The Mind under the Axioms: Decision-Theory Beyond Revealed Preferences (Perspectives in Behavioral Economics and the Economics of Behavior) 0128151315, 9780128151310

Table of contents :
The Mind Under the Axioms
Copyright
Dedication
Introduction
I.1 Axioms are not mindless
I.2 The limits of the behavioral turn in decision-theory
I.3 Working in the margins of the revealed preference paradigm
I.4 How to read and use this book
References
1 Cardinalism
1.1 The need for cardinal utility
1.1.1 Cardinalism and subjectivity
1.1.2 Allais’s cardinalism
1.1.3 Alternative hedonimetry
1.2 Utility functions as psychologies
1.2.1 Utility functions as primitives
1.2.2 Features of utility between ordinality and cardinality
1.2.3 The problem of loss-aversion
1.2.4 Standard neuroeconomics of utility functions
1.3 Informational and representational constraints
1.3.1 The informational basis of cardinal utility
1.3.2 Köbberling’s general result
1.3.3 Dispensing with representation theorems?
1.4 To the lab
1.4.1 Eyetracking rank-dependent utility
1.4.2 Does the lack of episodic experience lead to von Neumann-Morgenstern rationality?
References
Further reading
2 Incompleteness
2.1 Sources of incompleteness
2.1.1 Structural and cognitive sources of incompleteness
2.1.2 Incomparability and incommensurability
2.1.3 Preference for flexibility and incomplete preferences
2.1.4 Uncertainty and confidence about one’s preferences
2.1.5 On nudging incomplete preferences
2.2 Motives and models of incomplete preferences
2.2.1 Representing incompleteness by multiple-utility functions
2.2.2 Axiomatic interactions due to incompleteness
2.2.3 Status-quo bias and incomplete preferences
2.3 Incomplete preferences and incomplete beliefs
2.3.1 Bewley’s model
2.3.2 Symmetry between incomplete beliefs and incomplete preferences
2.3.3 Multiple-priors models and internal dialogs
2.4 Revealing incompleteness versus indifference
2.4.1 Choice-theoretic foundations of incomplete preferences
2.4.2 A revelation procedure of incompleteness versus indifference by means of choice-deferral
2.5 To the lab: temporal behavior of incomplete preferences
2.5.1 Measuring the degree of incompleteness through probabilistic deferral
2.5.2 Reaction times, indifference, and indecisiveness
2.5.3 Confidence and comparisons: what is inherited from tastes to preferences
References
Further reading
3 State-dependence
3.1 P3 and P4
3.1.1 Violations or intuitive pervasive behavior?
3.1.2 P3/P4 asymmetry and complementarity
3.1.3 Is state-dependence a remediable phenomenon?
3.2 What is a state-space good for?
3.2.1 Ontological fuzziness and common sense
3.2.2 States between pragmatics and metaphysics
3.2.3 Are states unambiguous observations?
3.2.4 Nonprimitive states
3.3 Hypothetical preferences
3.3.1 Modeling the basic intuitions of state-dependence
3.3.2 The representation of state-dependent preferences problem
3.3.3 The combination of local and global state-dependent preferences
3.3.4 The induced limits for the revealed preference paradigm
3.4 Drèze’s approach
3.4.1 Moral hazard as a potential revelator of state-dependence
3.4.2 Drèze’s model
3.4.3 The operationalization of the Reversal of Order criterion and virtual observations
3.4.4 Representation and separate identification
3.5 To the lab: can we observe state-dependence?
3.5.1 Health-states proxies and risk behavior
3.5.1.1 Experiment 1
Experimental template
3.5.1.2 Experiment 2
3.5.2 Measuring the impact of moral hazard on the nonadditivity of beliefs
References
Further reading
A conclusive remark: decision theory and the transition from the unconscious to the conscious
References
Index

Citation preview

The Mind Under the Axioms

The Mind Under the Axioms Decision-Theory Beyond Revealed Preferences

SACHA BOURGEOIS-GIRONDE Department of Economics II, University Paris 2, Paris, France Institut Jean-Nicod, Ecole Normale Supérieure, Paris, France

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-815131-0 For Information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

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Dedication

To my beloved wife and daughters, my most rational decisions. ------------

Introduction I.1 Axioms are not mindless In this book, we will deal with some aspects of the most well-known axiomatic systems of decision-theory: namely, expected utility theory (von Neumann and Morgenstern’s model) and subjective expected utility theory (Savage’s model). Only occasionally we will turn to the discussion of so-called alternative expected utility models, such as prospect theory. This can seem paradoxical for a book that appears in a series in behavioral economics. Prospect theory is often conceived as the birthmark of behavioral economics. We will try to convince the reader that it is not a counterintuitive or simple provocative move and that the most standard axiomatic systems in decision-theory raise a lot of behavioral and psychological issues as they stand, that the axiomatic work is in fact a work in formal psychology, and that the revision of an axiomatic system rather than the formulation of an alternative theory leads us to refine our view of the human mind. So we consider standard decision-theoretical axiomatizations as formal psychologies. But these formal psychologies do not stand in the void. They are connected with actual psychological capacities. They don’t just describe an ideal mind, and they don’t just prescribe either ideal mental operations that so-called rational minds ought to perform. We are not concerned with epistemic duties in this book but, really, with what it takes for a typical human mind to be like the mental structure the axioms delineate. Sometimes some strictures arise between these ideal delineations and what the typical human mind tends to be or how it operates. One way of labeling these strictures could be in terms of conflicts of intuitions. The question is then whether the axioms should accommodate these conflicts, internalize them, whether they should incorporate a phenomenology of conflict or continue to delineate a univocal mental life. Such an issue is beyond or besides behavioral economics and alternative nonexpected utility models as they have been developed and as a large community actively works in. However, they accompanied the infancy of these disciplinary changes. Early questioning of the intuitiveness of Savage’s axioms, of the theoretical depth of the Allais paradox, before the latter was accommodated in a purposefully designed behavioral framework, resorts to a seminal admission of a psychological relevance of classical decision-theoretical ix

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models. More exactly, the problem was raised in the mode of a possible cognitive adequacy between what these axioms normatively describe (such a phrase could sound as an oxymoron, but it is not; we are talking of an ideal description of the human mind) and their intuitiveness. This cognitive adequacy between axioms and the intuitive (in a Kantian, not a Kahnemanian sense) capacity of our minds is what this book is about. It selects a few themes for which cognitive adequacy between the most standard axiomatic systems in decision-theory and our intuitions seems to be under pressure: cardinalism, incompleteness of preferences, state-dependent utility. These themes have not especially interested (except for the first one to some larger extent) behavioral economists and behavioral decision-theorists. But they have triggered deep discussions in axiomatic decision-theory. We consider that these discussions have psychological import and our twofold aim is then (1) to make them sensible to the cognitive scientist or psychologist who is interested in understanding the current work in axiomatic decision-theory and (2), more presumptuously, to indicate to axiomatic workers how their work on these particular themes bear actual psychological, including experimental, implications. Aim (2) is far from obvious and generally admitted as a relevant one. An influential and clearly articulated reaction against behavioral economics and neuroeconomics has notably been given by Gul and Pesendorfer (2008) in their The Case Mindless Economics manifesto. The main argument of these authors is that cognitive and psychological data are not relevant for decision-theory and that decision-theory, conversely, makes no psychological or neurobiological predictions. This deserves a lot of qualification, but if the reaction was really expressed as we just did and if we did not look for more qualification, we would agree with it, because our aims fall outside its scope and because it sounds reasonable. But we need, on one side, to clarify what it would mean for some cognitive or neural data to be potentially relevant for decision-theory and, on the other side, to qualify what type of prediction, if any, decision-theory makes about cognitive or psychological processes. The claim of irrelevance of cognitive data (including neural ones) to decision-theory can be understood in different ways (we refer the reader to Cozic, 2012, for a finer and more distinct analysis (than ours) of this claim by Gul and Pesendorfer). Such data cannot confirm or disconfirm a decision-theoretical model. The presupposition is that these models by themselves do not make any predictions on actual psychological processes

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or states, so if they don’t, data won’t change them. This can be granted. Actually, it would be difficult to find how standard decision-theory has been affected by experimental data in psychology and neuroscience because, by definition, models that take this type of data into consideration are not standard. And, as we say, we focus on priority on standard decision models. This is where the difference between strictly behavioral economics or decision-theory and “cognitive economics” or “economics and psychology” (to name thus sometimes difficult to apprehend fields of research) lie. Standard decision-theory is linked to behavior. It makes behavioral predictions, if we read it in that sense. Violations of axioms of decision-theory such as the Allais or the Ellsberg paradoxes led to axiomatic refinements and to alternative models of decisions. The gain of these axiomatic revisions was, often, a gain in behavioral adequacy, if not in cognitive adequacy. Gul and Pesendorfer’s reaction can be interpreted, first, by a strict obedience to the revealed preference paradigm, for which the only acceptable data is observable choice-behavior. But accepting to limit the domain of predictivity of decision models in that way implies that we have a radical antipsychologist reading of the basic ingredients and constructs of decision-theory, which seems a too strong, or contrarian, position. It means that preferences, beliefs, the ability to comply with transitivity, completeness, independence, are just ways of speaking and that they are not actually preferences or beliefs in the mind of the individual but just dispositions to choose. In fact, we don’t even need them if they are metaphors or terms that are dissolved in behavior in a positivistic way. Why decision-theory should be encumbered by such psychological metaphysics? So, we could accept to some extent, if indeed we mainly confine our analysis to standard decision-theoretical models that data outside a choicebehavioral nature do not confirm or disconfirm these models, but we do not have to accept, and we do not, that they are dispensable terms and say nothing about actual minds. In what sense do they? There are, at least, two possible answers to this question. The first answer is the one we already suggested: Axiomatizations delineate ideal psychologies. The intuitiveness of these formal psychologies may, lead, in case of internal strictures, to axiomatic revisions. This is the case with the issues of incompleteness and state-dependence (our Chapter 2: Incompleteness and Chapter 3: State-dependence) in particular. It was the case with von Neumann and Morgenstern’s independence axiom and Savage’s Sure

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Thing principle at the beginning of standard decision-theory and before the appearance of behavioral economics. A second answer is that the basic axiomatic structures acquire some psychological dimension as soon as they are represented. A central, if not the central work in axiomatic decisiontheory is to obtain a representation theorem. A representation theorem is precisely what connects an axiomatic characterization of decisiontheoretical terms (such as preferences and beliefs) to (1) what we call “functional terms” and (2) conditions on observable data. The central functional terms in the representations that ensue from the standard axiomatic structures we focus on are “utility function” and “probability function.” Representation theorems, under their usual interpretation, state that an individual whose preferences and beliefs can be characterized by the axiomatic system under scrutiny can be functionally redescribed as if she were maximizing a utility function and complying with probability theory. Representation theorems do not give a psychological priority either to the ideal psychology characterized by the axioms (in fact, they are mute about it) or to the functional terms they introduce. They are not by themselves compelled to psychological realism with respect to the functional redescription they make possible. They just introduce another layer of psychological interpretation. One of the main questions that we will address, from Chapter 1, Cardinalism, on, is the informational continuity between axiomatic characterizations in themselves and their functional representations, with respect to their respective psychological implications. The connection of axioms, representations, and observable data is a very intricate issue. Once we have reached a functional representation of axiomatic structures, it is tempting to use it in order to rationalize observable data. Representation theorems do not indicate, in fact, what data are exclusively admissible. They are bound to a choice-theoretical paradigm in the sense that an individual who visibly violates, say, transitivity or independence cannot be redescribed as a utility maximizer, but it does not stipulate that only choices either can attest to such violations or are sufficient to confirm the theory. Saying the latter is, we deem, an overinterpretation of what decision-theory actually prescribes when we lend it a prescriptive role. It then prescribes what choices should be like, but it does not mean that, when they are such, the theory is confirmed. If we accept this type of confirmation by choices, why should not we also accept confirmation by nonchoice and nonbehavioral data (reaction times, eye movements, neural activity) as soon as they are clearly correlated to

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choices? We therefore accept the thesis, which was defended on other grounds by Hausman (2008), that subjective and psychological elements in decision-theory, whether they are implicit in the axiomatic structures or explicit at the level of functional representations, do not stand in a biunivocal relation with choices. On the other hand, we accept a form of homeomorphism between decision-theoretical models and human psychology. We borrow this term, as well as a good deal of the sense the author gives to it, from Wakker (2010). Wakker distinguishes between paramorphic and homeomorphic models. A paramorphic model does not have to stipulate any kind of correspondence between its theoretical constructs and reality. It is, of course, possible and perhaps safer to interpret and use decision models in this way, especially if the predictions we make on the basis of this model relate to a restricted set of phenomena. If decision models are there to predict choices, there is no need indeed to postulate any formal analogy between theoretical constructs such as preferences and beliefs, let alone particular axioms characterizing them, and choices. Preferences are then mere dispositions to choose. Representational notions such as utility and subjective probability help to rationalize those choices, but the question is whether they are committed towards a particular psychological interpretation of utility-maximization. Behavior, functional representation, axiomatic structures on preferences all remain separate levels, and the functional or implicit psychology that one can associate with the two latter levels is dispensable. Differently, we can continue to associate some level of predictivity to a homeomorphic level, but it relies on the postulate (that may be supposed to point to the source of its predictivity) that the terms of the model match underlying psychological data and processes. We want the terms of the model to have psychological plausibility. How this can fit with work within a revealed preference paradigm or impose conservative extension thereof is a major theme of this book. We would like to risk an analogy. Axiomatic systems deliver a sort of ideal syntax of decisions, but it is a shortcut or an excessive restriction to consider that its unique and even prevalent semantic domain of interpretation is provided by choice-behavior. Chomsky hesitated between adopting a realist and a nonrealist position syntax. “Syntactical realism” means that linguistic forms correspond to some psychological reality. Nonrealism in syntax assumes that the theoretical constructs of a theory of language are simply formal, have no connection with minds, and can simply help to make linguistic behavioral predictions. We definitely adopt a realist

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position by adopting as a working hypothesis in this book that decisiontheoretical syntaxes describe deep psychological structures and that a serious work in experimental decision-theory is to find out the conditions under which it can be revealed, while maintaining a revealed preference semantics, not exclusively in terms of choices and going conservatively beyond when necessary. The analogy goes further. One of the usual criteria of compliance with theoretical syntactic structures in a Chomskyan experimental methodology makes use of so-called intuitions of grammaticality (Chomsky, 1957, 1965; Cowart, 1997). Native speakers develop an “innate” sense of grammatical errors. It is far from clear that the same sense of error in the case of an axiomatic violation, when it arises, when the individual is faced with such a violation and a conflict of intuitions may arise, points to innate decision-theoretical structures. But this metacognitive phenomenon is in itself worthy of being accounted for in decision-theoretical terms. It can be in terms of a competition between models and hence axiomatic structures, in view of the best description of a human decision-maker (rather than, for instance, a machine), each model not necessarily capturing the whole psychological inner life of the subject.

I.2 The limits of the behavioral turn in decision-theory Behavioral economics, in its usual realizations, can summarily be defined as the integration of psychological parameters into models. It is quite different from extracting psychological implications from their axiomatic basis and submitting them to intuition and empirical tests, even though there is no contradiction between the two approaches: once a counterintuitive implication or a systematic, counter to the theoretical prediction, behavioral pattern is observed, the original axiomatic structure may be subject to revision. Prospect theory has been considered as the first and foremost behavioral model in decision-theory and beyond. It is interesting to briefly and schematically recall its evolution in order to understand the successive steps that also made it a paradigmatic homeomorphic model, which is not the case of most behavioral models in economics, for reasons we point out here. The original version of prospect theory (Kahneman & Tversky, 1979) incorporated in a descriptive model a series of psychological features: asymmetric risk attitudes in domains of gains and losses, loss aversion, and subjective distortion of probabilistic information. The main feature was

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that, instead of understanding preferences as bearing over final outcomes, they had to be understood as bearing on the prospect of changes relative to a current reference point. These features stood in contradiction with the standard psychological implications of expected utility, but, at this earlier stage, they could not represent a real theoretical alternative to it. External psychological features and internal psychological implications, the latter purely theoretical perhaps, had yet to meet. This is the condition for homeomorphism in the intended sense. What happened with cumulative prospect theory (Tversky & Kahneman, 1992) is not yet a fully satisfying step in that direction, even though it provided prospect theory with a deeper theoretical grounding. The psychological features of the first version were assimilated in a nonexpected utility model, namely rank-dependent utility, solving some of its theoretical shortcomings and providing it, consequently, with a valid functional representation. It then becomes possible to rationalize the distortion of probability, by means of a cumulative utility function, and the asymmetric risk attitudes in losses and gains, by means of a dual signed-valence utility function. Through functional representations of this sort, we can connect psychology and theory at a certain level, through an operational representation but not yet at the level of a corresponding axiomatic structure. This gap is filled by Wakker and Tversky (1993) through their axiomatization of prospect theory under uncertainty (prospect theory under risk is axiomatized by Chateauneuf & Wakker, 1999, in a subsequent exercise). Wakker and Tversky (1993) paradigmatically exemplifies a homeomorphic approach. Its axiomatization relies on the so-called tradeoff consistency approach, developed by Wakker (1989). It gives conditions under which value differences between prospects can be inferred from preferences. Given preferences that apparently violate expected utility theory—for example, h11, 20i is preferred to h10, 21i (if state 1 occurs, 11 or 10 is obtained, if state 2 occurs, 20 or 21 is obtained), but h30, 21i is preferred to h31, 20i under the same contingencies—a psychological explanation of this behavioral pattern is that the decision-maker performs trade-offs according to the psychological saliency of the concerned amounts. Conditions on the consistency of these trade-offs are spelled out by the authors, and they happen to directly connect the axioms to plausible psychological processes underscoring those apparent violations of expected utility theory. Some comonotonicity conditions, furthermore, relate to expected utility theory and cumulative prospect theory, affecting what value differences can be jointly inferred from these

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two models. Interestingly, the characterization of an axiomatic link between expected utility theory and cumulative prospect theory does not entail that we should have the same interpretation of their functional representations, which leaves intact their respective analytics. We can perfectly imagine individuals, ourselves for instance, who perform the trade-offs formalized at the axiomatic level and whose behavior could be rationalized partially by an expected utility function and more completely by a cumulative prospect theory functional. Alternative representations of preferences offer leeway in granularity, in diversity, in the choice of ordinal or cardinal aspects of utility (as we will analyze in Chapter 1: Cardinalism) to account for behavior and can yet induce the same underlying preference structure. Wakker and Tversky’s axioms lay out an abstract psychological model, that is, or not, intuitively and empirically validated, but that, for the matter, directly resonates with a widely evidenced psychological model of decision-making. This is one of the main points we will develop. Decision-theory is to a large extent foreign to decision-making sciences. Two research communities, on allegedly the same topic, work under completely distinct paradigms and aims. It is not simply that one is normative or theoretic and the other descriptive and experimental. Both can be normative or not, descriptive or not, etc. Neither is it that the first is not interested in psychological processes underlying decision-making and the second is. As we have already argued, axiomatic decision-theory formalizes possible psychological processes, even when it is not its primary purpose to do so. We will therefore think, throughout this book, about ways of connecting the hard core of decision-theoretical models and the testability of their ideal psychological implications. Our first move is thus not to integrate “external” psychological factors and parametrize decisiontheory accordingly. Only when psychological features of prospect theory were incorporated as “psychological implications” at the axiomatic level could cumulative prospect theory, the preeminent behavioral decisiontheoretical model, be integrated and accepted (to some extent) in decision-theory. So we do not think that we are developing a series of conceptual and psychological reflections from which decision-theorists could feel alienated. We are rather interested by a countermove in the field of behavioral economics, one that precisely consists in questioning the observability and testability of psychological and behavioral implications of formal decision-theoretical models, as they stand, and that favor

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revisions under strict criteria of intuitiveness, representability, and testability. It also means, conversely, that we are using decision-theory as an investigative tool in theoretical and experimental psychology, for the best use of decision-scientists at large. Behavioral decision-theory (and behavioral economics more generally), we deem, have invested most of its efforts in increased behavioral rationalizability through parametrized utility functions. To some significant extent, this move toward data-fitting has jointly alienated behavioral decision-theory from its axiomatic standard work and from psychology itself. A few instances, besides prospect theory and its eventual axiomatization, avoid this double pitfall, which amounts to confining behavioral theory within the bounds of an interaction between data and a functional curve. We can mention Regret Theory (Loomes & Sugden, 1982, for the first version), as a constant effort, until today, to combine psychological realism and axiomatic foundations. But what cumulative prospect theory and Regret Theory have in common, which may explain their successful compliance with the approach we are vindicating, is that they started from observable violations of behavioral axioms in decision-theory: the transitivity and independence axioms. From there, these homeomorphic models developed, trying to keep as tight as possible the conceptual links between observability (as a starting point then), axioms, representations, and a psychological account of the phenomenon at stake. Missing one of these links prevents behavioral decision-theory from being really acceptable by both axiomatic decision-theorists and cognitive scientists. The endeavor is all the more complicated when the observations likely to trigger a behavioral turn are not available. Such an absence can be due to too stringent constraints of the choice-theoretical revealed preference paradigm in order to obtain these observations or to the presupposition that some of the postulates of decision-theory are not testable. This may explain why, unlike transitivity and independence, axioms such as completeness, continuity, or postulates P3 (state-independence) and P4 (absence of moral hazard) (both axioms warranting the separability of beliefs and preferences) in Savage’s framework have not been the source of behavioral turns in decision-theory. We examine this situation in this book and account for some of the most important alternative models proposing to relax these axioms, as long as these models compel intuitions, express psychological implications, and call for an extension of observability criteria in decision-theory.

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I.3 Working in the margins of the revealed preference paradigm Constraining the semantics of a decision-theoretical syntax, in terms of choices revealing preferences, supposes that we also constrain the respective definitions of choices and preferences so that a correspondence between them can obtain, which is far from intuitive and in fact not always necessary. It is indeed possible that preferences are not revealed by choices, unless we adopt an oversimplified view of preferences, that choices do not reveal preferences, but that something loosely connected with them is a choice procedure, and, again, that preferences are revealed by observational data, which are more or less correlated with choices but which are usually discarded in a usual choice-theoretical revealed preference paradigm. Let’s envision these distinct possibilities. Decision-theorists usually model the agent’s preferences over a set of options, but the set of options over which the preference of the agent applies is modeled from the point of view of the theoretician, not the agent. It has an objective sense, and the presupposition is that the agent sees it the way the theoretician does. This structural assumption is too strong a constraint in some contexts wherein, precisely, we can assume that the agent cannot see the options as the modeler does. So we are not just making a point about perceptual relativity or a point about taking into account framing effects involving subjectivity, intentional contexts, or editing differences in granularities and partition at the level of the option sets (all points well remarked and many times discussed, as by Rubinstein, 1991, Ahn & Ergin, 2010, or Bourgeois-Gironde & Giraud, 2009), but we are raising the issue of an assumed assimilation of viewpoints between the modeler and the modeled individual. Rubinstein and Salant (2008), for instance, give the example of a 2 3 2 coordination game with action space {a,b} for the two players. Assuming that the two players subjectively rank the outcomes of the game (a,a) g (b,b) g (a,b) B (b,a). What each player can reveal through her choice-behavior is (a,a) g (b,a) and (b,b) g (a,b), but their behavior cannot reveal that (a,a) g (b,b) because none of the players never chooses between those two options (they are in the diagonal of the game matrix). Here, if we assume the complete order of these preferences, we cannot assume it corresponds to an observable choice-theoretical basis, and conversely. But the more general point is the robustness of the postulate that the model of the economic agent corresponds to the structural assumptions

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we make on her option set. It is not simply, we said, a matter of framing effects; neither is it only a matter of psychological awareness. For instance, in the standard model of expected utility, we define the preferences of the individual over a convex set of lotteries, but we cannot assume that the individual is aware that she is deciding over a convex set, let alone the convex set the theoretician or the experimenter has in mind. This point is addressed in depth, from a topological perspective, by O’Callaghan (2017). Our particular point is that a theoretical presupposition that the option set is objectively defined—that we can endow the individual with the perceptual and awareness assumptions of the modeler—is what facilitates the work of the choice-theoretical paradigm as a sound way of revealing preferences but that in many cases this presupposition is unwarranted. To be granted, we would have to define an endogenous (to the preferences of the agent) option set. But if we take it, more conventionally, as given, as an objective fixture of the world surrounding the decision-maker, discrepancies between the option set and the preferences must arise, and we have to admit mental (vs. purely behavioral as supposedly revealed by choices) preferences in our decision-theoretical ontology. Choices are said, in choice-theoretical parlance, to be rationalized by a preference relation when they respect certain typical conditions; for instance, when the choice function defined over a relevant choice-set respects a condition of independence to irrelevant alternatives and when what is selected by choices maximizes that preference relation. But we cannot a priori assume that what is thus maximized, which we define as the rationalizing preference relation, is tantamount to the mental preferences of the subject rather than something else uncorrelated to preferences. What choices reveal or correspond to can be a choice procedure, the application of an algorithm, that only partially (or not at all, rarely completely) capture preferences. For instance (we will discuss at greater length this type of issue in Chapter 2: Incompleteness), Manzini and Mariotti (2007) define a two-stage procedure over an option set that consists, first, in eliminating the dominated options in the set and, second, in ranking the undominated options by a complete and transitive preference relation. Such a procedure does not immunize a choice function from a dependence to independent alternatives and therefore is not rationalizable by a preference (mental or not), although it is a natural procedure to follow. Complex procedures can contain a preference relation and rationalize choice, but, in that case, the preference relation does not do it by itself. Choices, in that case, cannot reveal that preference relation. As we

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will see, this will be mainly the case when, furthermore, we admit some incompleteness at the level of the preference relation itself. In different ways and for different motives, the three themes that the successive chapters of this book will investigate bring us to these limits of the revealed preference paradigm, pointing to a discrepancy between the behavioral level (and the assumed behavioral preferences revealed through choices) and psychological preferences that we need to admit in order to model, in a meaningful way, a human rather than a mechanical or strictly procedural decision-maker. Let’s briefly state how and why. Chapter 1, Cardinalism, deals with the old issue of cardinalism. Perhaps the decision-maker gives a subjective significance and value to her preferences. How will the mere act of choice reflect this? And how can it be possible to (and why ought we) rationalize choice-behavior by means of a cardinal utility function when what can be induced is just an (often partial) order among options? Cardinalism is the acceptance (and demonstrability) that the notion of differences in preference or intensity of preferences can receive a utility representation. These natural requirements of cardinalism go beyond the cardinal utility function derived from von Neumann and Morgenstern’s expected utility representation. The latter is relative to a certain admitted structure and types of objects forming the options set (lotteries), showing precisely that there are bottom-up constraints from the definition of the options set to the type of subjective life it is allowed, in a given decision-theoretical structure, to endow the individual with. So in Chapter 1, Cardinalism, we discuss the threefold tension between the subjective and psychological significance of preferences, the operational use of a utility function (delimiting itself in its own psychological characterization of the individual that can be aligned or not with the intrinsic nature of the individual’s preferences), and the choicebehavioral basis that can be rationalized by a preference relation. Rationalization of choices by means of the preference relation or, independently and differently, by means of a utility function and representation of a preference relation by a utility function constitute the normal links between these different levels. Normally, too, the links are what we call “informationally conservative.” On the basis of a sample of choicebehavior, a behavioral preference relation is revealed (one that rationalizes those choices), and its representation by a utility function should not grant it a subjective property that conflicts with its rationalization of behavior. Cardinal preferences, to fit into this informationally conservative framework, therefore require an extended choice-theoretical informational

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basis, one that admits an elicitation of preference intensities and that will be subsequently informationally consistent with a cardinal utility measure. We discuss this possibility and more generally the admissibility of introspective data in decision-theory. Chapter 2, Incompleteness, discusses the issue of incomplete preferences. The same strictures between the axiomatization of a preference relation, a utility representation, and a choice-theoretical basis are met again. In that chapter, we discuss the main cognitive and behavioral sources that can motivate the weakening of the standard axiom of completeness and the possible extension of the revealed preference paradigm in order to accommodate incomplete preferences. We distinguish between incompleteness due to objective features of options and due to epistemic uncertainty. In the latter case, the individual is uncertain about her preferences, and this can transpose behaviorally into a deferred choice that reveals a preference for flexibility that then has to be axiomatized in relation with an (at least temporarily) incomplete preference relation. But in some other cases, the individual is certain that her preference is incomplete, and preference for flexibility does not offer a behavioral basis under which try to reveal that incomplete preference relation. Such a behavioral basis is not informative and relevant vis-à-vis the actual epistemic state (one of certainty) associated with the preference relation (an intrinsically or definitely incomplete one). In that situation, the purely behavioral discrimination between incompleteness and indifference is even harder than in the former one. We envision some experimental extensions of the choice-theoretical informational basis that can point toward a solution to this still open problem. Chapter 3, State-dependence, is dedicated to another problem, statedependence, at the junction of choice, preferences, and utility. It touches the core of the revealed preference paradigm in yet another way than the two former issues of cardinalism and incompleteness. State-dependence of preferences and utility prevents a separate elicitation of beliefs and preferences and contradicts the separability of probabilities and utilities, standardly performed in Savage’s framework, by the theoretical division of labor between axioms P3 and P4. We discuss two main models in the decision-theoretical literature that address this problem, by admitting state-dependence and defining a separable (between beliefs and preferences) utility representation: Karni’s and Drèze’s. Karni’s approach, however, is based on the admission of hypothetical preferences that cannot be based on observed behavior because they rely on comparisons of

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outcomes across states that cannot be jointly realized (somewhat as in the game-theoretical diagonal in Rubinstein and Salant’s coordination game example). What the individual mentally ranks that would serve to dissipate the collusion between beliefs and preferences cannot be part of the observational basis that normally separates (in Savage’s framework) beliefs and preferences in a strictly behavioral procedure. Drèze’s approach, differently, is in principle (but with a high level of complexity) compatible with a revealed preference paradigm, in the sense that state-dependent preferences can be revealed and neutralized, and allows for a subsequent separate elicitation of beliefs and preferences. In Chapter 3, Statedependence, the problems of measure (central in Chapter 1: Cardinalism) and observational discrimination (in Chapter 2: Incompleteness) submit the testability of models to serious pressure. We try to define indirect ways of identifying state-dependence preferences in the final (To-the-lab) section of that chapter.

I.4 How to read and use this book The aim of this book is to bridge the gap between axiomatic decision-theory and experimental psychology of decision, precisely in places where the canonical revealed preference paradigm is insufficient or unsatisfactory in terms of a plausible informational continuity between choices, preferences, and a functional representation of the latter. By the choice of topics and the way they are dealt with, we do not offer the reader a textbook. It is based on a selection of problems rather than on a systematic effort to introduce decision-theory. Yet we hope that the very first sections of each chapter will sufficiently clarify the standard background, in the contemporary decision-theoretical literature, from which these problems arise. The student in decision-theory may learn how some of the most theoretical work she is exposed to can be discussed in psychological terms and sometimes prolonged in experimental perspectives. Each chapter is closed by a to-the-lab section in which we sketch actual or possible experimental protocols in connection with the difficulties raised in the preceding theoretical sections. The student in psychology and cognitive sciences will find an informal discussion of models she is usually not inclined to consider, either by lack of familiarity with the formalism or because she cannot perceive the relevance of what is formalized there for her own investigation of the human mind. We hope that psychologists will come to appreciate how deeply in theoretical

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psychology these axiomatic models are in fact cut out, overcoming the all too common and uneducated prejudice that, ideal as they are, have nothing to do with real human behavior and mind. In decision-theory, the author of this book considers himself, at best, an in-outsider. Or a forager. Trying to grasp what intuitively lies beneath axiomatic systems and bring it back to his home community, cognitive sciences. The counteroffer he makes to wherefrom he borrows may be unbalanced. No alternative axiomatization or modeling is offered in the bounds of this book. The idea is not to pile up models over models but to deconstruct some models in discursive and conceptual glosses, which are far too talkative to be part of the usual (and required) streamlined writing style in decision-theory. We simplify theoretical papers, selected on the main criterion that they reflect our main problem as defined in the previous sections of this introduction, and we try to uncover their psychological implications when they are far from obvious. This is the method and style we have followed in order to build potential bridges and a partially common language between decision-theory and experimental psychology. Each chapter is independent. Each section is often organized around a small number of references on a specific problem, sometimes only one, if it is, at least in our eyes, the most central for the issue at stake. This leads to the effect that—in line with the fact that we have not written a textbook—each section most often displays an independent notation. This is justified by local dependence on literature but also because, as the models discussed deal with specific questions, they do not necessarily refer to the same concepts or modeling of the same concept in as other sections. No preliminary is required to understand what this book talks about, but its reading should be accompanied by the study of a real introduction to decision-theory, such as, in particular, Gilboa (2009), the classic Kreps (1988), and Wakker (2010) for reasons we have explicitly indicated here.

References Ahn, D. S., & Ergin, H. (2010). Framing contingencies. Econometrica, 78(2), 655695. Bourgeois-Gironde, S., & Giraud, R. (2009). Framing effects as violations of extensionality. Theory and Decision, 67(4), 385404. Chateauneuf, A., & Wakker, P. (1999). An axiomatization of cumulative prospect theory for decision under risk. Journal of Risk and Uncertainty, 18(2), 137145. Chomsky, N. (1957). Syntactic structures (10th ed.). The Hague: Mouton & Co. Chomsky, N. (1965). Aspects of the theory of syntax. MIT Press.

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Cowart, W. (1997). Experimental syntax: Applying objective methods to sentence judgements. London: SAGE Publications. Cozic, M. (2012). Economie « sans esprit » et données cognitives. Revue de philosophie économique, 13(1), 127153. Gilboa, I. (2009). Theory of decision under uncertainty (Vol. 1). Cambridge: Cambridge University Press. Gul, F., & Pesendorfer, W. (2008). The case for mindless economics. The foundations of positive and normative economics: A handbook (Vol. 1, pp. 342). Hausman, D. (2008). Mindless or mindful economics: A methodological evaluation. The foundations of positive and normative economics: A handbook (pp. 125155). Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decisions under risk. Econometrica, 47, 278. Kreps, D. M. (1988). Notes on the theory of choice (Underground classics in economics). Westview Press Incorporated. Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice under uncertainty. The Economic Journal, 92(368), 805824. Manzini, P., & Mariotti, M. (2007). Sequentially rationalizable choice. American Economic Review, 97(5), 18241839. O’Callaghan, P. H. (2017). Axioms for parametric continuity of utility when the topology is coarse. Journal of Mathematical Economics, 72, 8894. Rubinstein, A. (1991). Comments on the interpretation of game theory. Econometrica: Journal of the Econometric Society, 909924. Rubinstein, A., & Salant, Y. (2008). Some thoughts on the principle of revealed preference. The foundations of positive and normative economics: A handbook (pp. 115124). Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297323. Wakker, P. (1989). Transforming probabilities without violating stochastic dominance. Mathematical psychology in progress (pp. 2947). Berlin, Heidelberg: Springer. Wakker, P. P. (2010). Prospect theory: For risk and ambiguity. Cambridge University Press. Wakker, P., & Tversky, A. (1993). An axiomatization of cumulative prospect theory. Journal of Risk and Uncertainty, 7(2), 147175.

CHAPTER 1

Cardinalism 1.1 The need for cardinal utility 1.1.1 Cardinalism and subjectivity In economics, thanks to the concept of utility, we can compare not only between apples and between apples and pears but also between apples and clothes and apples and poetry or a good sleep. On the other hand, it is a controversial issue as to whether we can compare between such several comparisons. To do so, we would need a precise scale and measure of subjective utility that would in fact be objective enough to apply across individuals. What permits such inter-individual comparisons of utility is— the general saying goes—that the utility function is not subjective after all, that it does not capture real subjective aspects of the individual utility but nevertheless yields an objective indication of what this subjective utility is. Freed of essentially individual aspects in its construction, the standard (i.e., not significant in subjective terms) utility functions that represent individual preferences can be exported to social choice-theory and concerns. Many issues to be disentangled will constitute the successive topics of this chapter and determine its progressive inclination not only toward cardinalism but toward a significantly subjective admission of this doctrine. The first issue is to remember a clear definition of what counts as a cardinal utility function and what its relationship with respect to the subjective experience of utility is. Because the two notions are not equivalent (Section 1.1.1). The second issue is whether we can adopt a subjective conception of a cardinal utility function and yet pursue the significant goals that an apparently more parsimonious ordinal function achieves in decision-theory and economics (the rest of Section 1.1). Finally, we can wonder how sound the theoretical foundations of decision-theory can be when we adopt such a cardinalist and subjectivist point of view (Sections 1.2 and 1.3 will dwell on these issues). In particular, can we obtain a representation theorem—that is, the main conceptual link between preference orderings and utility—when we accept, in our informational basis a cardinal (and subjective) notion of preference intensities (Section 1.3)? Or should we and can we dispense in decision-theory with representation The Mind Under the Axioms DOI: https://doi.org/10.1016/B978-0-12-815131-0.00001-8

© 2020 Elsevier Inc. All rights reserved.

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theorems as foundational moves of the discipline (Section 1.3.3 in particular)? Finally, in our “To-the-lab” experimental Section 1.4, we suggest protocols that tackle psychological issues motivating the resort to a cardinal subjective utility function and the admissibility of introspective data in relation to a choice-based decision-theoretical paradigm. The first ambiguity to dispel is that the standard von NeumannMorgenstern (vNM) utility function is cardinal, not ordinal. To recall, the expected utility hypothesis that these authors have developed means that an individual having some preferences and exerting some choices on a space of risky objects (a set of lotteries) and satisfying a series of sufficiently intuitive axioms (see Fig. 1.1) can be described in terms of the maximization of the expected value of a utility function that is cardinal. It is cardinal in the sense that it is unique up to a positive affine transformation. It means that the properties we assigned to the preference relation represented in this way can be preserved through changes of interval scales (the “up to positive affine transformations” of an initial function). By contrast, an ordinal utility function is unique up to an increasing monotone transformation. A cardinal utility function is then by default ordinal, but the

Figure 1.1 vNM axioms.

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reverse is not true. A cardinal utility function is more demanding in terms of its informational basis, since it supposes that there are possible valuations of options on a cardinal space and not simply an ordering of these options. It more exactly supposes that the ordering of the options preserves the cardinal valuation of options and that we can continue to perform comparisons of preferences themselves in the form of “I prefer x to y more than I prefer w to z” [(xky) . (wkz)], implying that the intensity of preferences is preserved under the cardinal utility function (which it is not under a strict ordinal function). Cardinalism, in this sense, means that we do not simply order options by a preference relation (which, with no more informational basis, yields an ordinal utility function) but that we can rank the instances of that preference relation themselves. The main difficulty is not to distinguish cardinality and ordinality but to agree on what we call a “cardinal utility function.” The problem lies in the extent to which we want the cardinal utility function that represents the preferences of an individual to reflect her subjective experience. A classical statement in economics (see Baumol, 1951, 1958) holds that the vNM utility function has nothing to do with the neoclassical utilitarian idea that utility is the measure of the subjective satisfaction (let alone welfare or happiness) of the individual when she expresses her preferences through her choices. Suppose that an hedonimeter could measure degrees of satisfaction and that we obtain an objective measure of that subjective experience, it still would have nothing to do with what vNM utility measures or indicates. If this position is to be accepted, it would amount to saying that vNM utility is an indirect index of subjective utility, not a measure of it and that it therefore conveys richer information than ordinal utility functions. Many applied economists understand and use vNM utility in that way. Yet this position leaves an impression of ambiguity and dissatisfaction, and we may want a vNM utility function to have more subjective significance and correlate more directly with the experience of the decisionmaker. Ng (1984) identifies a psychological postulate that prevents a more subjective interpretation. It is worth analyzing Ng’s contribution in some details, as it forms a paradigmatic example of how axioms and the provability of a utility representation encapsulate implicit “psychologies,” that is, underlying views of the mind. Ng points out that the assumption of infinite sensitivity—the perfect ability to discriminate between the finest levels of subjective experience—is implicitly contained in vNM axioms. If we add to that unmodified list of axioms a psychologically more realistic

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postulate of imperfect experiential discrimination, we can grant the vNM utility function the subjective significance that classical utilitarianism associates with the very concept of utility. Classical utilitarianism suggests that the human perceptual apparatus is endowed with limited discriminative capacities but that it is possible to calibrate minimal perception thresholds and construct units of utility on this minimum. This is, of course, a postulate subject to objection and experimentation, but we postpone this discussion for now. Minimal sensitivity to differences is conceived in terms of maximal indifference between two stimuli. For Ng, a cardinal utility function acquires subjective significance if its unit of measurement is thus calibrated and if, by definition, it relies on a postulate of finite sensitivity. Finite sensitivity immediately implies that the indifference relation is no more transitive. In order to preserve vNM axioms, we then must differentiate and articulate between two levels of preferences and their corresponding axiomatic levels: one in which the indifference relation is transitive, supposing perfect discrimination, which is an implicit or ideal preference relation (the one modeled by vNM axioms), and one in which we accept finite sensitivity and that is attuned to the perceptual apparatus and the experience of the individual. The link between the two levels can be expressed in this way: ’x; yAL ða set of lotteriesÞxky2uðxÞ  uðyÞ $ 0;

yhx2uðyÞ  uðxÞ . k

xhy2u(y) 2 u(x) # k (where k is a positive constant corresponding to the intended perceptual threshold). . is the strict preference relation (no indifference), h signifies both preference and indifference, and k is the underlying, not perceptually restricted preference relation subject to vNM axioms as they stand. Under k, “x is preferred to y” means that x can be an infinitesimal utility increment to y or, when it is not the case, indifferent to y. But under h—the alternative preference relation axiomatized by Ng—finite sensitivity implies that x is preferred to y if the difference of utility associated with x and y is superior or equal to a constant k (the level of utility above which the utility of the two options can be discriminated). This level k is determined by applying the strict preference relation . to y and x (in this direction): “y is strictly preferred to x” means that the utility of y can be perceived as above the utility of x and then that that difference lies beyond the constant k. The constant k measures the utility difference of maximal indifference—which is the reference point needed to build a utility function grounded in subjective experience. It therefore supposes

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that we axiomatically take into account the finite sensitivity of real perceptual systems. This is not the case with vNM axioms from which we can deduce the following statement (an implicit axiom in a vNM framework): (1) ’r,x,y,zAL, (rBx and z.y) . (1/2r; 1/2z)g(1/2x; 1/2y) To clarify: B and g respectively denote the usual indifference and preference relations (the ones that we have previously called “underlying” above and that, precisely, imply infinite sensitivity). If we combine a strict (in the sense that no perceptual threshold is involved in the application of that relation) indifference between r and x with a strict preference . (including the constraint that this preference lies above a constant k) between z and y, and if we divide the chances by half (or any factor) and combine these outcomes (r,x,y,z) in composed lotteries, the preferred option under . continues to be intrinsically preferred (with no restriction due to the constant k through which . is construed) when it is mixed with a strictly indifferent outcome. It is an intuitive consequence of vNM axioms that an indifference can be outweighed by a yet perceptually constrained preference and yields an unrestricted (standard vNM) preference. By this axiom, the idea of limited sensitivity is lost, as far as division of chances and mixtures of lotteries are concerned. But we can see that, if the division of probability of the outcomes is indefinitely applied iteratively, only the utility of the outcomes makes a difference in the end, and the fact is that they can do so under the scope of an underlying (infinitely sensitive) preference relation, even though the initial preference is constrained by a threshold k. To avoid this self-defeating implication, Ng proposes an alternative formulation of statement (1): (10 ) ’r,x,y,z A L, (rBx and z.y) . (1/2r; 1/2z) . (1/2x; 1/2y) In (10 ) the implication does not change the nature of the preference concerned and holds only for the k-restricted one, .. What prevents the argument from collapsing in turn is the fact that the axiomatization of that preference relation departs from the vNM axiomatization of h on a crucial point. It is the case, in the vNM framework, that if the probability p is superior to probability q and w and b are respectively the worst and best outcomes of a set of alternatives, then we can say that p $ q 2 (pb; (1 2 p)w) h (qb; (1 2 q)w); the first lottery dominates the first. If we keep this axiom intact, we miss the difference between (1) and (10 ). We need to restate it in that way: ðpb; ð1 2 pÞw Þ:scnapðqb; ð1 2 qÞw Þp . q

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The implication is one-way: from the preferential discrimination of the composed lotteries to the superiority of probability p over q. In other terms, we cannot divide indefinitely iteratively the probabilities and continue to obtain a preference order between the mixed lotteries. The probability division (and, incidentally, one can speculate the perception of probability differences themselves) is constrained by the finite sensitivity of utility differences. The standard interpretation of vNM axioms implies or relies on a presupposition of infinite sensitivity. This limits the expressive and informational power of the cardinal utility function that can be derived from it. If the locus of this presupposition is identified and an alternative preference relation involving a constant perceptual threshold is instead presupposed, the vNM axiomatic structure can be maintained and from it a cardinal utility function with subjective significance derived. This has important consequences in economics, such as the fact that diminishing marginal utility (the concavity of the utility function in the domain of positive rewards, as well as the convexity of the utility function in the domain of loss, if cardinal utility and subjective experience carry over to that domain) and risk aversion (when the alternatives concerned are probabilistic objects such as vNM lotteries) also acquire subjective significance. This is, as we have said, an expressive advantage, but it raises supplementary issues when vNM utility functions are used beyond the modeling of individual decision-making, as one can still wonder why risk-aversion should be a concern in interpersonal utility comparisons or in the aggregation of individual preferences.

1.1.2 Allais’s cardinalism We can consider Allais1 the founder of behavioral (and perhaps experimental) decision-theory, not only because his celebrated and often misunderstood “Allais paradox” paved the way to the later quasi-industrial uncovering of counterexamples to standard decision-theory (and to the effort, maintained by some, to axiomatize decision-theory in the face of paradoxes and counterexamples) that shaped behavioral economics but mainly, we think, because he adopted a cardinalist understanding of the 1

In this subsection, we notably refer to the contributions by Allais in the book he edited with Ole Hagen on cardinalism, which sometimes stem back as far as to work he realized in the 1940s and 1950s, and mainly to his formulation of the so-called intrinsic invariant model.

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vNM utility function. Allais’s cardinalism in decision-theory (as well as Harsanyi’s cardinalism in social choice-theory) is driven not only by the empirical concern of an application of a vNM utility index to the measure of some aspects of reality but also by the fact that “economics has to do at least as much with quality (of life, of actions, of opportunities) as with quantity and ordering” (Allais & Hagen, 1994). The plausibility of cardinalism, for Allais, has indeed to do with the methodological admissibility of a behavioral shift and of nonstandard experimental data (fully reasserted in Allais, 1991): 1. Behavioral shift: A utility function can be constructed by taking into account the subjective intensity of preferences (revealed through questionnaires, verbal data, which, of course, is a direct and problematic assumed leap out of the revealed preference paradigm) of individuals. The model contains a few behavioral parameters. Each individual is endowed with a single (individual) cardinal utility function which is invariant in space and time. This invariant function can be, in a first approximation, described as the ratio X/U0, where U0 is named, by Allais, the “psychological capital” that the subject is accustomed to (his reference point) and X is a virtual change (it can be envisioned hypothetically) with respect to this reference point. Let’s briefly note that an atypical behavioral feature, in the characterization of a cardinal utility function, appears in this approach, which is a temporal aspect in the cardinal estimation of the change of utility. The present time is when the future change in utility is envisioned, mimicking, in that way, typical modeling of an intertemporal choice discounted utility function. 2. Experimental heterodoxy: Introspective data are deemed as real and admissible in the inductive construction of a utility function as revealed preferences through choices. Accepting hypothetical preferences yields a strong advantage over the type of function that can be based on actual choices. Introspective data can potentially generate all the comparisons we need (and comparisons of comparisons) between the psychological reference point at a certain time and all possible virtual changes. Actual choices yield only one point of comparison of utility with the choice actually made. Moreover, once that choice is made, it endogenously modifies the reference point, and the comparison with another choice at another time has lost its anchor. We see that the hypothetical and introspective methodology—in Allais’s approach at least—is consubstantial with the construction of a cardinal

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utility function. Only introspective data can yield the required invariance in view of a cardinal measure of utility. The cardinal utility function is normalized and assumed identical in this respect for every individual:
 uðxÞ 5 u 1 1 x=u0 =let y 5 x=u0 ; we have uðy 5 0Þ 5 0 and uðy 51 NÞ 5 1

rendering all cardinal utility function comparable. In a famous 1952 questionnaire, Allais elicited such cardinal utility functions from a group of individuals. One crucial aspect of this elicitation was the degree of risk aversion for each of these individuals. This raises again the question that we left open at the end of Section 1.1.1 about the relationship between cardinal utility and risk aversion. It is not intuitively obvious. One way a vNM utility index is typically used is to measure risk aversion. But this intuitively seems like another problem than valuing options, even though it is a tenet of the interpretation of a vNM utility function that diminishing marginal utility and risk aversion amount to two aspects of the same phenomenon. Let’s continue to assume that the identification of a measure of risk aversion, through a vNM utility function, and the elicitation of a cardinal utility function, in a sufficiently rich sense, is not so obvious. We need to clarify under which conditions the two can coincide. Let’s call a value function v a function that allows for a cardinal representation of preferences in the sense that it numerically encodes differences between instances of a basic preference relation. It is not clear that a vNM utility function u can perform this. Which suggests three distinct grades of cardinal involvement:  Null: The preference relation R over certain options of a set X (RCX2) can be represented by a function f, meaning: xRy 2 f(x) $ f(y), if R is complete and transitive. If f and g both numerically represent R, it is then the case that f and g are linked by a strictly increasing monotonic transformation. f and g are ordinal.  vNM-cardinal: If R satisfies the list of vNM axioms, then it can be represented by a function u that is a richer measure than f in the sense that it encodes interval scales. It captures the difference in preference between an option x and an option y on a certain scale. If another function u0 performs the same work, then it means that u and u0 are linked by a positive affine transformation: u 5 au0 1 b. An individual may be more interested by a certain scale than another. The typical use and interpretation of the vNM framework does not allow for this

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specification of a given utility function as being the most relevant one for the individual.  A function v encodes for differences in preferences in the sense that xRy and wRz can be compared. The question is whether v is something different from v, even though it seems to present a further grade of cardinalist involvement. Allais answers that question negatively: v and u are the same. A vNM utility function measures preferences and encodes differences in preferences. The coincidence of u and v amounts formally to uncover conditions that identify the two respectively represented preference relations: R2 C½P ðX Þ2 where P(X) defines a vNM space of lotteries, that is, probability measures P on the set of outcomes X, and R4CX4. Note that we defined R2 on a probabilistic space and R4 on a space of certain outcomes. It could be done otherwise, but the basic problem we raise would remain the same. Moreover, R4 takes as basic inputs preference relations that could be assimilated to instances of R2, in which case we should define more exactly R4 as R4C[P(X)]2 3 [P(X)]2 but still try to preserve the idea that R4 itself bears primarily on probabilistic objects (as R2 does) but rather on determinate preferences over probabilistic objects. For simplicity, here, let’s then consider that we are concerned with the equivalence of u and v over the space of outcomes X, where a cardinal measure of utility gets its primary sense in terms of valuation of outcomes and quaternary comparisons. Bouyssou and Vansnick (1990) prove a result concerning U and V defined over a probabilistic space (their result holding a fortiori for u and v and preference relations over degenerate lotteries, i.e., certain outcomes). Given the respective structure of R2 and R4, U and V are equivalent if and only if: ’p; qAP ðX Þ; ðp; p 1/2 qÞBðp 1/2 q; qÞ This condition simply implies that we can continue to estimate differences in preferences when mixing sure outcomes with the lotteries of which they are the certainty equivalents. If x is the certainty equivalent of a lottery yielding y and z with probability 1/2 each, this condition states that the utility difference between x and y must be the same as the utility difference between z and x. We have here a link between R4 on sure outcomes and R2 on lotteries.

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Experimentally, we have seen that Allais relied on hypothetical preferences and virtual comparisons. He was a vocal advocate of the admissibility of introspective data not only in order to test theoretical assumptions but also as possible grounds of theoretical assumptions. He deemed that decision-theory (and economics more generally) is concerned with the inner lives of individuals, at least as much as with the description of an objective structure in which their economic lives is inserted. But he did not create an alternative decision-theory, even though he was critical of the interpretation the promoters of decision-theory (in particular the anticardinalist Savage) gave to their creations. He just expressed the inherent cardinalism of a prevailing decision-theoretical framework or, in other words, started to uncover the mind under the axioms.

1.1.3 Alternative hedonimetry In his invariant model and his experimental elicitation of a cardinal utility function, Allais introduced a main difference with vNM utility: he supposed an invariant but nonarbitrary origin point, namely in terms of a reference point that is the “psychological capital” of the subject at the time of the experiment. This in fact involves a deeper grade of cardinalism, in the form of comparisons of intervals, than what the vNM utility function can imply, and it grants more expressivity to an induced utility function that would cease to be provable as equivalent to a vNM utility function. In particular, that alternative utility function would apply to a ratio scale and not to interval scales. It would incorporate the element of subjective relevance that is lacking in a vNM cardinal utility function. We sum up the main differences between interval and ratio scales in Table 1.1. Table 1.1 Interval scales versus ratio scales.

Operations Zero Statistics Measure

Type of utility function

Interval scale

Ratio scale

Addition, subtraction, multiplication Arbitrary Arithmetic mean Magnitude measure as multiple factors of a single unit Cardinal vNM

All operations 1 ratios Absolute zero Geometric mean Conversion of a unit into another Hedonic utility

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We label hedonic utility the measure of subjective utility in an absolute sense, that is, a measure that can reflect in a meaningful way the intensity of the individual’s experience. Allais took as the nonarbitrary origin (zero point) of utility the “psychological capital” of the subject. This could be interpreted as the subject’s current psychological state or as the state of mind he tends to be in when he is in a neutral mood, perhaps conceived as the average of all his past affective experiences. It will still be controversial whether any such psychological notions will really count as an absolute origin and how a meaningfully divisible measure can follow. It is then tempting to argue that the measure of hedonic utility goes beyond what a vNM utility function can do, that subjective hedonic experience cannot be observed and a fortiori measured. On the contrary, one could think that choices over lotteries provide sufficient information to build a cardinal utility function in a sufficiently rich sense and that it is implicit that agents, who have the sort of preferences that can be represented as if they were maximizing a utility function, actually maximize their hedonic utility. To get out of this interpretative limitation usually imposed on cardinalism, we need to depart from the background assumption that utility is derived from choices over options (whether lotteries or outcomes) and pursue the introspective move opened by Allais. We need, first, to revise our usual ontology of utility-theory and in fact replace the idea that utility is derived from decisions (choices that can be observed and are supposed to reveal preferences) by the idea that it is derived from experiences or hedonic episodes. Second, we must conceive of a plausible measurement of such episodes, on a ratio-scale, to obtain the cardinal expressivity we are looking for. Yet the question remains whether, by deciding to perform this drastic move, we are still preserving the advantages of a vNM utility function and building on it or we are suggesting a radically and potentially meaningless alternative hedonimetry. Skyrms and Narens (2018) attempt a conciliation between alternativeness and conservatism. They consider that the object of utility are episodes and that the relevant aspect of episodes to be measured is duration. Duration presents the advantage that it gives an absolute zero as the starting point of an episode in time. Episodes may vary in hedonic intensity, and, from the zero point to the end of the episode, the relevant measure is an integral. They think that this is not so irreconcilable a departure from vNM utility, on the basis of the argument that there is a sufficient structural analogy between prospects and episodes. Hedonic intensities are

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analogous to outcomes: the prospect of getting an amount of money can be formally (if not otherwise) compared to obtaining a degree of pleasure. Taking hedonic episodes over fixed periods of time (with an absolute zero and an absolute ending), you can consider the duration of episodes of various intensities during that whole period as outcomes with certain probabilities. Then the intrinsic structure of episodes is analogous to probability distributions, in fact subduration distributions, over various hedonic intensities. Skyrms and Narens rely on this analogy and assume they can derive a representation theorem, as is routinely done from vNM axioms on preferences on lotteries. This would suppose, however, that the same properties that vNM axioms apply to preferences over lotteries also apply to preferences over episodes, in particular transitivity (raising some questions about the order in which the episodes are experienced), continuity (supposing, again, a fine sensory apparatus that can discriminate between very similar hedonic experiences) and independence (meaning that the association of two distinct episodes with a third common one does not change the experience of the initial episodes). Instead of thinking in an analogical fashion, we may judge that vNM axioms are fit to describe inherent properties of preferences over lotteries (not over something else, in spite of some formal analogies) and that the type of cardinalism we can infer from their analysis (as done by Ng, 1984), is limited jointly by the nature of the objects over which preferences apply and by the axioms that model preferences over these objects. It is not enough to find a type of object for which it makes sense to have a ratio-scale with an absolute zero; it is also necessary to adapt the axiomatic structure to account for the way preferences apply to these objects, which may, certainly, lead to a richer type of cardinal utility function but may also become disconnected from the usual decisiontheoretical notion of utility. This is what Kahneman, Wakker, and Sarin (1997), when distinguishing between decision-utility (what is measured through a vNM utility function) and experience-utility (what traditional utilitarianism, Bentham in particular, had in mind). They set a list of axioms that fits the type of objects under consideration: episodes and hedonic experiences taking place and having instant values; the measure, hence the intended representation, being a Lebesgue integral taking instant hedonic values along the episode. The objects over which the preference relation is applied are called “utility profiles.” Formally, they are hedonic curves, that is, functions from a given interval to levels of instant utility. What makes the representation possible, at the level of

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axioms, is some conditions on the temporal location and the concatenation of episodes: Axiom 1: Adding a hedonically neutral episode to a utility profile does not affect the latter. It means that we sum (integrate) over the instant utilities independently of the duration of the episode. This is, of course, a counterintuitive condition to some extent. Some crucial points of this axiomatization by Kahneman et al. bear on the neutralization of different aspects of the role of time in the experience of utility, without which the possibility of representation is lost. Axiom 2: When an instant utility is increased within a utility profile, it increases, not decreases the total utility of that utility profile. It is a monotonicity condition, which, once again, in spite of its potential counterintuitiveness, requires some form of separability and integrability of experiences to be imposed between instant utility experiences within an episode. Axiom 3: A similar monotonicity condition applies when summing utility profiles, not only within utility profiles. Again, it means that temporally disjoint hedonic experiences are separated with respect to their total effect. Moreover, they are not subject to order considerations. Put in informational terms (to which we will devote closer attention in Section 1.3), it means that instant utility experiences contain all the relevant information in view of their measure. These three axioms grant the provability of the following theorem: KWS theorem: For a nonnull episode E . 0, and a relation k over utility profiles, formally defined as all measurable bounded mappings from subintervals [0,e[of [0,M[ (e being the open end of the considered subinterval) to an interval X (of instant utilities or of instant utilities associated with hedonic states at the intended instants of time) containing (an absolute) 0, the equivalence between (i) and (ii) holds: i. There exists a nondecreasing continuous value function v that assigns 0 to 0 (v is a ratio-scale) and orders utility profiles according to the value of instant utility over temporal episodes. ii. k is a continuous weak order. Axiom 3 makes clear that all information needed for a representation of hedonic utility of episodes must be contained in instant utility. Axiom 4 allows this informational basis to be cardinal: Axiom 4: The ordering (according to v) of the total utility of two utility profiles is not reversed if, for the two utility profiles, the instant utility level (of instants in those profiles) is increased by the same constant over an equally long time period.

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This yields a possible cardinal representation of hedonic experiences. As previously noted, these axioms impose separability constraints on preferences over episodes and thereby cancel out certain possible psychologically plausible interactions, affecting in principle overall perceived utility, between these episodes. It may be the case, however, that episodes themselves are psychologically impoverished within this framework. The surrounding effect of time is neutralized. Kahneman et al. (1997) (or originally Benthamite) alternative hedonimetry does not take into account subjective projections of utility in the past (memory effects) or in the future (in the guise of episodic simulation of future events, possibly generating savoring or utility discounting). The integration—that is, the representation—is temporally neutral. We may question to what type of psychology this model actually applies. And we may provide a surprising or expected (it depends) answer in the “To the lab” Section 1.4.2.

1.2 Utility functions as psychologies 1.2.1 Utility functions as primitives We have taken so far preference relations and their structural axioms as primitives. From there, we can deduce a utility representation. If we take as a basis vNM axiomatized preferences, we can infer a cardinal utility function and discuss under which strictures, or radical departure thereof, its degree of cardinalism can be modulated. But the main result is that we cannot get ratio-scales and then subjectively relevant cardinal measures, if we adopt as primitives vNM axioms as they stand, as well as the type of objects to which such axiomatized preferences usually applied (i.e., lotteries instead of episodes, for instance). But we can reverse the reasoning and decide to take as our primitives the utility functions themselves and from there modulate the nature of individual preferences according to some constraints that apply directly to the utility functions. Such a proposal, leading to defining, in another way than we did in Section 1.1, possible grades of involvement in cardinalism, has been advanced by Mandler (2006). We present, discuss, and dwell on his proposal, as it seems to us an unusual but very natural way to refine our understanding of the interplay between preferential structures and utility measures, especially when we keep in sight experimental procedures of cardinal measurement such as those tentatively employed by neuroeconomists (see Section 1.2.3). In Mandler’s framework, an individual is primitively endowed with a set of utility functions. We don’t detail, at that initial stage, the constraints

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that may link the utility functions in the set. They can be stronger or weaker in a sense that will be made precise. Endowing the individual with a single utility function (or with a set of functions derived from that unique function that are linked by monotonic linear or affine transformation clauses) is a restriction that is due to the fact that this function is induced from a preference axiomatic structure. We now try to envision the reverse move, and it is thus natural to start with a set of multiple utility functions rather than with a unique one. Another motive from this starting point could be that the preferences of the individual is incomplete. We will develop in the second chapter of this book how the standard modeling of incomplete preferences is framed in terms of a multiple utility set. Incompleteness is a natural fact, allegedly primitive, if the individual feels unable to make comparisons between some bundles of goods. In that case, we can include a good-specific utility function in the set of utility functions to describe the individual utility of an incomparable good. It is intuitive to think that if the individual has incomplete preferences that induce several utility functions, it is still possible to find transformation clauses that associate these functions. However, it may sound counterintuitive that a transformation clause indicating that the utility functions can be given an ordinal or a cardinal interpretation continues to reflect preference incompleteness, that is, incomparability between some bundles. But this is, precisely, an argument that is dismissed by the fact that, in this framework, we admit utility functions as primitives, not preferences. Incompleteness at the level of preferences has to be itself an induced fact, not a primitive. Mandler calls this primitive set of utility functions a “psychology.” This is to be taken literally: a psychology is the list of utility functions that characterizes the experience of consumption of bundles by the individual. This description can take place at different levels of granularity. There are more or less inclusive psychologies. A particular psychology, that is a characterization of a hedonic experience, is delimited by one or several properties that a set of utility functions have in common. Preference relations follow from psychologies. A psychology U is said to be weaker or nonstronger than another V, if it is more inclusive, if and only if U*V. Weaker psychologies make less assumptions about the utility experiences of the individuals and include fewer criteria (properties) for their measurement. Utility functions, hence psychologies, are primitives. It means that weaker psychologies describe a larger array of experiences than stronger ones. An ordinal psychology

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(a set of utility functions linked by a positive linear transformation) is weaker than a vNM cardinal one (utility functions linked by positive affine transformations), which is in turn weaker than experiences described by ratio-scales. Ultimately, we have a unique (not transformable) utility function that describes the idiosyncratic experience of one type of good by an individual. This approach allows to loosen the typical conclusion that vNMaxiomatized preferences over lotteries induce a cardinal utility function and permit a more flexible interpretation of the psychology of an individual having such preferences over such objects of choice. Imagine a subject having vNM preferences over a space of lotteries. It is clear that we can deduct from this sort of P preferences and objects of choices a utility funcn tion of the form u(p) 5 pi uðxi Þ and assert that it represents the preferi51 ences up to a positive affine transformation. But it does not mean that by deducting such a representation and by recognizing its cardinal nature, we have also derived the psychology of the individual. The psychology of the individual has been stipulated as a set of utility functions that can include this typical vNM representation. We have some leeway to delineate that psychology. Giving that u is present in the psychology of the individual and independently of the fact that we could have derived it from a certain preferential framework, we can still describe the psychology of the individual in terms of the positive linear transformation of u. It means that in that case we decide to describe the individual in terms of an ordinal psychology, which can be not only relevant but also sufficient and parsimonious in certain circumstances. Alternatively, we can endow the individual psychology with a dose of cardinalism. It means that we define her psychology as bounded at least by the maximal set of positive affine transformations of u. Again, this is relevant for a certain type of context—for example, choices under risk, contexts in which the objects of choice can be idealized in terms of lotteries but less clearly for episodes, as we have seen. So we decide to restrict even more the psychology of the individual. It is not clear that there is any transformation of u that could fit for further restriction and that describes, for instance, hedonic enjoyment of episodes, so we need to adopt a set of nonexpected utility functions. The psychology of the individual can be described by a set of functions that are neither ordinal nor cardinal if those properties are too strong to account for the type of experience and objects of experience of the individual. We accept here Mandler’s conclusion that individuals endowed with vNM preferences

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may perform cardinal comparisons of utilities, that this is representable by a cardinal utility function but that this is not a compelling conclusion, from the description of the psychology of these individuals, to be drawn from vNM axioms. To hone this argument, it is possible to frame it differently in terms of the properties that are inherited when one transits from one set of utility functions to another, from one psychology to another. Let’s suppose, abstractly, a property P. If two psychologies U and V coincide on P, they are, from that point of view, the same psychological characterization of an individual. Partial overlaps are also possible for some properties, like precisely between cardinality and ordinality, in the sense that cardinal functions are also ordinal but not the reverse. But convexity and concavity of the utility functions, for instance, characterize two disjoint psychologies. A psychology U is said to satisfy a property P. It satisfies maximally a property P (for instance, cardinality) if all uAU has that property, and there does not exist a larger psychology V*U such that each vAV satisfies P (for instance, V is an ordinal psychology). We can take properties as primitives as they coincide with maximal delimitations of psychologies. We can also partially detach the theoretical analysis and experimental testing of these properties without being constrained by underlying axiomatic characterizations of preferences and by their analytical links with special utility representations. It is possible to induce preference structures from psychologies as much as it is possible to induce specific utility representations from some preference structures. The fact that a special representation is demonstrable does not entail that the individual is psychologically limited by it or compelled to it. Axiomatic structures, when we realize we can take utility functions as primitives, are more psychologically underdetermined than usually admitted. Moreover, it opens the way to the analysis of specific features of utility function and of their locations within the ordinal-cardinal spectrum.

1.2.2 Features of utility between ordinality and cardinality An ordinal as well as a cardinal utility function can be concave. Concavity, which is standardly derived from the fact that preferences are convex, is a property of utility functions seemingly independent from ordinal or cardinal assumptions. Remember that, in this section, we take utilities and progressively their properties as primitives, not preferences. In other words, we state that if a psychology U, in Mandler’s sense, satisfies

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the property of (quasi-)concavity, then the induced preference relation kU is convex. Preferences are convex if, whenever xky, tx 1 (1 2 t) yky, for all tA[0,1]. Convexity of preferences is a psychological property stipulating that individuals prefer mixtures or averages to extremes. As a psychological property, we can state it in terms of utility functions by saying that, given xky, and k being convex, we have u(tx 1 (1 2 t)y) $ u (y). A utility function that satisfies this property is called quasi-concave (the inversion from convexity to concavity may be confusing for some readers, but think of the fact that convex indifference curves reflecting preferences for mixtures correspond to concave utility functions derived from an Edgeworth box). But even if there is a direct link from the convexity of preferences to a property of concavity of the utility function and despite that a convex preference is induced by a concave utility function, we can apply anew the argument about cardinality in Section 1.2.1 and emphasize that this deduction is not yet a characterization of the psychology of an individual endowed with such preferences. Again, it would depend on what transformation we associate with the deduced utility function and whether they preserve concavity or not. Under concavity preservation by a given transformation and the maximal limits of the set thereby generated, we have characterized a concave psychology in relation to which convex preferences have an essential role. But we want to focus on another aspect of the problem of the interplay between utility and preferences when we take properties of the utility functions as primitives. Namely, we are interested in the interaction between several properties themselves. In particular, what is the connection between utility concavity and ordinality or cardinality? The question is not purely formal but relates to observational and experimental issues. Suppose that we can infer, from raw experimental choices on a set of options, an individual’s utility function. What we can know, even before we can determine whether the individual has ranked the options based on ordinal or cardinal information, is whether she exhibits, across repeated trials, marginal decreasing utility over the consequences of the chosen options. We can infer from that experimental basis whether the individual exhibits a quasi-concave utility function or not, independently of the subjective informational basis on which the options were chosen. Concavity is not necessarily intrinsically linked with deep forms of cardinalism. The idea of marginally diminishing utility does not imply per se that the individual computes, each time he experiences an increment in

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the consumption of a good, by which degree that extra amount diminishes in utility compared to the previous consumption episode. He can simply say that he enjoys it less and less. But this latter point should not be understood as concavity being essentially ordinal. Taking concavity as a primitive property, we can try to locate it in the ordinality-cardinality spectrum and make more precise its alleged analytical links with these two poles. Mandler (2006) proves a two-fold theorem, stating that: Any ordinal property is no stronger than concavity [remember that this means that a set of utility functions that describes an ordinal psychology contains only concave utility functions]. Ordinal functions are by definition concave, but the reverse is not true. On the other hand, concavity is no stronger than any cardinal property. This means that there can be some cardinal utility functions that are not concave. Concavity is not essentially linked with cardinality.

If we are in a position to experimentally generate an individual utility function, we cannot interpret its possible marginal decreasing rate as indicating underlying cardinal trade-offs. Likewise, if we impose a cardinal evaluation of options by individuals across repeated trials, and we observe that they are best described by a concave utility function, we cannot infer an intrinsic link between those two facts. Cardinality is a far stronger psychological requirement than concavity since it would suppose that the individual is able to compute the ratio of utility increments between two successive consumption episodes, if the two properties coincided. Ordinality and cardinality are less immediately (at least we can presume so because this appreciation relies on the complicated and controversial fact that an individual utility function can be elicited from experimental data) observable features of the utility function than its concavity or its continuity. At least, imagine we are provided with a utility function that, we are told, describes the behavior of an individual. We observe that this utility function is concave and continuous. Of course, we cannot presume that the experimentalist has tested all the data points that would make this function really continuous. She has linearly interpolated it, but she observes in a surer sense that this function is concave. Now, she is a cautious experimentalist; she knows whether she has framed her questionnaire in view of collecting ordinal or cardinal judgments, but she does not draw undue inferences from this experimental methodology to the interpretation of the observable features of her elicited functions, say, concavity and continuity. A question remains open: is continuity a stronger property than concavity? Are all concave psychologies continuous ones?

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Formally, the answer is obvious. Let X be a convex set of options on which utility functions can be applied. The property of continuity of a utility function on X is weaker than the property of concavity. Any concave function is continuous but not the reverse. Let’s check at the level of induced preferences: a convex preference relation implies that the set of options over which it applies is closed, otherwise some combinations of preferences would fail to be included in that set. But there could be a continuous preference that applies to a nonconvex set. Moreover, any increasing transformation preserves continuity but not necessarily concavity. The properties of continuity and of concavity-continuity are associated with distinct transformations (respectively, increasing continuous and increasing concave functions), hence with distinct psychological characterizations of the individual. The cautious experimentalist, while performing a linear interpolation over her necessarily discrete data set, must remain aware that she is manipulating distinct psychologies by empirically approximating a utility function. But she is not mistaken in inferring that a concave psychology is continuous.

1.2.3 The problem of loss-aversion Convexity-concavity features of a utility function are traditionally associated with the risk attitudes of the individual (since Jensen, 1906): concavity of the utility function amounts to risk-aversion, convexity to risk-seekingness. The method through which this conclusion is reached relies on two combined facts, neither of which entails cardinalism. One is the fact that we can take the expected value or the means of two risky prospects, assign it a probability 1, and define it as the certainty-equivalent of the lotteries that would give the risky prospects a probability of 1/2 each. The second fact is that we project this certainty-equivalent on a concave utility function and find that the utility of this certain outcome is less than its expected value, meaning that the individual described by this utility function is ready to renounce some payoff in order to obtain this certain outcome rather than playing the original lottery. The individual is risk-averse, and this fact is implied by her concave psychology, with no commitment to ordinalism or cardinalism. We do not discuss here the issue that, from an intuitive cognitive point of view, we may certainly wish to separate concavity and risk-aversion and define an attitude toward risk that is independent of considerations about the diminishing marginal utility of the considered outcomes. After all, we have never mentioned any

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aspect of diminishing utility in the setting of Jensen’s method to infer risk-aversion, yet it depends on the admission of a concave utility function that is here because it reflects diminishing marginal utility. This point is well-known and was one of the motivations of rank-dependent utility models that can separate measures of risk-attitudes and measures of utility. So let’s admit that the measure of attitudes toward risk is at least neutral vis-à-vis the choice of a cardinal or an ordinal utility function. We may want to preserve this neutrality for other features of the utility function that lead to potential descriptive statements about the psychology of individuals. In particular, if one feature depends on another, we wish to remain consistent and see to it that the dependent feature does not entail links with other properties that are not entailed by the feature it depends on. The problem arises when we jointly consider risk and loss-aversion. Loss-aversion reflects the observed behavior that individuals are more sensitive to losses than to gains. Their utility functions, when prolonged backward to the domain of losses, present a steeper curve in the domain of losses than in the domain of gains (see Fig. 1.2). This is a feature that has been widely documented by Kahneman and Tversky (1979) and formally included in a rank-dependent utility model (cumulative prospect -theory) in Tversky and Kahneman (1992). In the domain of losses, the utility function becomes convex, signaling that the individual is risk-seeking in this context (by application of the usual Jensen’s method). In principle, these two phenomena, loss-aversion and risk-aversion or seekingness, should remain independent under this

Figure 1.2 Utility decreases faster in the domain of losses than in the domain of gains, a fact interpreted as loss-aversion in prospect-theory.

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observed phenomenon. But it is hardly conceived so. One can conceive of risk-aversion (or -seekingness) being caused by loss-aversion. This is because the individual fears of losing some certain gains or of incurring some certain loss that she is respectively risk-averse or risk-seeking. Moreover, this phenomenon is stronger in the domain of losses, which are proportionally more painful than equivalent counterpart gains. Conversely, one can conceive of loss-aversion as depending on riskaversion because if one takes the utility function (rather than choiceattitudes of the individual as in the former line of reasoning) as a primitive, she observes that in the domain of losses, the utility function being more convex than it is concave in the domain of gains, the individual is relatively more risk-seeking than she is risk-averse in the domain of gains, and she interprets this as the mark of loss-aversion. However, judgments of loss-aversion have a cardinal flavor. They are of the form “the individual is more sensitive to losses than to gains,” and this sensitivity is reflected in a steeper curvature of the utility function in gains than in losses, meaning that we give a meaning to what the utility function measures in these respective domains and compare it. The utility function in losses is interpreted not just as a positive linear (but not necessarily as an affine) transformation (modulo its inverse sign) of the utility function in gains but as providing cardinal information. Possibly this cardinal information is of a nonexpected utility nature; that is, it does not resort to the positive affine relations linking a set of utility functions that characterizes a typical vNM psychology. It is more restrictive or stronger than that. We can consider this leap from the possible neutrality of the risk-aversion notion, with respect to ordinal or cardinal aspects of the utility function, to the inherent cardinalism of the loss-aversion construct as presenting a form of a theoretically inconsistent psychological interpretation of manifest features of the utility function. In reaction to this inconsistency, one can either wish to preserve the conceptual link between risk-seekingness and loss-aversion but propose a noncardinal definition of the latter or disentangle the idea and measure of loss-aversion from that of other risk-attitudes. Köbberling and Wakker (2005) make significant moves in that direction, in our opinion. Namely, they suggest an index of loss-aversion that is neutral vis-à-vis ordinality and cardinality and that can be defined independently from the measure of risk-aversion, while the latter depends on the former. They note that Tversky and Kahneman (1992) adopt a scaling convention to measure loss-aversion that depends on the

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unit of payment. This is shown by the fact that if the utility function (in gains and losses) is normalized such that u(0) 5 0, u(1) 5 1, and u (21) 5 21, and if we adopt λ as the loss-aversion index such that we have the overall (in gains and in losses) utility function U decomposed as U(x) 5 u(x) when x $ 0 and U(x) 5 λu(x) when x , 0, we have λ 5 2U (21)/U(1), which depends on the unit of payment. Köbberling and Wakker suggest that a viable index of loss-aversion is the ratio of the derivatives of the U(x) function above 0 and below 0, that is, at the point where typically the overall utility function kinks (0 can be by convention any reference point where the overall utility function can be decomposed between its concave and convex subparts and where it is not apparently smooth): λ 5 U0 m(0)/U0 k(0). This index is invariant under changes of scale of U and can then be considered neutral vis-à-vis ordinal or cardinal psychology. Is the inversion of the utility function, both in terms of concavityconvexity and of steepness of its slope, at a point that we label the reference point of perceived utility, a feature intrinsic to the utility function or a more primitive phenomenon of loss-aversion? Once we have admitted a reference point, we can consider a function that spans the overall possible experiences of utility and disutility. It does not do it in terms of ordinal or cardinal utility judgments but in the format of a less structured and irregular mapping from an experience to a valence judgment. It may then primitively incorporate some element of asymmetry between losses and gains. Now we can associate this primitive experience with some more structure, but we can preserve some asymmetry at the level of the function we associate with positive and negative rewards. The experience of loss, in particular, can be less discriminative than the experience of positive utility, less easily associated to orderings, let alone to cardinal evaluations. This phenomenon is more primitive than utility itself, and this assertion can receive some descriptive plausibility from neurosciences studies (Asaad & Eskandar, 2011; Nioche, Bourgeois-Gironde, & Boraud, 2019). One of the great discoveries of prospect-theory, if we follow this thread of thought, would really be to have put at the core of decision-theory a basic psychological fact, more basic, perhaps, than utility itself. But it is a fact lacking some structure, apart from the observation of asymmetrical treatment around a reference point that offers the notion through which it is connected up to a utility function. Loss-aversion describes a general psychological fact to which utility functions lend some structure, both formally and psychologically.

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1.2.4 Standard neuroeconomics of utility functions Mindful economics can be paramorphic or homeomorphic, according to Peter Wakker’s distinction that we have recalled in the introduction of this book (Wakker, 2010). We oscillate between the two positions, depending on which aspects and types of axiomatic structures we discuss. Sometimes we can infer psychological implications from the axioms, while those axioms in no way imply some processual analogy with the mental realization of those psychological implications. However, one is tempted to adopt a homeomorphic standpoint when taking utility functions and their features as primitives. This is not compulsory. It depends on the degree of psychological structure we associate with sets of utility functions or, in the framework we have adopted in this section, of the strength of a psychology. This type of concern is clearly absent from standard neuroeconomic approaches of utility, which tend to be spontaneously homeomorphic (curiously, even in Mandler’s very paper from which we have elaborated our current discussions, in its references to then emerging neuroeconomics). The very idea of “neural correlates” of, say, loss-aversion, implies that there is a physically observable implementation of that abstract concept. It can remain an abstract concept, even if it is a basic psychological fact, but a homeomorphic approach states that there must be some correspondence between the concept and the fact. In neuroeconomics, this correspondence is expressed in terms of neural correlates (of a behavioral observation that itself correlates with an experimental implementation of the concept). Let’s illustrate this by prototypical examples of brain-imaging studies of the asymmetry between losses and gains, which we have analyzed above as a primitive psychological fact, on which finer utility structures can be imposed. Neuroeconomic studies can then differ one from the other on the same topic in terms of the coarser or finer structures that they (implicitly) admit from the onset on the utility function. Tom, Fox, Trepel, and Poldrack (2007) investigate neural correlates of loss-aversion when individuals accept or reject gambles that offer 1/21/2 chances of winning or losing money. They scan the brain of participants when they incur gains or losses and document distinct brain activities in the same regions of the brain in each case. They observe increased neural activities of midbrain dopaminergic targets in the event of gains and decreased neural activities of some of the same structures in the event of losses. We can relate these results to a very basic fact of loss-aversion with

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no further implication on the shape and nature of an underlying utility function. Even the idea that there could be two distinct utility representations for the domain of losses and the domain of gains is not implied by this study, simply the neural encoding of a valence asymmetry in the brain reward system on the whole, a basic psychological fact. Consider, by contrast, the study by Yacubian et al. (2006) on gainand loss-value prediction in the human brain. This study takes stock of the well attested fact that midbrain dopaminergic neurons encode rewards. But they give themselves, from the onset, a bit more utility structure than in Tom et al.’s study. They rely on the information that is available on how this brain system encodes different aspects of expected value: reward magnitude, reward probability, and the difference between actual and predicted outcome (prediction error). They confirm that the ventral striatum codes reward probability and magnitude and permits the computation of expected value. But they observe that this area of the brain represented only the gain-related part of expected value. Loss-related expected value (and its anticipation) was encoded in the amygdala, a disjoint brain structure. Thus, the ventral striatum and the amygdala distinctively process anticipated value and valence of a reward prediction. We interpret the fact that the Tom et al. (2007) and Yacubian et al. (2006) studies differ with respect to their uncovering of neural correlates of losses and gains in relation to the formal psychological framework that their respective studies adopt. Unlike Tom et al., Yacubian et al. put in evidence distinct brain structures supporting gains and losses because, we deem, they immediately relate this valence asymmetry to the neural basis of expected utility maximization and probability processing. When embedding the study of the loss/gain asymmetry with that of those other psychological concepts, we put ourselves in a position to observe that the neural correlation of utility associated with losses and of utility associated with gains differ, reflecting, in a homeomorphic way, that the properties associated with the utility function in losses and in gains may structurally differ. We more generally speculate that different results in the neuroeconomic approach to utility-theory depend on the structure of the utility function, that is, the formal psychology in the sense we have developed along this section, that is, most often implicitly implemented through experimental tasks. Trepel, Fox, and Poldrack (2005) and Fox and Poldrack (2009) develop clear arguments in that sense through reviews of neuroimaging experiments on prospect -theory. They recall that utility is a mathematical

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construct that may or may not reflect mental states. The case of prospect -theory is tricky for several reasons. First, in spite of having belatedly acquired an axiomatic foundation (Wakker & Tversky, 1993), it is essentially a behaviorally based and descriptive decision-theory. The mind has come before the axioms. While we are interested mainly in this book in spelling out the psychological implications of independent axiomatic structures, with prospect -theory we may have the feeling that axioms encode independent psychological facts. Having evidenced a particular utility function, with salient properties that moreover robustly correspond to psychological and behavioral observations, prospect-theory is an effort to unify this data in a single utility function (and then an axiomatic foundation thereof). An air of hybridity remains in the utility function, with the hesitation we have previously perceived whether the kink at the reference point should affect the smoothness of the utility function itself (and potentially compromise its local differentiability in view of a loss-index aversion) or just be the signal of a change of risk-attitudes between loss and gains. The prospect-theory utility function therefore embodies a variety of psychological phenomena. The interrelation between them is not easy to make precise in a single functional mathematical construct. Moreover, when testing utility in neuroeconomics, we must be very cautious in inferring the mental states of the subjects from neural observations, a lot of studies are tainted by the “reverse inference fallacy” (Bourgeois-Gironde, 2010; Poldrack, 2011). Kahneman et al. (1997) have underlined the difference between experienced utility and what they call “decision-utility.” Aspects of anticipation, savoring, or direct experience that define the former are in principle absent of the latter. Decision-utility is the weight of potential outcomes at the moment of decision. Decision-utility and experienced utility are distinct psychological phenomena and formally refer to different properties satisfied by sets of utility functions, in particular distinct degrees of cardinality. Neural results are ambiguous about the hypothesis of a simple twosystem model encoding gains and losses depending on the association with other features of the utility function. Likewise, if we consider hypothetical or anticipated gains and losses versus actual ones, the brain activities again differ. The utility concept that presumably describes the experience of the subject, the type of utility function that is implicitly tested, the implicit or explicit hypothesis about the fact that loss-aversion is a more primitive phenomenon than utility itself or not, the (related) hypothesis about the continuity of the utility function at the reference

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point (or alternatively whether two utility functions are involved in gains and losses and pasted together), and the neural explanation of the S-shape of the utility function reflecting opposed attitudes to risk in gains and losses are all testable features that prospect -theory has offered to neuroeconomic investigation. But, we insist, this legacy deserves to be better disciplined by a theoretical view about the definability of utility functions as formal psychologies and their availability to experimentation.

1.3 Informational and representational constraints 1.3.1 The informational basis of cardinal utility We have entertained two parallel views of utility functions, hence of their possible ordinal or cardinal nature. We have classically seen them as representing preferences. Given some axiomatic structure on the latter, we can derive (in favorable cases) a utility representation of the former that is, obviously, dependent on the axiomatic structure we have postulated. This fact means that representational possibilities lay informational constraints on the utility function. A fact that is not as obvious as it looks and would need clarification. But we have also considered utility functions as primitives from which preferences could be induced. In that alternative view, the link between the information that conveys the utility function and the representation of the preferences by a utility function (possibly the same) is looser. We can modulate the structure that we put on the utility function, which can directly or less directly correspond for the type of utility function that is strictly dependent on a representation theorem. It is useful to distinguish these two views of the utility function for a better grasp of what is involved in claims about cardinalism and ordinalism. A vNM utility function can be reached both ways. It is, of course, a representation of vNM axioms on a weak order over lotteries. But we can also decide to describe an individual psychology from the onset by means of the set of all positive affine transformations of a given vNM utility function. There is no observable difference between the two, but it means that in the first case we derive a cardinal utility function and in the second case we posit it. When we derive a particular utility function and this utility function has a particular property, like cardinality, we should be aware that this property is relative to the representation we have proved. Cardinalism in the case of the vNM utility representation is, then, not absolute but relative to that representation. It is not a property that we

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give ourselves at the onset as a general postulate to describe an individual’s psychology in all circumstances. This distinction, scholastic as it sounds, is nevertheless crucial to distinguish two roles of utility functions: representing preference relations and rationalizing choice-data. We consider that we can relax to some extent the classical revealed preference paradigm by distinguishing between these two roles, in the following sense. The fact that a utility function is derived from a set of axioms and represents a preference relation does not necessarily make it the best tool to account for and rationalize (i.e., try to find the particular justifications among which preferences thus axiomatized might certainly be one determining factor) choice data. We formulate the hypothesis that a standard representation-based approach to utility collapses these two roles and thus generates informational constraints on what counts as relevant data to reveal preferences. Enriching or slimming out the informational basis of utility functions can be guided by different concerns: generality or specificity of the induced utility function, parsimony, and indeed provability of a representation theorem. But another concern is that the utility function helps to rationalize the choice-data that are supposed to reveal the preference relation. The utility function plays two roles that need not be assimilated (as illustrated in Fig. 1.3). What decision-theorists want to represent through a utility function are preferences. But they use as input choices, not preferences themselves, for the reason that they consider choices as revealing those preferences and those preferences themselves to be unobservable. In much of the traditional debate around ordinalism and cardinalism, it is implicitly held that since preferences are rankings, and since utility functions represent preferences, this representational property of utility functions impose that they are by default ordinal. Moreover, this seems to coincide with the other role that we see the utility relation play, which is to account for choices. In that revealed preferences paradigm, utility functions do not take more, in terms of informational inputs, than what can be termed as binary rankings—not directly, though, because choices are binary piecewise and not simultaneous full rankings of options sets and also because choosing x over y is a fact and ranking x over y could be another one (so it is not even clear that choices reveal rankings). It is as if the logic of a representation procedure required this structural morphism from preferences to utility. But we can think that this morphism applies between choices (considered as rankings) and ordinal utility, not between preferences and utility, even when we accept that

Choices Preferences Reveal preferences Disposition to rank options Observable informational basis Unobservable

Utility function Represents preferences Rationalizes choices

Rationalizes

Utility function

Choices

Reveals Represents

Preferences Figure 1.3 The twofold role of the utility function.

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preferences are at least in part revealed through choices. Preferences may involve a richer structure than choices. But in the same way it is standard that representation theorems impose an interpretation of the nature (in terms of ordinality, cardinality, and type of cardinality) of the utility function and that the role of the utility function as rationalizing choice-data constraints back the interpretation of preferences; hence its axiomatization and its possible representation. It is sometimes not so easy to make everything cohere, which may retrospectively explain some ambivalence about the right interpretation of the vNM utility function. On the one hand, a vNM is cardinal (relative to the representation of a vNM preference), and, on the other hand, it is used to rationalize choices that, in terms of the usual informational basis we think they consist in, could be done by an ordinal function. A non-vNM-representation relative cardinal utility function has—to warrant its role of representing preferences—to represent quaternary relations of the form; wRzkxRy (R and k do not have to be two distinct preference relations, but at least we can say that k is of a logically higher order than R, justifying the difference in notation). It represents it by means of a utility differential: u(w)u(x) . u(z)u(y). The left part of the equivalence points to differences in preference intensities, or distances, and it remains to see how this intended interpretation of the quaternary relation is fully reflected in the subtraction of utilities of individual outcomes on the right side of the representation. We must decide whether the intended representation in terms of the preceding utility differences can be grounded in a choice-data basis, or if we have to search beyond it for the relevant informational basis of a rationalization by means of this utility representation. Going beyond a strict choice-theoretical paradigm, does not imply that the utility function loses its rationalizing power. It is an independent issue to decide what type of data are worth being rationalized by a utility function. The coherence clause bears on the fact that these data should reveal preferences. If, moreover, the informational and the representational roles of the utility function must continue to coincide, then the nonchoice-theoretical informational basis has to be part of the axiomatic characterization of preferences, so that it is also present in the possible utility-representation. A cardinal utility function presupposes precise comparisons if we impose continuity and transitivity axioms on preferences. Gilboa (2009) emphasizes the fact that privileging an ordinal utility may be an efficient

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and parsimonious but an undue informational impoverishment of preferences. It would be if we could effectively accrue observable data that would point to the actual processing of utility differences and comparisons of preferences intensities, if these data jointly reveal some inherent structure of preferences, and if the latter structure could be axiomatized and represented in these terms. Gilboa appears to be in favor of incorporating psychophysical considerations into decision-theory, meaning that he has in mind this dual representational and informational constraint. Beja and Gilboa (1992), in particular, propose a very refined and visual way of associating progressively more stringent conditions on orders of preferences with alternative representations corresponding to distinct levels of discrimination of utility (or other) differences (we refer the reader to this work and also warn that this is not what we present in the argument of the next paragraph). The general problem can be stated as associating utility representations to cases of limited discriminatory power (a level at which intensities cease to be perceived) and to cases in which utility preferences are perceived. Cases below discrimination should not be assimilated to cases of discrimination, or, in other terms, the joint representation of these cases by a single utility function should continue to express that difference, not pretending that nondiscrimination amounts to potential discrimination. In general, given a certain discriminatory power δ (below which an individual cannot tell the difference between two stimuli), we have equivalence classes of indifference. The indifference relation cannot be transitive for stimuli that stand below δ. We can furthermore postulate that, just above δ, the perceptual threshold, differences become progressively noticeable to some degree, which we capture by an increasing probability associated with a preference P, for example, {P # δ 5 0; Pδ1jnd 5 .5; Pδ12jnd 5 .75;. . .} (where jnd stands for “just noticeable difference”). At δ or below, the probability of discrimination is null, just above, at the level of just noticeable differences. Let’s assume its probability is .5; when we go one step further (Gilboa, interestingly, admits stepwise countability of jnds), the probability of discrimination increases, etc. At some level, discrimination is perfect. This gives some stochasticity to the preference relation, which is not essential to the argument but may allude to the fact that δ being deterministic is a limit-case. Suppose that we theoreticians know δ and the values associated with Pδ1jnd1. . .. We therefore model the individual’s preferences by means of a

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series of semiorders, as introduced by Luce (1956) for cases of intransitive indifference. The semiorders can be represented, under certain conditions, by the same utility function: x I # δ y2uðxÞ  uðyÞ 5 0 x Pδ1jnd y2uðxÞ  uðyÞ . 0 ðwith probability 0:5Þ etc. xRy2uðxÞ  uðyÞ . 0 ðwith certaintyÞ: (I denotes indifference, and R a preference relation that is not affected by a probability of discrimination. R is the usual preference relation that can be represented by a utility function unaffected by sensitivity issues). The function u jointly represents all the semiorders induced by successive probability of discrimination given δ. As in Ng (1984), a cardinal utility function (with “subjective significance”) is derivable, if we admit a finite discriminatory power of utility differences. Gilboa (2009, p. 70) reexpresses this fact in a very clear way: “Observe that the uniqueness result depends discontinuously on the jnd δ: the smaller δ, the less freedom we have in choosing the function u, since sup|u(x)u(y)| # δ. But when we consider the case δ 5 0, we are back with a weak order, for which u is only ordinal.” Ordinal utility is then a limit case, psychologically implausible but convenient. It relies on the idealization of transitive indifferences. Once we admit them, we do not need to bother with cardinal utility, even if we believe it can exist. Let’s also note that if we distinguish between indifferences due to a lack of perceptual discrimination and indifferences beyond this threshold, it intuitively amounts to interpreting the former as incomplete preferences rather than as real indifferences. If utility is a measure, we need to clearly distinguish between the limitations inherent to the measure and the nature of what is measured. In the absence of discriminatory threshold, we have transitive indifferences, and we can postulate completeness of preferences with no further ado. But a side effect following from the reasonable admission of finite sensitivity and from the vindication of cardinalism on this basis is that it leaves the nature of preferences more indeterminate than under ordinalism. One way to interpret the standard resistance to cardinalism in decision-theory is then to see it as a by-product of ordinalism, which avoids such retrospective axiomatic complications. Baccelli and Mongin (2016), in a very precise analytical reconstruction of the

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impediments to vindicating a cardinalist position, underline the apparent move from utility being ordinal to preferences having to be themselves ordinal. We can generalize by saying that ordinalism blinds us as to some possible characterizations of preferences that are intuitively available not only if we chose a cardinal benchmark but also independently, like considerations about their completeness. Finally, it is as if ordinalism was not only relative to particular representational possibilities (and their axiomatic bases) but adopted as an exclusive psychological assumption (constraining the axiomatic bases), which it cannot be. Baccelli and Mongin (2016) convincingly attribute to Suppes (Luce & Suppes, 1965; Suppes, 1956, 1961; Suppes & Winet, 1955) a position in decision-theory that combines the admission of the utility function as a formal representation of preferences and the rejection that preferences are mere disposition to rank options and therefore of a standard choicetheoretical foundation of utility. So, according to these authors, it means that Suppes accepts the conceptual possibility that a utility function is representational and nonetheless represents preference differences, which points to a coincidence of the informational and representation roles of utility beyond what choice-data can typically provide. At the same time, Suppes is lucid that deriving a utility function that would represent differences in preferences cannot but result from an enrichment of the axiomatic structure: if preference differences are not included in the axiomatic structure, the claim that the derived utility function can represent them is unwarranted. Relaxation of the theoretical admissibility of data beyond choice-behavior (in particular introspective judgments) does not loosen up, on the contrary, the logical constraints between an axiomatic characterization of preferences (including preference differences) and its representation. Suppes and Winet (1955) derive the set of axioms and characteristics of the preference relation that support a cardinal utility representation of preference differences. By the same token, they spell out the testability conditions that would ground this representation. Interestingly, although their hypothetical procedure would rely on introspective judgments, they do not seem to us really uncongenial to a potential choice-theoretical procedure. If an individual can rank his preferences of x relative to y and of y relative to z, and if he can state the degree of preference of x over y and of y over z, we can encode this information in a utility inequality u (x) . u(y) . u(z). We can further state that u(x)u(y) . u(y)u(z), if it is the case. Suppes and Winet introduce monetary amounts to be combined

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with the options x, y, and z. The objects of choices now become composite, which can raise some problems, but at the same time this is what makes this procedure potentially behavioral or choice-theoretically founded. By varying the amount of money associated with the options, one can reach equivalent points between x and y and y and z. The measure of preference intensities (through money) is remarkably simple: if more money was asked to yield xBy(1m) than to yield yBz(1m0 ) (if m . m0 ), then we grant that u(x)u(y) . u(y)u(z). We also can use m and m0 to generate the equality u(x)u(y) 5 u(y)u(z). We can vary the amounts of money involved by affine transformation, connecting thereby a set of utility functions that offer a cardinal representation in the sense of representing intensity differences of preferences. From the informational point of view, we have just extracted points of indifference, but we have enriched the domain of the preference relation by now applying it to composite couples (x,m). Consequently, the main problem in Suppes and Winet’s representation procedure may not be its resort to introspective data but its “in the middle of the way” modification of the preference domain. We have to decide to which structure—the intended initial one concerning preferences on options or the instrumental ones introducing option-money couples—the cardinal representation is actually relative to. Finally, we can list a series of problems that are raised by cardinalism in decision-theory with respect to the interaction of representational and informational issues. • One can see a stronger coherence in admitting a combination of representable preference differences in terms of a cardinal utility function and an informational basis that extends beyond choice-data than an attempt to support a cardinal rationalization of standard choicedata. The potential choice-theoretical foundations of cardinalism remain to be investigated. • We can continue to ask how much of a backward constraint ordinalism lays on the informational basis of admissible preference-revealing data and, thereby, on our psychological conception of what preferences amount to. Are they more than a disposition to order? • But also, in an opposed direction, could our disposition to order encompass our ability to higher-order preferences (express preferences over some preferences)? For instance, we could order intensities of preference without being compelled to measure those intensities. Can we obtain an as rich a structure of preferences as cardinalism supposes, in pure ordinal terms?

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•

•

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How can we better formulate and formalize the relationship between representational and informational issues at the level of demonstrations of representation theorems themselves? One subproblem would be to be able to conceive of representation theorems as more or less conservative informational channels. To what extent is an axiomatic characterization of preferences reflected in its representation by a utility function? The case of ordinalism presents a simple fact of informational conservation of rankings from preferences to utility, but we are not sure that an ordinal utility representation is fully conservative of all the structure and aspects of preferences under several possible axiomatizations and types of objects to which they apply (their semantic domain, so to say). It would be a theoretical effort beyond the scope of this book to envision representation demonstration procedures in information-theoretic terms. Basu (1982) emphasizes the relative cardinality of the utility function that one can derive from a “framework of analysis” (a domain for preferences and a set of axioms on preferences) that can lend itself to a cardinal representation. He underlines the role of constraints on the definition of the domain, which do not have the same scope as the constraints on preferences that the axioms impose. A cardinal (or an ordinal, for the matter) utility representation is absolute or general if we work within an “unrestricted domain” assumption (no topological restrictions, no a priori measurability assumptions in the description of the domain). It ceases to be if the domain is further characterized. Hence the relative—and only relative—cardinality of the vNM utility function. Basu shows that we can recover generality if the utility functions are continuous and defined on a connected topological space. It would be interesting to further clarify what informational constraints on the utility functions are inherited from axiomatic versus domainstructural characterizations. What theoretical or esthetical choices guide one way of rationalizing some choice-data rather than another? Parsimony concerns can be in order. But parsimony can be expressed in different ways. Kalai, Rubinstein, & Spiegler (2002), for instance, focus on the minimal number of orderings necessary to explain behavior by a choicefunction (we generalize the issue here to a rationalizing role assigned to a utility function when it is taken as a primitive, as we have explained). When the data are partitioned (for instance, in various indifferent equivalent classes), we can ask whether the same utility

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function is rationalizable in each of the cells or whether some coarser partitions would have the same property of being thus uniquely represented, implying less structure at the level of the rationalization of choices (and less expressivity of the preference relation). Uniqueness of utility, simplicity, and the fine-grainedness of the choice-revealing procedure can then be balanced. Kochov (2010) offers interesting forays on this issue.

1.3.2 Köbberling’s general result Köbberling (2006) uncovers the general assumptions that allow for a utility representation of preference differences. We present this result, which is guided by criteria of generality and intuitiveness. At the core of this result we point to a trade-off between psychological homeomorphism (the fact that the assumptions reflect some intuitive psychological processes) and operational homeomorphism (the fact that the assumptions are related to potential experimental measurement of the preference differences). Let X be a set of outcomes. The main primary assumption is that it is meaningful to order pairs of outcomes by a difference relation k, such that abkcd (’a,b,c,dAX) means that the improvement from a to b is as least as good as the improvement from c to d. Let’s briefly stop at this primary assumption to remark that the natural interpretation of the pair a-b we just gave implies a “displacement” in the option space. Supposing that all displacements between outcomes take place in the same way (that some displacements are not more difficult to achieve than others), a-b can be conceived as a distance. It is now an open issue whether this distance, which is one way of capturing preference differences, is also adequate to capturing preference intensities. Another point is that we expressed, so far, the idea of preference differences by formulas such as: aRbkcRd, meaning precisely that the preference of a over b is “more intense” than the preference of c over d. We have noted, however, that a unified version of preference differences could use the same concept for the preference between single outcomes akb and ckd and for the preferences difference (akb)k(ckd). This is not what we find in the literature because it is more intuitive to think of preferences difference as another relation taking its arguments in X2. Yet this conceptual and notational unification would more clearly express the idea of preference intensities comparison. In Köbberling’s framework, a basic underlying

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preference relation, k0 , is induced by the preference difference relation in a simple way: ak0 b if abkbb. We look for a function u: X-R, such that: u(a)u(b) $ u(c)u(d) iff abkcd. In that case, u would be called a difference representation, and since k implies k0 , it would also be a utility representation of the underlying preference relation. A first representation can be proved on the basis of four standard assumptions and a less standard crucial one, the intuitiveness of which is precisely our focus. The four standard assumptions are the following: 1. k is a weak order on X2, that is, complete and transitive. All pairs in X2 can be ordered by k. 2. k is weakly separable, that is, if ackbc, then adkbc (and if cakcb, then dakdb). Under weak separability, the properties of completeness and transitivity transfer from k to k0 . 3. Richness on X (solvability): k is solvable if xcgabgzc implies that there exists y such that ycBab and if cxgabgcz implies that there exists y such as cyBab. This amounts to a continuity assumption. 4. Neutrality, aaBbb. To these four standard conditions, one must be added to prove the existence of the desired preference difference representation (which is by the same token a cardinal utility representation of the underlying preference relation). 5. Concatenation condition: if abBa0 b0 and bcBb0 c0 , then acBa0 c0 , which is a basic segment addition postulate. On the basis of the four assumptions and the concatenation condition, which is then the crucial condition for a cardinal utility representation, the following equivalence between assumptions 1 and 2 can be proved (Köbberling’s theorem): 1. There exists a difference representation u: X-R with u(a)u(b) $ u (c)u(d) iff abkcd. 2. k is an Archimedean weakly separable weak order on X2 that satisfies neutrality and the concatenation condition. The function u in equivalence 1 is cardinal, unique up to positive affine transformations (scale changes) and also represents the underlying preference relation k0 . The interest of this first representation theorem for preference difference, involving the demonstrability of the existence and adequate form of uniqueness of a utility representation for a preference relation, is that it relies on standard conditions plus a very intuitive one, once we have

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admitted the psychological plausibility of preference comparisons. In that case, the utility function represents in an obvious way our introspective hedonic arbitrages. It is no more psychologically demanding, for instance, than a decision-theory of regret that requires post hoc or ex ante comparisons of the utility of outcomes (see Bourgeois-Gironde, 2017; Loomes and Sugden, 1982, 1987 for specific psychological implications under regret theory). Köbberling uncovers a more general condition than concatenation to support a preference difference representation. If one admits “standard sequences” of indifferent pairs such as a0f0 a1,. . .,an21f0 an and anan21B an21an22B. . . Ba2a1Ba1a0, we can consider a1a0 a measuring rod; so, given k steps in this sequence, we can always have am1k an1kBaman. Then one can prove, for a weakly separable weak order k on X2 such that there are a,bAX and af0 b (k0 is not trivial, and neither is k), the following equivalence between 10 and 20 : 10 k satisfies neutrality, the midpoint condition [abBbd implies baBdb], and the concatenation condition. 20 k satisfies the standard sequence condition. The standard sequence condition is thus a more general condition than the ones supporting Köbberling’s first formulation of a theorem for cardinal utility representation. Besides its generality, it connects the crucial formal assumption supporting cardinal representability to the idea of an empirical measure. It is, however, more psychologically demanding. In which way? One psychological operation that it would require is the stability of the measurement rod, of the basic unit, across all comparisons and across iterations of steps in the standard sequence of indifferences. It is a different question to wonder whether preferences are subject to measure and to consider that their measure is really independent and stable across their intensity variations. To plagiarize Elster, quoting Montaigne plagiarizing Petrarch, if preferences are desires, “He who can describe how his heart is ablaze is burning on a small pyre.” Köbberling’s generalization of cardinal utility representations of preferences, as implied by a representation of a preference-difference intuitive representation, has the side merit of pointing to the trade-off that may be inherent to all representational endeavor, between psychological realism and operationalization.

1.3.3 Dispensing with representation theorems? The connection between representation theorems and psychological underpinnings of preferences, beliefs, and decision-making on the basis of

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the prior two is far from direct and obvious. The main aim that we pursue in this book is to probe the descriptive implications of standard decisiontheoretical axioms. But these axioms are generally motivated not by psychological realism or accuracy but rather by providing the logical basis of a representation theorem. So, we can ask, how far a utility functional representation actually represents basic decision-theoretical ingredients such as preferences and beliefs? Some philosophers have addressed these questions and have inclined toward a negative assessment of the connection between representation theorems and psychology, whether this psychology is of a folk nature (like the very concepts of beliefs and preferences) or of a more scientifically oriented one (Meacham & Weisberg, 2011; Zynda, 2000). Meacham and Weisberg consider that representation theorems, in their usual conception by decision-theorists, aim at characterizing beliefs and preferences. For them, this is inevitable as soon as decision-theory formalizes these folk notions. It has to be noted that formalizing folk notions does not make them more scientific. But an argument about the connection between representation and psychology is independent of what type of psychology we adopt. The idea that representation theorems characterize psychological entities or processes has to be made explicit. These authors, classically, distinguish two sorts of characterization issues, descriptive and normative, meaning respectively what preferences and beliefs are and what they ought to be. Saying this, we could be led to think that a normative characterization presupposes a descriptive one because how do we tell what a psychological entity ought to be if we don’t know what it is. This is what the authors seem to assume, but by doing so they make the normative characterization dependent on the descriptive one, which goes against the general tendency in decision-theory to assume that a normative standpoint is independent of a descriptive perspective. When we frame the problem of the connection between representation and psychology in the terms of a characterization problem, we happen to put the descriptive issue on the fore. This is in fact what we also do in this book. We consider that normative accounts of decisions, preferences, and beliefs, in terms of rationality, cannot be detached from a descriptive perspective. But this is not because description logically precedes these normative accounts (we think that they can remain independent, if one wishes, and they actually are on the surface) but because normative decision-theoretical models involve

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psychological characterizations. Depending on the sensibility of the theoretician, they are more or less explicit, and our assigned work is to make them as explicit as possible. We therefore agree, but, unlike the mentioned philosophers, we positively defend that axiomatic structures and representation theorems, derived on their basis, characterize, in certain ways, psychological dispositions, entities, and processes. Let’s see and comment on their main arguments. Zynda (2000) identifies the steps, within a larger philosophical argument in view of the vindication of decision-theory in general, that underly the claim that representation theorems characterize beliefs and preferences and that acquire thereby a descriptive or realistic psychological dimension. For Zynda, decision-theory is justified because, it encompasses (1) a rationality condition, (2) a representability condition, and (3) a reality condition. Condition 1 simply means that the axioms of expected utility (the framework considered here) define rational preferences. But these axioms have, at least for some of them, a descriptive and above all an intuitive input. The fact we can intuit what those axioms normatively require does not, of course, entail their descriptive or psychological realism, but it means that we can mentally represent what it would amount to mentally or behaviorally have these preferences (complete, transitive, complying with the axiom of independence in particular). We can as well represent what it would mean to violate these axioms and to find these violations rather intuitive too, especially in the case when we have to justify an axiom-violating behavior. It means that the intuitiveness of axioms and their descriptiveness does not entail their rationality. Stating axioms that give conditions of rationality generates a mental intuitive layer, but this internal description of the way preferences and beliefs can stand vis-à-vis these axioms is not itself constrained by rationality concerns. The axioms simply delineate a mental (and associated behavioral) structure, an internal phenomenology, in which the preferences and beliefs can vary and be intuitively tested. The rationality constraints impose a format in which to think of preferences and beliefs more directly and compellingly, perhaps, than they imply admissible epistemic and motivational duties. Condition 2 states that a person complying with the axioms of expected utility-theory (i.e., not simply activating the mental format implied by the axioms but actualizing and approving them in thought and behavior) can be represented as having beliefs that agree with probability theory and preferences that agree with the maximization of a utility

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function. This representation condition is an as-if one. Representations can play the role of models. Those models are different, though, from the spontaneous intuitions and “mental format” that were involved in condition 1. They do not have to coincide, and in fact they don’t; otherwise, the proper “representational” moment would be eschewed. Representations induce a change of format. They redescribe what was implicit in the axiomatic structure. But they escape intuition in the sense that there is no more (unless the agent is a trained mathematician) intuitive link between the mental representations (in a naïve psychological sense) associated with the intuitive testing of the axioms and what is modeled by a representation theorem. The representation of preferences by a utility function and of beliefs by a probability function does not introduce entities that are not intuitive. In fact, they are easier to play with mentally than axioms. They represent, by expressing it in another format, what is implicit in rationality conditions given by axioms, but they break the intuitive link between preferences and their representation. In fact, they delineate, beyond intuition and spontaneous mental associations, what it operationally would mean to be rational. We consider representations to be less descriptive (even if they can be) than axioms and more normatively oriented. They deliver a utility maximization feature that expresses the normative content of axioms but have no direct grip on how the individual can think of her preferences and beliefs. A trained philosopher should be able to consider these two levels in parallel and refrain from forcing their interaction. Representation means a change of analytical level. However, we can continue to ask descriptive questions about representation theorems, in the way we have done in Section 1.3.1, by wondering how much of the descriptive content present in the axioms is still preserved in the characterization of the utility function (in terms of cardinality and ordinality and other analytical features). Taking utility functions as primitives implies a relative independence from the psychological content of axioms (focusing instead on a few structural features of the function), but what preserves the link between utility functions as rationalizing tools and the psychology of preferences and beliefs is the possibility of a representation, even though this link escapes intuition. Judgments about realism and descriptive and normative input should be formulated at the right level. Condition 3, the reality condition, precisely formulates such a judgment, by stating that decision-theorists are compelled to hold that if a person can be represented as endowed with beliefs complying with probability theory and preferences complying with utility maximization,

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then she has these preferences and these beliefs. But considering the comments we have provided in conditions 1 and 2, which suggest a clear dissociation and articulation of the descriptive input at the level of axioms (not negating their normative content) and the normative input at the level of representations (not negating their distinctive descriptive content), we are not at all forced to this conclusion. A person can have or have not such beliefs and preferences and be represented as a utility maximizer. In fact, we agree with Meacham and Weisberg (2011) that there is a possible dissociation between the two levels but not on the same interpretation of what decision-theory and its representational strategy do and mean. It is true that representation theorems do not characterize what our preferences and beliefs are, for the simple reason that they change the format of our folk concepts of beliefs and preferences and that, by doing so, they may lose some of the informational or characterization basis present in the axioms. The characterization of psychologies through representation theorems is looser than what these authors assume decision-theorists think it is, and representation theorems are precisely useful tools to constrain the realism or at least to help to point where and how the question of realism is raised in decision-theory. The key move, we insist, is that, having proved a representation theorem, decision-theorists do not claim, in general, that the individual “has” a utility and a probability function, whatever that would mean, but only that it is possible to reformat her beliefs and preferences in that more tractable model. There remains a sense, however, in which representation theorems can nevertheless characterize psychologies, that is, in functional terms. Without entering here into a long philosophical discussion, we can refer to an interesting position held by Dretske (2004), according to which contents of introspection are not especially unreliable, but mental attitudes toward them can be metacognitively indeterminate. We may fail to recognize them and correctly account for them. We tend to see a similar message in the experimental results, analysis, and interpretation by Abdellaoui, Barrios, and Wakker (2007), which we discuss in some detail in Section 1.4.1. They show that introspective judgments about preference intensities are reliable when analyzed in terms of a prospect-theory utility function. It does not imply that we know how we relate to those contents, that we are clear about our preferences (or beliefs) as attitudes. According to Dretske’s externalist position, our introspective judgments or any mental representations are not intentional and informational because of their intrinsic properties (the fact that they are judgments or

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representations of a certain type) but because they are, although indeterminate, reliably tracking their external content. They reliably fulfill their function in an informational scheme without our being (and needing to be) aware or lucid about the nature and functioning of this scheme. We can note that this function may be what a representation of preferences and beliefs in terms of utility and probability functions abstract out, without necessarily involving any intuitive psychological content at this level. The compatibility of choice-based utility and introspective judgments seen in this philosophical perspective indicates that experiments can reveal the adequacy between functional representations (here in the prospecttheory framework) and spontaneous psychological contents and evaluations. We therefore think that arguments against representation theorems do not bide equally well with respect to ordinalism and cardinalism.

1.4 To the lab 1.4.1 Eyetracking rank-dependent utility In a remarkable experiment, Abdellaoui et al. (2007) managed to demonstrate a correlation between a choice-based utility measure and introspective (choice-less) judgments of strength of preference. The choice-data are corrected for errors when analyzed in terms of prospect-theory. A cardinal utility index is derived from the application of this model that also accounts for the introspective judgments. This clearly militates in favor of the admission of introspective judgments in decision-theory and in economics. And it overcomes a negative appreciation such as the unreliability of introspective judgments compared to strict behavioral procedures. The measure of choice-based utility is performed by Abdellaoui et al. through a special so-called trade-off method that has the advantage of being valid both under expected utility- and prospect-theory. Prospecttheory is, in this experimental context, the utility paradigm that unifies their data on choice-based and introspective utility measures. We present the trade-off methods in some detail and sketch the use we plan to make of it with our colleagues Germain Lefebvre and Vincent Lenglin (a design owing principally to Vincent) in an eyetracking setting. Having elicited the utility function of subjects through the trade-off method, we subsequently elicit their subjective distortion of probabilities of risky lotteries. We submit them to a series of visual stimuli and estimate the attention they assign to each of the lotteries and their internal properties (payoffs and probabilities).

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Probabilities in Rank Dependent Utility (of which cumulative prospect-theory is a generalization to the cross-domains of gains and losses) are to be interpreted in terms of decision-weights once the lotteries are ranked according to increasing or decreasing payoffs. Our aim is to investigate which of the two factors of risky choices, either the processing of utility or the processing of probability, is the object of more cognitive attention on the part of the individuals. Adding this type of data to Abdellaoui et al.’s experiment can yield significant information about the different cognitive processes involved in the processing of prospect-theory ingredients and give more cognitive physiological ground to a homeomorphic view of that decision model. In Abdellaoui (2000)’s elicitation method, the individual faces two binaries lotteries: L1ðx0 ; p1 ; xb ; ð1 2 p1 ÞÞ versus L2ðx1 ; p1 ; xa ; ð1 2 p1 ÞÞ; with x1 . x0 . xb . xa : The individual has to choose for which value of x1 , she is indifferent between L1 and L2. Schematically, for instance, see Fig. 1.4. At the indifferent point, for any decision model, as, for instance, expected utility-theory and prospect-theory, we know that the subjective values of lotteries, UðLÞ, are equal. We have: UðL1Þ 5 UðL2Þ

(1.1)

Let’s take Rank Dependent Utility (RDU) (Quiggin, 1982), which in the context of Abdellaoui et al.’s experiment, is equivalent to prospecttheory. According to the equality 1.1 and the RDU formula, we have: w ðp1 Þuðx0 Þ 1 ð1 2 w ðp1 ÞÞuðxb Þ 5 w ðp1 Þuðx1 Þ 1 ð1 2 w ðp1 ÞÞuðxa Þ

(1.2)

We can extract uðx1 Þ: uðx1 Þ 5

p1=.5

1 2 w ðp1 Þ ½uðxb Þ 2 uðxa Þ 1 uðx0 Þ w ðp1 Þ

x0=200

L1

p1=.5

∼ Xb=50

(1.3)

X1=?

L2 Xa=10

Figure 1.4 Elicitation of indifference values in the trade-off method.

1 – p1 =.5

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To draw the utility function, we keep constant the utility difference between the best outcome of L1 and the best outcome of L2 [in the first iteration of this procedure, it is then u(x0) 5 u(x1)]. Applying Eq. (1.3),we then obtain: uðx1 Þ 2 uðx0 Þ 5

1 2 w ðp1 Þ ½uðxb Þ 2 uðxa Þ w ðp1 Þ

(1.4)

Graphically, this is shown in Fig. 1.5. We can continue to keep constant the difference between the best outcomes of two consecutive lotteries by assigning to x0 in each new lottery the value, elicited in L2, that generated the indifference between the two lotteries in the previous trial. In the two graphs in Fig. 1.7, we illustrate the successive iterative steps in the procedure. Let’s consider the example in Fig. 1.6. Now the difference between u(x1) and u(x2) should remain the same as between u(x0) and u(x1) because the equality in Eq. 1.4 depends only

Figure 1.5 First iteration in utility equivalent steps in the trade-off method. p1=.5

x1=400

∼

L1 Xb=50

L2 Xa=10

Figure 1.6 Finding the equivalent point in the trade-off method, iterated trial.

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on choice parameters that were exogenously fixed: xa, xb, and p1. Graphically, the next iterations are illustrated in Fig. 1.7. By this method, the nonparametric experimental induction of the utility function does not depend on w(p). We can take any psychological distortion of probability (concave, convex, S-shaped), and the utility

Figure 1.7 Iteratively eliciting the individual utility function by the trade-off method. The overall curve is the utility function that fits the data. It is obtained by linear interpolation performed between the data points. Iterations calculus is indicated along the y-axis.

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function remains the same. We can then plan on this theoretical and practical ground to investigate the separate cognitive underpinnings of probability and utility in RDU. We already know, thanks to Abdellaoui et al. (2007), that introspective judgments of utility coincide with a prospecttheory utility index of actual choices, meaning that these judgments are a reliable source of information tracking, in their own way, the utility of outcomes. We also know that, theoretically, there is no interference between a utility function and a probability distortion function in the applied RDU model. We want to check whether cognitive processes, furthermore, reveal this theoretical separation. If yes, RDU is exemplarily homeomorphic; if not, the measurement of choice-based utility reflects some introspective judgments of utility that are in part affected by probabilistic distortion, which the application of prospect-theory over Abdellaoui et al.’s data actually meant to correct but which in turn limits the neat homeomorphism of that decision model. The hypothesis to investigate, then, is that decision-weights, as formulated in RDU, transpose in attention-weights such as they can be measured in an eyetracking setting. It is a direct experimental transposition of Diecidue and Wakker’s definition of the cognitive core of RDU: “The intuition of rank-dependence entails that the attention given to an outcome depends not only on the probability of the outcome but also on the favorability of the outcome in comparison to the other possible outcomes” (Diecidue and Wakker, 2001, p. 284). The validity of this experiment relies on the subvalidation of some presuppositions. Only if they are validated do we get a clear correspondence between a clean (not affected by probability processing) cardinal utility index, reliable introspective judgments tracking utility per se, and cognitive processes of rank-dependent utility in terms of the cumulative probability of increasing one’s utility (the “favorability of the outcome”), following the predicted pattern of the same model that independently provides the cardinal utility index. Presupposition 1: The cognitive proxy of a decisional weight in RDU is the relative attention dedicated to the figuration of that weight in a graphically represented lottery. Presupposition 2: This cognitive proxy can be captured by a relative fixation duration in an eyetracking setting (see, initially, Just & Carpenter, 1976).

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We can parametrize the stimuli so that, for instance, 30% of allocated time for a stimulus visualization hypothetically corresponds to a decisionweight of 0.3. Moreover, extrema in RDU rankings have more important decision-weights than middle outcomes. This is a secondary hypothesis acting as a control for the validity of presuppositions 1 and 2. More details on this experiment and its conclusions should come in an independent study by Lenglin, Lefebvre, and Bourgeois-Gironde.

1.4.2 Does the lack of episodic experience lead to von Neumann-Morgenstern rationality? We have argued in Section 1.1.3, that the introduction of episodes as basic objects of preferences to be represented by a corresponding cardinal utility function would present a radical departure from the type of cardinalism that is implied by a vNM-utility function. The latter is relative to a framework, in terms of domain of preferences and axioms on preferences, which excludes episodes. The experience of episodes (a redundant phrase indeed) is pervasive. We can relate to events in two ways, semantically by their location in time (calendar time) or episodically by remembering their very experience (phenomenological time). This distinction is a staple of the psychology of memory (Tulving, 1972). More recently, it has been carried over to future events. Moreover, Schacter, Addis, and Buckner (2007) have shown that simulating future episodes in imagination relies on neural structures significantly overlapping with the neural basis of episodically remembering the past. Such episodic future simulation helps in daily decision-making, shaping and planning future goals, and, we add, performing experienced utility trade-offs of anticipated outcomes. Taking stock of this fact, we wanted to check whether the episodic simulation of future outcomes, which seems a critical ingredient of an experienced-utility model, was the cause of the deviation from vNM predictions. To do so, we tested a typical violation of the vNM framework, namely the Allais paradox (see Fig. 1.8) with KC, a now deceased patient who presented an exemplar case of episodic memory and future episodic simulation ability deficiency (see Rosenbaum et al., 2005, for a full clinical description of KC and Craver et al., 2014, for the details of our experiment). Despite his widespread brain damage in the hippocampal formation and other brain areas, KC’s semantic memory and implicit memory function were well preserved and had remained stable since the time of his accident. But he suffered from profound anterograde and retrograde

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Figure 1.8 The Allais paradox elicits inconsistent vNM preferences AgB in situation 1 and BgA in situation B, while it is clear from this table that once the similar contingencies associated with lottery tickets numbered 12100 are “rationally” ignored, choices A and B, in situations 1 and 2, are identical. Preference reversal is then an apparent violation of rationality.

amnesia for episodic memories, that is, memories for personal experiences, and was impaired in his ability to anticipate future events. It is important, in order to be able to interpret our results, to emphasize that, in contrast, the cingulate and orbitofrontal cortices were comparatively spared bilaterally. We administrated KC several versions of the Allais paradox and reported his performances. Processing the stimuli in the Allais paradox presumably supposes that we can anticipate gains or losses and make comparisons of outcomes. One salient aspect of this decision problem is that foregoing A and choosing B in situation 1 would lead to some regret if ticket 1 were be drawn. This type of simulation may lead people to prefer A. Strikingly, KC performed on the Allais paradox in a way perfectly similar to the majority of the population, presenting the same reversed preferences AgB and BgA across situations 1 and 2 respectively. In his case—although he was allegedly the most representative case of the absence of episodic mental life—violations of vNM utility could not be related to the notion that individuals’ decisions should be accounted for in terms of experienced utility—at least not under the usual and more realistic concept of episode. We have noted that KC’s orbitofrontal cortices were impaired. We suspected that this brain structure could be crucial in Allais behavior. It supports emotional and comparative processes already associated with typical expected utility decision problems (Camille et al., 2004). KC could process the stimuli at a semantic abstract level, as his cognitive abilities and semantic memory were preserved. We therefore hypothesize that he was able to relate “frozen episodes” to typical but detached comparisons leading to the usually observed behavior in these type of tasks.

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Figure 1.9 The absence of Allais behavior in DFT patients.

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With another group of colleagues and cohorts of 24 patients and healthy and clinical (Alzheimer) control subjects, we studied the behavior in the Allais paradox of bvFTD (behavioral variant of frontal temporal dementia) presenting focal lesions in the orbitofrontal cortex (Bertoux et al., 2014). These patients did not exhibit vNM-utility violations in the Allais paradox (see Fig. 1.9). Interestingly too, when tested over intertemporal choices stimuli, they tended to present a neutral utility discounting factor (Bertoux, de Souza, Zamith, Dubois, & Bourgeois-Gironde, 2015). We recall that in Kahneman et al.’s (1997) “episodic” cardinal utility approach, episodes are given in calendar time and that their model neutralizes the relation to future utility that is reflected in discounting utility models. Ironically, perhaps, a bvFTD brain appears to be a good biological model of expected utility-theory. The crucial issue, however, is not the ability to experience episodes but to emotionally relate to them, to be affected by them, which, is, after, all what experienced decision or utility means.

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Bertoux, M., de Souza, L. C., Zamith, P., Dubois, B., & Bourgeois-Gironde, S. (2015). Discounting of future rewards in behavioural variant frontotemporal dementia and Alzheimer’s disease. Neuropsychology, 29(6), 933. Bourgeois-Gironde, S. (2010). Is neuroeconomics doomed by the reverse inference fallacy? Mind & Society, 9(2), 229249. Bourgeois-Gironde, S. (2017). 11 How regret moves individual and collective choices towards rationality. In Handbook of Behavioural Economics and Smart Decision-Making: Rational Decision-Making Within the Bounds of Reason (p. 188). Edward Elgar Publishing. Bouyssou, D., & Vansnick, J. C. (1990). “Utilité cardinale” dans le certain et choix dans le risque. Revue économique, 9791000. Camille, N., Coricelli, G., Sallet, J., Pradat-Diehl, P., Duhamel, J. R., & Sirigu, A. (2004). The involvement of the orbitofrontal cortex in the experience of regret. Science, 304 (5674), 11671170. Craver, C. F., Cova, F., Green, L., Myerson, J., Rosenbaum, R. S., Kwan, D., & Bourgeois-Gironde, S. (2014). An Allais paradox without mental time travel. Hippocampus, 24(11), 13751380. Diecidue, E., & Wakker, P. P. (2001). On the intuition of rank-dependent utility. Journal of Risk and Uncertainty, 23(3), 281298. Dretske, F. (2004). Knowing what you think vs. knowing that you think it. The Externalist Challenge, 2, 389399. Fox, C.R., & Poldrack, R.A. (2009). Prospect theory and the brain. In Neuroeconomics (pp. 145173). Edward Elgar Publishing. Gilboa, I. (2009). Theory of decision under uncertainty (Vol. 1). Cambridge: Cambridge university press. Jensen, J. L. W. V. (1906). Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Mathematica, 30(1), 175193. Just, M. A., & Carpenter, P. A. (1976). Eye fixations and cognitive processes. Cognitive Psychology, 8(4), 441480. Kahneman, D., & Tversky, A. A. (1979). Prospect theory: An analysis of decisions under risk. Econometrica, 47, 278. Kahneman, D., Wakker, P. P., & Sarin, R. (1997). Back to Bentham? Explorations of experienced utility. The Quarterly Journal of Economics, 112(2), 375406. Kalai, G., Rubinstein, A., & Spiegler, R. (2002). Rationalizing choice functions by multiple rationales. Econometrica, 70(6), 24812488. Köbberling, V. (2006). Strength of preference and cardinal utility. Economic Theory, 27(2), 375391. Köbberling, V., & Wakker, P. (2005). An index of loss aversion. Journal of Economic Theory, 122, 119131. Kochov, A. (2010). The epistemic value of a menu and subjective states. Available at SSRN 2715001. Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice under uncertainty. The Economic Journal, 92(368), 805824. Loomes, G., & Sugden, R. (1987). Some implications of a more general form of regret theory. Journal of Economic Theory, 41(2), 270287. Luce, R. D. (1956). Semiorders and a theory of utility discrimination. Econometrica, Journal of the Econometric Society, 178191. Luce, R. D., & Suppes, P. (1965). Preference, utility, and subjective probability. In R. Luce, R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology (Vol. III, pp. 229441). New York: Wiley. Mandler, M. (2006). Cardinality versus ordinality: A suggested compromise. American Economic Review, 96(4), 11141136.

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Meacham, C. J., & Weisberg, J. (2011). Representation theorems and the foundations of decision theory. Australasian Journal of Philosophy, 89(4), 641663. Ng, Y. K. (1984). Expected subjective utility: Is the Neumann-Morgenstern utility the same as the neoclassical’s?. Social Choice and Welfare, 1(3), 177186. Nioche, A., Bourgeois-Gironde, S., & Boraud,T. (2019). An asymmetry of treatment between lotteries involving gains and losses in rhesus monkeys. Scientific Reports (accepted). Poldrack, R. A. (2011). Inferring mental states from neuroimaging data: From reverse inference to large-scale decoding. Neuron, 72(5), 692697. Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior & Organization, 3(4), 323343. Rosenbaum, R. S., Köhler, S., Schacter, D. L., Moscovitch, M., Westmacott, R., Black, S. E., . . . Tulving, E. (2005). The case of KC: Contributions of a memory-impaired person to memory theory. Neuropsychologia, 43(7), 9891021. Schacter, D. L., Addis, D. R., & Buckner, R. L. (2007). Remembering the past to imagine the future: The prospective brain. Nature Reviews Neuroscience, 8(9), 657. Suppes, P. (1956). The role of subjective probability and utility in decision-making. In J. Neyman (Ed.), Proceedings of the third Berkeley symposium on mathematical statistics and probability (Vol. 5, pp. 6173). Berkeley, CA: University of California Press. Suppes, P. (1961). Behavioristic foundations of utility. Econometrica, 29, 186202. Suppes, P., & Winet, M. (1955). An axiomatization of utility based on the notion of utility differences. Management Science, 1, 259270. Skyrms, B., & Narens, L. (2018). Measuring the hedonimeter. Philosophical Studies, 112. Tom, S. M., Fox, C. R., Trepel, C., & Poldrack, R. A. (2007). The neural basis of loss aversion in decision-making under risk. Science, 315(5811), 515518. Trepel, C., Fox, C. R., & Poldrack, R. A. (2005). Prospect theory on the brain? Toward a cognitive neuroscience of decision under risk. Cognitive Brain Research, 23(1), 3450. Tulving, E. (1972). Episodic and semantic memory. Organization of Memory, 1, 381403. Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297323. Wakker, P. P. (2010). Prospect theory: For risk and ambiguity. Cambridge University Press. Wakker, P., & Tversky, A. (1993). An axiomatization of cumulative prospect theory. Journal of Risk and Uncertainty, 7(2), 147175. Yacubian, J., Gläscher, J., Schroeder, K., Sommer, T., Braus, D. F., & Büchel, C. (2006). Dissociable systems for gain-and loss-related value predictions and errors of prediction in the human brain. Journal of Neuroscience, 26(37), 95309537. Zynda, L. (2000). Representation theorems and realism about degrees of belief. Philosophy of Science, 67(1), 4569.

Further reading Fiorillo, C. D., Tobler, P. N., & Schultz, W. (2003). Discrete coding of reward probability and uncertainty by dopamine neurons. Science, 299(5614), 18981902.

CHAPTER 2

Incompleteness 2.1 Sources of incompleteness 2.1.1 Structural and cognitive sources of incompleteness The axiom of completeness in van Neumann and Morgenstern’s and in Savage’s models has an immediate behavioral implication. Given two alternatives, however these alternatives are defined (lotteries or acts), the individual is normatively required to make a choice. But intuition opposes this systematic implication, both descriptively, because cases of indecisiveness are numerous, and normatively due to several intricate reasons that we enumerate in this section. Does the fact that choices necessarily imply that one option is picked and another foregone also imply that preferences follow that pattern and are complete? Is the normative content of an axiom inherited from choicetheoretical foundations or from independent considerations and intuitions? Is the comparability of options a requirement of rationality? What makes, in general, options comparable and subject to ordering? Are potential objects of choice, lying beyond that common ground and resisting preferential ordering, an exogenous limit to rationality? Is the completeness axiom absolutely required to obtain a representation theorem, the latter being the focus of normativity in decision-theory, that is, the link between rationality assumptions on the preference relation and their main behavioral implication in terms of utility-maximization? Is indecisiveness a mark of irrationality, or is it rather pointing toward a more complex underlying cognitive structure of the preference relation over a set of options that requires specific choice-theoretical settings in order to be revealed? We will dwell at length on these questions in this chapter. Choicetheoretical foundations raise a lot of questions about the relevance of the axiom of completeness. More generally, completeness issues point to the tension that may subsist in decision-theory between the adoption of choice-theoretical foundations of the preference relation and its

The Mind Under the Axioms DOI: https://doi.org/10.1016/B978-0-12-815131-0.00002-X

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axiomatization through behavioral and technical axioms in view of its utility-functional representation. It is not simply at the descriptive level that the problem lies but at the core of what we accept as rationality criteria. Hesitations, indecisiveness, unwillingness to make a choice are all perfectly legitimate attitudes, but it is not on such actual grounds that decision-theory must be theoretically questioned or at least that we question it. We again insist on the fact that our primary aim is not to ask how axiomatic decision-theory can be made closer to real choice or nonchoice behavior, but rather how standard decision-theory, which is purely normative, constrains our understanding of choice behavior, which we call its behavioral implications. And because it has such behavioral implications, it must be possible to delineate, at an observable level, ideal experimental settings in which these implications fail or are complied with. Interestingly, then, the case of the completeness axiom is probably the one in which this typical articulation between choice behavior and the axiomatization of the preference relation is under the deepest strictures. This is notably due to the fact that indifference and indecisiveness seem to be observationally indistinguishable. To salvage choice-theoretical foundations, the experimentalist will therefore try to devise a setting in which observational criteria for incompleteness can emerge. We will report on such attempts in our “To the lab” 2.5 section. But as our list of questions makes clear, it is not only at this juncture that the issue of incompleteness is relevant. Even though we could trace back a complete or an incomplete preference relation to relevant observations of choice or nonchoice behavior, we still could entertain some doubts about the normative relevance of this axiom in a deeper sense than for transitivity or independence, for instance. This is acknowledged from the origins. In the Foundations of Statistics (1954, Section 2.6), Savage is explicit on the fragile link between a preference ordering and a choice behavior: Of two acts f and g, it is possible that the person prefers g to f. Loosely speaking, this means that, if he were required to decide between f and g, no other acts being available, he would decide on f. This procedure for testing preference is not entirely adequate, if only because it fails to take account of, or even define, the possibility that the person may not have any preference between f and g, regarding them as equivalent; in which case his choice of f should not be regarded as significant.

In what directly follows this remark, Savage underlines a difference between equivalence and indifference between two options, equivalence being more or less synonymous with incompleteness but incompleteness

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being the name that we can give to this particular state of indeterminacy punctually affecting the preference relation. Two ways of establishing distinguishability between indifference and “equivalence” are suggested and dismissed. One is that, given two options, if adding a small bonus to one of them triggers a preference in its favor, it would be the sign of a previous indifference. This is far from obvious. The other is that, by introspection, the individual knows whether he is indifferent or indecisive. But at this point Savage reaffirms the revealed preference paradigm: “I think it of great importance that preference, and indifference, between f and g be determined, at least in principle, by decisions between acts and not by response to introspective questions.” We discuss recent strategies, within the revealed-preference paradigm, to elicit incompleteness as opposed to indifference in Section 2.4. If we admitted feelings of preference as a criterion for preference itself, we should also admit feelings of probability as a basis of subjective probability. But note that, in one sense, concerning the latter, Savage bases his elicitation of subjective probability on the same material—choice behavior between well-defined options—as for preferences. Yet what is a prior if not the “feeling” that such an event has this amount of probability? We will get back to this contentious point in Section 2.3, when we discuss the joint possibility of incomplete preferences and incomplete beliefs. As we have said, salvaging choice-theoretical foundations for utility theory is a deep methodological move connecting, in a two-way manner, human behavior to a consistent theoretical model of rationality and the set of hypotheses making up this model to empirical verifiability. But this two-way relation in itself is not what provides a foundation or justification for rationality, in either a normative or a descriptive sense. Choices are not normative sources, and axioms, on the other hand, are not meant to be descriptive. The focus of the articulation between norms of rationality and behavioral implications traditionally lies in the possibility of demonstrating a representation theorem. If there is a utility representation that can continue to be maximized under a characterization of the individual’s preference relations modeled without the completeness axiom, we have preserved the core of rationality from the point of view of standard decision-theory. Typical strategies to that effect are discussed in Section 2.2. Some of them directly inherit Savage’s recommended move in the passage we discussed, which is to consider, instead of the usual “preferred to” relation, the weaker one “not preferred to.” This is the one that representation constructions will target.

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For the nonce, we can note that Savage dismisses two very distinct ways of probing incompleteness: either by manipulating the options in terms of their relative utility weights or by resorting to introspective criteria. He opts to weaken the preference relation with no specific concern about the external or internal source of the phenomenon of incompleteness. We will, however, distinguish between “cognitive” and “structural” sources of incompleteness. In principle, the axiomatic approach and the proof of a representation theorem in the absence of the completeness axiom can remain neutral about sources of incompleteness. Typical solutions, such as adopting partial orders that preferentially structure the set of options and multiutility representations, can indifferently reflect indecisiveness due to either uncertain preferences or the incomparability of options. On the other hand, we can wonder whether this neutrality does not miss something interesting about understanding the nature of the preference relation and the construction of a theory of preferences. Axiomatic neutrality about the sources of incompleteness would miss out on fundamental discussions through which a better grasp of what normatively connects the possibility of a representation theorem to human psychology and behavior can be gained. The “mindless economics” position can amount to deliberate blindness and, finally, theoretical insufficiency. For instance, what degree of homogeneity, and under which dimension(s), is required between options so that they can be comparable, and, conversely, what type of heterogeneity is present when a preference relation no longer applies (a structural issue)? Or is indecisiveness a matter of preferences not yet shaped, a metapreference to the later expression of a latent preference relation, a type of epistemic uncertainty (cognitive issues)? We further assume that it is not irremediable that an interest in those aspects of preferences and their possible theoretical representation precludes a choice-theoretical approach, keeping the traditional link between choices and the evaluation of preferences. But we are also reminded that the main focus is between the axiomatization of preferences and the possibility of their utilitarian representation, as a core normative behavioral implication. We explore cognitive and structural sources of incompleteness within this twofold bound.

2.1.2 Incomparability and incommensurability Structural motives for incompleteness are related to the nature of options rather than to cognitive factors. But at this level too a distinction can be

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made between features of the options that make them objectively incomparable or categories under which they are subsumed and that may prevent their comparability. We keep the term “incomparability” for the first case and adopt “incommensurability” to refer to the lack of a common criterion for comparison between two options. Incommensurability may correspond to the presence of a conflict of values in categorizing various alternatives. One value can prevail for an option and not for another in that set. Note that it means that in this case the set is not homogenized in terms of a prevailing value. From the axiological point of view, the set is just a juxtaposition of alternatives, and the exercise of applying a preference relation over that set may reveal internal axiological divides within the sets. It is not the preference that is then revealed through the choice of an option but the absence of choice that reveals conflicting values affecting the plain application of the preference relation. Incommensurability in that sense may be accommodated in terms of Levi’s v-admissibility (Levi, 1990). Individuals may maximize their utility with respect to one particular value operating over the set of options, leaving aside options that may be optimal under an alternative axiological appreciation. This value-dependence of preferences, therefore, seals the internal splits of the option sets, which is the same phenomenon as incomplete preferences. Typically, “commodities” can be categorized in terms of utilitarian trade-offs, a type of mutual comparison that cannot apply, for axiological motives, to noncommodities such as friendship or faith, for instance. In this particular example, one can even say that it is the possibility or impossibility of trade-offs, of comparisons under the criterion of utility, that defines the category to which the good belongs. An extensive philosophical literature has addressed this issue (see Chang, 1997). However, it does not fit well with the type of problems that decision-theorists seem to have in mind when they model incomplete preferences. Incomplete preferences concern well-defined domains of choice, unsplit by axiological barriers. In principle, there is a presupposed common measure to apply among the considered alternatives, and yet a preference between them cannot be expressed in some cases. It is, at least, a conceptual possibility: analyze incompleteness when apparently nothing in the options themselves and in the categories or values under which they are subsumed prevents their comparison. One can certainly start this analysis by relying on Sen’s association between maximization (vs optimization) procedures and the parameterization

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of the act of choice itself affecting the value of options to be compared (Sen, 1997). The first distinction is one concerning the conceptual and analytical priority between choice and preferences and how this affects the question of the completeness of preferences. One can have preferences over a set of options given the act of choice (forced or not) leading to an optimal outcome in that set. Those are preferences that are conditional on choices. Differently, one can have comprehensive preferences over an enriched set of options in the sense that that set now includes considerations about the choice process or about parameters affecting the act of choice itself. In simple terms, either the choice process conditions preferences (and imposes completeness, as in the standard revealed-preferences approach), or the choice process is itself submitted to preferences. This analytic reversal suggested by Sen is, of course, radical in terms of bypassing the standard choice-theoretical foundations of preferences. We are tempted to maintain them as long as possible and to see whether, in their own terms, completeness itself could be revealed; we will discuss attempts in that sense in Sections 2.4 and 2.5. But Sen offers a characterization of the sources of incompleteness that relates to the preceding axiological considerations and maintains sufficient a priori homogeneity in the option set, so that it constitutes a distinct approach from that in terms of maximization over a particular value (Levi’s v-admissibility). The internalization of the act of choice itself or of some of its parameters can be associated with the presence of norms or values but also, more basically, with contextual factors such as menu-dependence (the extension of the set of options) and chooser-dependence. The frontier is subtle between the presence of values and these contextual parameters. Standard choice-theoretical foundations of preference, which by definition impose the completeness of the preference relation, impose contextual neutrality: the choice is independent of the contraction, of the extension, and of the temporal situation (now or in the future) of the option set (menu-independence). It is also independent of who chooses and for whom (chooser-independence). The introduction of contextual dependence along those parameters is easily associated with ethical norms: Menu-dependence can indicate a metapreference for the freedom of choice, and chooser-dependence can be associated with strategic nobility (if somebody leaves me the choice, I may favor her preferences over mine) or to forms of politeness. This type of contextual dependence locates the act of choice itself as a

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potential value-bearer and as a parameter of the options to be considered, within the set of alternatives, which otherwise would be a readymade and normatively neutral space immediately fit to unimpeded utilitarian trade-offs. In Sen (1997), the act of choice is the focus of normative evaluation, not the preference relation itself that is applied to value-laden objects. This is the main difference from a philosophical analysis in terms of v-admissibility that directly connects preference and axiological divides, in the sense that values drive preferences. Those are two very distinct philosophical strategies. Incommensurability indicates that my values prevent me from comparing two options, that I cannot express any preferences between them because this would entail an axiological conflict. But when the act of choice itself becomes endowed with value and associated with norms, the preference can remain normatively neutral while the types of objects upon which it applies may not be. When considered in a nonparametric way, choices operate over a homogeneous and full-fledged set of options and consist in picking up an optimal option that in the end is tantamount to optimizing a utility object-function. The introduction of contextual parameters—that is, the internalization of aspects of the act of choice into the evaluation of the options themselves, making the choices partially dependent on preferences—undermines the approach of choice in terms of optimization. Now, choices depending on preferences can help in picking a maximal element within the set of options that does not necessarily coincide with the nonparametric optimal one. For instance, if my choice is dependent on a menu such that I prefer not to pick up a mango when there is only one in the option set and the other two fruits are apples, but I know that my partner’s preference is also for mangoes but will willingly go for a mango if there is another mango in the set, my change from apple in the first case to mango in the second set may look like a preference reversal, but it is not. It is just that the maximal choice (apple) in the first case, driven by contextual factors, does not coincide with the optimal choice (mango), which prevails either when the menu does not create this type of choice-shift or when the choice parameters are ignored (optimization whatever the menu is). In Sen’s approach, the difference between optimization and maximization, which ensues from his reversal of the analytical priority order between choice and preference, corresponds, respectively, to a (forced) completeness of preferences and (possibly) incomplete preferences.

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An optimal best-response function B(S, R) over a set of alternatives S and a preference relation R can be expressed as such: BðS; RÞ 5 fxjxAS & ’yAS: xRyg: Introducing menu-dependence, we need to restrict a revealed preference relation RS of a choice function C(S) over a particular menu S (a menu is an intuitive parameter to manipulate extensions to other parameters can follow similar lines). We now start from the observed menudependent preference: xRS y3fxAC ðSÞ & yASg: It is clear that RS can be incomplete by definition. Two options that happen not to be compared in S by this preference relation are de facto not ranked together. The universal quantifier in the definition of the optimal best-response function avoids this possibility. The type of choice process that corresponds to menu-dependent preferences and choice function is maximization in the sense that the process is satisfied when an option is found that is not known to be worse than another in S. By contrast, B(S,R) supposes that the set S is wholly scanned and that the best element is selected. This calls for two remarks, one on the source of incompleteness and the other on a conceptual stricture in Sen’s approach. As long as an option is not known to be better than the one currently selected by the menu-dependent choice function, the preference relation is satisfied. Under such a formulation, we characterize only what Sen himself labels a form of “tentative incompleteness.” If more effort were put on the scanning of the option set and on an as complete as possible ranking of all options obtained, we would “complete” the preference relation. In that respect, the maximization perspective is then akin to a cognitive satisficing procedure in the spirit of Simon. Yet this is not the conceptual result we were expecting. Such a result concerns what Sen calls “assertive incompleteness,” that is, when some pair of alternatives (which are then the focus of comparison and not just in the nonranked background of an option-set) is asserted to be nonrankable. This type of incompleteness is not solved by putting more cognitive or time resources into the ranking process. Rather, it is the case when optimization cannot be reached but maximization has been performed over the full set of options. If maximization is extended to the limits of the option set, mimicking the usual optimization approach, then assertive

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incompleteness can be observed. If no incompleteness is observed, the difference between optimization and maximization is canceled out. Two central theorems proved by Sen show precisely that maximization can be replicated by an optimizing procedure but not the reverse. Only an acyclic preference relation (not a complete or a transitive one) is needed for maximization over an option-set while optimization requires completeness and transitivity. A conceptual stricture now arises. By extending a maximization procedure over the full option-set (all comparisons being made), we attempt to turn the menu-dependent choice function that defines maximization into a menu-independent revealing-preference choice function. If, under that extensive procedure, we meet with apparent “assertive incompleteness” that we then cannot account for in terms of menu-dependence or by any other contextual parameters of the act of choice, having procedurally frozen them out, what we meet with indeed is a principled but seemingly unexplainable failure of the choice-theoretical revealed-preference procedure. The picture, at this stage, is as follows. We can adopt a parametric view of choices that combines well with a maximization view of choice and preference incompleteness. Assertive incompleteness can be met and explained, in that perspective, in terms of principles (values, or, more neutrally, contextual parameters) of choices that are internalized in the option set. The point is then to disambiguate between tentative incompleteness (due to the application of satisficing rules in the completion of choices) and assertive incompleteness (due to the principled and extensive application of a choice-parameter, for instance menudependence). Or we want to be sure that we meet a form of incompleteness that goes beyond the assertion of a principle or a value, thereby connecting preference incompleteness to choice-theoretical procedures. But, in that case, the extension of a typical choicetheoretical procedure, in order to complete as far as possible an initially contextually dependent incomplete preference relation, may lead to uninterpretable failures of that procedure and, if not that, may induce forced choices, obfuscating incompleteness.

2.1.3 Preference for flexibility and incomplete preferences David Kreps, in a seminal 1979 article, proves the following representation theorem for a type of so-called preference for flexibility relation over opportunity sets.

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If Z, a set of outcomes (or primary options), is finite, if a binary relation _k on X (the set of menus, nonempty sets of Z) is complete and transitive and satisfies basic conditions, then: • xðAXÞ+x0 ðAXÞ.x 6 x0 • x Bx , x0 .’xv; x , xvBx , x0 , xv if and only if there exists a finite set S, a function U: Z 3 S-R, and a strictly increasing function u: RS-R, such that if w: X-RS is defined by: ðw ðxÞÞðsÞ: 5 max U ðz; sÞ: Then u3w (: X-R) represents _k. Let’s paraphrase in order to clarify the conceptual connection we are interested in between preference for flexibility and incompleteness of preferences. As such, those notions do not appear in this spelling of Kreps’s representation theorem, but they play a crucial underlying role. Kreps is interested in finding consistency conditions for deferred choices. Choices that cannot be made at time t—and we interpret this by saying that the preference is temporarily incomplete at that time—are deferred to a later time, say t 1 1 (if we consider two periods, without specifying the intermediary delay). The basic logic of such deferrals is to leave the options open for t 1 1. Nothing in Kreps’s account and representation theorem forces the final choice for a particular option to be made; the choice, in principle (although Kreps does not envision this situation), could be indefinitely deferred (if we consider choices over meals and the open set of options as restaurant menus, this, of course, means death by starvation, but this is another problem). It means that the basic preference relations over final options (meals) can remain incomplete, but the preceding theorem aims at providing classical consistency conditions (especially completeness and transitivity) over opportunity sets (menus), over which choices can be made at some point. Preference for flexibility is then a classical preference relation driven by an incomplete basic one. The idea is to express a basic incomplete preference relation, in terms of a standard preference relation regarding menus, by specifying logical links between incompleteness and flexibility. An example of inconsistency between choices of options and menu would be the following case: when a client asks for pineapple for dessert and is told by the waiter that they also serve peaches. The client maintains her choice for pineapple, but when the waiter further informs her that they also have bananas, she then switches to peaches. It is not clear that the initial preferences of the client were complete or not (even though a

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choice has been made spontaneously)—or rather just exemplified a form of inconsistency that goes beyond what can be intuitively expected from choices and menu-flexibility. Preference for flexibility means that the union of two opportunity sets x and x0 cannot be less preferred than either x or x0 . Deferring one’s choice—or keeping one’s instrumental freedom of choice—makes one better off than choosing now. This is the situation modeled by Kreps. It does not have to be psychologically overgeneralized. In some cases, the fact that I know clearly my preferences can lead me to prefer to choose now once and for all. It is not sure what I can gain by keeping my choices for later if I do not anticipate a change in my tastes. In slightly different cases, I can be to some degree uncertain about my current preferences, but I can prefer to precommit to one’s specific option rather than postponing the pain of choices for later. In Gul and Pesendorfer’s (2001) model, for instance, small menus are preferred to large ones for this reason. An interesting result by Pejsachowicz and Toussaert (2017) is that the conceptual association between incomplete preferences and preference for flexibility imposes a systematic resort to monotonic choice-theoretical preferences; that is, x will be systematically chosen over x0 as long as x + x0 . What is shown by the authors is an incompatibility between incomplete preferences over options, preference for flexibility, and the fact that in some cases the inclusion of menu x0 into menu x does not systematically entail a preference for x. We certainly wish this not to be the case, if the individual is in fact certain about her choice of x0 . It means that, in this model, precommitment as an option cannot be deduced. As long as there is indecisiveness, even very local, the whole choice-theoretical relation becomes monotonic, or, in other terms, a general preference for flexibility is induced. We may deem this connection between incomplete preferences and preference for flexibility too strong, even though it seems to rely on mainstream views in the literature since Kreps and enjoys some psychological plausibility. Making a link between incomplete preferences and preference for flexibility has become relatively standard, especially in an effort to establish possible choice-theoretical foundations of incomplete preferences. On the one side, indecisiveness, as a psychological state of mind, can be formalized by an incomplete preference relation. On the other side, preference for flexibility is an observable behavioral phenomenon. It is then tempting to connect the two notions and use a monotonic preference relation over opportunity sets, as a model for preference for flexibility. We can thereby

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obtain a regular utility representation of that preference relation. We now briefly discuss several conceptual and formal aspects on which such a result is built. First, both at a very intuitive level and also as a theoretical consequence of Pejsachowicz and Toussaert’s model, it is not clear that flexibility is a consequence of incompleteness rather than incompleteness being a side effect of flexibility. In particular, if a very local level of genuine incompleteness (of the primary preference relation supposed to capture my inner tastes) entails a generalized nonmonotonic choice-theoretical preference relation, then the systematic observation of a preference for flexibility cannot provide an observational basis to discriminate between incomplete and complete primary preferences. How can we know then, theoretically, that incompleteness is not a side effect of a generalized preference for flexibility? The notion of incompleteness and the generalization of a preference for flexibility make an interesting use of other standard rationality axioms. In particular, completeness can be associated with the presence (as we have seen in the previous Subsection 2.1.2) of several preference relations (and utility criteria under which the preference relation is maximized), leading to disjoint subparts in the overall option set. It means that the preference relations themselves are standardly continuous but that some discontinuity is induced in the option set by the juxtaposition of several partial orders. An overall single preference relation over the whole option set is then discontinuous if the primary preference relation is incomplete. Interestingly then, continuity, which is not reputed to be a testable behavioral axiom, can be instrumentalized in a conceivable preference-revealing device for incomplete preferences. It is sensible to think that if A t B (a usual notation for incompleteness), then every option in a continuous or sufficiently finely discrete set, in the vicinity of A or B, will also give rise to incompleteness. The use of independence in a classical way is, for its part, a source of generalization over the whole set of options of a nonmonotonic choice-theoretical preference relation. If we want to avoid the incompatibility that these authors point out between nonmonotonicity, incompleteness, and preference for flexibility (as a potential incomplete preference-revealing device), it might be interesting to adopt a weakened form of independence. These internal axiomatic links show how classical axioms over the preference relation are interdependent and that seeking behavioral foundations to either support or weaken one of them irremediably involves other

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axioms, thereby rendering indeterminate the arbitrary split between behavioral and nonbehavioral axioms, according to a classical DuhemQuine phenomenon. A remaining issue is that most current models of a connection between a choice-theoretical basis for incomplete preferences and a preference for flexibility à la Kreps (by means of intermediary axioms) supposes building a link between two distinct preference relations. In particular and rather obviously, that link must be established between a choice-theoretical relation over menus and an “inner” preference relation that we can suppose or infer to be incomplete. (In the models discussed in this subsection, we can note that for Kreps this primary preference relation bears on final options, consumption items, while for Pejsachowicz and Toussaert, it continues to be on menus.) This is also the case for other influential models on the same issue, such as Gilboa, Maccheroni, Marinacci, and Schmeidler (2010), Kopylov (2009), Danan and Ziegelmeyer (2006), Lehrer and Teper (2011), and Nehring (2009) (in the latter case, however, the incompleteness is deferred to beliefs, and we will envision this particular case in Section 2.3). The underlying issue is that the choice-theoretical relation (on menus) is conceived as a means to potentially complete the primary preference relation or, if not to complete it at the psychological level, to make it consistent, jointly (and here lies the tension) with a revealed preference framework and with a rationalization through a classical axiomatic structure. Several remarks can be made about this typical way of addressing the problem. First, it has been convincingly argued by Mandler (2005) that connecting psychologically incomplete primary preferences with an intransitive revealed preference relation does not expose individuals to irrational behavior (such as money-pump manipulations). But, at a more fundamental level, we may wish that the preference relation we assume or think we observe incomplete on final options be also the one we keep analyzing when applying it to menus. After all, the individual continues in both cases, in front of single items or menus, to express his preferences. We don’t want or need to imagine him exerting another cognitive faculty when he chooses over meals and chooses over menus. It may be the case that menus help him deal with his incomplete preferences over items, but it is the same preference relation that is exerted on menus, and, even if behavior is not the same over meals and menus, the preference relation must have the same properties in both contexts; in a basic sense, it is not complete in one case and incomplete in the other. This is an intuitive

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constraint from which models of incomplete preferences in terms of preference for flexibility depart, and we should try to think along other lines. We can indirectly point to an attempt in this direction based on an article by Arlegi and Nieto (2001). If the typical psychological hypothesis that preference for flexibility stems from uncertainty about one’s (present or future) tastes, one can directly exploit levels of uncertainty, in particular the certain part of the primary preference relation, when connecting incomplete preferences and preference for flexibility. We can isolate the certain part of the preference relation and not apply to it the clause of monotonicity. For those preferences for which I am certain, there is indeed no extra utility gained from the fact that I keep a larger set than needed. A disutility can even be induced by these larger menus when they are not needed (a psychological of incompleteness that may counterbalance, in some situations, the advantage of keeping flexibility of one’s choice). It is not a matter of self-commitment but just the idea that the certain (and as such complete, if uncertainty is the source of incompleteness) part of my primary preference relation should be internally isolated in the set X of subsets of Z. Let’s call P this certain subset; the next question is whether we can conceive the less certain preferences over X\P 5 U as necessitating another preference relation than P. Arlegi and Nieto envision a second binary relation for this uncertain subpart U of X. But, interestingly, they do not need a strong monotonicity axiom to apply to the preference over menus in U, which may open the way to conceive of a single preference relation over X (encompassing primary complete and incomplete instances of the preference relation in Z, as well as preferences over opportunity sets, elements of X) with which different levels of certainty are associated.

2.1.4 Uncertainty and confidence about one’s preferences Uncertainty about one’s preferences is not necessarily solvable by extending deliberation time or acquiring more information. I may have taken my time and gathered all the relevant information, and still my preferences remain indeterminate. In that sense, it is not that the terms of the decision are vague, imprecise, or ambiguous. This latter possibility has been formally considered by several authors. For instance, Salles (1998) gives choice-theoretical foundations for a fuzzy preference relation but at the same time critically discusses the very concept of preference fuzziness. Take a binary relation xRy, which may have a certain

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amount of indeterminateness. What it could mean, subjectively, is that the individual is not quite sure how she prefers x to y to some extent. A fuzzy set-theoretical representation of this indeterminateness will typically consist in projecting xRy into an interval [0,1] representing the amount of vagueness of the preference relation. This mimics perceptual judgments. The property of vagueness is indeterminate, conceptually, between the nature of a concept (like color concepts, size concepts, etc.), objective features (estimating the number of sand grains on a beach), and discrimination power of a perceptual or cognitive system. First of all, it may seem awkward to represent vagueness as a specific numerical value in an interval [0,1], as is generally proposed. Vagueness is itself a matter of interval and may justify that an imprecise level of preference xRy may be projected onto a set of preference relations at different levels of confidence. We will come back to this type of proposal. Second, it is not clear whether a binary preference relation actually follows the same patterns and motives of imprecision as perception. The concept of preference is not especially imprecise in itself. And it is possible that the judgments of precision we make over our preferences are more imprecise than our preference themselves. We can keep in mind, in consideration of the uncertainty of preferences as a source of preference incompleteness, that such uncertainty comes from the fact that introspection of one’s preferences may be a difficult matter. In general, it is not clear how much accuracy is gained by raising an issue of vagueness associated with a cognitive ability to a higher metacognitive level. I am not certain that I know the capital of Zambia, my knowledge is vague and uncertain, but I am even more unable to say how much it is vague. What would be the worth of my saying this if my ability to produce such metacognitive assessments tends to be less accurate than my knowledge itself? This fact is not, to our knowledge, considered when metacognitive abilities, such as confidence in one’s preferences (or beliefs), are used to account for indetermination in the preference relation. Yet if a modeler ignores this fact and pursues her line of investigation, as if the metacognitive ability could be sufficiently accurately exerted, she would formally suggest to associate a preference relation with a measure of its imprecision. For instance, she can offer to represent an imprecise preference by a set of precise preference relations (complete orderings), one of which is likely to represent correctly the basic imprecise relation or all of which could potentially represent the basic preference relation. The size of that set would constitute

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a measure of the basic preference relation imprecision. But that size itself cannot be vague without falling into a painful regress. This criticism applies, to some extent only, to the following recent proposal to capture imprecise preference relations in terms of confidence levels in our particular preferences (Hill, 2012, 2013, 2016). In a sense it does fall under that critique, because, although cautiously noncommittal vis-à-vis any form of psychological realism, it assumes that we can get a precise representation of preference indeterminateness in terms of confidence levels, as if we had this ability to finely emit such judgments. But in another way, that model can accommodate imprecision at this level too without necessarily regressing, due to some particular feature in its construction, leaving open the possibility of unresolved indeterminateness. Moreover, it is precise in another way by allowing us to distinguish as clearly as possible between confidence about preferences and beliefs about preferences. The distinction is subtle. The representation that is eventually obtained preserves the separation of beliefs and preferences, which would not be obviously the case if confidence in one’s preferences were represented by means of belief-like or probabilistic assessments of one’s primary preferences, in the guise, for instance, of confidence judgments in one’s knowledge. So it is better to try to conceive confidence in one’s preferences in terms of how confident would be the choice based on that preference if I had to make it rather than in terms of a judgment about how confident I judge my preference to be, which is, of course, subtly distinct but jointly avoids the previously mentioned metacognitive difficulty and does not collapse together belief and preference. Hill (2016) replaces the usual representation of an incomplete preference by a quasi-order (reflexive and transitive) by a representation by means of a set of weak orders (and therefore complete). There is an asymmetric correspondence between the two sorts of representation, but a set of weak orders is more expressive than a quasi-order, in the sense that a single quasi-order can be associated with several weak orders. This plurality of weak orders associated with a partial one can therefore offer a richer representation of a quasi-order representing an incomplete preference. This type of representation, however, leads only to binary judgments as to whether I have confidence in my basic preference (only one weak order corresponds to a quasi-order, so the quasi-order is immediately completed) or several weak orders are needed to represent my preference (lack of confidence), but a measure of confidence levels is not immediately offered, for instance, from the cardinality of the set of weak orders that

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can correspond to a quasi-order. What is required is a nested set of weak orders. We can represent it by a centered (or a noncentered) series of concentric rings, each ring representing a level of confidence associated with preference orders. For instance, if my preference xRy is more confident than my preference wR0 z, R is located in a ring farther from the center than R0 . It means that x would be chosen at R’s level of confidence but w would not be at R’s level. A measure of confidence in terms of the minimal cardinality of preference relations pertaining to a given level or lesser is possible. Hill makes the further assumption that levels of confidence in one’s preferences vary with the stakes involved in the corresponding choices. The intuition is that the higher the stake involved in a decision, the more confidence is required in the preference that would lead to an actual choice on its basis. To each choice situation is then attached not only a set of available alternatives (menus) but a degree of importance of that choice. Choicetheoretical foundations that weaken Sen’s choice-theoretical axioms of choice (Subsection 2.1.2) can be provided. It is indeed possible that the choice function associated with a set of alternatives S, which normally is constrained to yield a singleton (in the case of determinate choice or artificially constrained choice) or a menu (in terms of a model allowing flexibility of choice deferral) can yield the empty set; in that case, the level of confidence to make that choice is not met. A choice function can be rationalized by a confidence order (a distance or an order in the nested set of weak orders) and a stake-level inducing a form of cautiousness in the act of choice. The choice function yields an empty choice if there is no weak order (and not only one order, as when we usually analyze incomplete preferences in terms of a quasi-order) at the confidence and importance levels granting that choice. We dedicate some space to this proposal because it presents distinct features compared to the previous analyses of incomplete preferences. First, in its approach of choice-theoretical foundations of incomplete preferences, it gives the preference relation the primary role, as it should be, in what indeed determines choices, rather than having to deal with the difficulty of getting a revealed incomplete preference through behaviorally complete choices (under typical conditions of revelation). Second, it does not collapse beliefs and preferences. The criterion of nested weak orders as determining a confidence level is a matter of how the preference feels, not a probabilistic judgment. It can itself be considered a metapreferential judgment, to some extent, about how much I prefer to make or not

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make the choice proposed to me. For this reason, it is natural to consider preferences, potentially incomplete, as determined by choice parameters such as the stake and the associated confidence level to that stake. And finally, this approach suggests possibly revealing procedures of incomplete preferences on the basis of levels of confidence in preferences and of the importance of decisions rather than resorting to the usual choice-deferral devices.

2.1.5 On nudging incomplete preferences In popular philosophy, Buridan’s fable of a donkey starving to death because, either being equally thirsty and hungry it cannot decide on a pail of water or a pile of hay or being simply hungry but unable to decide between two equidistant and similar stacks of hay, sometimes illustrates the problem of incomplete preferences. Jean Buridan never conceived that example, but his critics used it against his conception of free will, resuming the same example that had appeared long before in Aristotelian philosophy, for instance in Al Ghazali’s parable: Suppose two similar dates in front of a man, who has a strong desire for them but who is unable to take them both. Surely, he will take one of them, through a quality in him, the nature of which is to differentiate between two similar things.

Several issues are entangled. First, the absence of choice is related to the consideration of two similar options (pithy, sizeable medjool dates) for which we would rather envision indifference between them. Indifference can be solved by the use of a random device because there is no dimension on which to differentiate the options. It is indifferent which one is chosen and it is regret free that that one is chosen and the other foregone. All this suggests tests for discriminating incompleteness and indifference, which is one of the issues that the revelation of an incomplete preference relation meets (see Section 2.5). Presumably, solving incompleteness by a random device is not regret free. Moreover, one basic structural source of incomplete preferences in the comparison of the options along several intrinsic dimensions or background values, which is lacking in the choice between two dates of the same species (it could be different if we were speaking of trading off medjool against deglet noor). For this reason, the hesitation between water and hay gets closer to a case of incompleteness, but let’s notice that what is at stake, in the example, is that such a case is implausible. The implausibility stems from the fact that the will (supposed

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to aspire to the better good, from a moral point of view, but also in practical issues as fit to a donkey’s will) predetermines the choices. When the subject faces two identical options from the point of view of their value (therefore not presenting a conflict of values), the will cannot determine which one to choose, and the judgment and the action are suspended. The parable refutes that it is absurd as it is unlikely that a donkey will starve to death, a clearly dominated option. It is clear that a donkey won’t starve to death, but humans often behave like Buridan’s asses when dealing with their incomplete preferences and difficult choices. In fact, this Buridanic pattern of behavior may turn into a revelatory device for the presence of incomplete preferences. The absurdity or paradoxicality of Buridan’s ass parable is due to the fact that it is pushed beyond the limit of reasonableness. The deadly consequence is in flagrant opposition to what the will (which determines choices according to Buridan’s moral philosophy) can will. Death is too much of a dominated consequence. But, precisely, in contexts wherein the hard choice between two (or more) options is completed by a third dominated one, the latter can solve the ass’s predicament. Weakness of the will that comes by degree (contrary to Buridan’s moral philosophy) can be admitted and makes us divert from the optimal choice when this choice is embedded in a structure (similarity, equidistance, difficult tradeoffs) leading to incomplete preferences. In that case, the choice is not (potentially indefinitely) postponed or deferred, as we have seen in most models of uncertain preferences, but recoils into a dominated option. Psychologists (Tversky, Sattath, & Slovic, 1988) have documented our unwillingness to perform trade-offs between options presenting multiple (even two) characteristics. They would rather go for an option that presents strictly fewer advantages along the concerned dimension but that avoids computing a compensatory assessment (of how much the quality Q1 of an option A compensates for the quality Q2 of an option B, and vice-versa). Bachi and Spiegler (2018) have recently envisioned the market equilibrium properties in a competitive game involving trade-off-averse individuals. These individuals will typically have incomplete preferences, in a Buridan’s ass way, due to their aversion to compute compensatory tradeoffs (all the more if they are under time and competition pressures) and prefer options, albeit dominated, that are easier to cognitively process. This work has the particular characteristic of being the rarest, to our knowledge, to envision equilibrium properties of incomplete preferences

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of a certain sort. It also relates to choice-structures pointing to environmental or mechanism design solutions to deal with deficiencies in preference expression psychological ability, deficiencies that are ignored in the usual choice-theoretical approach. However, the choice-theoretical literature has studied the impact of a noncompensatory option on the violation of rationality or of the independence axiom. Rubinstein (1988) as analysed this phenomenon in terms of the use of similarity heuristics. Taking a usual common-ratio Allais problem such as the following: L1 5 ð4000; 0:8Þ; L2 5 ð3000; 1Þ; L3 5 ð4000; 0:2Þ; L4 ð3000; 0:25Þ: We observe the modal choice of L2 over L1 and L3 over L4, a violation of the expected utility theory, which can be accounted for by the fact that people use similarity and dissimilarity heuristics (probability 0.2 is sufficiently similar to 0.25 to consider only the amount of money at stake, whereas 1 is sufficiently different from 0.8 not to compute the expected gains). If one characteristic is processed at a time rather than a trade-off computed, the expected utility relying on trade-offs between utility and expectation, it is expected that it is violated. But not only the independence axiom is then likely not to be complied with; completeness too, in the sense that sometimes there will be no resort to a particular saliently prevailing dimension (as in the Allais problem) in its absence, leaving the choice indeterminate if the trade-off is eschewed. Bachi and Spiegler (2018) point to switches in the choice situation with three options, two of them giving rise to a compensatory trade-off supposedly hard enough to compute and a third noncompensatory dominated option. The “opt-out” design refers to the fact that, facing a difficult trade-off, the individual can opt out for a third noncompensatory one (this is the case of Buridan’s ass, unless the opting out option is postponed or equivalent to death). The “opt-in” design is when the individual can consume a noncompensatory option or engage in a trade-off between two dominating options. This noncompensatory option is then a defaultalternative, and choosing to trade off means breaking the status quo, which is more unlikely when the dominance gap with the default-option is small. Masatlioglu and Ok (2005) have modeled this situation. Choosing dominating options—because of a scarce use of cognitive resources—has for an immediate consequence that some options—those requiring more cognitive load to be compared—will not be dealt with

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and that the likeliness of a determinate preference relation encompassing them in the option set will decrease. Individuals restrict their attention to a reduced subset of the whole option set. The equilibrium and welfare implications of such a realistic cognitive phenomenon are ambivalent. On the one hand, processing an excess of options incurs a cost, and rational inattention might in principle lead to discarding them, in the case. On the other hand, such a reduction of the option set does not imply a loss of welfare by excluding dominant opportunities. All depends, too, on whether the individual is given, first, the reduced set as a default and then an opt-in option (the opportunity to deal with difficult choices that are potentially profitable) or the whole set first and then resorts to a selfimplemented opt-out option by being content with processing cognitively easy options. Bachi and Spiegler prove in their paper the interesting fact that, expectedly, the switch from an opt-in to an opt-out option increases “market participation” (in our environment, it increases the consideration of options in the whole set of opportunities; in their setting, it means that more interactions and actual transactions between competitors will take place). However, such a switch has a negative equilibrium implication, weakening the pressure to consider dominant alternatives and on the whole, in spite of the incurred cognitive cost to deal with these options, leading to a loss of welfare. This should raise questions for proponents of default-architectures associated with an opting-in option, which is the main import so far in terms of so-called nudging policies. Nudges should be better anchored in behavioral mechanism design (of which they are a sort of popular instance) and, in particular, in a more theoretical appreciation of the equilibrium and welfare implications of the meeting of a cognitive system and the design of a default choice architecture. Facilitating choices—because people procrastinate, exhibit generally decision-avoidance, fail to consider what would be optimal for them, etc.—does not necessarily mean that a default-setting, which may yet contain what is good for them in principle and make them avoid the costs of decision, will in fact lead to their situation being better off. Many normative reasons could be brought forward to justify this opposition to “soft paternalism,” but the reduction of the set of opportunities could be the main one. This is a bone of contention, of course, since soft paternalists precisely assert that they leave the set of options intact, just changing its framing in terms of opting-in and optingout. But we pointed out that this is not as innocuous a change as it seems and that it actually corresponds, from a cognitive point of view, to a

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reduction of the set of options that, in the case of opting-in, induces a lesser engagement with trade-offs and a reduction of market participation (in the setting of Bachi and Spiegler) and of the contemplation of one’s opportunity set (in our case). Finally, incomplete preferences have been shown to correspond either to incommensurable values creating incomparability, to the presence of choice parameters beyond what has to be chosen, to a metapreference for flexibility and choice-deferral, to uncertainty about one’s tastes and a lack of confidence given stakes, or to avoidance of cognitive costs in decisionmaking. This is an array of cognitive motivations to be potentially incorporated in axiomatic decision-theory. In some cases, the theoretical response was to preserve the choice-theoretical foundations of such preferences by exhibiting a potential completing mechanism (especially through the resort to flexibility and choice-deferral), which grants the preservation of core axiomatic properties of preferences on the whole and the possibility of a representation theorem but which does not grant a transparent and unambiguous behavioral basis to identify incompleteness. We have also seen a different possible approach that inverts the order of priority between choices and preferences, bringing preferences and how we cognitively proceed with them (be it in terms of confidence in Subsection 2.1.4 or in terms of cognitive cost in the present subsection) to the fore and making choices as determined by them. The main models of incomplete preferences discussed in the next section reflect these tensions.

2.2 Motives and models of incomplete preferences 2.2.1 Representing incompleteness by multiple-utility functions Aumann (1962) was the first, after some remarks by van Neumann and Morgenstern themselves, to consider the axiom of completeness of preferences as psychologically unrealistic, counterintuitive, and normatively excessive. The preference relation cannot apply just to a subset of the option-set but to all its possible pairs (typically of lotteries). Or, in ordinary parlance, the decision-maker, under the classical decision-theoretical assumptions, cannot refrain from deciding. Aumann asks the question, “Does ‘rationality’ demand that an individual make definite preference comparisons among all possible lotteries (even on a limited set of alternatives)?” We cannot restrain ourselves from feeling that nothing would prevent a preference relation to be complete with respect to such a

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homogenized option-set (a set of lotteries). If it were the case, the type of incomparability that we have envisioned in the previous section, due notably to heterogeneous aspects of options, would no more be what is at stake here but would relate, for instance, to a different aspect of the comparison between options, for example, the perceptual threshold of possible discrimination between them. We could consider that there is a form of incompleteness due to some sort of intrinsic failure of the preference relation itself, conceived in perceptual terms. However, Aumann escapes from this possibility by envisioning in fact heterogeneous cases, reinstating a usual option-dependent form of incompleteness: “I prefer a cup of cocoa to a 75-25 lottery of coffee and tea, but reverse my preference if the ratio is 25-75.” But it is not clear how to fix the indifference point between those lotteries. Interestingly, the absence of an indifference point between composite lotteries might behaviorally reveal incompleteness. There could have been a direct approach, but it would have led to some difficulty that we have incidentally pointed out in Section 2.1.3. Under that approach, called the RichterPeleg representation, we try to represent the incomplete preference relation by a unique utility function u. Thus, for an incomplete preference relation k on a set X, x k y will imply u(x) . u(y) and xgy will imply u(x) . u(y), but there is no equivalence between the preference order and the order on the utility function values. What this representation does is extend an alternatively weak or strict preference relation to an order that is definitionally complete (numbers), but it does not offer a characterization of the preference relation through the tracing back of the latter by the former (see Majumdar & Sen, 1976; Evren & Ok, 2011). The cost for admitting a partial ordering of the option-set is that it imposes a weaker utility representation than usual. If x is preferred to y, we still have u(x) . u(y), but the reverse does not systematically hold anymore if we aim at a single-utility representation. This fact gives rise to the commonly accepted modeling approach of incomplete preferences in terms of a set of utility functions rather than a unique one. To explain the logic of a multiple utility representation of an incomplete preference relation, we can rely on several models such as Dubra, Maccheroni, and Ok (2004), Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2003), or Bewley (2002); they respectively fit with the van Neumann and Morgenstern, Savage, and AnscombeAumann frameworks. Bewley presents a notably distinct framework, some aspects of which we explore in the dedicated 2.2.3 section. As everywhere else in

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this book, we present the basic model while emphasizing some of its explicit or implicit psychological and behavioral implications, as its internal construction can be understood as defining an ideal mental frame prescribed or, at least, sketched by the modeling choices. Let’s consider a set of options X and k a preorder (a reflexive and transitive binary relation over X), interpreted as a preference relation between mutually exclusive options, and define its strict part g as x k y and not y k x. To define the incomplete part of k, we state that not x k y and not y k x, denoted by x ⨝ y. Note that the definition of incompleteness stems from the weak preference relation and its double symmetrical negation. The double symmetrical negation of the strict relation would not discriminate between indifference and incompleteness. Let’s consider now a set of utility functions U. As we want, in most cases, only a finite set, U is in fact a subset of ℝX, and we want k to be represented, in general, by the smallest set U defined over X that contains or induces all other possible such utilitarian set-theoretical representations of k. A result in the literature is that the multiple-utility representation of a preference relation is unique, meaning that k is represented by U if and only if, having defined ΓU: X-ℝU, which associates a value u(x) for a possible u in U to all x in X, another subset of utility functions V on ℝX and a function f: ΓU: X-ℝV are such that for every u(x) and u(y) in ΓU(X), we have u(x) . u(y) iff f(u(x)) . f(u(y)). It means that the usual uniqueness up to monotonic transformation, guaranteeing an ordinal reading of utility, is structurally preserved in the context of representing an incomplete preference relation by a set of utility functions. Geometrically, it means that U is the basis of a cone and that V is a narrower slice of that cone. We need to recover the intuitive feeling of where the correspondence, or reciprocal characterization, stands between the preference order and its multiutility representation. The natural representation we can form is that the individual becomes endowed with multiple Bayesian selves, maximizing the expected utility or subjective expected utility in the connected subparts of her partially ordered preference domain. We don’t need the moment to renounce Bayesianism in the sense of her probabilistic aptitudes and sophistication (more on that in Section 2.3); the individual is probabilistically precise and consistent over her disconnected preference orders. Authors working on incomplete preferences indeed attenuate the disconnection between distinct partial orders by setting some overarching coherence constraints over the utility functions set.

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Dubra et al. (2004) hold that the preference relation obtains between two options when the preferred option is ranked higher by all utility functions. When that is not the case, the preference relation is incomplete, and those options cannot be univocally ranked by the set of utility functions. This is a maximally conservative extension of the usual van Neumann and Morgenstern expected utility framework: x k y iff Ex(u) $ Ey(u) for all uAU. It is also the mirror-image of current standard models of Knightian uncertainty or ambiguity in terms of maximizing a single-utility function with respect to a set of probability measures over uncertain outcomes. Lotteries are definite probabilistically objective outcomes, but the individual is uncertain about her utility when comparing between some lotteries in her option set, as in Aumann’s seminal example. She then maximizes a unique probability function with respect to a set of utility measures. This approach solves the characterization problem left open by a RichterPeleg representation of an incomplete preference relation. The latter type of representation loses information about the structure of the preferences. It imposes the condition that to represent an incomplete preference k, a unique maximal element in the set of option X must be found, while maximizing over a set of utility functions requires that all maximal elements in S be recovered, which equates the informational content of the preference relation and its utility representation. We meet here the same type of problem discussed in Chapter 1, Cardinalism, namely that a representation by a utility function be informationally conservative, which is made possible in this multiutility ordinal framework. Multiutility representations for incomplete preferences have also been achieved in the context of a Savagean subjective expected utility model or AnscombeAumann both objective and subjective expected utility models. Most of these models exploit a symmetry between incomplete beliefs and incomplete preferences. We discuss those in the next section. Bewley (2002), in particular, while modeling incomplete preferences per se, locates its source in incomplete beliefs. A general model of incompleteness would therefore encompass and try to isolate sources of incompleteness and to capture their possible reciprocal implications. It would generically look like the following: A Savagean act f is preferred to an act g if and only if: ΣπðsÞU ðf ðsÞÞ . ΣπðsÞU ðgðsÞÞ’ðπ; U ÞAΦ;

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with Φ being a nonempty set of pairs ðπ; U Þ being respectively a set of priors and a set of utility functions representing the incomplete beliefs and incomplete preferences of the individual (see Galaabaatar & Karni, 2013 for details). Allowing multiple-priors π as a source of incomplete preferences, which in turn must be represented by a set of utility functions U, may be deemed a form of violation of the separability of beliefs and tastes or of state-dependent utility. But, in a Savagean context, it is to be noted that the same behavioral data, observed choices on a set of options, yields a joint representation in terms of a utility and a subjective probability function. A model that captures incomplete preferences in a subjective utility framework would need, by construction, to allow for multiplicity at the levels of probability and utility functions. We indicate two more approaches that shed light on particular modeling options of the incompleteness of preferences, insofar as we want to represent them in utility terms. Dubra and Ok (2002) focus on the possible completion of a partial preference relation. We find this model interesting in the sense that it describes a sort of axiomatically based ideal procedure that takes an incomplete core-preference relation and sequentially rationalizes and completes it within a VNM framework. The rationalizing means is the standard axiom of independence. The individual starts with one determinate preference he can hold between two lotteries p and q. He can then deduce the superiority of a mixing p with another option r over the same mixing between q and r, which is a mere application of the independence axiom. The individual repeats that procedure until he obtains his full (not necessarily complete but maximally complete) preference relation that in turn can be represented by a multiple-utility function. It is interesting to note that procedural rationality is serving standard axiomatically grounded rationality when one of its ingredients is weakened. Another psychologically interesting approach (one equally based on a procedural use of independence) is the one proposed by Manzini and Mariotti (2008). It radically differs, though, from the model discussed so far in this section in the sense that it does not offer a multiutility representation of an incomplete preference relation. In their model, an incomplete preorder can be associated jointly to a unique utility function and to a vagueness function. The vagueness function allows the preference to be represented by intervals of utilities rather than by several utility functions giving pointwise real values. If we gloss over the preceding parallel between multiple-prior accounts of subjective

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uncertainty and multiple-utility accounts of incomplete preferences, we could envision the conjunction between a single-utility function and a vagueness function in terms of capacity. A generic representation of a Savagean act f for an individual endowed with imprecise beliefs and preferences would then look like this: ð0 ðN     vðϑðf Þ , t Þ 2 1 dt 1 v ðϑðf Þ . t Þ dt; 2N

0

with v being a Choquet capacity (as in Schmeidler, 1989) and ϑ its utilitarian equivalent, expressing through a unique vague utility function the incomparability of some acts in the Savagean option set vis-à-vis some probability reference cumulative point t. In all cases, the main concern is how much the representation preserves the informational content and the type of informational limits associated with (preferably) the internal assessment or the behavioral expression of an individual’s preferences. The main modeling trade-offs that we have reported in this section concern two main aspects: (1) multiple precise utility functions versus an imprecise single one and (2) strictly utilitarian sources of preference incompleteness (easier to capture when probabilities are exogenously given as in a van Neumann and Morgenstern framework) or interaction between preference incompleteness and probability imprecision (as is more naturally expected in a Savagean framework). However, these choices about modeling incompleteness cannot be made by simply tampering with the completeness axiom in isolation and rather involving internal connections within the axiomatic structure that, only taken as a whole with a not so neat distinction between technical and behavioral axioms, bears interpretable psychological implications.

2.2.2 Axiomatic interactions due to incompleteness Weakening completeness, while maintaining the possibility of a utility representation of a preference relation, entails interactions with other axioms. It is rather obvious in which sense transitivity is affected. In fact, there is a positive correlation between incompleteness and intransitivity. Mandler (2005) has convincingly but nonstandardly uncovered the reasons why completeness and transitivity are standardly considered joint staples of rational behavior. But their “positive correlation” involves two very different levels of analysis, one of psychological preferences and the other of

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preferences revealed by observable choices, which is precisely the presupposition of the standard view. At one level, if preferences are psychologically intransitive, they will, when confronted with sequences of choices, be subject to money pumps, which is implausible if the individual is rational. At another level, there cannot be incomplete preferences at the behavioral level, the standard approach goes, because it is always possible to constrain and observe a choice between two options. In that case, completeness may be seen as the cause of the irrationality of intransitivity, since the choice is constrained, and it yields a spurious positive correlation between completeness and transitivity, by going back and forth between the two levels of analysis: psychological preferences and revealed preferences. What Mandler shows is that the admission of psychological incompleteness, when transferred into behaviorally observed preferences that are intransitive (which is therefore an inversion of the two levels at which completeness and transitivity were respectively considered in the standard approach), does not yield self-harmful behavior such as supposedly exemplified by money pumps. It is in this sense that incompleteness and intransitivity are positively correlated and consistent with an account of human behavior in terms of weakened rationality axioms. When, however, incompleteness and transitivity are considered jointly in a psychological and behavioral approach, transitivity may be used to complete the preference relation by using alternative criteria of comparability. I may be unable to compare two states of affairs with respect to, for instance, how much they instantiate fairness between individuals, but when those states affairs are considered in terms of, say, efficiency, the comparison becomes possible. This is, of course, quite unsatisfactory since it is transitivity trivially regained by clustering options under different categories. What works for social functions (to some extent) may not transpose to a welcome individual rationality criterion. Manzini and Mariotti (2007), through their use of a vagueness function associated with a single-utility function, raise a relevant question with respect to the issue of criterion change. Instead of changing the criterion (and being led again to a multidimensional maximizing model), they inquire about what level of vagueness in the use of unqualified (by a specific criterion) transitivity can be admitted without losing rationality. Interestingly, they set no limit on the level of vagueness that is compatible with rationality, as long as some standard decision-theoretical axioms play a structural and rationalitymaintenance procedural role, as we have seen in Subsection 2.1.2 when discussing their use of the independence axiom in a similar perspective.

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When two options are incomparable and the preference is locally incomplete, the idea is to keep the choice open (a sort of deferral). Now classical axioms can be applied to higher-order options (built from locally vague comparisons). At some definite level of comparison, when having left some choices open and reconfiguring the option set by creating higherorder comparisons, vagueness is dissolved and the two classical axioms of transitivity and completeness can be applied. It seems to us quite psychologically plausible that axioms apply standardly at some level of procedural shaping of the option set and less standardly when the option set is homogeneously and exogenously given, that is, in the absence of psychological reconfiguration, as is normally the case in standard decision-theoretical approaches that first define the class of objects to which the preference relation uniformly applies. Most of the strictures that we observe between axioms, when admitting incompleteness, come from the fact that, on the one side, homogeneity and exogeneity assumptions are maintained and that, on the other side, two clearly distinct levels, psychological and behavioral preferences, are yet conflated. Another important feature that should be emphasized in admitting incomplete preferences is the structural link with a continuity axiom. Schmeidler (1971) was possibly the first to analytically uncover this. Interestingly, he also uncovered the potentially intuitive appeal of essentially a mathematical fact. Namely, Schmeidler states that a connected topological space X (i.e., a space that does not admit two—or more— separated subsets whose union is the set defining that space) and a binary relation on X that is (1) transitive; (2) open for the strict part of that relation (i.e., for each x in X, we can find a superior part of x (y|y . x) and an inferior part of x (y|x . y) that are open); (3) closed for the weak part of that relation (i.e., replaces open by closed in the definition of assumption 2; and (4) nontrivial (in the sense that there exists some elements x and y in X that are ordered by the strict relation) implying that the weak relation k is complete. In other terms, 1 and 2 (in conjunction with 1, transitivity, and 4, nontriviality) hold if and only if the relation is complete. Schmeidler connects this with intuitiveness and behavioral implications when he emphasizes that “order properties of preference relations have intuitive meaning in the context of the behavioral sciences. This is not the case with topological conditions. (. . .) They may imply, however, a very restrictive condition of plausible nature.” The restriction is that, to obtain completeness, we must assume continuity in the sense defined.

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When X is further specified as a space of lotteries (similar to the space envisioned in Dubra et al., 2004), Dubra (2011) shows that if the preference relation is nontrivial and complies with the independence axiom, then the admission of two of the three axioms of Archimedean axiom, mixture continuity, and completeness entails the third. Mixture continuity is a strengthening of commonly assumed continuity. It formally amounts to the following definition: Mixture continuity: For all p, q, r in a lottery space (the set of probability distributions on X, Δ(X)), given a probability αA[0,1], αp 1 (1α)q k r, and r k αp 1 (1α)q are closed sets. While the Archimedean axiom states only that if we have p g q and q gr, we can find a value αA[0,1], such a ponderation of p and r by that value can raise the standing of r or lower the standing of p in the initial order; α is a lever: p g q g r and for some αA[0,1] αp 1 (1α)r g q g (1α)p 1 αr. Mixture continuity is stronger than the Archimedean axiom. If we have both, we also have completeness. But the logic is more intricate than that, and we can uncover, at this juncture, an issue of level of analysis as before with the connection between transitivity and completeness. It is not as obvious, though. Karni and Safra (2015) show that a preference relation (a preorder as in Schmeidler and Dubra) that is nontrivial and satisfies the independence and Archimedean axioms can be incomplete and yet satisfy mixture continuity. The explanation of such a flagrant formal contradiction lies in the choice of the primitives of the analysis: Dubra takes the weak preference relation k as the primitive of his analysis, while Karni and Safra take g. The standard approach is to take the weak relation as a primitive. If, the argument goes, k is assumed incomplete, then some of its continuity properties are lost, but if the weak preference relation is derived from the strict one taken as primitive, incompleteness of the former can be assumed without loss of any continuity. There is, in that sense, no contradiction between the two approaches as they employ different starting points in the analysis. But this choice of primitives is what makes apparent the behavioral implications of continuity assumptions when formalizing incompleteness. It is possible, if one starts with a strict preference relation rather than a weak one, to preserve continuity (and leverage on this property in view of achieving a utilitarian representation of an incomplete preference relation) and yet have a

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formally admissible incomplete preference relation. Aside from the formal compatibility between continuity and incompleteness it supports, this approach has, in its turn, an intuitive appeal, as it allows to distinguish from the start between incomparability (as incompleteness of the weak preference relation is derived from constraints on the strict one) and indifference (when the weak preference relation goes both ways between x and y).

2.2.3 Status-quo bias and incomplete preferences Indecision is not the only motive of incomplete preferences, although most of the models just discussed consider it in possible different guises and aspects, as its main source. An alternative source could be the fact that two alternatives in the option set are not worth being compared (psychological incompleteness due to lack of interest) or are never compared and become the objects of actual choices (behavioral incompleteness due to irrelevance for the decision-maker of such a comparison). This appears clearly when considering psychological phenomena such as the status quo bias and more generally reference-dependence of choices. The status quo bias entails incomplete preferences in an obvious way. If I have the choice between my present situation where I earn h50,000 a month by going twice a week to an office near my home, and two less well-paying jobs requiring heavy commuting, not only will I never choose one these dominated options, but I will never compare between them and reveal my preferences through an actual choice (we don’t consider hypothetical choices here). When the status quo is inside the option set and dominates the other options, incompleteness follows from the fact that choices will never reveal the preferences. This does not mean that hypothetical preferences could not be expressed here, but it is not immediately clear that they would be well behaved either with respect to completeness. In that case, the status quo is temporarily frozen out of the option set but may remain as a reference point. Going mentally from the status quo to one or the other of the available options according to one dimension may represent a loss for these options and a gain when considering another dimension of comparison. On the other hand, if there no multiple criteria were involved in that type of choice, it is still true that the choice between the dominated options would not take place. Incompleteness deriving from the status quo bias then stems from the presence of a status quo within the option set, the continued use of the

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status quo as a reference point when outside of the option set, and the fact that the actual or hypothetical comparisons involved would not operate along a single dimension. Bleichrodt (2007, 2009) proposes a model of reference-dependent expected utility with incomplete preferences for the case in which the status quo is in the option set. This model distinguishes itself from other models that consider the status quo effect as simply generating indecisiveness, collapsing in a direct way the two phenomena, and using models of incompleteness of the type presented in the former sections in order to account for that more particular phenomenon. Masatlioglu and Ok (2005), for instance, model the status quo bias as an agent having an incomplete preference relation to compare the status quo with another option and when the agent is indecisive, she prefers the status quo, which trivially completes her preference. Given the inclusion of the status quo in the option set, Bleichrodt differently defines upper completeness if at least one of the options dominates the status quo. Such domination means that the option matters for the individual, and this entails comparison and completeness. When two or several options lie “under” the status quo, choices are not made between them, and we have an incomplete preference relation in that set. When the reference point shifts because in the option, alternative perspectives (in terms of gains vs losses, typically) are taken, incompleteness can also ensue. Bleichrodt derives a representation for those two contexts of incompleteness, providing, then, the formal conditions for an incomplete reference-dependent preference relation, unlike most models of reference-dependence that assume completeness and thereby lack choicetheoretical grounding in spite of their behavioral flavor (e.g., Prospect Theory in that case). The question we want to focus on is whether indecision and reference-dependence are two perfectly dissociable sources of incompleteness. If not, a more general modeling of incompleteness using an approach like Bleichrodt’s and other models that capture incompleteness as indecisiveness, such as the one just discussed above, should be combined. For instance, upper completeness due to options lying above the reference point in Bleichrodt’s model can be, as he himself notes, subject to indecisiveness, but what we have in mind here is something that makes these two notions of incompleteness less orthogonal and in fact stemming from a similar source. We suggest that a common source could be restricted attention to a subpart of the option set. Trivially, of course, if not all options are considered and submitted to choice-procedures, there will be

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some choice-theoretically based incompleteness. But we mean something else. Incompleteness due to reference points is due to the fact that dominated options are not considered. Incompleteness due to indecisiveness may be due to the fact that too much information concerning options or too many options are contained in the option set. There is a psychological cost to performing all comparisons that results in an incomplete preference relation (whether psychological or behavioral). Moreover, that psychological cost is not offset by possibly other psychological costs of being indecisive. A potentially more general model, precisely, could play along these two types of cost: the cost of the burden of too many choices and the cost of indecisiveness. The situations presented by the status quo bias and by the abundance of options are symmetrical with respect to these two types of cost. When the reference point dominates the other options and entails incompleteness, the burden of choice is null as the status quo is systematically preferred and there is no indecisiveness. When the choices are too burdensome to be made, we can speak of too high a cost of choices, which entails indecisiveness. In a sense, the case in which there is no choice and no indecision is the situation that an individual, overwhelmed by too many options, would dream to tend. On the other hand, the individual facing a trivial choice situation because of a favorable reference point has no choice to perform, and may be frustrated and wish to tend toward a larger, more interesting option set. It is interesting to lay those two extreme situations on a continuum when the same analysis of incompleteness would apply. There is, in other terms, an internal cognitive tension between the preference for flexibility and the potential costs of incompleteness. The main idea is to consider that, in spite of the fact that referencedependent incomplete preferences may receive a sui generis formalization and that the advantage of that formalization is to obtain referencedependent expected utility compatible with incomplete preferences, it is also plausible to consider reference-dependent incompleteness as a not essentially distinct phenomenon from what has been considered so far under incompleteness in terms of indecisiveness models. The common ground is the restriction of attention to a consideration set. Such a common ground, as defined by Lleras, Masatlioglu, Nakajima, and Ozbay (2017), may encompass such variegated motives of incompleteness (due to the induced restriction of the option set), such as focus on top options (Rubinstein & Salant, 2011), short-listing options (Manzini & Mariotti,

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2012), and in general narrowing down the option set, avoiding seeking in its extremely far reaches, or using a criterion (possibly irrelevant as in post hoc rationalization of choices) that makes the options more similar than they are and discards those that cannot be thus subsumed. Following Lleras et al., completeness over the restricted consideration set is recovered because either inattention to the discarded options is complete (unawareness) or because we can apply a consistency choicetheoretical principle when restricting the option set. Namely, when y is preferred in some large option set wherein x is also present, but x is preferred in a smaller option set where y is also present, the idea is to consider the choice over the smaller set as revealing the actual preference relation rather than inducing a cycle. It thereby relates the degree of “revelatoriness” due to choices to the amount of attention required by an individual over his opportunity sets. What distinguishes a reference point is the degree of saliency, or attention it attracts, in whatever option set it is included. A full model correlating well behaved preferences, in terms of noncyclicity, and completeness should consider the relative saliency of options within a set, because now the attention is not over opportunity sets on the whole but is distributed unequally among options inside it. This is the absolute saliency of the status quo, when it is contained in the option set, that makes the preference over dominated options incomplete. When there are no issues of saliency but an abundance of choices, restricting the consideration set may reveal true preferences. When there are saliency effects in the option set due to some reference point, restricting the consideration set but still retaining the status quo inside may always lead to a consistent choice of that reference point under that restriction. A general psychological model of incomplete preferences, combining reference-dependency and indecisiveness, could be a model of attention allocation over and within opportunity sets.

2.3 Incomplete preferences and incomplete beliefs 2.3.1 Bewley’s model Bewley’s model combines two elements that we have discussed. It considers status quo as a source of observed incompleteness (Bewley, 2002). And it normatively relates this status quo form of incompleteness, which he calls “inertia,” to the fact that some beliefs, under situations of status quo, can be incomplete and, as such, insufficient to motivate a decision

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against the current status quo. The source of incompleteness, differently from the models considered so far, will then be located at the level of beliefs and subjective probabilistic anticipations. Bewley distinguishes three technical senses of inertia; each of them entails different criteria on beliefs and their completeness to make a decision that differs from the status quo. Before discussing those senses and associated criteria, let’s be precise about what a typical decision problem is in Bewley’s view. It somewhat differs from the type of choice situations we have so far explicitly or implicitly considered. It, in particular, involves a central dynamic element. A typical decision problem will be a full-fledged life program. It encompasses a lot of more typical situations like the choice between tomatoes and potatoes at dinnertime when I am 63 years old, but, basically, before some margin of action and surprise is introduced, it is a deterministic program from the onset. Each event E in that program is, classically, a set of possible states of nature, temporally situated from the event of my birth E0 (which is a singleton) to Et. St denotes E0 3 E1 3 . . . 3 Et, which is the event tree of one’s life. For each sAS, a set of action A(s) are available. For each action a, there is a reward r(a) in a state. Three possible descriptions of that program yield different senses of inertia. Those different senses relate to the beliefs the individual can make about the underlying probabilities of a state arising in a future time of her life and thus about the degree of determinism of the program in which she is existentially embedded. We can already see, at this point, that what is extremely intriguing in Bewley’s model is the formal connection he establishes between motives for incompleteness and the degree of freedom we expect to encounter in our existences. We now present the three senses of inertia according to the alternative specifications of the basic decision problem. The underlying probabilities of the unfolding of states in a life program is unknown to the individual. She is not necessarily a good predictor of what will befall her, but she forms priors about that. In some cases, then, some unexpected alternatives arise, which are those, in the first specification of Bewley’s decision problem, to which the individual has assigned a prior probability of 0. The individual had made plans corresponding to these priors. Bewley’s inertia assumption, in its first sense, applies when a new alternative arrives under that definition. From a subjective point of view, this is as if a new decision problem had arisen (whereas objectively it has not). So now the individual is in a position to

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compare between two different programs: the one she has planned to follow and the new that came by surprise. More formally, a decision problem comprises the following variables P 5 (S, A, r, Ω, Δ) where S, A, r, are as before, Ω is the set of all possible states of nature, and Δ is a probability distribution over states. Suppose that at time t an expected event s0 6¼ s arises. Simply put, it means that now the individual has to compare between P(st) 5 (S, A, r, Ω, Δ) and P(s0 t) 5 (S0 , A0 , r0 , Ω0 , Δ0 ). It is not a change to consider lightly because of a simple surprise! So the inertia assumption of type 1 applies here, and the individual pursues her preconceived life program, if and only if the program associated with P(st) is completely dominated by the program associated with P(s0 t). Domination is not to be understood in terms of an expected utility criterion, since it is clear that the type information that would motivate its application is not available in the circumstance. The individual has in fact acted under a multiple-prior assumption. A program is an allocation of rewards through choices of actions over states across time according to the variables, known or unknown, of a decision problem. A program γ with the association of reward r [Σaγ(a)r(a,st)] until time t when the surprise arrives dominates a program γ0 if Eπr(γ) . Eπr (γ0 ) for all πAΔ. So, psychologically, it means that the individual has no doubt about the fact that, under any distribution of probabilities over states, she can anticipate at time t that she will be better off by adopting γ 0 (now that the surprise event has occurred) than persisting with γ or, actually, a, her deterministic action program. Because of the incompleteness of her beliefs on this particular domination criterion, the individual persists in her initial choices. The second sense Bewley gives to inertia is when a programmatic change in one’s life is not due to a proactive reaction to surprise due to unexpected events. It is rather admitted from the beginning of life that new choices may arise. So the psychological component of surprise, which may hinder a choice due to extreme cautiousness or aversion to uncertainty, is here dissipated because the individual knows and anticipates that novelty can occur. Due to this lucid anticipation of novelty, the new alternatives that will present themselves are now part of the initial program. But to what extent are novel events acceptable as a basis for behavioral change? Inertia, in that context, means that an individual chooses a program that makes use of new alternatives only if not doing so would lead him to adopt a more conservative program that would be dominated in the sense just specified. In spite of this anticipation, aversion to uncertainty prevails.

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The reader can be referred to Bewley’s article for formal details, but one can, under a different theoretical framework, understand the difference between the type of program that gives rise to inertia in its first sense and what is being discussed in terms of the game-theoretical difference between playing a mixed strategy and a behavioral strategy in an extensive form game. Bewley’s decision problems are extensive games, and a mixed strategy assigns a probability distribution over pure strategies (fixed sequences of nodes in the game tree) while a behavioral strategy assigns, independently for each information set (i.e., events at every time t in Bewley’s model), a probability distribution over actions. However, Bewley’s idea departs from a clear-cut distinction such as between mixed and behavioral strategies in a game. The inertia assumption states that an individual will adopt a behavioral strategy only if the latter supersedes the initial program more properly defined by an ex ante choice of a mixed strategy. Moreover, since it is known in advance that novelties can arise, it is possible to define ex ante that some behavioral strategies can be adopted. Bewley’s model is then a mix between deterministic and behavioral assumptions. What has to be adopted in advance is an adaptive plan and in fact a behavioral program anticipating possible behavioral changes, ones for which some deviation from the initial behavioral plan leaves no ambiguity that the new behavioral program starting from that changing point dominates the one followed so far. The particular sense of inertia in that context is that the individual anticipates that the changes she will willingly undertake are those acceptable under the former criterion of domination. Once again, if the individual has no complete information about the probability distribution that can define unambiguous domination, he will abstain from change, and the initial behavioral program that he adopts is one that complies with this inertia assumption. The third notion of inertia that Bewley develops in his seminal article is closer to what we can properly consider a behavioral strategy in gametheory and constitutes, above all, a concession to bounded rationality in the sense that we do not model anymore an individual with perfect foresight concerning all the contingencies that can come up. (Note, however, that in the two previous definitions of inertia, the individual could make prior assignation errors leading to surprise and, in the second sense, was still uncertain about what novelty will arise, just knowing that some will arise). Under bounded rationality, the individual is led to make constant approximate plans, and the inertia assumption takes on a looser sense. He is no more required to anticipate which changes he will have to undertake

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and can apply a local adaptive rule whenever necessary. So inertia simply means not to change when insufficiently motivated. Interestingly, Bewley expresses little satisfaction with that third definition of inertia inasmuch as it stems from considerations on bounded rationality that exceed the limits of the notion of rationality he had considered so far. By bounded rationality, one has to understand that decision problems can be too complex for the decision-maker to informationally process. Under too imprecise information, inertia, in the sense of behavioral habits, can be optimal. But this sense of optimality is too vague compared to the strict dominance criterion of probability distributions associated with the consequences of alternative courses of action. Bewley’s model connects status quo and incompleteness in a very different way from the models discussed in Section 2.2.3. Preferences of the individual are no more incomplete in themselves but as a side effect of insufficient probabilistic domination of a set of priors associated with a potential alternative act when ex ante is compared to an ongoing behavioral program. This points to a difference in the source of incompleteness. Second, the interpretation of the status quo bias that Bewley provides is not in terms of the incomparability of options due to an absence of choice made for the options that lie under the status quo but, on the contrary, due to inconclusive but extensive comparisons of all the possible contingencies associated with different decision problems. This is a difference in methods and in underlying norms of completeness. Incompleteness is thereby linked to the application of a strict criterion of rationality and nonchoice due to status quo that is grounded rationally rather than behaviorally.

2.3.2 Symmetry between incomplete beliefs and incomplete preferences Bewley’s model uses a multiple-prior approach as a criterion to motivate an inertia assumption leading to incompleteness of preferences due to incomplete beliefs. How much symmetry can be considered in the modeling of incomplete beliefs in terms of comparing sets of the probabilities associated with uncertain events and the modeling of incomplete preferences in terms of a set of utilities functions that have to be jointly maximized? To question that symmetry further, does it make as much sense to motivate incomplete preferences because beliefs are uncertain as it would to try conceiving of an incompleteness of beliefs because some particular preferences would be incomplete? Following a model like

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Bewley, we see a clear line of argumentation that goes from the incompleteness of beliefs to the incompleteness of preferences (through an inertia assumption), but we hardly conceive of the reverse. Prior to suggesting an answer to that question, it is worth trying to disentangle how one can behaviorally distinguish between doxastic incompleteness or indecisiveness in beliefs and incompleteness of preferences or indecisiveness in tastes. Ok, Ortoleva, and Riella (2012) propose an axiomatic characterization of this distinction. They use, crucially, the AnscombeAumann framework in which, please recall, acts are defined as functions from a set of states of nature to a set of lotteries (a probability distribution on final prizes or consequences). In that framework, preferences over acts comply with the usual standard axioms of transitivity, continuity, independence, and completeness, leading to a single-priorsingle-utility representation of that preference relation. We have seen how a multiutility representation can accommodate a notion of incomplete preference when it is understood as indecisiveness in tastes and assumes incomparability between the utility of consequences or lotteries. Here, there is no need to suppose that the individual is doxastically uncertain. Conversely, Bewley’s model shows that incompleteness of preferences can be expressed in terms of a multipriorssingle-utility representation. They form very different conceptual and cognitive presuppositions about what failure of completeness amounts to. Ok and his colleagues formally investigate what is common to both approaches by pointing to an interesting cognitive ability (which they axiomatize) of uncertainty reduction. Technically, it consists in finding for an act whose outcomes probabilistically varies over states of affairs (assigning different probabilities to different lotteries at different states) another act that is constant (delivering the same lottery with probability 1 over the set of states) and for which the individual would be indifferent. They define this property as motivating a reduction axiom: for an act f in F and any probability distribution α in Δ(X) (intended as usual), fα denotes the constant act that yields the lottery {ΣωAΩα(ω)f(ω)}1Ω (the indicator function 1Ω signifying the fact that the lottery occurs with certainty across the whole state-space). The reduction axiom stipulates that for any act f, it is possible to find an αf (i.e., a suitable mixture of the outcomes over a given act) such that fαf B f. Ok et al. shows that a preference relation satisfying independence, continuity, and the reduction axiom admits a single-prior expected multiutility representation. It is the dual of Bewley’s representation and

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tantamount to representations we have seen in the chapter (Dubra et al., 2004; Galaabaatar & Karni, 2013), except that it is reached through quite a different insight. Bewley’s representation admits a unique source of incompleteness located in beliefs. It means that, from the point of view of the preferences-as-tastes of the individual, the preference relation would be complete, and, actually, it is when a set of priors entirely dominates another in his model. We can rephrase this fact as a property that therefore consists in the dual of the reduction axiom. Ok et al. name this property a C-completeness axiom, meaning that preferences must be complete at least over constant acts (acts for which there is no subjective uncertainty). In an AnscombeAumann setup, this amounts to saying that preferences are complete over final lotteries, such that there is no indecisiveness in tastes. Note, however, at this point, that there remains a source of incompletenessbearing, in a particular way (other ways are conceivable), at the same time on beliefs and tastes that consist in being indecisive or uncertain about one’s future tastes. In Bewley’s framework, wherein individuals adopt behavioral programs, it would mean that individuals know in advance whether or not they are likely to change their tastes (preference-based inertia). Is it possible to connect the two axiomatic stipulations of uncertainty reduction and constant-act-completeness to a single insight that would alternatively motivate the choice between a modeling of incompleteness in terms of beliefs or in terms of taste indecisiveness? What is the nature of completeness or incompleteness that is shared by these two properties and, hence, these two approaches? Intuitively, Ok and his colleagues speak of partial completeness. When considering the singlepriormultiutility model of incompleteness, completeness is present at the level of priors. Reciprocally, for the multipriorsingle-utility model of incompleteness, completeness is at the utility level. In both cases, full completeness is partially recovered, respectively, when an uncertain act over lotteries is converted by mixing their outcomes to a constant act, yielding the same lottery over all states, so that no more incomparability can remain (reduction axiom), or when subjective uncertainty is reduced to objective uncertainty when, in Bewley’s model, only unambiguous domination cases between alternatives are considered. The common ground of both approaches is then a possible reduction of uncertainty to risk so that one act becomes decisively preferred to another. As far as incompleteness of preferences-as-tastes is concerned, this is a weakening

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of the reduction axiom because now: fαf k f. Ok et al. name this condition a “weak reduction axiom” and show that a preference relation that complies with it, together with independence and continuity assumptions, can satisfy either a single-priormultiutility representation or its dual but not both at the same time. To understand this important last restriction, it suffices to emphasize that the common property expressed by the weak reduction axiom derives from the assumption of a partial completeness shared by these two models. It is, of course, conceptually possible to imagine an individual plagued by the two types of incompleteness at the same time. Stated in the formal terms advanced by Ok and his colleagues, the cognitive predicament of that individual means that she cannot reduce subjective uncertainty to objective uncertainty in any way but would still try to continue to make comparisons among the alternatives that lie before her. As weak reduction jointly entails C-completeness and reduction—or the two forms of incompleteness—that individual will never find a constant act that is preferred to a multiple-prior and a multiple-utility associated with her actual choice for an act. Another problem arises at a deeper level, which may break the symmetry between beliefs and preferences envisioned so far. In a Savagean perspective, preferences over bets, revealed by choices, determine beliefs. It is well-known that Ellsberg paradoxes undermine this direct determination and that, instead of clearly individualized probabilistic beliefs thus revealed, in decision-contexts presenting ambiguity parameters, the beliefs of the individual are better captured in terms of a set of multiple-priors (each one behaving like a regular probability) or in terms of nonadditivity, beyond probability. In that case, it is not any more clear that preferences revealed through choices between acts continue to determine beliefs. Under a multiple-prior representation of beliefs, it is now unclear which beliefs (a criterion has to be set at the level of beliefs) are determined by preferences, but it is rather the case that preferences over bets under ambiguity will depend on the structure of beliefs of the individual. The more primitive notion appears to be, especially when we continue to adopt a Savagean choice-theoretical setting based on revealed preferences over acts, the indecisiveness of beliefs. It entails in its wake indecisiveness of preferences, either due directly to the impossibility of reducing subjective uncertainty to an objective one (in Ok et al.’s approach) or due to inertia about what my actual preferences would be if I could access unambiguous information about contingencies and form complete beliefs (in Bewley’s model).

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2.3.3 Multiple-priors models and internal dialogs Multiple-priors representations of incomplete beliefs do not necessarily entail incomplete preferences if a determining principle is found to complete the preference relation. Bewley’s principle is the strictest we can conceive of since it requires unanimity: Domination between two alternatives is assured when all priors associated with the new alternative dominate those associated with the current choice. Unanimity, therefore, is tantamount to a complete reduction of subjective uncertainty. Now the preference bearing on constant acts, that is, lotteries, is “naturally” complete, since the only source of incompleteness here was the indecisiveness of beliefs and not of tastes. A less strict determining principle is one telling which prior in a set of several priors reflecting the individual’s subjective uncertainty is to be considered with respect to maximizing the utility of the associated outcome. In that case, there is equally no reason to consider that the preferences are incomplete, if the only source of incompleteness was supposed to lie in beliefs. In other terms, as long as there is a way to generate unanimity or to select a relevant prior in a multiprior set, incompleteness of preferences induced by the structure of beliefs vanishes. Those two ways of dissipating belief-incompleteness, unanimity or a single-prior selection criterion, are very distinct in spirit. A famous model corresponding to the second approach is initially given in Gilboa and Schmeidler (1989) through the application of a maxmin rule to the set of priors expressing the individual’s uncertainty. That rule may be deemed to reflect too high a level of ambiguity aversion, but clearly it is the simplest alternative to the unanimity criterion. Adopting this rule, Gilboa and Schmeidler, unless they wanted to incorporate indecisiveness in tastes as a cognitive factor to be modeled, have eschewed the issue of incompleteness. However, this modeling strategy, even in the absence of indecisiveness in tastes, does not so simply solve any issue of incompleteness of preferences. Another issue is in fact how the individual can form such complete preferences—the completion problem—depending on the source (beliefs or tastes) of incompleteness of her preference relation. In a rare and deep recent attempt in the literature at both metaanalyzing principles of rationality and incorporating the meta-analysis in the object-language of decision-theory, Gilboa et al. (2010) relate those two approaches to distinct views of rationality, of rationalizing preferences, and of completing them. We can conceive of their way of setting the issue as a game, or—as we prefer, going somewhat beyond what the

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authors may have in mind—as an internal dialog. They distinguish between what they label “objective” and “subjective” rationality. And they envision these two types of rationality (or rationalization) as guiding the formation of complete preferences. When the preference is not complete, it is yet to be formed. It can, of course, stop short of a completion process for good reasons. By objective rationality, they mean that the individual can convince others that she is right in making the choices she makes. By subjective rationality, they instead emphasize the idea that the individual cannot be convinced that she is wrong in making her choices, that she will feel no embarrassment if confronts her with an alleged behavioral contradiction. Before we go into further formal detail, let’s note that this approach is reminiscent—but in a developed meta-analytical form—of Allais’s conception of his own paradox and of some of the literature in its wake concerning the intuitiveness of rationality axioms and the conflict of intuitions when apparent behavioral violations arise (Allais, 1953; MacCrimmon & Larsson, 1979; Slovic & Tversky, 1974). We can define rationality constructively (Gilboa and his colleagues would say “objectively”) as a way of engaging in choices on the basis of well pondered principles (for instance, on the basis of some decision-theoretical axioms) or structurally (“subjectively”) as a way of making sense (for instance, in reference to some decision-theoretical axioms) of a set of behavioral data. Taking the axiom of transitivity first, it can be used objectively, indeed constructively, as an inference rule to expand on one’s piecemeal choices and complete one’s preference relation. We have seen a similar use of the independence axiom in Dubra and Ok (2002)’s procedural approach in Section 2.2.1. By such a use of axioms, the individual can perform a rhetorical act of convincing an interlocutor (or herself) of the rationality of her choices. When the interlocutor (or an inner devil advocate) conversely flags an axiom to point to some behavioral incoherence of the individual, the latter can accept or reject the argument on the basis of her intimate conviction or counterarguments. If she cannot be convinced of her incoherence or wrongness, she is deemed subjectively rational. Now the problem is that there is more leeway in subjective than in objective rationality. Transitivity may be deemed harder to reject (although one can conceive of arguments motivating intransitivity), and independence is precisely the type of axiom that motivated the conflict of intuitions raised by Allais’s paradox. The same axioms can thus be involved in the two tasks, but they will not be used

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and thought about in the same way. In particular, the noncompliance with the axioms cannot entail the same (internal or external) judgments of irrationality. What is the minimal axiomatic overlap that corresponds to an agreement between those two views of rationality? The answer to this question is what we mean by incorporating a metatheoretical question about rationality into the language-object of decision-theory. Interestingly, the language used by Gilboa et al. to address this question is the expression of the common ground, the compatibility conditions, between Bewley’s and Gilboa-Schmeidler’s approaches to the completion of preferences. It means that the determining or completion principles that underlie those two models of preferences can be said to correspond, respectively, to an objective and a subjective conception of rationality. Saying so consists, formally, in endowing the individual with two preference relations, one objective, k , one subjective, k^, each associated with, respectively, a Bewley’s or a Gilboa-Schmeidler’s axiomatization, and find out what their compatibility conditions are. In particular, one condition on k is Bewley’s C-completeness, which stipulates that the relation is complete on constant acts. This condition and other basic conditions, such as preorder, monotonicity, Archimedean continuity, and nontriviality, ensure its multipriorsingle-utility representation. In turn, k^ accepts completeness, the same basic conditions as k , and independence, in order to achieve a multipriorsingle-utility representation. A common axiomatic ground between the two preference relations can be obtained by replacing independence, which holds for k^, by an axiom of uncertainty aversion (for all f and g A F, if f B^ g, then 1/2(f) 1 1/2(g) k^ g. / hedging). We have seen the exact equivalent of this axiom with Ok et al.’s model in the previous section, under the guise of the weak reduction axiom, and the equivalent of it plays in their system as C-completeness in Bewley’s in terms of the subjective reduction of uncertainty. Gilboa et al. then show that a limitation of independence to constant acts (Cindependence) and the axiom of uncertainty aversion, in accompaniment with the basic conditions, provide the basis for expressing the compatibility conditions between the two models. Those two conditions will therefore be the two means by which an internal agreement can be reached between subjective and objective rationality. It may be worth, at this point, underlining that those conditions, when set over the minimal axiomatic overlap between k and k^ as just

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defined, form two means of completing one’s preferences. Objective rationality means that from a set of complete preferences (for instance those on constant acts), I will try to constructively expand my preference relations to other alternatives. While subjective rationality means that, given a set of choices that I have realized, the axioms apply to them in a way immune to subjective refutation. In fact, Bewley’s k and GilboaSchmeidler’s k^ allow us to express just that. Precisely, f k  g implies f k ^g. Clearly, I am all the more subjectively convinced of my preference if there is no objective reason to convince me that I am wrong in holding it, provided that k and k^ have a common axiomatic ground to be thus compared. This condition is called “consistency” by Gilboa and his colleagues. Conversely, g \ /  f implies f k ^g. This condition (called “caution” by the authors) indicates that having no objective ground to prefer g to f, implies that f is preferable to g due to ambiguity aversion, that is, in relation to the axiom that precisely constitutes the axiomatic overlap between the two preferences’ relations and their associated views of rationality. We think that the interplay between k and k^ schematizes an internal dialog (between intuitions of rationality) that has some cognitive grip. Saying this does not amount to pretending that the set of priors, as used in Bewley’s or Gilboa-Schmeidler’s models, are “cognitive probabilities,” They can be inferred from the individual’s behavior (in the way Savage’s unique priors can in principle), but it would be too much to “cognitivize,” beyond what is actually cognitive in the present discussion, the two preference relations with which a theoretical individual is endowed and by means of which her internal dialog can be modeled. But the internal dialog is not simply a metaphor. It is something that takes place in the individual’s mind when she thinks about the rightness or wrongness of her actual preferences, when she tries to rationalize it, and can use one of the two ways described by Gilboa et al., possibly finding reconciling conditions. This is a metatheoretical issue, to some extent, from the point of view of decision-theory, but it is not from the point of view of cognitive sciences. It helps modeling what in cognitive sciences has, when raised in connection with rationality axioms, been put in terms of metacognitive feelings, due to the impression not to comply with rationality axioms that are accepted otherwise (see, for instance, Gangemi, Bourgeois-Gironde, & Mancini, 2015, on this issue). The lingering presence of such feelings could be an introspective (with some possible behavioral manifestations) signal of noncompletable preferences.

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2.4 Revealing incompleteness versus indifference 2.4.1 Choice-theoretic foundations of incomplete preferences The adoption of the classical revealed-preference paradigm appears to stand in direct contradiction with the admission of an incomplete preference relation. This negative conclusion stems simply from the fact that, since preferences are revealed by choices, a choice, when it is made, selects, by necessity, a given option in the choice-set. By default, indifference is revealed if the constraint is relaxed and choice can be deferred or refused (so nonchoice is, at least temporally, revealed). In one case, we constrain choice, and we decide to call the selected option the preferred one. In the other case, the choice is less constrained, and we decide to interpret nonselection in terms of indifference. It is not obvious that nonchoice due to the relaxation of the constraint of making a choice is revelatory of the weak part of the preference relation in the same way that the full constraint reveals its strict part. Or, to it differently, there is no a priori correspondence between strong and weaker constraints at the level of choices and the revelation of, respectively, the strict and the possibly weak part (indifference) of a preference relation k. In both contexts, what is revealed by stronger or weaker constraints on the act of choice, is a matter of interpretation and, as such, of exclusion of alternative explanations. To wit, in the case of the strict constraint, it is clear that what is revealed may well be not chosen (or exclusively chosen) in the case of a weaker constraint. If x and y are actually indifferent, the selection of x does not mean that x is strictly preferred to y. Hence the use of a weaker constraint through which we may observe whether x continues to be chosen over y, revealing then the preference relation x k y. But if {x,y} is selected under the weaker constraint, we still don’t know whether x is indifferent to y or the individual is indecisive between the two alternatives. Deciding that this case is revelatory of indifference versus strict preference is then excluding an alternative interpretation of what we observe under the applied choice-theoretical constraints. Theoretically, one may want to preserve completeness of preferences. Instead of choosing the modeling strategies of incompleteness discussed in this chapter, one of two possible ways to preserve completeness is to restrict the applicability of the preference relation to comparisons that we know do not yield indecisiveness. But since it may seem like this way excessively limits the set of alternatives, a second way of obtaining this restriction without any limitation of the option set is to apply the

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preference-revealing procedure through choices and interpret it as grounding a complete preference relation. But if one refuses this type of procedure because of the hidden restriction it actually amounts to, one is then faced with a practical problem: How can this procedure be used, possibly amended, to reveal incompleteness instead of “constrained indifference?” So the question is whether it is possible to reveal incomplete preferences while maintaining the practical use of a congenial version of the revealed preference paradigm. Eliaz and Ok (2006) developed a positive answer to this question. They clearly disentangled the implications that follow from practically adopting a revealed preference paradigm in choice-theory and from the theoretical assumptions and interpretations that bear on the observation of choice patterns. They show that it is possible to be conservative on the idea that an individual is maximizing over a preference relation but that this relation preference can also be revealed to be incomplete through a particular way of looking at his choices. Classical choice-theoretical foundations of a complete preference relation are based on the so-called WARP (weak axiom of revealed preference). This is where the interpretation that connects revealed preferences and completeness lies: WARP: Be S any choice-set taken from a choice space (X, Ωx) (where X is the set of individual alternatives, and Ωx a choice-field of 2x \{[}). And let c be any choice-correspondence on Ωx (c(S) 5 x means, for instance, that x has been selected by c in S. WARP states that for any SAΩx, and yAS, if there is xAc(S) and yAc(T) for some other TAΩx with xAT, then yAc(S). Clearly, when x and y both belong to a choice-set T, y is selected by the choice-correspondence applied over T, and x is selected by the choice-correspondence applied over another set S (we are not mentioning here inclusion relations between S and T), then y must be selected by c( ) from S too. Such behavior is compatible with the maximization of a complete preference relation over Ωx. If, indeed, we are not considering inclusion or whatever relation between S and T; a behavior that is consistent with a complete preference relation is one that interprets y being selected from T, when x is also present in T as y being as least as good as x. So, when x is selected from S, and y is present in S, it means in turn that x is at least as good as y. But given that y is at least as good as x, it should be selected from S too. The “at least as good” interpretation of choice behavior carries over to S and T. Or, in terms that are closer to

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our present interest, it means that x and y continue to be comparable when they are present in different choice-sets, which is compatible with maximizing by one’s choice behavior a complete preference relation (under that interpretation of WARP). Sen (1971) has proposed to analyze WARP in two subproperties. This standard decomposition helps us to pinpoint where the alleged conceptual transition between choice behavior under WARP and the maximization of a complete relation lies and how it can be blocked: Sen’s α-axiom: for any S and T A Ωx, if xATDS and xAc(S), then xAc(T); Sen’s β-axiom: for any S and T AΩx with TDS, if x, y A c(T) and xAc(S), then yAc(S). These two subaxioms do not bear the same normative appeal. The monotonicity principle in α seems an intuitive component of rational behavior, which is less obvious with β. Eliaz and Ok suggest a very simple example to emphasize this point: Just keep its formal structure. Suppose an individual is faced with a choice-set (x,y,z). That individual can be faced with a social choice problem (taking into account the preference of two other individuals and jointly maximizing their preferences over that set) or a multiple criteria individual choice problem (presenting two different attributes ranked differently by that individual). Suppose the two rankings are the following: (1) ,y,z,x. and (2) ,x,y,z.. When faced with the all set (x,y,z), the individual discards z because it is dominated by both y and x. More precisely, it is the only option that is dominated twice across the two rankings. So c(x,y,z) 5 (x,y). Having eliminated z, the choice of the individual between y and x can be uncertain: c(x,y) 5 (x,y). Suppose now that alternatively y or x are made unavailable, which yields c(x,z) 5 (x,z), and c(y,z) 5 (y). Such a choice behavior cannot be rationalized by WARP because c(x,z) 5 (x,z) being interpreted as xBz, z should also be selected from (x,y,z), which is incompatible with the fact that it is twice dominated in the two rankings. Another subtler point is that c(x,y) (once z is eliminated) equals (x,y). But this cannot be interpreted as indifference either because it would be possible to think that y is chosen over x given it performs somewhat better than x, taking into consideration the two rankings on the whole. But we suppose here an individual who does not use this type of reasoning. And he may have good reasons, too, not to do so if, for instance, by choosing y only, he frustrates the individual with ranking 2. In fact, c(x,z) 5 (x,z) and c(x,y,z) 5 (x,y)

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cannot be rationalized by the same reasoning under WARP because the usual reading would be in terms of indifference, although they are actually based on two inconsistent interpretations: one is in terms of simple dominance, while the other introduces extra reasons. Elias and Ok suggest a minimal departure from WARP. Their weakening of this choice-theoretical framework incorporates the basic rational notion of dominance, which motivates the elimination of z in c(x,y,z). They do so rather than enriching WARP with extra psychological procedures that would make the choice more determinate than it is at first sight. To account for this behavioral pattern, they therefore propose an alternative WARNI (weak axiom of revealed noninferiority) axiom. It precisely consists in weakening the β-subpart of WARP. β is where the interpretation of having x and y being jointly selected over T and over S (when TDS) yields indifference between x and y. WARNI states alternatively that: WARNI: For any SAΩx and yAS, if for any xAc(S) there exists TAΩx with yAc(T) and xAT, then yAc(S). The intended reading is that when an alternative is selected by a choice-correspondence, it is revealed as noninferior (rather than superior) to the other alternatives present in the choice-set. yAc(T) means that y is revealed as noninferior to x in T. But this does not carry over S in the same way as under WARP. Under WARP, when we extend the choiceset to S, y continues to be chosen, only if it is considered “superior to all other options” except x, with which it is indifferent. But if y is simply deemed noninferior to x in T, it is not necessarily the case that it continues to be chosen in S because it might be inferior to some other option z. WARNI states that y can be continued to be chosen when it is revealed noninferior to all options in S. In the preceding schematic example, we have the individual’s choice-correspondence yielding c(x,y,z) 5 (x,y) and c(x,z) 5 (x,z), z is judged noninferior to x in the second case, but it is not selected in the first case because of the presence of y by which it is dominated. Under the usual WARP interpretation, c(x,z) would constrain c(x,y,z) 5 (x,y,z), which would not reflect a choice pattern that has a clear intuitive appeal. The immediate conclusion is that c(x,z) 5 (x,z) cannot be interpreted as xBz, but as xtz, x is incomparable to z. Likewise, c(x,y) 5 (x,y) means xty. At this point, we wish to note that Eliaz and Ok, due to the minimal departure of WARNI from WARP (which they impose on

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themselves in order to preserve the compatibility of a revealing procedure with the maximization of some preference relation), collapses under t, which seems to us to consist of two distinct phenomena: one positive in terms of domination and the other negative in terms of a nonresort to extra reasons (that would be to recoil into procedural rationality) that would possibly dispel the incomparability. Bounding themselves in such a way, they remain as close as possible to the revealed-preference paradigm as one can be. This collapse of the two ideas (domination and nonprocedural rationality) allows for a richer domain of incompleteness to be formally captured than if we allowed for reason-based disambiguation of t. We note I(c), this domain of incomparability, for a choice-correspondence c. How can I(c) be choice-theoretically distinguished from the equivalence part of the preference relation k? Eliaz and Ok’s approach is to operationalize WARNI under the intended interpretation that when an individual is such that c(x,y) 5 (x,y), it signals indifference only in the case when for any choice problem T, the individual would choose x whenever he chooses y and would choose y whenever he chooses x. Consistent selection of the pairs across all choice problems attest that xBy. However, if for some choice-set T0 , it is no more the case that both options are chosen, and it becomes natural to think that the individual finds them incomparable, at least to some extent. More precisely, let’s have c(x,y) 5 (x,y). Let’s then take a set S to which x belongs but not y. We sometimes retrieve x from S and introduce y instead (noted Sy,2x). I(c) admits xty cases for one of the following choice-theoretical motives: xAc(S), but y= 2c(Sy,x) / x= 2c(S) but yA c(Sy,x) / c(S)\{x} 6¼ c(Sy,x) \{y}, the latter being the general case. This provides a possible choice-theoretical basis of an incomplete preference relation, in the sense that the maximization of a preference relation is compatible with the observation of such behaviors that reveal incompleteness versus indifference. However, an incomplete preference relation based on choice-theoretic foundations still remains to be tested.

2.4.2 A revelation procedure of incompleteness versus indifference by means of choice-deferral Experimental attempts to reveal incomplete preferences are anchored in the interpretation of choice-theoretical foundations. The interpretation is itself driven by the purpose of proving that the behavior is compatible

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with maximizing a preference relation. The preference relation can be incomplete, and the maximizing will bear on a multiutility vector, as the source of indecisiveness is at the level of tastes. Even within those confines, it is possible to implement different procedures of revelation, still linked to the main hypothesis one formulates about the source and possible resolution of incompleteness (see Deparis, Mousseau, Öztürk, Pallier, and Huron (2012) for a discussion about alternative criteria of incompleteness. The authors, though, do not adopt, in their own experiment, a revealed-preference paradigm, allowing their subjects to express judgments of incomparability). Danan (2003) and Danan and Ziegelmeyer (2006), on the other hand, adopt the revealed preference paradigm as well as the hypothesis (one very prevalent in the literature as we have seen in Section 2.1) that incompleteness involves a preference for flexibility. But they manage to operationalize choice-deferral in a way that can in principle discriminate between indifference and incompleteness, if we can observe that the individual is ready to pay a small price in order to postpone his decision. Like Mandler (2005), Danan postulates a distinction between behavioral and cognitive preferences. Cognitive preferences are typically not revealed under a usual choice-theoretic approach. Once we have accepted this distinction, it is clear that only behavioral preferences are. So the idea is to indirectly reveal cognitive preferences that can be incomplete through behavioral preferences that are constrained to be complete. This is possible when choices over menus can be postponed provided a fee be paid for that purpose. We refer the reader to Danan (2003) for a fuller characterization of the relation between cognitive (kc) and behavioral (kb) preference relations; we are content to report here that, according to that proposal, incompleteness is revealed if we have: For X and X0 menus in Ωx, X t X0 iff ['ε,ε0 . 0 such that X , X0 kb X 1 ε and X , X0 kb X0 1 ε0 ]. It means that the individual is ready to incur a cost ε in order to keep his choice flexible. It differs from cognitive indifference in that X Bc X0 implies that [’ε . 0, X0 1 ε gX]. ε is then a disambiguating device between cognitive incompleteness and indifference that can be behaviorally implemented. The experimental implementation consists in using bracketing procedures when choices are actually realized over different lotteries (under risk). Choices between lotteries l 5 [z1,z2] and sure payoffs c are offered.

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Bracketing amounts to a calibration procedure for cognitive preferences. Flexibility is costly, and in Danan and Ziegelmeyer (2006) that cost is ε 5 10 cts. To recover cognitive indifference, first, the choice between the lottery l and the menu {l,c} is offered; then, between l 1 ε and c, and finally between c 1 ε and the menu {l,c}. The authors state that if preferences are complete, then there exists a certain payoff c lying between z1 and z2 such that c g l for any c .c and l g c for any c , c . Relaxing  this completeness assumption is tantamount to determining lower (clow )  and upper (cup ) certainty equivalents of the lottery l 5 [z1,z2] lying in the  interval (z1,z2), such that c g l for any c . cup and l g c for any c ,   clow. We have ltc for any c between clow and cup . Observing incompleteness means observing such an interval through the bracketing procedure, which also amounts to observing a postponement of choice (to which we come back in Section 2.5.1). Completeness is then characterized by   clow 5 cup 5 c , and l is then indifferent to c . Very ingeniously, Danan and Ziegelmeyer also suggested a method to measure the degree of incompleteness between two lotteries. They present their subjects with lotteries having different spreads σ of risk. Given a lottery l, its degree of incompleteness normalized in [0,1] is: h i 3   =σ: inc 5 cup  clow   °inc ranges between 0 (when cup 5 clow ) and 1 (full incompleteness) when the individual is indecisive between a lottery and a sure payoff c . There remain open questions with this procedure. In particular, as indifference and incompleteness both give in principle a rise in postponement, only the addition of some tipping point linked to a cost ε discriminates between the two. The authors rely on the fact that the asymmetrical introduction of such a cost would break indifference but not incompleteness. But it could be the case—in a certain psychological interpretation of the two phenomena—that indifference extends beyond a point of equivalence and that ε is not enough to disrupt it. If this were the case, indifference, thus understood, and incompleteness would be the same phenomenon within a certain bound. But on the other hand, it is possible that incompleteness might never be resolved by postponing the choice, that the individuals anticipate that she won’t become decisive at some point (the authors consider this psychological possibility) and that the individual, under that form of lucidity about her cognitive preferences, prefers to randomize her choice now, a behavioral feature that is typically

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associated with the practical resolution of indifference. The possibility of a completion process or resolution of incompleteness seems to be presupposed by Danan and Ziegelmeyer. But this resolution can follow two very different paths—to the extent that the subject in the end has to make a choice: if the subject (1) acknowledges indecisiveness but has the metacognitive ability to tell whether it will be resolved through more cognitive effort over time (or simply more time, when a situation will eventually impose choice, as in the classical Krepsian restaurant example), or (2) has the subjective conviction that the choice will not be solved. We try to find solutions to this type of cognitive issue in the following to-the-lab section, some of them extending, though, beyond the revealed-preference paradigm.

2.5 To the lab: temporal behavior of incomplete preferences 2.5.1 Measuring the degree of incompleteness through probabilistic deferral Following up on Danan and Ziegelmeyer (2006), we suggest an addition to their experimental procedure, having in mind our remark at the end of Section 2.4.2 regarding some possible subjective certainty or uncertainty by the individual with respect to the possible resolution of her indecisiveness. This modification also intends to provide an alternative or, rather, a complementary measure of the degree of incompleteness than theirs. The idea is as follows. Choice-deferrals are mostly conceived as oneshot experimental devices. We propose to make them sequential. Choices can be deferred over several periods. The sequence can be finite or infinite. It means that a choice, in the case of an infinite sequence, can be postponed for ever. We want to use repeated postponement as an alternative means to disentangle indifference and indecisiveness when a choice is deferred. Let’s assume for the moment that we do not resort to a cost for flexibility as another means to discriminate between the two states of mind. The trick is to make probabilistic the occurrence of the next step in the sequence. The more we step in the sequence, the less likely it is that there will be another opportunity to make the same choice. This is another way to introduce a cost for indecisiveness, which reflects exactly the cost one can pay in real life for not being able to make choices. The second opportunity to choose is not certain to show up. It does not matter here whether the probabilities of further occasions to choose are

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known or ambiguous. This is a refinement that can come later and help to characterize the nature of indecisiveness in a particular way. It could mean, for instance, that if I postpone under risky conditions (I know the exact probability with which a next occasion to choose will present), I enter in a certain conscious subjective measurement, a computing disposition, vis-à-vis my degree of indecisiveness. Under ambiguity conditions, I just have to admit that I take a certain risk of not having another chance to resolve my uncertainty. So, to put it simply, to give specific probabilities about the likeliness of other occasions to choose is a proxy to measure the degree of subjective uncertainty of the individual vis-à-vis her preferences, while the reverse is less obvious. We imagine a sequence like that in risky conditions in Fig. 2.1. We assume that cognitive indifference will behaviorally transpose into the immediate choice (typically random) not to take the risk of not obtaining one of the involved payoffs in the choice. Of course, indecisive individuals will also tend to avoid not obtaining a payoff or a reward. In Danan and Ziegelmeyer’s setting, they know that if they pay a small amount of money, they will have with certainty the possibility to make that choice again. Here the cost is psychological, but the observation is purely behavioral (it is a choice): We suppose there is a probability Pε with which the individual is ready to postpone her choice in case of indecisiveness. However, it is possible to think that the nature or psychological source of the indecisiveness that is thereby revealed is subtly different from what is captured under costly deferral. I hypothesize that what we capture by means of probabilistic deferral is indecisiveness as current impossibility to choose, whereas costly deferral taps into preference for flexibility without specifying the nature of the underlying incompleteness. If an individual cannot currently choose, she will accept to postpone her choice, even if there is a slight chance that the choice won’t present itself again. She then engages in a trade-off between losing the opportunity of a reward and constraining herself to make a decision she presently does not want to

Choice 1

• Transition to choice 2 with probability P1 Instruction: choose or defer.

Choice 2

• Transition to choice 3 with probability P 2 Instruction: choose or defer.

P1