COGNITIVE DISSONANCE AND CHOICE∗

Larry G. Epstein Igor Kopylov

February 3, 2006

Abstract

People like to feel good about past decisions. This paper modelsself-justification of past decisions. The model is axiomatic: axiomsare defined on preference over ex ante actions (modeled formally bymenus). The representation of preference admits the interpretationthat the agent adjusts beliefs after taking an action so as to be moreoptimistic about its possible consequences. In particular, the ex postchoice of beliefs is part of the representation of preference and not aprimitive assumption. Behavioral characterizations are given to thecomparisons “1 exhibits more dissonance than 2” and “1 is more self-justifying than 2.”

1 INTRODUCTION

1.1 Objective

There is considerable evidence in psychology that people like to view them-selves as being smart, and in particular, as having made correct decisionsin the past. Thus they may change beliefs after taking an action and be-come more optimistic about its possible consequences, in order to feel betterabout having chosen it. Such behavior is a special case of an affinity for

∗Epstein is at Department of Economics, University of Rochester, Rochester, NY 14627,lepn@troi.cc.rochester.edu; Kopylov is at Department of Economics, UC Irvine, Irvine, CA92697, ikopylov@uci.edu. We would like to acknowledge comments by seminar participantsat BU, and helpful conversations with Emanuela Cardia, Faruk Gul, Jawwad Noor, Wolf-gang Pesendorfer and especially Massimo Marinacci.

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cognitive consistency - for example, an affinity for consistency among beliefsor opinions (Festinger [13]). Here the two cognitions are “I have taken anaction that could lead to unfavorable outcomes” and “I am a smart personwho would not make poor choices”; adopting a more optimistic belief aboutfuture outcomes serves to reduce this dissonance. Though the term cognitivedissonance is often used more broadly, we use it here to refer to ex post self-justification of past actions. Our objective is to model an agent who exhibitssuch cognitive dissonance.

Models of cognitive dissonance in economics treat beliefs as choice vari-ables, on a par with other more standard choice variables, such as consump-tion and savings. Thus Akerlof and Dickens [1, p. 307] propose as basicpropositions of their model of cognitive dissonance that preference is definedover beliefs and that beliefs are subject to choice. While a more optimisticoutlook makes one feel better about the past decision, the agent recognizesthat adopting more optimistic beliefs would take her further from the “truth”and thus would lead to suboptimal choices in decisions still to be made. Theoptimal belief is determined by making this trade-off. Similarly, in Rabin[28], utility depends directly on beliefs. This modeling approach is nonstan-dard in economics and may make one uncomfortable for a number of reasons.First, it begs the question “what is the feasible set from which beliefs are cho-sen?” Unlike other choice variables for which the market determines feasible(budget) sets, the feasible set of beliefs is presumably subjective (in the mindof the agent) and thus invariably requires an ad hoc specification. A possi-bly more important concern is the observability of chosen beliefs and hencetestability of the model. While in psychology it is standard to take beliefs asobservable through interviews or questionnaires, many economists adhere tothe choice-theoretic approach to beliefs, initiated by Savage, whereby beliefsare observable only indirectly through choices among actions.

In this paper, we propose a choice-theoretic and axiomatic model of cog-nitive dissonance. Preferences are defined over actions (modeled formally bymenus) and axioms are imposed on these preferences. Thus empirical testa-bility relies only on the ranking of actions being observable. The functionalform for utility admits an interpretation whereby the agent behaves as if shechooses beliefs ex post in the manner described above, but this is a result- part of the representation of preferences over actions. Finally, the abovequestion about the feasible set of beliefs is answered automatically by therepresentation.

We emphasize that our agent is not boundedly rational or myopic. Rather

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she is sophisticated and forward-looking - when choosing an action ex anteshe is fully aware that she will later experience cognitive dissonance and thatthis will affect her later decisions. She has this sophistication in commonwith agents in most economic models, but one may wonder whether indi-viduals outside those models are typically self-aware to this degree. We arenot familiar with definitive evidence on this question and in its absence, weare inclined to feel that full self-awareness is a plausible working hypothesis.Even where the opposite extreme of complete naivete seems descriptivelymore accurate, our model may help to clarify which economic consequencesare due to cognitive dissonance per se and which are due to naivete. In addi-tion, the assumption of sophistication is vital for a choice-theoretic approach:because she anticipates her cognitive dissonance, it affects her current choiceof actions. This makes it possible to infer cognitive dissonance from her (inprinciple observable) choice of actions, consistent with the choice-theoretictradition of Savage. Thus sophistication seems justifiable also on the method-ological grounds of permitting the exploration of modest departures fromstandard models.

1.2 Model Outline

As described above, cognitive dissonance implies changing beliefs, hencechanging preferences, which poses difficulties for modeling behavior. Onepossible modeling route is to specify dynamically inconsistent preferencesand then to tackle the questions of to what degree the agent anticipatesfuture changes in preference and how intrapersonal conflicts are resolved.These are the issues familiar from Strotz [33]. We follow instead the routeadvocated by Gul and Pesendorfer [16] (henceforth GP) whereby behaviorthat indicates changing preferences over underlying alternatives can alter-natively be viewed as coming from stable preferences over menus of thesealternatives.

A brief outline is as follows: uncertainty is represented by the (finite)state space S. Time varies over three periods. The true state is realized andpayoffs are received at the terminal time. The intermediate time is called theex post stage. Physical actions chosen then are identified with Anscombe-Aumann acts, maps from S into lotteries over consumption. A physicalaction is chosen also at the initial ex ante stage. Each such action is modeledby a menu of acts - the idea is that any action taken ex ante limits optionsex post. The agent understands when choosing a menu that ex post she will

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choose an act from that menu. She also knows, when ranking menus, thather beliefs about S will change ex post so as to make the previously chosenmenu seem more attractive. She will try to resist the temptation because sheviews her prior beliefs, formed with the detachment afforded by the ex antestage, as being ‘correct.’ Whether or not she succeeds in exerting self-control,however, temptation is costly, and this affects her ranking of menus. Thusthe latter reveals her expected change in beliefs, or her cognitive dissonance.

As a concrete illustration of the relevance of choice of menus and thebehavioral manifestation of cognitive dissonance, consider a job choice modelalong the lines of Akerlof and Dickens [1]. Ex ante the worker chooses a job,either in a hazardous industry or in a safe one. If he chooses the hazardousindustry, then ex post he can select between two kinds of safety equipment(high quality h and low quality `). Each kind affects the likelihood of anaccident but may not eliminate the risk entirely. Thus h and ` each imply arandom payoff, net of cost of the equipment, that depends on the exogenousstate of the world. In other words, each can be viewed as an act and the jobcorresponds to the menu {h, `}. For the safe industry, there are no choices tobe made ex post and the ultimate payoff is certain and given by c (a constantact). Therefore, the safe industry corresponds to the singleton menu {c} andthe choice of job corresponds to the choice between the menus {h, `} and{c}.

If the worker can commit to safety equipment at the same time that hechooses the job, then ex ante beliefs are such that he would prefer the highquality equipment, that is,

{h} Â {`}. (1)

In the standard model, menus are valued according to the best alternativethat they contain, and thus the worker would also exhibit the indifference

{h} ∼ {h, `}.

However, an agent who exhibits cognitive dissonance, and knows this ex ante,may exhibit the ranking

{h} Â {h, `}.The intuition is as follows: after accepting the job in the hazardous industry,the worker faces the two cognitions - “my job is dangerous” and “I am a smartperson and would not choose a precarious job”. He relieves this dissonance,and reduces doubts about his job choice, by changing his prior beliefs, as

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reflected in the ex ante ranking (1), and believing instead that the job isnot so dangerous after all. This creates the temptation to choose ` ratherthan h. The worker anticipates this temptation. Accordingly, if he dislikestemptation, he would rank {h, `} as strictly worse than {h}.

If commitment to high quality equipment can be made simultaneouslywith choice of the hazardous job, the worker would so commit. Becausethat would leave no decisions left to be made ex post, cognitive dissonancewould not be behaviorally relevant. Assume that such commitment is notpossible (Akerlof and Dickens give reasons why commitment may not bepossible). Then there remains the question of whether given the menu {h, `}ex post, he yields to the temptation and chooses `. He feels that his priorbeliefs were “correct” and thus “should” be used to guide decisions - in otherwords, h is the correct choice. The balance between what he ought to doand the tempting alternative depends on the worker’s self-control. With highself-control, he may resist the temptation and choose h. Following GP, wesuppose that this case (or rather the ex ante expectation thereof) is capturedby the ranking

{h, `} Â {`}.The expectation of yielding and choosing ` is captured by the ranking

{h, `} ∼ {`}. (2)

Rational expectations about cognitive dissonance may lead to choice ofthe safe industry. However, if

{h, `} Â {c},

then the worker chooses the hazardous industry and, assuming (2), lateradopts the poor safety equipment corresponding to `. To an outsider, orfrom the perspective of (1), the worker may appear careless or overconfident.

To this point, we have suggested that cognitive dissonance could explainthe ranking

{h} Â {h, `} º {`}. (3)

This ranking is a special case of GP’s central axiom of Set-Betweenness.While GP argue that such rankings reveal the presence of temptation andself-control problems, the reason for temptation is unspecified. Put anotherway, the ranking under commitment may conflict with choice behavior out ofthe menu available ex post, but the reason for this difference is not clear given

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only (3). For example, (3) could be due to underlying preferences (taste orrisk aversion) changing with the passage of time, rather than beliefs changingin order to justify the previous choice of menu. But there is other behavior,described via axioms in our formal model, that would exclude such interpre-tations and support an interpretation in terms of cognitive dissonance.

1.3 Related Literature

It has been argued that a moderate degree of (optimistic) illusion can bepsychologically beneficial even net of the loss in efficacy of decisions; seeTaylor and Brown [35], Taylor [34] and Baumeister [4], for example.

The psychological theory of cognitive dissonance is due to Festinger [13].Dissonance originates with an action and the subsequent evaluation of thataction. Where there exists dissonance between having taken that actionand subsequent beliefs, the theory posits that those beliefs will be changedto match or justify the past action. Aronson [3] is an excellent textbooktreatment and overview of the supporting evidence from psychology. Someof this evidence is strongly suggestive that cognitive dissonance has economicconsequences; for example, the efficacy of the “foot-in-the-door-technique”,whereby a small commitment by individuals makes it easier to persuade themlater to commit further in that direction, suggests the efficacy of two-stagemechanisms, possibly including an entry fee at the first stage. Several otherapplications have been developed in formal economic models as we describebelow.

Akerlof and Dickens suggest that cognitive dissonance can play a rolein explaining some economic phenomena that are arguably puzzling fromthe perspective of more standard models. These include the existence ofsafety regulation (based on the job-choice model sketched above), why non-informational advertising is effective (it gives external justification for anindividual to believe she is making a smart decision in buying the product),and why persons often fail to purchase actuarially favorable disaster (floodor earthquake) insurance. The story here is analogous to that concerningsafety equipment and fits naturally into our modeling approach: after choos-ing a house (or menu), it reduces dissonance to believe that a flood is sounlikely as to not justify buying insurance (the choice of a particular act),even though she would have bought insurance simultaneously with the housepurchase. Similarly, cognitive dissonance can explain why researchers mayappear “overly optimistic” in their pursuit of a previously chosen project (a

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menu). It feels good to believe that the research project previously embarkedon is a promising one and thus ongoing efforts may be guided by otherwiseunwarranted optimism.

Rabin [28] models the choice of an enjoyable but immoral activity inlight of dissonance between one’s beliefs about what is moral and the chosenlevel of activity. Haagsma and Koning [17] show how cognitive dissonancecan generate barriers to exiting an unproductive industry. Smith [32] showshow cognitive dissonance can explain why wages tend to rise faster thanproductivity. The worker justifies his job situation ex post by adjusting hisbeliefs about the cost of effort needed to fulfill his duties - the need for self-justification, and the adjustment in beliefs, are greater the lower is his pastwage. The employer can exploit this by offering a contract with an increasingwage profile. Goetzmann and Peles [15] argue that cognitive dissonanceleads investors to justify remaining in mutual funds that consistently performpoorly; and that such inertia can help to explain why money flows in morerapidly to mutual funds that have performed well than flows out from thosethat have performed poorly. See Dickens [8] and Oxoby [27] for furtherapplications of cognitive dissonance.

With regard to modeling, we have already acknowledged our debt to GP.Their model does not apply directly, however. One difference is that whilethey study preferences over menus of lotteries, it is important for our storythat menus consist of (Anscombe-Aumann) acts. Kopylov [19] has extendedthe GP theorem from (menus of) lotteries to abstract mixture spaces, in-cluding, in particular, the space of Anscombe-Aumann acts. A second andmore important formal difference from GP, and also from Kopylov’s exten-sion, and the major source of technical difficulty in our model, is that werelax the Independence axiom - the latter is not intuitive given cognitive dis-sonance. Finally, we note that Dekel, Lipman and Rustichini [7] generalizeGP’s model of temptation. However, their motivation is much different thanours - in particular, they assume Independence.1

1In their concluding remarks about possible directions for further research, they men-tion that accommodating guilt may be a reason for relaxing Independence when modelingtemptation. This rationale is obviously much different than ours. There exist other repre-sentation results in the menus-of-lotteries/acts setting that do not rely on Independence.Epstein and Marinacci [11] study an agent who is not subject to temptation, but rathervalues flexibility because she is uncertain about the future; she violates Independencebecause her conception of the future is coarse. More recent results, with still differentobjectives, appear in Ergin and Sarver [12] and in Noor [26].

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There are two final but important connections to the literature. Themore optimistic beliefs held ex post by our agent come about in our modelbecause she uses a (nonsingleton) set of probability measures, and whenevaluating a prospect, she chooses the measure that maximizes its utility.This recalls Dreze’s [9] model of choice between Anscombe-Aumann actsunder moral hazard. It recalls also Gilboa and Schmeidler [14] - they modelagents who are averse to ambiguity, in the sense illustrated by the EllsbergParadox, by assuming that they minimize (rather than maximize) over a setof priors, but their model has an obvious counterpart for ambiguity loving.Our model differs from both of these primarily through its focus on the time-varying nature of beliefs and the corresponding value of commitment. Inprinciple, one could reinterpret our model in terms of a change from ex anteprobabilistic beliefs to ex post multiple-priors reflecting ambiguity loving,but then there is no apparent reason for the agent to exert self-control as shedoes in our model. Thus we disregard this interpretation of the model.

2 UTILITY

The model has the following primitives:

• time t = 0, 1, 2

• finite state space S

• C: set of (Borel) probability measures over a compact metric space

refer to c ∈ C as a lottery over consumption, or more briefly as con-sumption

C is compact metric under the weak convergence topology

• H: set of acts h : S −→ C, with the usual mixture operation

• compact sets of acts are called menus and denoted A, B, ...

K (H) is the set of all menus

it is compact metric under the Hausdorff metric2

• preference º defined on K (H)

2See [2, Theorem 3.58], for example.

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The interpretation is that a menu A is chosen ex ante (at time 0) accordingto º. This choice is made with the understanding that at the unmodeledex post stage (time 1), the agent will choose an act from A. Uncertainty isresolved and consumption is realized in the terminal period t = 2. Cognitivedissonance and choice behavior at time 1 are anticipated ex ante and underliethe ranking º.

Menus are natural objects of choice.3 The consequence of a physicalaction taken at time 0 is that it determines a feasible set of physical actionsat time 1, and these actions can be modeled by acts in the usual way. Thuseach physical action at time 0 corresponds to a menu of acts.

Our model of utility has the form4

U (A) = maxh∈A

[(1− κ) U (h) + κV (h)]− κ maxh′∈A

V (h′) , (4)

whereU (h) = p · u(h), and (5)

V (h) = maxq∈Q

q · u(h). (6)

Here 0 ≤ κ ≤ 1, p is a probability measure on S, Q is a closed and convexset of probability measures on S containing p, and u : C −→ R1 is mixturelinear and continuous.

The standard model of subjective expected utility maximization is thespecial case where κ = 0 or Q = {p}. More generally, the functional form canbe interpreted along the lines suggested by GP. When restricted to singletons,U coincides (ordinally) with U ; thus expected utility with prior p representspreference over consumption lotteries when the agent can commit ex ante.When she does not commit, then the new (temptation) utility function V overlotteries becomes relevant. Temptation utility is computed by maximizingover probability measures in the set Q. Since p ∈ Q, V imputes higherexpected utility to the menu at hand than was the case ex ante using p,corresponding to cognitive dissonance. She is tempted to maximize V expost. Though she views p as “correct”, there is a self-control cost of resisting

3Kreps [21, 23] was the first to propose menus as a way to model physical actions inan ex ante stage.

4For any real-valued random variable x on S, and probability measure q, q · x is short-hand for the expected value

∫S

xdq.

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the temptation given by

κ

(V (h)−max

h′∈AV (h′)

)≤ 0.

Thus a compromise is struck between maximizing U and maximizing V -choice out of A is described by maximization of the weighted sum, or bysolving

maxh∈A

maxq∈(1−κ){p}+κQ

q · u(h). (7)

which balances ex ante realism and ex post cognitive dissonance. The natureof the compromise is further illustrated by the fact that

p ∈ (1− κ) {p}+ κQ ⊂ Q,

so that the set of beliefs underlying the choice of an act ex post lies “between”the prior view p and the optimistic view represented by Q.5

Since ex post choice out of the menu maximizes the utility functionmaxq∈(1−κ){p}+κQ q ·u(h), which does not depend on the menu, one may won-der whether the model captures beliefs that adjust to make the previouslychosen menu attractive. To see a sense in which this is true, note that (byreversing the order of the maximizations),

maxh∈A

maxq∈(1−κ){p}+κQ

q · u(h) = maxh∈A

q∗A · u(h),

for any q∗A that solves maxq∈(1−κ){p}+κQ maxh∈A q · u(h). Thus ex post choiceconforms with SEU and probabilistic beliefs given by q∗A. Evidently, q∗A de-pends on the menu A and is chosen to make the value of the menu, given bymaxh∈A q · u(h), as large as possible.

We can say something about the qualitative difference between ex postchoice and the “correct” choice. Given any menu A ex post, the choice out ofA is determined by maximizing maxq∈(1−κ){p}+κQ q ·u(h), while the “correct”choice would maximize p · u(h). Suppose for concreteness that consumptionis real-valued (C consists of lotteries over [a, b] ⊂ R1) - typically, one assumesthat u (·) is concave on [a, b] corresponding to risk aversion. On the other

5As is familiar from GP-style models, this interpretation in terms of ex post choice issuggested by the functional form, and by intuition for the underlying axioms, but ex postchoice lies outside the scope of our formal model. See Noor [25] for a model of temptationwhere ex post choice is part of the primitives.

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hand, the maximization over q in the ex post utility function introduces someconvexity. Thus it may not be concave in h and may even exhibit risk loving.(For example, if one restricts attention to Savage acts h : S → [a, b], thenex post utility is a convex function of h if u is linear.) Consequently, expost choice may appear extreme - for example, it may correspond more toboundary optima.

A final comment is that both subjective and objective probabilities arepresent in the model - the latter underlie consumption lotteries - but theyare treated differently: while the agent chooses new beliefs ex post about hersubjective uncertainty (the state space S), she does not distort or modify ob-jective probabilities. For example, both U and V agree about the ranking oflotteries in that, for every lottery c, U (c) = V (c) = u (c), the vNM expectedutility of c. Because an objective probability law is based on undeniable fact,distorting it to a more favorable one, is folly or ignorance that would notbe undertaken by the sophisticated individuals that we model. But wherefacts alone do not pin down beliefs uniquely, an agent is free to choose be-liefs and feeling good about oneself is one possible consideration in doing so.As an illustration of the difference, note that Knox and Inkster [18] reportthat persons leaving the betting window after placing bets at a race-trackare more optimistic about “their horse” than persons about to place bets.On the other hand, it is more difficult to imagine someone being similarlyoptimistic about a coin, which is known to be unbiased, after choosing thatcoin for a game of chance.

3 AXIOMS

The first two axioms require no discussion.

Axiom 1 (Order) º is complete and transitive.

Axiom 2 (Continuity) º is continuous.

Menus can be mixed via

αA + (1− α) B = {αf + (1− α) g : f ∈ A, g ∈ B} .

Formally, the indicated mixture of A and B is another menu and thus whenthe agent contemplates that menu ex ante, she anticipates choosing out of

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αA+(1− α) B ex post. It follows that one should think of the randomizationcorresponding to the α and (1− α) weights as taking place at the end - aftershe has chosen some mixed act αf +(1− α) g out of the menu. In fact, sincethe mixture of acts is defined by (αf + (1− α) g) (s) = αf (s)+ (1− α) g (s)for each s, the randomization occurs after realization of the state.

The above mixture operation permits one to state the Independence ax-iom, which is adopted by GP. However, Independence is not intuitive undercognitive dissonance.6 To see this, suppose for concreteness that A ∼ B andconsider whether the mixture αA + (1− α) B should also be indifferent to Aas required by Independence. Indifference between A and B is based on theanticipation that, in each case, the agent will choose beliefs ex post to makethe menu in hand attractive, and that these beliefs will tempt her to chooseout of the given menu differently from what she would have prescribed exante. Evaluation of the mixture αA + (1− α) B can be thought of similarly,but the important point is that beliefs for the mixed menu must be chosenbefore the randomization is played out. Since also beliefs chosen given Agenerally differ from those chosen given B, optimistic beliefs for the mixedmenu bear no simple relation to those for A and B. A similar disconnectapplies to anticipated temptation and ex post choices across the three menus.For example, it is possible that the acts f and g be chosen out of A and Brespectively, while αf + (1− α) g not be chosen out of αA + (1− α) B. Asa result, the agent will generally not be indifferent between the mixed menuand A, violating Independence. (The deviation from indifference could goin either direction: αA + (1− α) B Â A and αA + (1− α) B ≺ A are bothpossible.)

However, suitable relaxations of Independence are intuitive. To proceed,for any act f ∈ H, let

Hf = {tc + (1− t)f : t ∈ [0, 1], c ∈ C}.

If h = tc + (1− t)f is an act in Hf , then for any mixture linear u and for allprobability measures q,

q · u (h) = tu (c) + (1− t) q · u (f) .

6The reason is essentially that because the agent anticipates that she will adjust herbeliefs ex post to the menu at hand, the situation is analogous to that of choice between“temporal risks”. As explained by Machina [24], for example, preferences over temporalrisks typically violate Independence even at a normative level.

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Because the first term on the right is independent of q, it follows that anymenu A that is a subset of Hf is rendered attractive by beliefs that makef attractive. In particular, for any two menus A and B in Hf , when theagent chooses beliefs to fit the menu, there is an optimistic measure that iscommon to both A and B. But this invalidates the reason given above forviolating Independence. Thus we adopt:

Axiom 3 (Collinear Independence) For all α ∈ (0, 1), for all f ∈ H,and for all menus A′, A, B ⊂ Hf ,

A′ Â A =⇒ αA′ + (1− α) B Â αA + (1− α) B.

Acts h′ and h in Hf are naturally called collinear, which explains the nameof the axiom.7

When ranking singleton menus, there is no choice to be made ex post.Thus cognitive dissonance is not relevant and there is no reason for Inde-pendence to be violated. This motivates the following second relaxation ofIndependence:

Axiom 4 (Commitment Independence) For all f, g, h ∈ H and α ∈(0, 1),

{f} Â {g} =⇒ {αf + (1− α) h} Â {αg + (1− α) h}.

In the standard model, a menu is as good as the best alternative that itcontains. Then

A º B =⇒ A ∼ A ∪B,

a property called strategic rationality by Kreps [22]. Such a model excludestemptation. Temptation is an integral part of cognitive dissonance becausethe agent changes beliefs to make the menu at hand look attractive and thenis tempted to make subsequent choices accordingly (see the discussion ofutility in Section 2). In seeking a suitable relaxation of strategic rationality,we begin with GP’s central axiom Set-Betweenness.

7For any collinear acts h′ and h, it is easy to see that for every s′ and s,u (h′ (s′)) > u (h′ (s)) =⇒ u (h (s′)) ≥ u (h (s)), that is, the real-valued func-tions u (h (·)) and u (h′ (·)) are comonotonic. Collinearity implies the stronger restriction(1 − t) (u(h′ (s′))− u(h′ (s))) = (1 − t′) (u(h (s′))− u(h (s))) for some t and t′. Thuscollinearity can be viewed as a cardinal counterpart of comonotonicity.

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Set-Betweenness (SB): For all menus A and A′, if A º A′, then A º A∪A′ ºA′.

An equivalent and perhaps more revealing, though less compact statementis that if A º A′, then one of the following conditions holds: (i) A ∼ A∪A′ ∼A′,or (ii) A Â A ∪ A′ Â A′, or (iii) A ∼ A ∪ A′ Â A′, or (iv) A Â A ∪ A′ ∼ A′.

Following GP (p. 1408), we may interpret these conditions intuitively.The underlying assumptions are that: unchosen acts can only reduce utility,acts can be ranked according to how tempting they are, and only the mosttempting act affects utility. Consider an agent having the menu A ∪A′, andwho expects to choose g though she finds f most tempting. (i) is the residualcase. (ii) indicates that g is in A′ (hence A Â A ∪ A′), and that f is in A(hence A ∪ A′ Â A′). The next two cases are our main interest.

In (iii), she still plans to choose out of A which now also contains themost tempting act. Confront her next with the larger menu A∪A′∪B. Themost tempting act lies in A ∪B. What about her choice out of A ∪A′ ∪B?Suppose that her expected choices satisfy the Nash-Chernoff condition (orSen’s property α); defer discussion of possible objections to this assumption.Then having rejected acts in A′ when facing A ∪ A′, she would (expect to)reject them also when facing A ∪ A′ ∪ B. Thus A ∪ B contains both theact to be chosen and also the act in A ∪ A′ ∪ B that is most tempting. Theindifference A ∪B ∼ A ∪ A′ ∪B follows.

Finally, consider (iv), which indicates that both f and g lie in A′. Againconfront the agent with the larger menu A∪A′ ∪B. The most tempting actlies in B ∪A′ and, assuming the Nash-Chernoff condition, so does the act tobe chosen. Deduce the indifference A′ ∪B ∼ A ∪ A′ ∪B.

The preceding provides intuition for the following axiom:

Axiom 5 (Strong Set-Betweenness (SSB)) For all menus A and A′, ifA º A′, then: (i) A ∼ A ∪ A′ ∼ A′, or (ii) A Â A ∪ A′ Â A′, or

(iii) A ∪ A′ Â A′ and A ∪B ∼ A ∪ A′ ∪B for all menus B, or(iv) A Â A ∪ A′ and A′ ∪B ∼ A ∪ A′ ∪B for all menus B.

SSB implies Set-Betweenness: let B = A in (iii) and B = A′ in (iv). Thetwo axioms are equivalent given Independence (and Order and Continuity)- this follows from counterparts of the representation results in GP [16] and

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Kopylov [19] - and thus SSB is not invoked explicitly in those papers. How-ever, we do not adopt Independence, and we show below that SSB is strictlystronger than Set-Betweenness even given all our other axioms.

Our intuition for SSB assumed the Nash-Chernoff condition, which doesnot appear in GP’s rationale for Set-Betweenness. The Nash-Chernoff condi-tion can be criticized in a model of temptation - the addition of the acts in Bshould not affect the normative appeal of A versus A′, but it may change theself-control costs associated with various choices, and this may lead to thechoice of an act in A′ when facing A∪B∪A′ even where she chooses an act inA when facing A∪A′. The Nash-Chernoff condition is weaker than the weakaxiom of revealed preference (WARP), which requires also Sen’s condition β(see [22]). Noor [25] provides an example to illustrate why WARP may beproblematic in a model of temptation, and in [26] he develops a model oftemptation and self-control that does not impose WARP for ex post choice.Note, however, that since SSB and WARP for ex post choice are impliedwhen one assumes also Independence, such objections apply equally to theGP model. Moreover, while they may be important for guiding developmentof a general model of temptation, they do not seem germane to temptationgenerated by cognitive dissonance.

One can raise other objections to Set-Betweenness, and hence a fortiorito our stronger axiom. Dekel, Lipman and Rustichini [7] argue that Set-Betweenness excludes some forms of temptation, for example, where thetemptation generated by different alternatives is cumulative, or where there isuncertainty ex ante about which alternatives will be tempting. Once again,we do not view these objections as especially relevant for cognitive disso-nance.

Say that f ∈ H dominates g ∈ H if {f (s)} º {g (s)} for every s ∈ S. Ifthe evaluation of a lottery does not depend on the state, then a dominatingact should be preferred under commitment. Similarly, if f dominates g , wewould not expect f to be tempted by g. Thus we assume:

Axiom 6 (Monotonicity) If f dominates g, then {f} ∼ {f, g} º {g}.

Our axioms thus far have for the most part been concerned with modelingtemptation in general, that is, not tied specifically to cognitive dissonance.A partial exception is Collinear Independence, the intuition for which did

15

rely on the assumption that temptation arises because of an ex post choiceof beliefs to “fit the menu” in hand. However, Collinear Independence issatisfied even if the agent becomes more pessimistic ex post and adopts beliefsthat make the menu less attractive ex post. The final two axioms build inex post optimism and hence cognitive dissonance.

Axiom 7 (Constants-Do-Not-Tempt) For all c ∈ C and f ∈ H,

{f} Â {c} =⇒ {f} ∼ {c, f}.

Temptation is due to a change in beliefs (as opposed to a change in riskaversion, for example), which leaves the evaluation of constant acts unaf-fected. In addition, the noted change is always to become more optimisticex post about the available menu, rendering it even more attractive relativeto any constant act c than it was ex ante. Therefore, constant acts cannottempt. Note that, in contrast, {c} Â {c, f} º {f} is both permitted by theaxiom and intuitive given our story.

Axiom 8 (Convex Temptation) The set {f ∈ H : {c} ∼ {c, f} Â {f}}is convex for every c ∈ C.

Suppose that f and g both lie in the indicated set, that is, each is worsethan c under commitment and neither tempts c, and consider the mixtureαf + (1− α) g. By Commitment Independence, {c} Â {αf + (1− α) g}.We now argue that in addition, αf + (1− α) g should not tempt c, thuscompleting intuition for the axiom. We are given that {c} ∼ {c, f}. Becauseex post beliefs are chosen to make the menu {c, f} attractive, and becausethe expected utility of c does not depend on beliefs, we can interpret theindicated indifference as follows: the act f , when matched with the beliefsthat make it attractive, does not tempt c. A similar statement applies forg. Consider now the menu {c, αf + (1 − α)g}. Beliefs to render this menuattractive are chosen ex post (time 1), before the randomization is completed(which, as noted earlier, occurs only at the terminal time after the true statein S is realized). Since the beliefs that make f attractive may differ fromthose that make g attractive, matching beliefs with the mixed act is moredifficult. Therefore, one would expect the mixed act not to tempt c.

16

4 REPRESENTATION RESULT

Our main result is that the preceding axioms characterize the functional formdescribed in Section 2.

Theorem 4.1 The binary relation º on K (H) may be represented as in(4)-(6) if and only if it satisfies Axioms 1-8. Moreover, u is unique up to apositive linear transformation, and if º is not strategically rational, then p,Q and κ are unique.

Convex Temptation is used only at the very end of the sufficiency proofin order to prove that V has the form given in (6). If the axiom is deleted,then the remaining axioms characterize the representation (4)-(5), for someV : H → R1 that is continuous, monotone (V (f) ≥ V (g) if f dominates g),satisfies certainty additivity (V (αf + (1− α) c) = αV (f)+ (1− α) V (c) forall c in C), and that satisfies V (f) ≥ p · u (f) for all f with equality if f isconstant. (See Example 3 below.)

We present some examples to demonstrate the tightness of the charac-terization in the theorem. Each of the first three examples satisfies Order,Continuity, Commitment Independence, Strong Set-Betweenness and Mono-tonicity, and violates precisely one of the axioms that relate more specifi-cally to cognitive dissonance - Collinear Independence, Constants-Do-Not-Tempt and Convex Temptation. The final example violates only Strong Set-Betweenness, though it satisfies GP’s Set-Betweenness, thus proving that ouradoption of the stronger axiom is necessary.

Example 1 : Let

U (A) =maxh∈A [U (h) V (h)]

maxh′∈A V (h′),

where U and V are as in (5)-(6), and where u > 0. Then º violates onlyCollinear Independence. (See Appendix C for some details.)

There exist simpler examples violating only Collinear Independence -these retain (4)-(5) but modify the specification of V . However, becausethe above ratio form deviates from the GP functional form, we find it morerevealing about the power of Collinear Independence.8

8The example is inspired by weighted utility theory [5], a model of risk preferencein which the utility function over lotteries equals a ratio of expected utility functions.Readers familiar with the ‘non-expected utility’ literature will not be surprised by the

17

Example 2 : Assume (4)-(5), but take

V (h) = q · u(h),

for some probability measure q 6= p. Then º violates only Constants-Do-Not-Tempt.

Example 3 : Modify Example 2 by taking

V (h) = max

{p · u (f) ,

∫u (f) dν

},

where ν is a capacity on S and the integral∫

u (f) dν is in the sense ofChoquet (see Schmeidler [31]). Then º violates only Convex Temptation.

Example 4 : This example violates only SSB. Let S = {s1, s2}, and fix avNM utility function u such that u(C) = [0, 1]. For every f ∈ H, let u1(f) =u(f(s1)), u2(f) = u(f(s2)), and

ν(f) = max{0, u1(f)− u2(f)− 45},

γ(f) = max{0, u2(f)− u1(f)− 45}.

Let º be represented by U , where, for all menus A,

U(A) = maxf∈A

[u2(f)− γ(f) maxg∈A

ν(g)].

Then º satisfies Order and Continuity. By construction, γ(f)ν(f) = 0 forall f , so that U({f}) = u2 (f), implying Commitment Independence. Inaddition, γ(f)ν(g) = 0 holds in each of the following cases (i) f or g areconstant; (ii) f and g are collinear; (iii) f dominates g; (iv) g dominates f .9

Thus º satisfies Collinear Independence, Monotonicity, Constants-Do-Not-Tempt, and Convex Temptation. For Set-Betweenness, take any menus Aand B and the acts f, g ∈ A ∪ B that deliver the maxima in the definitionof U(A ∪ B), so that U(A ∪ B) = u2 (f) − γ (f) ν (g). Wlog f ∈ A. ThenU(A) = U(A∪B) if g ∈ A, and U(A) ≥ U(A∪B) ≥ U(B) if g ∈ B. However,

observation that º satisfies the following alternative relaxation of Independence: A ∼B =⇒ αA + (1− α) B ∼ A.

9If γ (f) · ν(g) > 0, then: u1 (f) , u2 (g) < 1/5 and u2 (f) , u1 (g) > 4/5. But this isimpossible given any of the conditions (i)-(iv).

18

º violates SSB: if u1(f) = 0, u2(f) = 1, u1(g) = 0, u2(g) = 12, u1(h) = 1 and

u2(h) = 0, then U({f}) = U({f, g}) = 1 > U({f, g, h}) = 2425

> U({g}) =U({g, h}) = 1

2> U({h}) = 0.

A tuple (u, p,Q, κ) as in the theorem is said to represent º. The repre-senting tuple is unique (up to cardinal equivalence for u) if the degeneratecase of strategic rationality is excluded. Thus it is meaningful to ask aboutbehavioral interpretations of its components. We have already noted those ofu and p: u ranks lotteries (constant acts) and p is the “commitment prior”- it underlies the ranking of singleton menus. Turn to Q and κ. In whatfollows, we adopt variants of GP’s comparative notions “greater preferencefor commitment” and “greater self-control”, renamed so as to reflect betterthe psychological motives we have in mind.

Say that º∗ has greater dissonance than º if for all acts f and g,

{f} Â {f, g} =⇒ {f} Â∗ {f, g}. (8)

The ranking {f} Â {f, g} indicates that though f is better than g ex ante, g isbetter ex post when holding the menu {f, g}. Then there is dissonance for theagent with preference º between the ex ante ranking under commitment (orthe underlying beliefs) and the distinct ex post ranking (or ex post beliefs).If º∗ has greater dissonance, then she should strictly prefer {f} to {f, g}.

Theorem 4.2 Suppose that both º and º∗ have utility representations (4)-(6), with components (u, p, Q, κ) and (u∗, p∗, Q∗, κ∗) respectively, and thatneither is strategically rational. Then º∗ has greater dissonance than º ifand only if

(u, p) = (au∗ + b, p∗) for some a > 0 and some b, and (9)

Q = (1− ε) {p}+ εQ∗, for some 0 < ε ≤ 1. (10)

The characterizing conditions assert both that the commitment rankingsinduced by º and º∗ coincide (this is (9)) and that Q is “closer to p” than isQ∗ in the sense of an epsilon contamination (this is (10)). Since Q∗ is convexand contains p, (10) implies in particular that Q ⊂ Q∗, but it implies more.Note that if º is strategically rational, then any º∗ has greater dissonance

19

- the defining condition is satisfied vacuously - and no restrictions on com-mitment preferences are implied. If º∗ is strategically rational, then (8) issatisfied if and only if º is also strategically rational, and again, condition(9) is not implied.

We are interested not only in how much dissonance an agent experiences(or expects to experience), but also in what she does about it, or moreprecisely, in the extent to which ex post choices are distorted by dissonance.Say that º∗ is more self-justifying than º if it has more dissonance than ºand

{f} Â {f, g} ∼ {g} =⇒ {f} Â∗ {f, g} ∼∗ {g}.The hypothesized rankings for º indicate not only that there is dissonancebut also that given {f, g} at the ex post stage, the agent succumbs andchooses g, even though f was optimal ex ante under commitment. She doesthis because the choice of g better justifies her previous choice of {f, g}. Ifº∗ is more self-justifying, then she should also choose g out of {f, g}.

Theorem 4.3 Suppose that both º and º∗ have utility representations (4)-(6), with components (u, p, Q, κ) and (u∗, p∗, Q∗, κ∗) respectively, and thatneither is strategically rational. Then º∗ is more self-justifying than º ifand only if (u∗, p∗, Q∗, κ∗) and (u, p, Q, κ) satisfy (9), (10) and κ∗ ≥ εκ.

It follows that a change from κ to κ∗ > κ, keeping other components ofthe functional form fixed, renders º∗ more self-justifying than º but leavesthe two preference orders equally dissonant (each has greater dissonance thanthe other).

5 EXTENSIONS

To conclude, we outline two generalizations of the above model.10

5.1 Effort and Dissonance

An intuitive prediction of dissonance theory is that cognitive dissonance ismore pronounced when past actions are “difficult”. As Aronson writes (p.175), “if a person works hard to attain a goal, that goal will be more attractive

10The extensions are in terms of functional forms. We have not provided axiomaticfoundations, though we believe it would be straightforward to do so.

20

to the individual than it will be to someone who achieves the same goal withlittle or no effort.” See [3, pp. 175-8] and [1, p. 310] for discussion andreferences to supporting experimental evidence. Here we outline an extensionof the model that can accommodate this prediction.

Modify the time line described in Section 2 only by supposing that thechoice to made at the ex ante stage is of a pair (e0, A), where e0 ∈ E denoteseffort in period 0 and A is, as before, a menu of Anscombe-Aumann acts oneof which will be chosen in the following period. Ex ante choices are assumedto maximize preference º, which is defined on E × K (H).

Let utility have the form

U (e0, A) = maxh∈A

[U (e0, h) + κ(e0)

1−κ(e0)V (e0, h)

]− κ(e0)

1−κ(e0)maxh′∈A

V (e0, h′) , (11)

whereU (e0, h) = −v (e0) + δ p · u(h), and (12)

V (e0, h) = −v (e0) + δ maxq∈Q

q · u(h). (13)

Here 0 ≤ κ (e0) < 1, p is a probability measure on S, Q is a convex andcompact set of probability measures on S containing p, u : C −→ R1 ismixture linear and continuous, v : E → R1 gives the utility cost of effort, and0 < δ < 1 is a discount factor. When restricted to singletons,

U (e0, {h}) = −v (e0) + δp · u (h) .

For nonsingletons, ex post choice out of A solves

maxh∈A

maxq∈(1−κ(e0)){p}+κ(e0)Q

q · u(h),

which depends on e0 via κ (·).Suppose that

κ (e0) = κ̂ (v (e0)) ,

where κ̂ (·) is increasing. Then an increase in v(e0), corresponding to greatereffort, renders the agent more self-justifying, but leaves the level of dissonanceunchanged.11 More generally, we could also specify Q as a function of v(e0),for example,

Q = (1− ε (v(e0))) {p}+ ε (v(e0)) ∆ (S) .

11We are using the formal comparative notions defined in the preceding section appliedto the preferences on K (H) induced by º and the two levels of consumption.

21

If κ̂ (·) ε (·) is increasing, then greater effort implies both greater dissonanceand greater self-justification.12

5.2 Response to Information

The justification of a past decision may also influence the reaction to in-formation - dissonance theory predicts that information is interpreted in away that is favorable to past choices. By adding a signal realized at time 1and building on Epstein [10], we can extend our model to capture also theresponse to information.

An outline of the model is as follows: let S1 denote the (finite) spaceof signals, one of which is realized at time 1. Ex ante, the agent choosesa contingent menu - a mapping F from signals into menus of Anscombe-Aumann acts. At time 1, she observes the realized signal, updates her beliefsabout S, and then chooses an act from the realized menu F (s1). Denote byp prior beliefs on S1 × S, by p1 its first marginal, and, for each signal s1,let Qs1 be a (closed and convex) set of probability measures on S containingp (· | s1), the Bayesian update of p. Then the utility of any contingent menuF is given by

W (F ) =

∫

S1

U (F (s1) ; s1) dp1 (s1) ,

where, for any menu A,

U (A; s1) = maxh∈A

[(1− κ) U (h; s1) + κV (h; s1)]− κ maxh′∈A

V (h′; s1) ,

U (h; s1) = p (· | s1) · u(h), and

V (h) = maxq∈Qs1

q · u(h).

The interpretation is clear given the parallel with our model (4)-(6). The keyis that at the ex post stage, the agent does not rely simply on the Bayesianupdate p (· | s1) of her prior beliefs, but rather behaves as though she adjuststhe latter in a direction that renders the realized menu F (s1) attractive, asindicated by the maximization over Qs1 . As a result the signal is interpretedso as to justify the past choice of an action (that is, F ).13

12If Qi = (1− εi) {p}+ εi∆(S), i = 1, 2, with ε1 ≥ ε2, then Q2 = (1− ε) {p}+ εQ1 withε = ε2/ε1. Thus Theorem 4.3 implies that preference 1 is more self-justifying (and hasgreater dissonance) than preference 2 if ε1κ1 ≥ ε2κ2.

13A closely related bias, called confirmatory bias, states that people tend to interpret

22

A Appendix: Proof of the Representation The-

orem

For necessity, verification of the axioms is straightforward.The proof of sufficiency proceeds roughly as follows: apply the Anscombe-Aumann

Theorem to derive an expected utility function U : H → R1 for preference restricted tosingleton menus. This delivers a linear utility index u : C → R1 and a prior p on S, suchthat U (f) = p · u (f). Next, for any f ∈ H, let

Hf = {tc + (1− t)f : t ∈ [0, 1], c ∈ C},

and let Af be the class of menus in Hf . Then Hf is a compact mixture space, and ºrestricted to Af satisfies Independence (because º satisfies Collinear Independence) andSet-Betweenness. Thus, by Kopylov’s [19, Theorem 2.1] extension of GP’s theorem tomixture spaces, one obtains a continuous and linear function Vf : Hf → R such that

U(A) = maxh∈A

(U(h) + Vf (h))−maxh∈A

Vf (h)

represents º on Af . The critical step is to extend the local functions Vf to a globaltemptation function V . The remaining step is to show that V has the form (6) for some Q,which is done by analogy with the proof of Gilboa and Schmeidler [14] (suitably modifiedfor the maxmax model rather than maxmin).

Turn to the detailed proof. Throughout abbreviate the domain K(H) by A, andassume that º is non-degenerate, that is, A Â B for some A,B ∈ A. (Otherwise, thedesired representation holds trivially with u ≡ 0.)

Lemma A.1 There exist a continuous function U : A → R, a probability measure p onS, and a non-constant expected utility function u : C → R such that U represents º and

U({f}) = p · u(f) for all f ∈ H. (14)

Such p is unique, and u is unique up to a positive linear transformation.

Proof. By the Anscombe–Aumann Theorem, the axioms of Order, Continuity, Mono-tonicity, and Commitment Independence imply that the preference º restricted to sin-gleton menus can be represented by U({f}) = p · u(f), where p is a probability measureon S, and u : C → R is a continuous vNM expected utility function. As C is compact,there exist lotteries c+, c− ∈ C such that u(c+) ≥ u(c) ≥ u(c−) for all c ∈ C. Then{c+} º {f} º {c−} for all f ∈ H. By Set-Betweenness, {c+} º A º {c−} for all finitemenus A; by Continuity, {c+} º A º {c−} for all menus A ∈ A. As º is non-degenerate,{c+} Â {c−} and hence, u is non-constant.

evidence in ways that confirm prior beliefs, as opposed to past actions (see [29], for exam-ple).

23

By Continuity, for any A ∈ A, there exists a unique α ∈ [0, 1] such that A ∼ {αc+ +(1− α) c−}. Let

U(A) = u(αc+ + (1− α) c−).

Then U represents º on A and inherits continuity from º. ¤

Hereafter, fix c+, c− ∈ C as in the proof of the above lemma, and fix the unique u (andthe unique corresponding U) such that u(c+) = 1 and u(c−) = −1. Let c0 = c++c−

2 ; thenu(c0) = 0.

For every act f ∈ H, let

• U(f) = U({f}) = p · u(f)

• e (f) = 1+U(f)2 c+ + 1−U(f)

2 c−; then e(f) ∈ C and {f} ∼ {e (f)}• f + α = αc+ + (1− α)f and f − α = αc− + (1− α)f

Take an arbitrary act f ∈ H and invoke [19, Theorem 2.1]: Hf is a compact mixturespace satisfying properties M1–M4 in [19], and º restricted to Af satisfies Order, Conti-nuity, Binary Independence, and Set-Betweenness, the axioms in the cited theorem. Thusº can be represented on Af by

Uf (A) = maxg∈A

(Uf (g) + Vf (g))−maxg∈A

Vf (g),

where Uf : Hf → R and Vf : Hf → R are continuous and linear functions normalized byUf (c+) = 1 and Uf (c0) = Vf (c0) = 0. The normalization of Uf implies that Uf ≡ u ≡ Uon C ⊂ Hf , and hence, Uf ≡ U on Hf . It follows that Uf ≡ U on Af . (To see this, for anyA in Af , let e (A) ∈ C satisfy A ∼ {e (A)}. Then U (A) = u (e (A)) = Uf (e(A)) = Uf (A).)Thus

U(A) = maxg∈A

(U(g) + Vf (g))−maxg∈A

Vf (g) = maxg∈A

Wf (g)−maxg∈A

Vf (g), (15)

where Wf (·) = U(·) + Vf (·) on Hf .Show that Vf is monotonic. Take any h, h′ ∈ Hf such that h dominates h′. For all

α ∈ (0, 1), Monotonicity and Lemma A.1 imply {h + α} ∼ {h + α, h′ −α} Â {h′ −α} andhence, Vf (h + α) ≥ Vf (h′ − α). Let α → 0; then Vf (h) ≥ Vf (h′) by Continuity.

Say that f ∈ H is never tempting if {c, f} º {c} for all c ∈ C; otherwise call fpotentially tempting.

Lemma A.2 If f ∈ H is never tempting, then Vf (f)Vf (e(f)), and U(A) = maxg∈A U(g)for all A ∈ Af .

If f ∈ H is potentially tempting, then Vf (f) > Vf (e(f)), the representation (15) isunique, and there exists a unique κf ∈ (0, 1) such that Vf (c) = κf

1−κfU(c) for all c ∈ C.

Proof. The act f must satisfy exactly one of the following three cases.Case 1. Vf (f) = Vf (e(f)). The linearity of Vf and U imply that Vf (g) = Vf (e(g)) for

all g ∈ Hf . By monotonicity of Vf , for all A ∈ Af ,

maxg∈A

(U(g) + Vf (g)) = maxg∈A

U(g) + maxg∈A

Vf (g),

24

that is, U(A) = maxg∈A U(g). Thus º satisfies Strategic Rationality on Af , and f is nevertempting.

Case 2. Vf (f) > Vf (e(f)). The monotonicity of Vf implies that Vf (c+) ≥ Vf (f) >Vf (e(f)), and that for all c, c′ ∈ C, if {c} º {c′}, then Vf (c) ≥ Vf (c′). By the vNMtheorem, Vf is a positive linear transformation of U on C. As U(c0) = Vf (c0) = 0, thereexists a unique κf ∈ (0, 1) such that Vf (·) = κf

1−κfU(·) on C. The inequality Vf (c+) ≥

Vf (f) > Vf (e(f)) implies further that U(c+) > U(e(f)) and

1 ≥ Vf (f)−Vf (e(f))Vf (c+)−Vf (e(f)) >

Vf (f)−Vf (e(f))Vf (c+)−Vf (e(f))+U(c+)−U(e(f)) = Wf (f)−Wf (e(f))

Wf (c+)−Wf (e(f)) > 0. (16)

Take any α ∈ (0, 1) such that Vf (f)−Vf (e(f))Vf (c+)−Vf (e(f)) > α >

Wf (f)−Wf (e(f))Wf (c+)−Wf (e(f)) . Then Wf (e(f)+α) >

Wf (f) and Vf (f) > Vf (e(f) + α) because Wf and Vf are linear. By (15), {e(f) + α} Â{e(f) + α, f} Â {f}. This ranking implies that f is potentially tempting, and by [19,Theorem 2.1], that the representation (15) on Hf is unique.

Case 3. Vf (e(f)) > Vf (f). Then by Continuity, there exists α such that Vf (e(f)−α) >Vf (f +α). By (15), {f +α} Â {f +α, e(f)−α} º {e(f)−α}, which contradicts Constants-Do-Not-Tempt. So this case is impossible.

It follows that f is never tempting if and only if Case 1 holds, and f is potentiallytempting if and only if Case 2 holds. ¤

Lemma A.3 If f, g ∈ H are potentially tempting, then κf = κg.

Proof. Given the potentially tempting acts f, g ∈ H, let

f0 =

1U(f)+1f + U(f)

U(f)+1c− if U(f) ≥ 0

11−U(f)f + −U(f)

1−U(f)c+ if U(f) < 0,g0 =

1U(g)+1g + U(g)

U(g)+1c− if U(g) ≥ 0

11−U(g)g + −U(g)

1−U(g)c+ if U(g) < 0.

Then f0 ∈ Hf , g0 ∈ Hg, and e(f0)e(g0) = c0. By Lemma A.2, Vf (f0) > Vf (c0) = 0 andWf (f0) > 0. Fix γ ∈ (0, 1) such that Vf (f0 − γ) > 0 and Wf (f0 − γ) > 0. By (15),U({c0, f0 − γ} = U({f0 − γ} and U({c0, f0 − 1}) = U({c0, c−}) = 0.

Define a function φ on (0, 1] by

φ(α) =U({c0, f0 − α})U({f0 − α}) =

U({c0, f0 − α})−α

.

Then φ is continuous and satisfies φ(γ) = 1 and φ(1) = 0. By continuity, there existsαf ∈ (γ, 1) such that φ(αf ) = 1

2 . Then U(f0 − αf ) = −αf and Vf (f0 − αf ) = αf

2 .Analogously, find αg ∈ (0, 1) such that U(g0 − αg) = −αg and Vg(g0 − αg) = αg

2 .Let f ′ = αg(f0 − αf ) + (1− αg)c0 and g′ = αf (g0 − αg) + (1− αf )c0. Then

Vf (f ′) = Vg(g′) = αf αg

2 > 0 > Wf (f ′) = Wg(g′) = −αf αg

2 > U(f ′) = U(g′) = −αfαg.

Set-Betweenness and the representation (15) imply the rankings

{c0} Â {c0, f′} ∼ {c0, f

′, g′} ∼ {c0, g′} Â {f ′} ∼ {g′}.

25

Take ε > 0 such that Vf (c0 + ε) < Vf (f ′) and Vg(c0 + ε) < Vg(g′). By NC,

{c0 + ε, c0, f′} ∼ {c0 + ε, c0, f

′, g′} ∼ {c0 + ε, c0, g′}.

By (15), U({c0 + ε, c0, f′}) = ε

1−κf− αf αg

2 , and U({c0 + ε, c0, g′}) = ε

1−κg− αf αg

2 . Thusε

1−κf− αf αg

2 = ε1−κg

− αf αg

2 , that is, κf = κg. ¤

Let κ ∈ (0, 1) be such that κh = κ for all potentially tempting acts h ∈ H (κ exists byLemma A.3). For every f ∈ H, let W (f) = U(f) + V (f), where

• V (f) = Vf (f) if f is potentially tempting,

• V (f) = κ1−κU(f) if f is never tempting.

For every menu A ∈ A, let

UWV (A) = maxg∈A

W (g)−maxg∈A

V (g).

Later we show that both W and V are continuous and hence, the maxima in the abovedefinition are obtained even if A is not finite.

Lemma A.4 If f ∈ H is potentially tempting, then

(i) V (f) > κ1−κU(f),

(ii) V (·) = Vf (·) and W (·) = Wf (·) on Hf ,

(iii) for all finite menus A ∈ Af , U(A) = UWV (A).

Proof. Let f be potentially tempting. (i) By Lemma A.2,

V (f) = Vf (f) > Vf (e(f)) = κ1−κU(e(f)) = κ

1−κU(f).

(ii) Let g = αf +(1−α)c ∈ Hf . If α = 0, then g = c ∈ C and V (g) = κ1−κU(c) = Vf (g). If

α > 0, then g is potentially tempting because there exists c′ ∈ C such that {c′} Â {c′, f},which implies {αc′ + (1 − α)c} Â {αc′ + (1 − α)c, g} by Collinear Independence. ByLemma A.2, the function Vg in representation (15) is unique, and hence, Vg(·) = Vf (·) onHg ⊂ Hf . In particular, V (g) = Vg(g) = Vf (g).(iii) If A ∈ Af , then, by Lemma A.2,

U(A) = maxg∈A

Wf (g)−maxg∈A

Vf (g) = maxg∈A

W (g)−maxg∈A

V (g) = UWV (A)

because V (g) = Vf (g) and W (g) = Wf (g) for all g ∈ A. ¤

Lemma A.5 For all finite menus A ∈ A and for all acts f, g ∈ H, if U(f) = U(g) andV (f) = V (g), then U({f} ∪A) = U({g} ∪A).

26

Proof. Fix A, f and g as in the hypothesis and consider two possible cases.Case 1. f is never tempting. By Lemma A.2, for all α ∈ (0, 1),

{f + α} ∼ {f + α, e(f)− α} Â {e(f)− α},{e(f) + α} ∼ {e(f) + α, f − α} Â {f − α},

and by NC, {f + α} ∪ A ∼ {f + α, e(f) − α} ∪ A and {e(f) + α} ∪ A ∼ {e(f) + α, f −α} ∪ A. Let α → 0; by Continuity {f} ∪ A ∼ {f, e(f)} ∪ A ∼ {e(f)} ∪ A. The equalityV (g) = V (f) = κ

1−κU(f) = κ1−κU(g) implies, by Lemma A.4(i), that g is never tempting.

Therefore, a similar argument proves that {g} ∪ A ∼ {g, e(g)} ∪ A ∼ {e(g)} ∪ A. Finally,{f} ∪A ∼ {e(f)} ∪A = {e(g)} ∪A ∼ {g} ∪A, that is, U({f} ∪A) = U({g} ∪A).

Case 2. f is potentially tempting. By (16) and Lemma A.4.(ii),

1 ≥ V (f)−V (e(f))V (c+)−V (e(f)) > W (f)−W (e(f))

W (c+)−W (e(f)) > U(f)−U(e(f))U(c+)−U(e(f)) = 0.

Let c = e(f) + W (f)−W (e(f))W (c+)−W (e(f)) . Then U(c) > U(f), V (c) < V (f), and W (c) = W (f)

because the functions V and W are linear on Hf . Thus for any sufficiently small γ > 0,U(c) > U(f + γ), V (f + γ) > V (f − γ) > V (c), and W (f + γ) > W (c) > W (f − γ). ByLemma A.4(iii),

{c} Â {f + γ, c} ∼ {f + γ} ∼ {f + γ, f − γ} Â {f − γ}. (17)

By Lemma A.4(i), g is potentially tempting. Therefore, V and W are linear on Hg aswell; hence, V (g − γ) = V (f − γ) and W (g − γ) = W (f − γ). By Lemma A.4(iii) andSet-Betweenness,

{c, f − γ} ∼ {c, f − γ, g − γ} ∼ {c, g − γ} Â {g − γ}. (18)

It follows from NC and the rankings (17) and (18) that

{f +γ} ∼ {f +γ, c, f −γ} ∼ {f +γ, c, f −γ, g−γ} ∼ {f +γ, f −γ, g−γ} ∼ {f +γ, g−γ}.

Thus {f +γ} ∼ {f +γ, g−γ} Â {g−γ}. Analogously, {g +γ} ∼ {g +γ, f −γ} Â {f −γ}.By NC, {f + γ}∪A ∼ {f + γ, g− γ}∪A and {g + γ}∪A ∼ {g + γ, f − γ}∪A. Let γ → 0;then {f} ∪A ∼ {f, g} ∪A ∼ {g} ∪A by Continuity. ¤

Lemma A.6 For all finite menus A ∈ A, U(A) = UWV (A).

Proof. Fix a finite menu A ∈ A and consider several possible cases.Case 1. A = {f, g}, where both f and g are never tempting. Wlog U(f) ≥ U(g). As

U(g) = U(e(g)) and V (g) = κ1−κU(g) = V (e(g)), then by Lemmas A.5 and A.2,

U({f, g}) = U({f, e(g)}) = max{U(f), U(e(g))} = U(f).

On the other hand, the equality UWV ({f, g}) = W (f) − V (f)U(f) follows from the defi-nitions of the functions V , W , and UWV . Thus, U(A) = UWV (A).

27

Case 2. A = {f, g}, where f is potentially tempting, and g is never tempting. ThenU(g) = U(e(g)) and V (g) = V (e(g)). By Lemmas A.5 and A.4 (iv),

U({f, g}) = U({f, e(g)}) = UWV ({f, e(g)}) = UWV ({f, g}).Case 3. A = {f, g}, where both f and g are potentially tempting. Wlog U(f) ≥ U(g).

Consider three possible subcases.Subcase 3.1. U(f) = U(g). Wlog V (f) ≥ V (g). By Set-Betweenness,

U({f, g}) = U({f}) = W (f)− V (f) = UWV ({f, g}).Subcase 3.2. U(f) > U(g) and V (g) ≥ V (f). We claim that there exist α ∈ (0, 1) and

c ∈ C such that U(αg +(1−α)c) = U(f) and V (αg +(1−α)c) = V (f). To construct suchα and c, let Y (f) = (1 − κ)V (f) − κU(f) and Y (g) = (1 − κ)V (g) − κU(g). Then Y (f)and Y (g) are both positive by Lemma A.4(i) and satisfy the identity

κ[U(f)Y (g)− U(g)Y (f)] = (1− κ)[V (f)Y (g)− V (g)Y (f)].

The inequalities U(f) > U(g) and V (g) ≥ V (f) imply that Y (g) > Y (f), and hence,

−1 ≤ U(g) < U(f)Y (g)−U(g)Y (f)Y (g)−Y (f) = 1−κ

κ · V (f)Y (g)−V (g)Y (f)Y (g)−Y (f) ≤ 1−κ

κ · V (f) ≤ 1.

Take α = Y (f)Y (g) ∈ (0, 1) and c ∈ C such that U(c) = U(f)Y (g)−U(g)Y (f)

Y (g)−Y (f) ∈ [−1, 1]. Then

U(αg + (1 − α)c) = U(f) by linearity of U , V (c) = κ1−κU(c) = V (f)Y (g)−V (g)Y (f)

Y (g)−Y (f) , andhence, V (αg + (1− α)c) = V (f) by linearity of V on Hg.

Conclude by Lemmas A.5 and A.4(iv) that

U({f, g}) = U({αg + (1− α)c, g}) = UWV ({αg + (1− α)c, g}) = UWV ({f, g}).Subcase 3.3. U(f) > U(g) and V (f) > V (g). As V (g) > V (e(g)) and V is linear onHf ,

there exists α ∈ (0, 1) such that V (αf + (1− α)e(g)) = V (g). Let f ′ = αf + (1− α)e(g).As 0 < α < 1, f ′ is potentially tempting and satisfies U(f) > U(f ′) > U(g), V (f) >V (f ′) = V (g), and W (f) > W (f ′) > W (g). By Lemma A.4(iii) and by Subcase 3.2,

U({f, f ′}) = UWV ({f, f ′}) = W (f)− V (f) = U(f) > U(f ′) andU({f ′, g}) = UWV ({f ′, g})W (f ′)− V (f ′) = U(f ′) > U(g).

By NC, {f, g} ∼ {f, f ′, g} ∼ {f, f ′} ∼ {f}. Thus, U({f, g}) = U(f) = W (f) − V (f) =UWV ({f, g}).

Case 4. A is an arbitrary finite menu. Take gA ∈ arg maxf∈A W (f) and hA ∈arg maxf∈A V (f). Then for all f ∈ A,

UWV ({gA, f}) ≥ UWV ({gA, hA}) ≥ UWV ({f, hA}).Cases 1–3 imply that U({gA, f}) ≥ U({gA, hA}) ≥ U({f, hA}), that is, {gA, f} º {gA, hA} º{f, hA}. From Set-Betweenness, it follows by induction with respect to the size of the setA that

A =⋃

f∈A

{gA, f} º {gA, hA} º⋃

f∈A

{f, hA} = A,

that is, A ∼ {gA, hA}. Thus, U(A) = U({gA, hA}) = UWV ({gA, hA}) = UWV (A). ¤

28

Lemma A.7 There exists a convex and closed set Q of probability measures on S suchthat for all f ∈ H,

V (f) = κ1−κ max

q∈Qq · u(f). (19)

Moreover, Q is unique and p ∈ Q.

Proof. First show that V is monotonic, continuous, and quasi-convex.Monotonicity: Take any f, f ′ ∈ H such that f dominates f ′. For all α ∈ (0, 1), Mono-tonicity and Lemma A.1 imply that {f + α} ∼ {f + α, f ′−α} Â {f ′−α}. It follows fromLemma A.6 that V (f + α) ≥ V (f ′ − α), that is,

αV (c+) + (1− α)V (f) ≥ αV (c−) + (1− α)V (f ′).

Take α → 0 to deduce that V (f) ≥ V (f ′).Continuity. Let a sequence of acts fn converge to f as n →∞. There exist sequences αn

and βn both converging to zero such that f + αn dominates fn, and fn dominates f − βn.As V is monotonic,

αnV (c+) + (1− αn)V (f) ≥ V (fn) ≥ βnV (c−) + (1− βn)V (f).

It follows that V (f) = limn→∞ V (fn).Quasi-Convexity. Suppose that V (αf + (1 − α)g) > V (f) = V (g) for some f, g ∈ H andα ∈ (0, 1). Take c ∈ C such that

V (αf + (1− α)g) > V (c) > V (f) = V (g).

Then V (c) > V (e(f)) and V (c) > V (e(g)). By monotonicity of V , U(c) > U(f), U(c) >U(g) and hence, U(c) > U(αf + (1 − α)g). By Lemma A.6, {c} ∼ {c, f} Â {f} and{c} ∼ {c, g} Â {g}. However,

{c} Â {c, αf + (1− α)g} º {αf + (1− α)g},contradicting Convex Temptation.

The preceding shows that the ranking on H represented by V satisfies all the axiomsof the maxmax model—these are the axioms of Gilboa and Schmeidler’s multiple-priorsmodel [14], with the exception that “Uncertainty Aversion”, which is convexity of weaklybetter-than sets, is replaced by convexity of weakly worse-than sets. It follows from [14]that V has the form (19), and that Q is unique. The inclusion p ∈ Q follows from the factthat for all f ∈ H, V (f) ≥ V (e(f)) = p · u(f). ¤

The continuity of V and W implies that UWV is continuous on A. Lemma A.6 assertsthat U ≡ UWV on the set of all finite menus, which is dense in A. Thus U ≡ UWV on allof A.

To show the required uniqueness of (u, p, κ, Q) in representation (4)-(6), suppose thatthis tuple can be replaced by (u′, p′, κ′, Q′). Then u′ is a positive linear transformationof u, and hence, (u, p, κ, Q) can be replaced by (u, p′, κ′, Q′) as well. The uniquenessstatements in Lemmas A.1, A.2 and A.7 imply that if º is not strategically rational, thenp = p′, κ = κ′, and Q = Q′.

29

B Appendix: Proofs for Comparative Disso-

nance

Proof of Theorem 4.2: Let º∗ and º conform to our model with corresponding tuples(u∗, p∗, Q∗, κ∗) and (u, p, Q, κ). Suppose that neither preference is strategically rational.Then κ, κ∗ > 0 and sufficiency of (9) and (10) is immediate:

{f} Â {f, g} ⇒ [p · u(f) > p · u(g) ∧Q · u(g) > Q · u(f)] ⇒[p∗ · u∗(f) > p∗ · u∗(g) ∧Q∗ · u∗(g) > Q∗ · u∗(f)] ⇒ {f} Â∗ {f, g}.

For necessity, let º∗ have greater dissonance than º. For all vectors a ∈ RS , let

Q · a = maxq∈Q

q · a and Q∗ · a = maxq∈Q∗

q · a. (20)

Lemma B.1 (i) u and u∗ are cardinally equivalent.(ii) For all a, b ∈ RS,

p · a > p · b and Q · b > Q · a ⇒ p∗ · a > p∗ · b and Q∗ · b > Q∗ · a. (21)

(iii) p = p∗.

Proof. First, show that for all c, c′ ∈ C,

u(c) = u(c′) ⇒ u∗(c) = u∗(c′). (22)

Suppose to the contrary that u(c) = u(c′) and u∗(c) > u∗(c′) for some c, c′ ∈ C. Takef, g ∈ H such that {f} Â {f, g}. The equality u(c) = u(c′) implies

{αf + (1− α)c} Â {αf + (1− α)c, αg + (1− α)c′}.

Because º∗ has greater dissonance, {f} Â∗ {f, g}. Therefore, the inequality u∗(c) > u∗(c′)implies that for sufficiently small α > 0,

{αf + (1− α)c} ∼∗ {αf + (1− α)c, αg + (1− α)c′}.

But this contradicts the hypothesis that º∗ has greater dissonance than º.Take c+, c− ∈ C such that u(c+) > u(c−) and u(c+) ≥ u(c) ≥ u(c−) for all c ∈ C.

Then for all c ∈ C,

c ∼ u(c)− u(c−)u(c+)− u(c−)

c+ +u(c+)− u(c)

u(c+)− u(c−)c−, and by (22),

u∗(c) =u∗(c+)− u∗(c−)u(c+)− u(c−)

u(c) +u∗(c−)u(c+)− u∗(c+)u(c−)

u(c+)− u(c−).

Note that u∗(c+) 6= u∗(c−) because º∗ is not strategically rational and hence non-degenerate. Thus, either u∗ is a positive linear transformation of u, or u∗ is a negative

30

linear transformation of u. Next, we show that the former case implies statements (ii) and(iii), and that the latter case is impossible.

Case 1. u∗ is a positive linear transformation of u. Wlog assume that u = u∗ andu(C) = u∗(C) = [−1, 1]. Fix any a, b ∈ RS such that p · a > p · b and Q · b > Q · a. Takeα > 0 such that | αa(s) |, | αb(s) | ≤ 1 for all s ∈ S. Then αa = u(f) and αb = u(g) forsome f, g ∈ H. (Here u(f) and u(g) are vectors in RS .) Then

p · a > p · b and Q · b > Q · a ⇒ p · u(f) > p · u(g) and Q · u(g) > Q · u(f) ⇒{f} Â {f, g} ⇒ {f} Â∗ {f, g} ⇒

p∗ · u(f) > p∗ · u(g) and Q∗ · u(g) > Q∗ · u(f) ⇒ p∗ · a > p∗ · b and Q∗ · b > Q∗ · a,

which proves (ii).To show (iii), suppose that p 6= p∗. Let

R = {q ∈ RS : q = p + α(p− p∗) for α ≥ 0} = {q ∈ RS : p ∈ [q, p∗]}.Consider two subcases.

(1) Q 6⊂ R: Let p′ ∈ Q\R. Take a hyperplane b ∈ RS that separates the singleton pand the segment [p′, p∗]:

p · b < 0, p′ · b > 0, p∗ · b > 0.

These inequalities violate (21) for a = 0.

(2) Q ⊂ R: Then Q is a segment with end points p and p′ = p + α(p − p∗) for someα > 0. Note that p is an interior point of the segment [p∗, p′]. Take a hyperplanea ∈ RS that separates p∗ and p′ and passes through p:

p∗ · a > 0, p · a = 0, p′ · a < 0.

Take a hyperplane b ∈ RS that separates p′ and the segment [p, p∗]:

p′ · b > 0, p · b < 0, p∗ · b < 0.

Wlog p∗ · a > Q∗ · b (multiply a by a positive scalar if needed). Thus p · a > p · b,Q · b ≥ p′ · b > 0 = max{p · a, p′ · a} = Q · a,

but Q∗ · a ≥ p∗ · a > Q∗ · b. This contradicts (21).

Case 2. u∗ is a negative linear transformation of u. We show this is impossible.Wlog assume that u∗ = −u and u(C) = u∗(C) = [−1, 1]. Then, paralleling (21) in the

previous case,

p · a > p · b and Q · b > Q · a ⇒ p∗ · (−a) > p∗ · (−b) and Q∗ · (−b) > Q∗ · (−a). (23)

for all a, b ∈ RS . It follows that for some a, b ∈ RS , p · a > p · b but p∗ · a < p∗ · b. Thusp 6= p∗. Consider two subcases.

31

(1) Q 6⊂ [p, p∗]: Let p′ ∈ Q\[p, p∗]. Take a hyperplane b ∈ RS that separates p′ and[p, p∗]:

p′ · b > 0, p · b < 0, p∗ · b < 0.

This contradicts (23) for a = 0.

(2) Q ⊂ [p, p∗]: Then Q is a segment with end points p and p′ = αp∗ + (1 − α)p forsome α > 0. Take a hyperplane a ∈ RS that separates p and [p′, p∗]:

p · a = 0, p′ · a < 0, p∗ · a < 0.

Take another hyperplane b ∈ RS that separates p and [p′, p∗]:

p · b < 0, p′ · b > 0, p∗ · b > 0.

Wlog p∗ · (−a) > Q∗ · (−b) (multiply a by a positive scalar if needed). Thenp · a = 0 > p · b,

Q · b ≥ p′ · b > 0 = max{p′ · a, p · a} = Q · a,

but Q∗ · (−a) ≥ p∗ · (−a) > Q∗ · (−b). This contradicts (23). ¤

The following method of proof is analogous to the one used by Kopylov [20]. Let Dbe the set of all points a ∈ RS at which the convex functions Q · a and Q∗ · a are bothdifferentiable. By [30, Theorem 25.5], the complement of the set D has measure zero. ThusD is dense. For every a ∈ D, let

q(a) = ∇(Q · a) and q∗(a) = ∇(Q∗ · a)

be the derivatives of Q · a and Q∗ · a respectively. Let ~1 = (1, . . . , 1) ∈ RS .

Lemma B.2 The functions q(·), q∗(·) : D→ RS have the following properties:

(i) For all a ∈ D and q ∈ Q, q = q(a) iff Q · a = q · a.(ii) For all a ∈ D and q ∈ Q∗, q = q∗(a) iff Q∗ · a = q · a.(iii) If a ∈ D, α > 0 and γ ∈ R, then

αa + γ~1 ∈ D, q(αa + γ~1) = q(a), q∗(αa + γ~1) = q∗(a).

(iv) For any a ∈ D, there exists εa ∈ [0, 1] such that q(a) = εaq∗(a) + (1− εa)p.

(v) There exists ε ∈ [0, 1] such that q(a) = εq∗(a) + (1− ε)p for all a ∈ D.

Proof.

(i) Fix a ∈ D and q ∈ Q such that Q · a = q · a. For all b ∈ RS and δ ∈ R,

Q · a + δ(q · b) = q · (a + δb) ≤ Q · (a + δb) = Q · a + δ(q(a) · b) + o(δ).

Then q · b = q(a) · b for all b ∈ RS , that is, q = q(a). Similarly for (ii).

32

(iii) Fix a ∈ D, α > 0 and γ ∈ R. Then αa + γ~1 ∈ D because the superpositionQ · b = αQ ·

(b−γ~1

α

)+ γ is differentiable at αa + γ~1. By (i), q(αa + γ~1) = q(a)

because Q · (αa + γ~1) = α(Q · a) + γ = q(a) · (αa + γ~1). Similarly for Q∗ and q∗ (·).(iv) Suppose that for some a no such εa exists. Let b separate q (a) from the segment

[q∗ (a) , p], so that q∗(a) · b < 0, p · b < 0, but q(a) · b > 0. Then for sufficiently smallδ > 0, Q∗ · (a + δb) = Q∗ · a + δ(q∗(a) · b) + o(δ) < Q∗ · a, but also

p · a > p · (a + δb) and Q · (a + δb) ≥ q (a) · (a + δb) > q (a) · a = Q · a.

By (21), Q∗ · (a + δb) > Q∗ · a, a contradiction.

(v) Let a, b ∈ D be such that q∗(a) 6= p and q∗(b) 6= p, and prove εa = εb. (Note thatif q∗(a) 6= p, then εa is unique, and if q∗(a) = p, then εa ∈ [0, 1] is arbitrary. ) Asq∗(a) 6= p and p = p∗ ∈ Q∗, then by (iii), Q∗ · a > p · a. Similarly, Q∗ · b > p · b. Let

a′ =a− (p · a)~1

Q∗ · a− p · a and b′ =b− (p · b)~1

Q∗ · b− p · b .

By (iii) and (iv), a′, b′ ∈ D, q∗(a′) = q∗(a), q∗(b′) = q∗(b), and

q(a′) = q(a) = εaq∗(a) + (1− εa)p and q(b′) = q(b) = εbq∗(b) + (1− εb)p.

By construction, p · a′ = p · b′ = 0, Q∗ · a′ = Q∗ · b′ = 1, Q · a′ = εa, and Q · b′ = εb.Suppose that εa 6= εb; wlog let εa < εb. Then for sufficiently small γ > 0,

p · (a′ + γ~1) = γ > p · b′, Q · (a′ + γ~1) = εa + γ < εb = Q · b′,but Q∗ · (a′ + γ~1) = 1 + γ > Q∗ · b′. This contradicts (21). Thus εa = εb. ¤

Conclude that Q · a = ε(Q∗ · a) + (1− ε)(p · a) for all a ∈ D and hence, by continuity,for all a ∈ RS . It follows that Q = εQ∗ + (1 − ε)p; ε > 0 because º is not strategicallyrational. This completes the proof of Theorem 4.2.

Proof of Theorem 4.3: Let º∗ and º conform to our model with corresponding tuples(u∗, p∗, Q∗, κ∗) and (u, p, Q, κ). Suppose that neither preference is strategically rational.

Let P = (1 − κ){p} + κQ and P ∗ = (1 − κ∗){p∗} + κ∗Q∗. The conditions (9), (10),and κ∗ ≥ εκ imply

P =(1− εκ

κ∗

){p}+

εκ

κ∗P ∗.

Sufficiency of these conditions now follows from:

{f} Â {f, g} ∼ {g} ⇒ [p · u(f) > p · u(g) ∧ P · u(g) > P · u(f)] ⇒[p∗ · u∗(f) > p∗ · u∗(g) ∧ P ∗ · u∗(g) > P ∗ · u∗(f)] ⇒ {f} Â∗ {f, g} ∼∗ {g}.

For necessity, let º∗ be more self-justifying than º. Then º∗ has more dissonancethan º, and Theorem 4.3 implies (9) and (10). Moreover, for all a, b ∈ RS ,

p · a > p · b and P · b > P · a ⇒ p∗ · a > p∗ · b and P ∗ · b > P ∗ · a. (24)

33

To prove this claim, fix any a, b ∈ RS . Take α > 0 and f, g ∈ H such that αa = u(f) andαb = u(g). Then

p · a > p · b and P · b > P · a ⇒ p · u(f) > p · u(g) and P · u(g) > P · u(f) ⇒{f} Â {f, g} ∼ {g} ⇒ {f} Â∗ {f, g} ∼∗ {g} ⇒

p∗ · u(f) > p∗ · u(g) and P ∗ · u(g) > P ∗ · u(f) ⇒ p∗ · a > p∗ · b and P ∗ · b > P ∗ · a.

Use the condition (24) to replace Q and Q∗ by P and P ∗ in Lemma B.2 and obtain0 < θ ≤ 1 such that P = (1− θ){p}+ θP ∗. In particular, P ⊂ P ∗ and therefore also

(1− κε){p}+ κεQ∗ ⊂ (1− κ∗){p}+ κ∗Q∗.

As Q∗ is a nonsingleton, κε ≤ κ∗. ¤

C Appendix: Examples

We provide some details for Example 1.

Set-Betweenness: suppose that U (A) ≤ U (B). Then

U (A ∪B) =maxh∈A∪B [U (h)V (h)]

maxh′∈A∪B V (h′)

=max {maxh∈A [U (h)V (h)] , maxh∈B [U (h) V (h)]}

max {maxh∈A V (h) , maxh∈B V (h)}

≤max

{maxh∈A V (h)maxh∈B V (h) maxh∈B [U (h)V (h)] , maxh∈B [U (h)V (h)]

}

max {maxh∈A V (h) , maxh∈B V (h)} .

If maxh∈A V (h)maxh∈B V (h) ≤ 1, then above equals maxh∈B [U(h)V (h)]

maxh∈B V (h) = U (B). If maxh∈A V (h)maxh∈B V (h) ≥ 1, then

above equalsmaxh∈A V (h)maxh∈B V (h) maxh∈B [U (h) V (h)]

maxh∈A V (h)= U (B) .

Thus U (A ∪B) ≤ U (B) .The verification of U (A) ≤ U (A ∪B) is symmetric:

U (A ∪B) =max {maxh∈A [U (h) V (h)] , maxh∈B [U (h)V (h)]}

max {maxh∈A V (h) , maxh∈B V (h)}

≥max

{maxh∈A [U (h)V (h)] , maxh∈B V (h)

maxh∈A V (h) maxh∈A [U (h)V (h)]}

max {maxh∈A V (h) , maxh∈B V (h)}If maxh∈B V (h)

maxh∈A V (h) ≤ 1, then above equals maxh∈A[U(h)V (h)]maxh∈A V (h) = U (A). If maxh∈B V (h)

maxh∈A V (h) ≥ 1, thenabove equals

maxh∈B V (h)maxh∈A V (h) maxh∈A [U (h)V (h)]

maxh∈B V (h)= U (A) .

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Thus U (A ∪B) ≥ U (A) .

NC: {f ′} Â {f, f ′} ∼ {f} =⇒ [U (f ′) > U (f) ∧ UV (f ′) < UV (f)].But choice out of {f, f ′, g} solves maxh=f,f ′,gU (h)V (h) - it follows that f ′ could not bechosen. Thus {f, f ′} ∼ {f, f ′, g} as argued in the text.

Similarly if {f} ∼ {f, f ′} Â {f ′}. The bottom line is that NC is satisfied because thetemptation ranking is represented by V and choice out of menus is rationalizable by UV .

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