11915.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11915.2 Axioms and result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12115.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

viii

[20:11 2001/5/8 G:/TeX/Keyword/Gilboa/gilboa.tex] Ref: 3751 GILBOA: A Theory of Case-Based Decisions Frontmatter Page: ix 1–199

Contents

6 Repeated choice 12516 Cumulative utility maximization . . . . . . . . . . . . . . . . . . . . . . . . . . 125

16.1 Memory-dependent preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12516.2 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12716.3 Model and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12916.4 Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13316.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

17 The potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13617.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13617.2 Normalized potential and neo-classical utility. . . . . . . . . . . . . 13817.3 Substitution and complementarity . . . . . . . . . . . . . . . . . . . . . . . . . 141

7 Learning and induction 14618 Learning to maximize expected payoff . . . . . . . . . . . . . . . . . . . . . 146

18.1 Aspiration-level adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14618.2 Realism and ambitiousness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14718.3 Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15018.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15318.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15618.6 Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15818.7 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

19 Learning the similarity function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17419.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17419.2 Counter-example to U-maximization . . . . . . . . . . . . . . . . . . . . . . 17719.3 Learning and expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

20 Two views of induction: CBDT and simplicism .. . . . . . . . . 18320.1 Wittgenstein and Hume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18320.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Bibliography 189

Index 197

ix

[20:11 2001/5/8 G:/TeX/Keyword/Gilboa/gilboa.tex] Ref: 3751 GILBOA: A Theory of Case-Based Decisions Chapter 1 Page: 1 1–199

C H A P T E R 1

P ROLOGUE

1 The scope of this book

The focus of this book is formal modeling of decisionmaking by a single person who is aware of the uncertaintyshe is facing. Some of the models and results we proposemay be applicable to other situations. For instance, thedecision maker may be an organization or a computer pro-gram. Alternatively, the decision maker may not be awareof the uncertainty involved or of the very fact that a decisionis being made. Yet, our main interest is in descriptive andnormative models of conscious decisions made by humans.

There are two main paradigms for formal modeling ofhuman reasoning, which have also been applied to deci-sion making under uncertainty. One involves probabilisticand statistical reasoning. In particular, the Bayesian modelcoupled with expected utility maximization is the mostprominent paradigm for formal models of decision makingunder uncertainty. The other employs rule-based deduc-tive systems. Each of these paradigms provides a conceptualframework and a set of guidelines for constructing specificmodels for a wide range of decision problems.

These two paradigms are not the only ways in whichpeople’s reasoning may be, or has been, described. In par-ticular, the claim that people reason by analogies datesback at least to Hume. However, reasoning by analogies hasnot been the subject of formal analysis to the same degreethat the other paradigms have. Moreover, there is no gen-eral purpose theory we are aware of that links reasoning byanalogies to decision making under uncertainty.

1


A theory of case-based decisions

Our goal is to fill this gap. That is, we seek a general pur-pose formal model, comparable to the model of expectedutility maximization, that will (i) provide a frameworkwithin which a large class of specific problems can be mod-eled; (ii) be based on data that are, at least in principle,observable; (iii) allow mathematical analysis of qualitativeissues, such as asymptotic behavior; and (iv) be based onreasoning by analogies.

We believe that human reasoning typically involves acombination of the three basic techniques, namely, rule-based deduction, probabilistic inference, and analogies.Formal modeling tends to opt for elegance, and to focuson certain aspects of a problem at the expense of others.Indeed, our aim is to provide a model of case- or analogy-based decision making that will be simple enough tohighlight main insights. We discuss the various ways inwhich our model may capture deductive and probabilis-tic reasoning, but we do not formally model the latter. Itshould be taken for granted that a realistic model of thehuman mind would have to include ingredients of all threeparadigms, and perhaps several others as well. At this stagewe merely attempt to lay the foundations for one paradigmwhose absence from the theoretical discussion we findtroubling.

The theory we present here does not purport to be morerealistic than other theories of human reasoning or ofchoice. In particular, our goal is not to fine-tune expectedutility theory as a descriptive theory of decision making insituations described by probabilities or states of the world.Rather, we wish to suggest a framework within which onecan analyze choice in situations that do not fit existingformal models very naturally. Our theory is just as idealizedas existing theories. We only claim that in many situationsit is a more natural conceptualization of reality than arethese other theories.

This book does not attempt to provide even sketchysurveys of the established paradigms for formal modelingof reasoning, or of the existing literature on case-based

2


Prologue

reasoning. The interested reader is referred to standard textsfor basic definitions and background.

Many of the ideas and mathematical results in this bookhave appeared in journal articles and working papers (Gilboaand Schmeidler 1995, 1996, 1997a,b, 1999, 2000a,b, 2001).This material has been integrated, organized, and inter-preted in new ways. Additionally, several sections appearhere for the first time.

In writing this book, we made an effort to address readersfrom different academic disciplines. Whereas several chap-ters are of common interest, others may address more spe-cific audiences. The following is a brief guide to the book.

We start with two meta-theoretical sections, one devotedto definitions of philosophical terms, and the other to ourown views on the way decision theory and economic theoryshould be conducted. These two sections may be skippedwith no great loss to the main substance of the book. Yet,Section 2 may help to clarify the way we use certain terms(such as “rationality”, “normative science”, and the like),and Section 3 explains part of our motivation in developingthe theory described in this book.

Chapter 2 of the book presents the main ideas of case-based decision theory (CBDT), as well as its formal model.It offers several decision rules, a behaviorist interpretationof CBDT, and a specification of the theory for predictionproblems.

Chapter 3 provides the axiomatic foundations for thedecision rules in Chapter 2. In line with the tradition indecision theory and in economics, it seeks to relate the-oretical concepts to observables and to specify conditionsunder which the theory might be refuted.

Chapter 4, on the other hand, focuses on the epistemo-logical underpinnings of CBDT. It compares it with theother two paradigms of human reasoning and argues that,from a conceptual viewpoint, analogical reasoning is prim-itive, whereas both deductive inference and probabilisticreasoning are derived from it. Whereas Chapter 3 providesthe mathematical foundations of our theory, the present

3



chapter offers the conceptual foundations of the theory andof the language within which the mathematical model isformulated.

Chapter 5 deals with planning. It generalizes the CBDTmodel from a single-stage decision to a multi-stage one, andoffers an axiomatic foundation for this generalization.

Chapter 6 focuses on a special case of our general model,in which the same problem is repeated over and over again.It relates to problems of discrete choice in decision theoryand in marketing, and it touches upon issues of consumertheory. It also contains some results that are used later inthe book.

Chapter 7 addresses questions of learning, dynamicevolution, and induction in our model. We start with anoptimality result for the case of a repeated problem, whichis based on a rather rudimentary form of learning. We con-tinue to discuss more interesting forms of learning, as wellas inductive inference. Unfortunately, we do not offer anyprofound results about the more interesting issues. Yet,we hope that the formal model we propose may facilitatediscussion of these issues.

2 Meta-theoretical vocabulary

We devote this section to define the way we use certainterms that are borrowed from philosophy. Definitions ofterms and distinctions among concepts tend to be fuzzyand subjective. The following are no exception. These aremerely the definitions that we have found to be the mostuseful for discussing theories of decision making underuncertainty at the present state. While our definitions aregeared toward a specific goal, several of them may facilitatediscussion of other topics as well.

2.1 Theories and conceptual frameworks

A theory of social science can be viewed as a formalmathematical structure coupled with an informal interpre-tation. Consider, for example, the economic theory that

4


Prologue

consumer’s demand is derived from maximizing a utilityfunction under a budget constraint. A possible formalrepresentation of this theory consists of two components,describing two sets, C and P . The first set, C, consists of allconceivable demand functions. A demand function mapsa vector of positive prices p ∈ Rn++ and an income levelI ∈ R+ to a vector of quantities d(p, I) ∈ Rn+, interpretedas the consumer’s desired quantities of consumption underthe budget constraint that total expenditure d(p, I) · p doesnot exceed income I. The second set, P , is the subset of Cthat is consistent with the theory. Specifically, P consists ofthe demand functions that can be described as maximizinga utility function.1 When the theory is descriptive, the setP is interpreted as all phenomena (in C) that might actuallybe observed. When the theory is normative, P is interpretedas all phenomena (in C) that the theory recommends. Thus,whether the theory is descriptive or normative is part ofthe informal interpretation.

The informal interpretation should also specify theintended applications of the theory. This is done at two lev-els. First, there are “nicknames” attached to mathematicalobjects. Thus Rn+ is referred to as a set of “bundles”, Rn++ – asa set of positive “price vectors”, whereas I is supposed torepresent “income” and d – “demand”. Second, there aremore detailed descriptions that specify whether, say, theset Rn+ should be viewed as representing physical commodi-ties in an atemporal model, consumption plans over time,or financial assets including contingent claims, whether ddenotes the demand of an individual or a household, and soforth.

Generally, the formal structure of a theory consists of adescription of a set C and a description of a subset thereof, P .The set C is understood to consist of conceivably observable

1 Standard (neo-classical) consumer theory imposes additional con-straints. For instance, homogeneity and continuity are often part ofthe definition of demand functions, and utility functions are requiredto be continuous, monotone, and strictly quasi-concave. We omit thesedetails for clarity of exposition.

5



phenomena. It may be referred to as the scope of the theory.A theory thus selects a set of phenomena P out of the setof conceivable phenomena C, and excludes its complementC\P . What is being said about this set P , however, is speci-fied by the informal interpretation: it may be the predictionor the recommendation of the theory.

Observe that the formal structure of the theory does notconsist of the sets C and P themselves. Rather, it consistsof formal descriptions of these sets, DC and DP , respec-tively. These formal descriptions are strings of charactersthat define the sets in standard mathematical notation.Thus, theories are not extensional. In particular, two differ-ent mathematical descriptions DP and D′

P of the same setP will give rise to two different theories. It may be a non-trivial mathematical task to discover relationships betweensets described by different theories.

It is possible that two theories that differ not only inthe formal structure (DC , DP ) but also in the sets (C, P )may coincide in the real world phenomena they describe.For example, consider again the paradigm of utility maxi-mization in consumer theory. We have spelled out aboveone manifestation of this paradigm in the language ofdemand functions. But the literature also offers other the-ories within the same paradigm. For instance, one maydefine the set of conceivable phenomena to be all binaryrelations over Rn+, with a corresponding definition of thesubset of these relations that conform to maximization ofa real-valued function.

The informal interpretation of a theory may also be for-mally defined. For instance, the assignment of nicknamesto mathematical objects can be viewed as a mapping fromthe formal descriptions of these objects, appearing in DCand in DP , into a natural language, provided that the latteris a formal mathematical object. Similarly, one may for-mally define “real world phenomena” and represent the(intended) applications of the theory as a collection of map-pings from the mathematical entities to this set. Finally, thetype of interpretation of the theory, namely, whether it is

6


Prologue

descriptive or normative, can easily be formalized.2 Thusa theory may be described as a quintuple consisting of DC ,DP , the nicknames assignment, the applications, and thetype of interpretation.

We refer to the first three components of this quintu-ple, that is, DC , DP , and the nicknames assignment, as aconceptual framework (or framework for short). A concep-tual framework thus describes a scope and a description ofa prediction or a recommendation, and it points to a typeof applications through the assignment of nicknames. Buta framework does not completely specify the applications.Thus, frameworks fall short of qualifying as theories, evenif the type of interpretation is given.

For instance, Savage’s (1954) model of expected utilitytheory involves binary relations over functions defined ona measurable space. The mathematical model is consis-tent with real world interpretations that have nothing to dowith choice under uncertainty, such as choice of streamsof consumption over time, or of income profiles in a soci-ety. The nickname “space of states of the world”, which isattached to the measurable space in Savage’s model, definesa framework that deals with decision under uncertainty.But the conceptual framework of expected utility theorydoes not specify exactly what the states of the world are, orhow they should be constructed. Similarly, the conceptualframework of Nash equilibrium (Nash 1951) in game theoryrefers to “players” and to “strategies”, but it does not spec-ify whether the players are individuals, organizations, orstates, whether the theory should be applied to repeated orto one-shot situations, to situations involving few or manyplayers, and so forth.

By contrast, the theory of expected utility maximiza-tion under risk (von-Neumann and Morgenstern 1944), as

2 Our formal model allows other interpretations as well. For instance,it may represent a formal theory of aesthetics, where the set P isinterpreted as defining what is beautiful. One may argue that such atheory can still be interpreted as a normative theory, prescribing howaesthetical judgment should be conducted.

7



well as prospect theory (Kahneman and Tversky 1979) areconceptual frameworks according to our definition. Still,they may be classified also as theories, because the scopeand nicknames they employ almost completely define theirapplications.Terminological remark: The discussion above implies

that expected utility theory should be termed a frameworkrather than a theory. Similarly, non-cooperative games cou-pled with Nash equilibrium constitute a framework. Still,we follow standard usage throughout most of the book andoften use “theory” where our vocabulary suggests “frame-work”.3 However, the term “framework” will be used onlyfor conceptual frameworks that have several substantiallydistinct applications.

2.2 Descriptive and normative theories

There are many possible meanings to a selection of a setP out of a set of conceivable phenomena C. Among them,we find that it is crucial to focus on, and to distinguishbetween, two that are relevant to theories in the socialsciences: descriptive and normative.

A descriptive theory attempts to describe, explain, or pre-dict observations. Despite the different intuitive meanings,one may find it challenging to provide formal qualitativedistinctions between description and explanation. More-over, the distinction between these and prediction may notbe very fundamental either. We therefore do not dwell onthe distinctions among these goals.

A normative theory attempts to provide recommenda-tions regarding what to do. It follows that normativetheories are addressed to an audience of people facingdecisions who are capable of understanding their recom-mendations. However, not every recommendation qualifiesas normative science. There are recommendations that maybe classified as moral, religious, or political preaching.

3 We apply the standard usage to the title of this book as well.

8


Prologue

These are characterized by suggesting goals to the decisionmakers, and, as such, are outside the realm of academicactivity. There is an additional type of recommendationsthat we do not consider as normative theories. These arerecommendations that belong to the domain of social plan-ning or engineering. They are characterized by recommend-ing tools for achieving pre-specified goals. For instance, thedesign of allocation mechanisms that yield Pareto optimaloutcomes accepts the given goal of Pareto optimality andsolves an engineering-like problem of obtaining it. Our useof “normative science” differs from both these types ofrecommendations.

A normative scientific claim may be viewed as animplicit descriptive statement about decision makers’ pref-erences. The latter are about conceivable realities that arethe subject of descriptive theories. For instance, whereas adescriptive theory of choice investigates actual preferences,a normative theory of choice analyzes the kind of prefer-ences that the decision maker would like to have, that is,preferences over preferences. An axiom such as transitivityof preferences, when normatively interpreted, attempts todescribe the way the decision maker would prefer to makechoices. Similarly, Harsanyi (1953, 1955) and Rawls (1971)can be interpreted as normative theories for social choicein that they attempt to describe to what society one wouldlike to belong.

According to this definition, normative theories are alsodescriptive. They attempt to describe a certain reality,namely, the preferences an individual has over the real-ity she encounters. To avoid confusion, we will reservethe term “descriptive theory” for theories that are notnormative. That is, descriptive theories would deal, bydefinition, with “first-order” reality, whereas normativetheories would deal with “second-order” reality, namely,with preferences over first-order reality. First-order realitymay be external or objective, whereas second-order realityalways has to do with subjective preferences that lie withinthe mind of an individual. Yet, first-order reality might

9



include actual preferences, in which case the distinctionbetween first-order and second-order reality may become arelative matter.4,5

Needless to say, these distinctions are sometimes fuzzyand subjective. A scientific essay may belong to sev-eral different categories, and it may be differently inter-preted by different readers, who may also disagree withthe author’s interpretation. For instance, the indepen-dence axiom of von-Neumann and Morgenstern’s expectedutility theory may be interpreted as a component of adescriptive theory. Indeed, testing it experimentally pre-supposes that it has a claim to describe reality. But itmay also be interpreted normatively, as a recommendationfor decision making under risk. To cite a famous exam-ple, Maurice Allais presented his paradox (see Allais 1953)to several researchers, including the late Leonard Savage.The latter expressed preferences in violation of expectedutility theory. Allais argued that expected utility maxi-mization is not a successful descriptive theory. Savage’sreply was that his theory should be interpreted normatively,and that it could indeed help a decision maker avoid suchmistakes.

Further, even when a theory is interpreted as a recom-mendation it may involve different types of recommen-dations. For instance, Shapley axiomatized his value forcooperative transferable utility games (Shapley 1953). Wheninterpreted normatively, the axioms attempt to capturedecision makers’ preferences over the way in which, say,cost is allocated in different problems. A related result by

4 There is a temptation to consider a hierarchy of preferences, and to askwhich are in the realm of descriptive theories. We resist this temptation.

5 In many cases first-order preferences would be revealed by actualchoices, whereas second-order preferences would only be verballyreported. Yet, this distinction is not sharp. First, there may be first-orderpreferences that cannot be observed in actual choice. Second, one mayimagine elaborate choice situations in which second-order preferencesmight be observed, as in cases where one decides on a decision-makingprocedure or on a commitment device.

10


Prologue

Shapley shows that a player’s value can be computed as aweighted average of her marginal contributions. This resultcan be interpreted in two ways. First, one may view it as anengineering recommendation: given the goal of computinga player’s value, it suggests a formula for its computation.Second, one may also argue that compensating a player tothe extent of her average marginal contribution is ethicallyappealing, and thus the formula, like the axioms, has anormative flavor.

To conclude, the distinctions between descriptive andnormative theories, as well as between the latter and engi-neering on the one hand and preaching on the other, arenot based on mathematical criteria. The same mathemat-ical result can be part of any of these types of scientificor non-scientific activities. These distinctions are basedon the author’s intent, or on her perceived intent. It isprecisely the inherently subjective nature of these distinc-tions that demands that one be explicit about the suggestedinterpretation of a theory.

It also follows that one has to have a suggested inter-pretation in mind when attempting to evaluate a theory.Whereas a descriptive theory is evaluated by its conformitywith objective reality, a normative theory is not. On thecontrary, a normative theory that suggests to people to dowhat they would anyway do is of questionable value. Howshould we evaluate a normative theory, then? Since wedefine normative theories to be descriptive theories deal-ing with second-order reality, a normative theory shouldalso be tested according to its conformity to reality. Butit is second-order reality that a normative theory shouldbe compared to. For instance, a reasonable test of a nor-mative theory might be whether its subjects accept itsrecommendations.

It is undeniable, however, that the evaluation of nor-mative theories is inherently more problematic thanthat of descriptive theories. The evidence for norma-tive theories would have to rely on introspection andself-report data much more than would the evidence for

11



descriptive theories. Moreover, these data may be subjectto manipulation. To consider a simple example, supposewe are trying to test the normative claim that incomeshould be evenly distributed. We are supposed to find outwhether people would like to live in an egalitarian soci-ety. Simply asking people might confound their ethicalpreferences with their self-interest. Indeed, a person mightnot be sure whether she subscribes to a philosophical ora political thesis due to pure conviction or to self-servingbiases. A gedanken experiment such as putting oneselfbehind the “veil of ignorance” (Harsanyi 1953, 1955, Rawls1971) may assist one in finding one’s own preferences,but may still be of little help in a social context. Fur-ther, the reality one tries to model, namely, a person’sethical preferences over the rules governing society, orher logico-aesthetical preferences over her own prefer-ences are a moving target that changes constantly withactual behavior and social reality. Yet, the way we definenormative theories admits a certain notion of empiricaltesting.

2.3 Axiomatizations

In common usage, the term “axiomatization” refers to atheory. However, most axiomatizations in the literatureapply to conceptual frameworks according to our defini-tions. In fact, the following definition of axiomatizationsrefers only to a formal structure (DC , DP ).

An axiomatization of T = (DC , DP ) is a mathematicalmodel consisting of: (i) a formal structure T ′ = (DC , DP ′ ),which shares the description of the set C with T, but whoseset P ′ is described only in the language of phenomena thatare deemed observable; (ii) mathematical results relatingthe set P ′ of T ′ to the set P of T. Ideally, one would like tohave conditions on observable phenomena that are neces-sary and sufficient for P to hold, namely, to have a structureT ′ such that P = P ′. In this case, T ′ is considered to be

12


Prologue

a “complete” axiomatization of T. The conditions thatdescribe P ′ are referred to as “axioms”.6

Observe that whetherT ′ = (DC , DP ′ ) is considered to be anaxiomatization of T depends on the question of observabil-ity of terms in DP ′ . Consequently, the definition above willbe complete only given a formal definition of “observabil-ity”. We do not attempt to provide such a definition here,and we use the term “axiomatization” as if observabilitywere well-defined.7 Throughout the rest of this subsectionwe assume that the applications of the conceptual frame-works are agreed upon. We will therefore stick to standardusage and refer to axiomatizations of theories (rather thanof formal structures).

Because human decisions involve inherently subjectivephenomena, it is often the case that the formulation of atheory contains existential quantifiers. In this case, a com-plete axiomatization would also include a result regardinguniqueness. For instance, consider again the theory statingthat “there exists a [so-called utility] function such that,in any decision between two choices, the consumer wouldopt for the one to which the [utility] function attaches ahigher value”. An axiomatization of this theory should pro-vide conditions under which the consumer can indeed beviewed as maximizing a utility function in binary decisions.Further, it should address the question of uniqueness ofthis function: to what extent can we argue that the utilityfunction is defined by the observable data of binary choices?

There are three reasons for which one might be interestedin axiomatizations of a theory. First, the meta-scientificreason mentioned above: an axiomatization provides a linkbetween the theoretical terms and the (allegedly) observable

6 As opposed to the original meaning of the word, an “axiom” need notbe indisputable or self-evident. However, evaluation of axiomatic sys-tems typically prefers axioms that are simple in terms of mathematicalformulation and transparent in terms of empirical content.

7 Indeed, people who disagree about the definition of observability mayconsequently disagree whether a certain mathematical result qualifiesas an axiomatization.

13



terms used by the theory. True to the teachings of logicalpositivism (Carnap 1923, see also Suppe 1974), one wouldlike to have observable definitions of theoretical terms, inorder to render the latter meaningful. Axiomatizations helpus identify those theoretical differences that have observ-able implications, and avoid debates between theories thatare observationally equivalent.

One might wish to have an axiomatization of a theory fordescriptive reasons. Since an axiomatization translates thetheory to directly observable claims, it prescribes a way totest the empirical validity of the theory. To the extent thatthe axioms do rule out certain conceivable observations,they also ascertain that the theory is non-vacuous, thatis, falsifiable, as preached by Popper (1934). Note also thatthere are many situations, especially in the social sciences,where it is impractical to subject a theory to direct empiri-cal tests. In those cases, an axiomatization can help us judgethe plausibility of the theory. In this sense, axiomatizationsmay serve a rhetorical purpose.

Finally, one is often interested in axiomatizations for nor-mative reasons. A normative theory is successful to theextent that it convinces decision makers to change the waythey make their choices.8 A set of axioms, formulated inthe language of observable choice, can often convince deci-sion makers that a certain theory has merit more than itsmathematical formulation can. Thus, the normative role ofaxiomatizations is inherently rhetorical.

It is often the case that an axiomatization serves allthree purposes. For instance, an axiomatization of utility

8 Changing actual decision making is the ultimate goal of a norma-tive theory. Such changes are often gradual and indirect. For instance,normative theories may first convince scientists before they sift to prac-titioners and to the general public. Also, a normative theory may changethe way people would like to behave even if they fail to implement theirstated policies, for, say, reasons of self-discipline. Finally, a normativetheory may convince many people that they would like society to makecertain choices, but they may not be able to bring them about for polit-ical reasons. In all of these cases the normative theories definitely havesome success.

14


Prologue

maximization in consumer theory provides a definition ofthe term “utility” and shows that binary choices can onlydefine such a utility function up to an arbitrary mono-tone transformation. This meta-scientific exercise savesus useless debates about the choice between observation-ally equivalent utility functions.9 On a descriptive level,such an axiomatization shows what utility maximizationactually entails. This allows one to test the theory that con-sumers are indeed utility maximizers. Moreover, it rendersthe theory of utility maximization much more plausible,because it shows that relatively mild consistency assump-tions suffice to treat consumers as if they were utilitymaximizers, even if they are not conscious of any utilityfunction, of maximization, or even of the act of choice.Finally, on a normative level, an axiomatization of utilitymaximization may convince a person or an organizationthat, according to their own judgment, they should betteradopt a utility function and act so as to maximize it.

Similarly, Savage’s (1954) axiomatization of subjectiveexpected utility maximization provides observable defini-tions of the terms “utility” and “(subjective) probability”.Descriptively, it provides us with reasons to believe thatthere are decision makers who can be described as expectedutility maximizers even if they are not aware of it, thusmaking expected utility maximization a more plausibletheory. Finally, from a normative point of view, decisionmakers who shrug their shoulders when faced with thetheory of expected utility maximization may find it morecompelling if they are exposed to Savage’s axioms.

While our use of the term “axiomatization” highlightsthe role of providing a link between theoretical concepts andobservable data, this term is often used in other meanings.Both in mathematics and in the sciences, an “axiomatiza-tion” sometimes refers to a characterization of a certainentity by some of its properties. One is often interested in

9 This, at least, is the standard microeconomic textbook view. For anopposing view see Beja and Gilboa (1992) and Gilboa and Lapson (1995).

15



such axiomatizations as decompositions of the concept thatis axiomatized. Studying the “building blocks” of whicha concept is made typically enhances our understandingthereof, and allows the study of related concepts. Axiom-atizations in the sense used in this book will typicallyalso have the flavor of decomposition of a construct. Thus,on top of the reasons mentioned above, one may also beinterested in axiomatizations simply as a way to betterunderstand what characterizes a certain concept and whatis the driving force behind certain results. For instance, anaxiomatization of utility maximization in consumer theorywill often reduce utility theory to more primitive “ingredi-ents”, and will also suggest alternative theories that shareonly some of these ingredients, such as preferences that aretransitive but not necessarily complete.

2.4 Behaviorist, behavioral, and cognitive theories

We distinguish between two types of data. Behavioral dataare observations of actions taken by decision makers. Bycontrast, cognitive data are choice-related observations thatare derived from introspection, self-reports, and so forth.We are interested in actions that are open to introspection,even if they are not necessarily preceded by a deliberatedecision-making process. Thus, what people say about whattheir choices might be, the reasons they give for actual orhypothetical choices, their recollection of and motivationfor such choices are all within the category of cognitivedata. In contrast to common psychological usage, we donot distinguish between cognition and emotion. Emotionalmotives, inasmuch as they can be retrieved by introspec-tion and memory, are cognitive data. Other relevant data,such as certain physiological or neurological activities, willbe considered neither behavioral nor cognitive.

Theories of choice can be classified according to the typesof data they recognize as valid, as well as by the types oftheoretical concepts they resort to. A theory isbehaviorist ifit only admits behavioral data, and if it also makes no use of

16


Prologue

cognitive theoretical concepts. (See Watson 1913, 1930 andSkinner 1938.) We reserve the term “behavioral” to theoriesthat only recognize behavioral data, but that make use ofcognitive metaphors. Neo-classical economics and Savage’s(1954) expected utility theory are behavioral in this sense:they only recognize revealed preferences as legitimate data,but they resort to metaphors such as “utility” and “belief”.Finally, cognitive theories make use of cognitive data, aswell as of behavior data. Typically, they also use cognitivemetaphors.

Cognitive and behavioral theories often have behavior-ist implications. That is, they may say something aboutbehavior that will be consistent with some behavioristtheories but not with others. In this case, we refer to thecognitive or the behavioral theory as a cognitive specifi-cation of the behaviorist theories it corresponds to. Onemay view a cognitive specification of a behaviorist theoryas a description of a mental process that implements thetheory.

2.5 Rationality

We find that purely behavioral definitions of rationalityinvariably miss an important ingredient of the intuitivemeaning of the term. Indeed, if one adheres to the behav-ioral definition of rationality embodied in, say, Savage’saxioms, one has a foolproof method of making rationaldecisions: choose any prior and any utility function, andmaximize the corresponding expected utility. Adopting thismethod, one will never be caught violating Savage’s axioms.Yet, few would accept an arbitrary choice of a prior asrational.

It follows that rationality has to do with reasoning aswell as with behavior. As a first approximation we sug-gest the following definition. An action, or a sequence ofactions is rational for a decision maker if, when the deci-sion maker is confronted with an analysis of the decisionsinvolved, but with no additional information, she does

17



not regret her choices.10 This definition of rationality mayapply not only to behavior, but also to decision processesleading to behavior. Observe that our definition presup-poses a decision-making entity capable of understanding theanalysis of the problems encountered.

Consider the example of transitivity of binary prefer-ences. Many people, who exhibit cyclical preferences, regretsome of their choices when exposed to this fact. For thesedecision makers, violating transitivity would be consid-ered irrational. Casual observation shows that most peoplefeel embarrassed when it is shown to them that they havefallen prey to framing effects (Tversky and Kahneman 1981).Hence we would say that, for most people, rationality dic-tates that they be immune to framing effects. Observe, how-ever, that regret that follows from unfavorable resolution ofuncertainty does not qualify as a test of rationality.

As another example, consider the decision maker’sautonomy. Suppose that a decision maker decides on anact, a, and ends up choosing another act, b, because, say,she is emotionally incapable of forgoing b. If she is surprisedor embarrassed by her act, it may be considered irrational.But irrationality in this example may be due to the intent tochoose a, or to the implicit prediction that she would imple-ment this decision. Indeed, if the decision maker knowsthat she is incapable of forgoing act b, it would be rationalfor her to adjust her decisions and predictions to the actualfeasible set. That is, if she accepts the fact that she is techno-logically constrained to choose b, and if she so plans to do,there will be nothing irrational in this decision, and therewill be no reason for her to regret not making the choice a,which was never really feasible. Similarly, a decision makerwho has limitations in terms of simple mistakes, failingmemory, limited computational capacity, and the like, maybe rational as long as her decision takes these limitationsinto account, to the extent that they can be predicted.

10 Alternatively, one can substitute “does not feel embarrassed by” for“does not regret”.

18


Prologue

Our definition has two properties that we find necessaryfor any definition of rationality and one that we find usefulfor our purposes. First, as mentioned above, it relies on cog-nitive or introspective data, as well as on behavioral data,and it cannot be applied to decision makers who cannotunderstand the analysis of their decisions. According to thisdefinition it is meaningless to ask whether, say, bees, arerational. Second, it is subjective in nature. A decision makerwho, despite all our preaching, insists on making frame-dependent choices, will have to be admitted into the hallof rationality. Indeed, there is too little agreement amongresearchers in the field to justify the hope for a unified andobjective definition of rationality. Finally, our definitionof rationality is closely related to the practical questionof what should be done about observed violations of clas-sical theories of choice, as we explain in the sequel. Assuch, we hope that this definition may go beyond capturingintuition to simplifying the discussion that follows.11

2.6 Deviations from rationality

There is a large body of evidence that people do not alwaysbehave as classical decision theory predicts. What shouldwe do about observed deviations from the classical notionof rational choice? Should we refine our descriptive theo-ries or dismiss the contradicting data as exceptions that canonly clutter the basic rules? Should we teach our normativetheories, or modify them?

We find the definition of rationality given above usefulin making these choices. If an observed mode of behavior isirrational for most people, one may suggest a normative rec-ommendation to avoid that mode of behavior. By definitionof irrationality, most people would accept this recommen-dation, rendering it a successful normative theory. Bycontrast, there is a weaker incentive to incorporate this

11 For other views of rationality, see Arrow (1986), Etzioni (1986), andSimon (1986).

19



mode of behavior into descriptive theories: if these the-ories were known to the decision makers they describe,the decision makers would wish to change their behavior.Differently put, a descriptive theory of irrational choice isa self-refuting prophecy. If, however, an observed mode ofbehavior is rational for most people, they will stick to iteven if theorists preach the opposite. Hence, recommend-ing to avoid it would make a poor normative theory. Butthen the theorists should include this mode of behavior intheir descriptive theories. This would improve the accuracyof these theories even if the theories are known to theirsubjects.

2.7 Subjective and objective terms

Certain terms, such as “probability”, are sometimes clas-sified as subjective or objective. Some authors argue thatall such terms are inherently subjective, and that the term“objective” is but a nickname for subjective terms onwhich there happens to be agreement. (See Anscombe andAumann 1963.) A possible objection is raised by the follow-ing example. Five people are standing around a well thatthey have just found in the field. They all estimate its depthto be more than 100 feet. This is the subjective estimate ofeach of the five people. Yet, while they all agree on the esti-mate, they also all agree that there is a more objective wayto measure the depth of the well.

Specifically, assume that Judy is one of the five peoplewho have discovered the well, and that she recounts thestory to her friend Jerry. Compare two scenarios. In thefirst scenario, Judy says, “I looked inside, and I saw thatit was over 100 feet deep.” In the second scenario she says,“I had dropped a stone into the well and three seconds hadpassed before I heard the splash.” Jerry is more likely toaccept Judy’s estimate in the second scenario than he is inthe first. We would also like to argue that Judy’s estimateof the well’s depth in the second scenario is more objectivethan in the first. This suggests a definition of objectivity

20

A Theory of Case-Based Decisions -...

Documents

Transcript of A Theory of Case-Based Decisions -...