Imperfect Memory and Behavior under Risk

Daniel Gottlieb�

June 20, 2008

Abstract

This paper proposes a model of choice under risk based on imperfect memory and self-deception. The model assumes that people have preferences over their perceived charac-teristics and can, to some extend, manipulate their memories. It leads to a non-expectedutility representation and provides a uni�ed explanation for several empirical regularities,including non-linear probability weights, �rst-order risk aversion, the uncertainty e¤ect, theendowment e¤ect, and the sunk cost fallacy.

1 Introduction

Choices with uncertain outcomes are an important part of a person�s life. Most of the times,the outcomes are at least partially determined by the person�s characteristics. Therefore, theya¤ect how one views himself and how he is viewed by others. Choices that turn out to bewrong typically lead to self-doubts while choices that turn out to be right enhance the person�sself-image.

A person that cares about self-image has an incentive to manipulate recollections and beliefs.Of course, this incentive does not matter if memory is perfect. This paper analyzes how theconcern for self-image a¤ects an individual�s behavior under risk when memory is imperfect.The individual is assumed to behave as a standard expected utility maximizer except for thefact that he has imperfect memory.

Memory management leads the decision-maker to avoid lotteries whose outcomes generatememory manipulation. Theorem 1 (Section 2) shows that preferences over signals � whoseoutcomes � 2 fL;Hg are correlated with the person�s characteristics can be represented by autility function of the form

U (�) = w (q)uH + [1� w (q)]uL;

where u� is the expected utility given � and q is the probability of observing � = H: w is aprobability weighting function such that w (q) � q; with strict inequality whenever there is mem-ory manipulation. Therefore, outcomes that lead to memory manipulation are worth less thanpredicted by the standard expected utility model. Subsection 2.5 employs this representation todiscuss the demand for information.

Section 4 augments the model to allow for lotteries over money. Theorem 2 shows thatpreferences over lotteries $ with monetary outcomes t 2 fH;Lg can be represented by

U ($) = w (q) vH + [1� w (q)] vL;�Department of Economics, MIT. Preliminary version.

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where vt is the decision-maker�s utility in the state where t occurs. The probability weight w (q)di¤ers from the probability q in two respects. First, w (q) is lower when outcomes lead to memorymanipulation. Second, the larger the degree of complementarity between ability and money, thelarger w (q) will be. These preferences provide an explanation for non-linear weighting functionsbased on memory manipulation.

As in other models that admit representations by non-linear weighting functions, the decision-maker may exhibit �rst-order risk aversion (Subsection 4.2). An agent with �rst-order riskaversion rejects small gambles with small but positive expected value. The agent may also exhibita gap between the maximum willingness to pay for a good and the minimum compensationdemanded for the same endowment (endowment e¤ect). However, di¤erently from other non-expected utility models, the departure from linear weighting functions in our model is directlyrelated with the decision-maker�s perceived characteristics. This is consistent with experimentalevidence suggesting that departures from expected utility theory are associated with the lotteriesbeing correlated with the decision-maker�s skill or knowledge [e.g. Heath and Tversky (1991),Fox and Tversky (1995), Goodie (2003), and Goodie and Young (2007)].1

Subsection 4.3 presents a self-deception explanation for the endowment e¤ect. The mainidea is that successful trading often requires certain skills or knowledge. At the very least, theagent must form expectations about how much each good is worth. In more complex markets, hemust also estimate the future prices of the goods. Thus, the outcome of the trade is informativeabout the person�s skills. Since decision-makers avoid information correlated with skills, theywill only accept to trade if the expected bene�t from trade is above a certain positive threshold.

In Subsection 4.4, it is shown that memory manipulation may cause people to value a lotteryless than the worst possible monetary outcome from that lottery. Known as the uncertaintye¤ect, this result was documented by Gneezy, List, and Wu (2006), who claim that it contradicts�virtually all models of risky choice�.

Subsection 4.5 provides a self-views rationale for the existence of sunk cost e¤ects. Accord-ing to this explanation, which is consistent with arguments made in the psychology literature,abandoning a project usually involves admitting that a wrong decision was made. Thus, revisingone�s position is often informative about the decision-maker�s skills or knowledge. Sunk decisionsmay in�uence current behavior if the agent wants to avoid information about his characteristics.

Section 3 considers a repeated environment. The agent observes a sequence of signals thatare informative about her characteristics and engages in memory manipulation after each real-ization. It is shown that the agent�s behavior and attitude towards risk converge to that impliedby expected utility theory as the number of signals grows. This result is consistent with theargument that people do not exhibit ambiguity aversion over events that have been observedseveral times and that experts are subject to much less biases than beginners [e.g. List (2003)].

Subsections 1.1 and 1.2 brie�y review the psychological evidence on memory and the relatedliterature in Economics.

1.1 An Overview

�Ego-involvement, or its absence, makes a critical di¤erence in human behavior.When a person reacts in a neutral, impersonal, routine atmosphere, his behavior isone thing. But when he is behaving personally, perhaps excitedly, seriously com-mitted to a task, he behaves quite di¤erently. In the �rst condition his ego is notengaged; in the second, it is.�

1See Subsection 4.1 for a more detailed discussion.

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Gordon W. Allport (1943, pp.459).

Psychologists have largely documented a human tendency to deny or misrepresent reality tooneself (i.e., engage in self-deception). In general, people consider themselves as being �smart�,�knowledgeable�, and �nice�. Information con�icting with this image is usually ignored or de-nied.

People are more likely to remember successes than failures [Korner (1950)]. After choosingbetween two di¤erent options, they tend to recall the positive aspects of the chosen option andthe negative aspects of the forgone option [Mather, Sha�r, and Johnson (2003)]. Relatedly,individuals overestimate their achievements and readily �nd evidence that they possess charac-teristics which they believe to be correlated with success in professional or personal life [Kundaand Sanitioso (1989)]. Failure is usually attributed to external factors while success tends to beattributed to one�s own actions [Zuckerman (1979)].2

Self-assessments and the memory are intrinsically connected. In his Essay Concerning Hu-man Understanding, Locke (1690) identi�ed the self with memory. Mill (1829) argued that�[t]he phenomenon of Self and that of Memory are merely two sides of the same fact�. Moderncognitive psychologists de�ne the self as the �mental representation of oneself, including all thatone knows about oneself� [Kihlstrom et. al, (2002)]. Therefore, a model of self-views shoulddevote considerable attention to memory.

In Psychology, the memory is typically viewed as imperfect and manipulable. Rapaport(1961), for example, conceived �memory not as an ability to revive accurately impressions onceobtained but as the integration of impressions into the whole personality and their revivalaccording to the needs of the whole personality.�Allport (1943) believed that self-deception wasa mechanism of ego defense and the maintenance of self-esteem. Hilgard (1949, pp. 374) arguedthat �the need for self-deception arises because of a more fundamental need to maintain or torestore self-esteem. Anything belittling the self is to be avoided.� Festinger (1957) suggestedthat individuals have a tendency to seek consistency among their cognitions (i.e., beliefs andopinions). The discomfort felt when one is presented with evidence that con�icts with his or herbeliefs and the resulting e¤ort to distort one�s beliefs or opinions was called cognitive dissonance.

There are several reasons why people may want to believe in things that are not true. First,there may be a hedonic value of positive self-views so that they simply like to think that they havethese attributes.3 Second, as argued by Compte and Postlewaite (2004), a person may bene�tfrom having overcon�dent beliefs in situations where emotions a¤ect performance. Third, since�the best liar is the one who believes his own lies,� there may be a signaling value.4 It mayalso play a role as a credible self-promotion or self-exaggeration device. Fourth, there may be amotivational value of belief manipulation. As argued by Benabou and Tirole (2002), con�dencein one�s ability may help the person take more ambitious goals and persist in adverse situations.

In this paper, I abstract from the exact reason why people may value a positive self-image.The model developed here is based on the two basic premises discussed above. First, thatindividuals have preferences over their perceived abilities. Second, that they can (to someextent) a¤ect what they will remember. Apart from these assumptions, individuals are assumedto behave as in standard economic models. Their preferences satisfy the standard expected

2See Van den Steen (2004) for a model of rational agents with di¤ering priors that generates these biases.3See Schelling�s (1985) theory of the mind as a consuming organ.4As argued by Trivers (2000), �[b]eing unconscious of ongoing deception may more deeply hide the deception.

Conscious deceivers will often be under the stress that accompanies attempted deception.� This argument isformally modelled by Byrne and Kurland (2001).

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utility axioms. Furthermore, they follow Bayes�rule and, therefore, are aware of their memoryimperfection.5 The main focus of the paper is on how memory manipulations a¤ect the person�schoice under risk.

As demonstrated by the opening quote from Allport, psychologists have long realized thatself-deception may change a person�s behavior. Festinger (1957, pp. 3), for example, arguedthat �[w]hen dissonance is present, in addition to trying to reduce it, the person will activelyavoid situations and information which would likely increase the dissonance�. More recently,Heath and Tversky (1991) argued that the consequences of bets include the credit and blameassociated with the outcome as well as with monetary payo¤s. Therefore, psychic costs resultingfrom the person�s self-evaluation may be an important component of behavior under risk.

In this paper, it is shown how the idea of self-deception can lead to a uni�ed theory of choiceunder risk that is consistent with economic phenomena such as ambiguity averse/seeking behav-ior, �rst-order risk aversion, the endowment e¤ect, the sunk cost fallacy, and the uncertaintye¤ect.

1.2 Related Literature

The economic literature on imperfect memory can be divided in two strands. The �rst assumesthat decision makers are naive and act as if they have not forgotten anything (Mullainathan,2002). The other strand assumes that decision makers are sophisticated so that they drawBayesian inferences given that they might have forgotten things. In this paper, I will follow thelatter approach and consider the case of rational decision makers subject to imperfect recall.

As suggested by Piccione and Rubinstein (1997), the resulting game of imperfect recall issolved by the principle of "multiself consistency", whereby decisions made in di¤erent stages areviewed as being made by di¤erent incarnations of the decision maker.

An important special case of imperfect memory are models of limited memory. They wereoriginally proposed by Robbins (1956) in the Mathematical Statistics literature. He suggested adecision rule for choosing between two lotteries with unknown distributions that was conditionalon a �nite number of outcomes (�nite memory). In a series of papers, Cover and Hellmancharacterized optimal solutions to some �nite memory problems.6

More recently, economists have independently studied optimal decision making subject tolimited memory. Dow (1991) considered the behavior of a consumer looking for the lowest price.Wilson (2004) studied how limited memory leads to certain biases in belief formation. Hirshleiferand Welch (2002) considered informational cascades generated by players who observe actionsbut not the information leading to such actions. Bernheim and Thomadsen (2005) showed thatmemory imperfections and anticipatory emotions may lead to a resolution of Newcomb�s Paradoxand sustain cooperation in the Prisoners Dilemma.

In a sequence of papers, Benabou and Tirole have used imperfect memory frameworks tostudy questions from the Psychology literature. Based on the assumption that agents recalledactions but not their motivations, they have proposed theories of personal rules and internalcommitments [Benabou and Tirole (2004)], prosocial behavior [Benabou and Tirole (2006b)],and identity and taboos [Benabou and Tirole (2006c)]. Using a model of self-deception, Benabouand Tirole (2002, 2006a) analyzed the provision of self-motivation and the formation of collectivebeliefs and ideologies.

The model of memory presented here is general enough to allow for an agnostic view ofthe behavior of the memory system. It encompasses both Benabou and Tirole�s self-deception

5 In Appendix C, I consider the case of non-Bayesian individuals.6See Hellman and Cover (1973) for a review of the main results in this literature.

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framework and a static version the limited memory framework as special cases. This paper isalso connected to the economic literature on cognitive dissonance [Akerlof and Dickens (1982),Rabin (1994)]. This literature assumes that agents derive utility from their beliefs and that theycan, at some cost, choose their beliefs. Separately, the idea of anticipatory emotions has beenstudied by Lowenstein (1987), Caplin and Leahy (2001 and 2004), and K½oszegi (2006).7

2 Basic Framework

Consider a decision-maker (DM) who has preferences over her characteristics �. The agent�scharacteristics � may be interpreted as her skills or knowledge as well as a parameter of an-ticipatory utility. Let � be a non-empty subset of R representing the possible values of � anddenote by F (:) the agent�s prior distribution of �.8 Preferences satisfy the standard axiomsof expected utility so that there exists a strictly increasing von Neumann-Morgenstern utilityfunction u : �! R representing DM�s preferences over characteristics �.

The individual acts in two periods: 1 and 2: In period 1, a signal �, which can be either high(H) or low (L), is observed. Denote the probability of observing a high signal by q 2 (0; 1) : Ahigh signal is assumed to be more favorable than � = L in the sense of �rst-order stochasticdominance:

F (�j� = H) � F (�j� = L) for all � 2 �;with strict inequality for some value of �:

The informational structure is represented in Figure 1. Following Benabou and Tirole (2002,2006a) and Rabin (1994), I assume that the individual can, at a cost, manipulate her recollec-tions. The DM remembers a signal i 2 fL;Hg with probability

�i + ri � fi:

The parameter �i 2 [0; 1] is the agent�s "natural" rate of remembering signal i:9 It determinesthe probability that DM recollects the signal if she does not employ any manipulation e¤ort.However, the DM is able to depart from the natural rate of forgetting the signal by exertinge¤orts ri and fi; where ri 2 [0; 1� �i] denotes an e¤ort to remember and fi 2 [0; �i + ri] denotesan e¤ort to forget the signal. Engaging in memory manipulation ri and fi leads to a cost of r (ri) � 0 and f (fi) � 0:

The costs of memory manipulation r and f can be related to psychic costs (stress fromrepression), time (searching for reassuring information or excuses, lingering over positive feed-back), or real resources (avoiding certain cues and interactions or eliminating evidence).10 Theycan also be interpreted as the shadow costs of memory in a limited information framework.Remembering one signal with probability above the natural rate �i requires the individual tofocus on it and on information correlated with it. In turn, this restricts the amount of attentionavailable to other information (which has shadow cost r). Similarly, forgetting a signal withprobability above the natural rate 1��i requires the individual to focus on confronting evidencewhich, again, restricts the amount of attention available to other potentially useful information.

7See also Philipson and Posner (1995), Caplin and Eliaz (2003), and K½oszegi (2003) for speci�c applications.8� can be continuous or discrete, as long as it contains at least two elements (otherwise, � cannot be random).

Note that we have not assumed that the agent has a correct prior distribution over �: Therefore, agents are allowedto hold optimistic or pessimistic beliefs about their characteristics.

9The natural rates of learning and forgetting were initially studied by the German psychologist HermannEbbinghaus.10 In Hvide (2002) and Brunnermeier and Parker (2005), the costs of manipulation arise endogenously through

suboptimal decision-making.

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Figure 1: Informational Structure

Assumption 1 The cost of memory manipulation i (:) is strictly increasing, convex, twice-continuously di¤erentiable, and satis�es i (0) = 0; i 2 fr; fg :

Assumption 1 leads to a cost of memory manipulation as depicted in Figure 2. Becausememory manipulation is costly, the DM will never simultaneously choose ri > 0 and fi > 0:11

Furthermore, I assume that there is some positive probability of forgetting a high signal if theagent does not spend e¤ort in order to remember it: �H < 1:12

Throughout the paper, I will also consider two alternative assumptions about the cost ofmanipulation:

Assumption 2a 0i (0) = 0; i 2 fr; fg and �L > 0.

Assumption 2b 0i (0) > 0; i 2 fr; fg :

Assumption 2a implies that the cost of a small amount of memory manipulation is of order2 and that the space of feasible fL�s is non-trivial. 2b implies that the cost of a small amountof memory manipulation it is of order 1. As will be described in Subsection 2.2, Assumption2a ensures that all equilibria have positive amounts of manipulation whereas there might beequilibria with no manipulation under 2b:

11Lingering over positive feedback may also be pleasant for a certain amount of time. In that case, �H 2 [0; 1]can be interpreted as the rate that maximizes utility from lingering over positive feedbacks. The results from thepaper remain unchanged if the pleasure from lingering over good news is "not too high". Otherwise, the agentmay be information-seeking.12 If �H = 1; then the model becomes trivial. Since the agent always recalls high signals, she will perfectly infer

that � = L was observed if she recollects � = ?: Therefore, she will never engage in memory manipulation.

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Figure 2: Cost of Memory

Note that the assumption that � = H �rst-order stochastically dominates � = L implies thatthe agent will never spend e¤ort in order to forget a high signal or remember a low signal (i.e.,rL = fH = 0). Hence, with no loss of generality, we can restrict the set of feasible manipulatione¤orts to rH 2 [0; 1� �H ] and fL 2 [0; �L] :

The model can be interpreted as a formalization of the neurophysiological argument putforth by Trivers (2000). He notes that it takes about 20 ms for a nervous signal to reach thebrain, although it is only registered in consciousness after 500 ms. Furthermore, stimuli receivedup to 100 ms before the event reaches consciousness can a¤ect the content of the experience.Trivers (2000) argues that �this is all time in the world, so to speak, for emendations, changes,deletions, and enhancements to occur.�Thus, the date-1 self in the model can be interpreted asthe person�s unconscious process of information manipulation.13

However, not all belief manipulation occurs between the moment where a signal is receivedand when it enters consciousness. By allocating attention and rehearsing, an individual isconstantly involved in some sort of memory manipulation. Tirole (2002) provides an interestingexample of self-deception. After receiving a hostile referee report, one typically tends to searchfor contradictory evidence or excuses: �The referee is either biased or incompetent.�Then, theperson (consciously or not) tends to avoid negative cues later on, hides the report, and does nottalk about the paper for a while.

The model can also be seen as a con�ict between a �hot�or �impulsive�self and a �cold�self.The hot (date-1) self wants to minimize current losses from negative information and maximizethe current gains from positive information. The cold (date-2) self, wants to circumvent themanipulations made by the other self in order to make a correct inference. The hot self exertse¤orts rH and fL in order to manipulate the beliefs of the cold self. Then, the cold self appliesBayes�rule in order to �lter these manipulations.14

13This interpretation assumes that the agent�s unconscious process is rational in the sense of taking into accountthe bene�ts and costs of memory manipulation. Prelec (2008) shows experimental evidence suggesting thatmemory manipulation seems to respond positively to its expected bene�ts.Similarly to the interpretation above, Bodner and Prelec (2002) present a signaling model between an agent�s

privately informed gut and the agent�s uninformed mind.14 In the context of intertemporal choice, several papers have proposed dual self models [c.f. Thaler and Shefrin

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Figure 3: The Forgetfulness Model

As the following examples show, the model described above encompasses other models ofimperfect memory.

Example 1 (The Forgetfulness Model) If �L = 1; �H = 0 and r (r) = +1 for all r > 0;the informational structure reduces to the one in Benabou and Tirole (2002). As shown in Figure3, it can be interpreted as a model of bad news or no news. If the agent receives bad news, shecan exert an e¤ort fL 2 [0; 1] in order to forget them.

If we reinterpret the state ? as the recollection of a high signal, then the model becomesone where the agent is able to convince herself that a low signal was a high signal.15 Hence,memory manipulation would allow the DM to believe that she received a signal � = H: Thisreinterpretation is compatible with neurological evidence from Prelec (2008), who showed thatsubjects experience heavy brain activity only when they try to convince themselves that a badsignal was actually a good one. In the other states (both when they acknowledge that thesignal was bad news or when they observe a good signal), no such activity is detected. Hence,Example 1 can be interpreted as the agent incurring in psychological costs when she tries toconvince herself that a bad signal was actually a good one.

Example 2 (The Limited Memory Model) Let �L = �H = 0 so that the DM forgets anysignal if she does not employ memory e¤orts. Then, the framework becomes a model of limitedmemory.

In this model, the DM must allocate a limited amount of memory in order to store informa-tion. By spending a memory cost r (ri) ; she remembers the signal with probability ri: Since, in

(1981), Fudenberg and Levine (2006), and Brocas and Carrillo (2008)].15 In this model, the agent would never choose to believe that a high signal was actually a low signal.

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Figure 4: The Limited Memory Model

our model, the agent would never choose to remember a low signal, there is no loss of generalityin assuming that e¤ort can only be spent after a high signal.16

The following example consists of a model where memory is exogenously imperfect.

Example 3 (Exogenous Memory Model) Let r (ri) = +1 for all ri > 0 and f (fi) =+1 for all fi > 0 so that the agent does not engage in endogenous memory manipulation:f�i = r�i = 0; i = L;H: Let �i < 1 so that the agent forgets signals with (exogenous) probabilities1 � �i > 0: If �H > �L; memory is selective in the sense that good news are more likely to beremembered than bad news.

2.1 Multiself Game

I follow Piccione and Rubinstein (1997) in modeling a decision problem with imperfect memoryas a game between di¤erent selves. The decision maker is treated as a collection of selves, each ofthem unable to control the behavior of future selves. The perfect Bayesian equilibrium (PBE) ofthis game between selves corresponds to the decision made by an agent with imperfect recall.17

Because DM has preferences over �, she has an interim incentive to manipulate her beliefsby exerting e¤orts fL and rH : However, the set of possible beliefs that an agent can hold is

16The main di¤erence with respect to Dow (1991) and Wilson (2004) is that I allow the DM to remember asignal with any probability rL; rH 2 [0; 1] : Hence, a higher ri is related to having higher memory resources usedto store the information. In their models, however, the agent either remembers or forgets the information for sure.17For the games considered here, the set of sequential equilibria coincides with the set of PBE. Piccione and

Rubinstein (1997) also propose a �modi�ed multiself consistency�condition which, for the games considered here,leads to the same equilibria as the ones obtained by applying the multiself consistency approach.

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Figure 5: Extensive Form

restricted by the assumption that recollections are interpreted according to Bayes�rule. Thus,the agent makes correct inferences about her characteristics � given the recollections �:

Equivalently, we can conceptualize a �second-period self�that tries to make a correct inferenceabout the agent�s characteristics. The period-2 self chooses beliefs so as to minimize a quadraticloss function:

u� = argminu2R

Z 1

�1[u� u (�)]2 f (�j�) d�:

The solution to this program is u� =Ru (�) f (�j�) d�; which is the Bayes estimator of u (�)

given the recollection �. Thus, by minimizing the quadratic loss function, the period-2 selfconstrains the decision-maker to be Bayesian given her memory imperfection.

The extensive form of the multiself game is represented in Figure 5. In period 0; nature plays� = H with probability q and � = L with probability 1 � q: Then, the date-1 self decides theamount of memory manipulation. Given � = L and manipulation e¤ort fL; nature plays � = Lwith probability �L�fL and � = ? with probability 1��L+fL. Similarly, given � = H and rH ;nature plays � = H with probability �H + rH and � = ? with the complementary probability.Then, the date-2 self makes an inference about the characteristics � so as to minimize a quadraticloss function given the recollections �:

An important assumption for the multiself approach is that DM cannot commit to an ex-

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ante strategy. In the present model, memory manipulation leads to externalities among di¤erentselves. Forgetting a low signal increases the payo¤ of the agent after the signal was realized.However, this strategy would have left the agent worse o¤ had a high signal occurred since,in this case, the expected payo¤ when a signal is forgotten is lower. The multiself approachassumes that, given a low realization, the self does not take into account the e¤ect of memorymanipulation on her utility had she observed a high signal instead.

De�nition 1 A perfect Bayesian equilibrium (PBE) of the game is a strategy pro�le r�H ; f�L and

posterior beliefs F (:j�) such that:

1. f�L 2 argmaxfL

(�L � fL)Ru (�) dF (�j� = L)+ (1� �L + fL)

Ru (�) dF (�j� = ?)� f (fL)

2. r�H 2 argmaxrH(�H + rH)

Ru (�) dF (�j� = H)+(1� �H � rH)

Ru (�) dF (�j� = ?)� r (rH)

3. 8x 2 fL;H;?g ; F (�j� = x) is obtained by Bayes�rule if Pr (� = xjf�L; r�H) > 0.

1 and 2 are the standard perfection conditions. Condition 3 implies that beliefs are given byBayes�rule. Denote the expected utilities given � = H and � = L by

uH �Zu (�) dF (�j� = H) ; and uL �

Zu (�) dF (�j� = L) :

Given the recollection � = H; the date-2 self infers that a high signal was observed in date 1:Hence, the expected utility of the date-1 self conditional on � = H is uH : Similarly, the expectedutility of the date-1 self conditional on � = L is uL: If the DM forgets which signal was observedat date 1 (i.e., she recollects � = ?); then there is a probability (1� q) (1� �L + fL) that � = Lwas observed and a probability q (1� �H � rH) that � = H was observed. Thus, the expectedutility given � = ? is

u; � � (fL; rH)� uH + [1� � (fL; rH)]� uL; (1)

where � (fL; rH) � q(1��H�rH)q(1��H�rH)+(1�q)(1��L+fL)

is the conditional probability of � = H impliedby Bayes�rule.

Remark 1 Denote the expected ability conditional on the observed signal � by �� and the ex-pected ability conditional on the recollections � by ��: �� is "less variable" than �� in the sense ofsecond-order stochastic dominance (see Appendix A). Therefore, since �� is the Bayes estimateof � given the signal �; forgetfulness implies that the decision-maker updates observed signals �less than implied by Bayes�rule. This result is consistent with experimental evidence from Falk,Hu¤man, and Sunde (2006).

2.2 Memory manipulation

After observing a low signal, the date-1 self has expected utility (�L � fL)uL+(1� �L + fL)u;� f (fL) : Let f

�L and r

�H denote the amount of memory manipulation that the date-2 self believes

that was employed in time 1. Note that f�L and r�H are taken as given by the date-1 self when

choosing the amount of memory manipulation to exert.Using equation (1), the expected utility after a low signal can be written as

uL + � (f�L; r

�H) (1� �L + fL)�u� f (fL) ; (2)

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where �u � uH � uL > 0 denotes the utility gain from observing a high signal instead of a lowsignal. The period-1 self chooses the amount of manipulation e¤ort fL so as to maximize theexpression in (2).

Analogously, the expected utility of the date-1 self after observing a high signal is (�H + rH)uH+(1� �H � rH)u; � r (rH) : Substituting equation (1), we obtain

uL + f(�H + rH) [1� � (f�L; r�H)] + � (f�L; r�H)g�u� r (rH) : (3)

The date-1 self chooses the amount of e¤ort rH that maximizes the expression in (3).Condition 3 for a PBE implies that the period-2 self correctly infers the amount of self-

deception exerted in period 0 so that f�L = fL and r�H = rH . The following proposition establishesthe existence of a PBE and considers the possibility of equilibria at the boundary.

Proposition 1 There exists a PBE. Suppose that i is strictly convex, i 2 fL;Hg : Then, inany PBE:

1. If Assumption 2a holds, then f�L > 0; and r�H > 0,

2. If Assumption 2b holds, then

0f (0) � �u =) f�L = 0; and

0r (0) � �u =) r�H = 0:

Memory manipulation is ex-post desirable because the DM increases the probability of eitherremembering a high signal or forgetting a low signal, which increases her payo¤. The incentives toengage in memory manipulation are higher the higher the payo¤gain from a high signal comparedto a low signal, �u: The following lemma shows that whenever 0r (0) > 0 and 0f (0) > 0 wecan split the range of �u in three intervals. When �u is su¢ ciently low, there is a uniqueequilibrium where the DM does not engage in memory manipulation. For intermediate values of�u; there exist both equilibria where she engages in memory manipulation and equilibria whereshe does not. And, when �u is high enough, only equilibria where the DM exerts memorymanipulation exist.

Lemma 1 Suppose Assumption 2b holds. There exist � and � with � � � > 0 such that in anyPBE,

�u < � =) max ff�L; r�Hg = 0; and�u > � =) max ff�L; r�Hg > 0:

Furthermore, when � � �u � �; there is a PBE where max ff�L; r�Hg > 0 and a PBE wheremax ff�L; r�Hg = 0:

Next, we characterize the PBE in the models of Examples 1 and 2.

2.2.1 The forgetfulness model

Consider the forgetfulness model of Example 1. Because r�H = 0 we rede�ne the Bayesianweighting function as � (fL) � q

q+(1�q)fL with some abuse of notation. Given a low signal indate 1; the agent solves

maxfL

(1� fL)uL + fL f� (f�L)� uH + [1� � (f�L)]� uLg � f (fL) : (4)

12

Figure 6: Self-Deception in the Forgetfulness Model

Applying Kuhn-Tucker�s theorem, and substituting the equilibrium condition fL = f�L; we obtain�q

q + (1� q) f�L

��u = 0f (f

�L) ; (5)

in any interior equilibrium.Let f�L be implicitly de�ned by equation 5. From the implicit function theorem, such f�L 2 R

exists and is unique (see Figure 6).The following proposition characterizes the PBE and presents some comparative statics

results:

Proposition 2 In the forgetfulness model, there exists a unique PBE given by equation (5) if

0f (0) < �u < 0f (1)

q ;

f�L = 0 if �u � 0f (0) ; and

f�L = 1 if �u � 0f (1)

q:

Furthermore, the equilibrium amount of belief manipulation f�L is:

1. increasing in the bene�t of manipulation �u (for uL �xed),

2. decreasing in the marginal cost of manipulation, and

3. increasing in q; the probability of not observing a signal.

The results above follow from simple cost-bene�t comparisons. When the marginal bene�tof self-deception is higher or the marginal cost is lower, the agent chooses to engage in more self-deception. Moreover, when the probability of not observing a signal (q) is higher, it becomesmore credible that the individual has not manipulated her beliefs into forgetting the signal.

13

Hence, higher q�s increase the marginal bene�t of self-deception which lead to an increase inmanipulation f�L:

18

2.2.2 The limited memory model

Consider the limited memory model of Example 2. Given a high signal, the date-1 self solves

maxrH

rHuH + (1� rH) f� (r�H)uH + [1� � (r�H)]uLg � r (rH) ;

where, with some abuse of notation, I have de�ned the Bayesian weighting function as � (rH) �q(1�rH)

q(1�rH)+1�q : In an interior equilibrium, we must have"1� q

q�1� r�H

�+ 1� q

#�u = 0r (rH) : (6)

Proceeding as in 2.2.1, we obtain:

Proposition 3 In the limited memory model, the PBE are characterized by equation (6) if

0r (0) ��

1�qq(1�r�H)+1�q

��u � 0r (1) ;

r�H = 0 if �u � 0r (0)

1� q ; and

r�H = 1 if �u � 0r (1) :

An interesting feature of the limited memory model is the possibility of multiple equilibria.Since both sides of Equation (6) are increasing in r�H , there may be multiple interior equilibria.It may also simultaneously feature interior equilibria and corner equilibria or equilibria at bothcorners rH = 0 and rH = 1.19

A person that believes she often forgets good signals is not hurt much by not recalling a goodsignal. Therefore, she will not manipulate her memory enough and, in equilibrium, she will oftenforget good signals. On the other hand, a person that usually remembers good signals is severelyhurt by recollecting � = ?: Therefore, she will have much more incentives to remember goodsignals. As we show in the next section, these equilibria are welfare ranked (from an ex-anteperspective): the equilibrium with the lowest amount of memory manipulation is preferred. Theindividual may be caught in a self-trap where she exerts more manipulation e¤ort because thedate-1 self believes that she will have engaged in more memory manipulation.20

2.3 Ex-ante utility

In this subsection, we compute the decision-maker�s expected utility from observing the signal.In equilibrium, when the DM forgets the signal (� = ?), she knows that there is a probability18Consistently with Claim 1 from Proposition 2, Prelec (2008) presents experimental evidence suggesting that

self-deception is increasing in the bene�ts of manipulation.19For example, if 0r (1) � �u �

0r(0)1�q there exist both an equilibrium with r�H = 1 and one with r

�H = 0:

20See Benabou and Tirole (2002) for a discussion of self-traps. The existence of multiple equilibria is interestingsince there seems to be a large heterogeneity in the amount of self-deception accross di¤erent people [Prelec(2008)].

14

� (f�L; r�H) that there was a high signal and 1 � � (f�L; r

�H) that there was a low signal. There-

fore, Bayesian updating implies that the agent�s posterior beliefs take into account the e¤ectof memory manipulation. On average, the only e¤ects of engaging in self-deception are themanipulation costs f (f

�L) and r (r

�H) : Of course, there is still an ex-post incentive to manip-

ulate beliefs after she observes the signal. Nevertheless, the inability to commit not to engage inself-deception leads to a loss in (ex-ante) expected utility.

The decision maker�s ex-ante expected utility from observing the signal is the expected utilityfrom characteristics minus the expected costs of memory manipulation:

quH + (1� q)uL � q r (r�H)� (1� q) f (f�L) : (7)

The following result shows that, because the agent cannot commit not to manipulate hermemory after the signal, she is better o¤ by not observing it:

Proposition 4 Let U s denote the expected utility of observing the signal and Uns the expectedutility of not observing the signal. Then, Uns � U s: Furthermore,

� under Assumption 2a, Uns > U s;

� under Assumption 2b, there exist � and � with � � � > 0 such that in any PBE,

�u < � =) Uns = U s;

�u > � =) Uns > U s;

and there is a PBE where Uns > U s and a PBE where Uns = U s when � � �u � �:

In order to observe the signal, the individual requires a "participation premium" of q r (r�H)+

(1� q)� f (f�L).

As in other decision problems with imperfect recall, the timing of decisions has importantimplications for the solution. If the agent could commit to a strategy at an ex-ante stage, shewould choose not to engage in memory manipulation (r = f = 0) : However, after receiving eithera good or a bad signal, she would like to revise her previous choice and engage in manipulation.Hence, the ex-ante optimal strategy is time-inconsistent. This contrasts with decision problemswith perfect recall, where ex-ante optimal strategies are time-consistent.21

2.4 Probability weights

In this subsection, I will show that the model leads to a non-expected utility representation,where the decision-maker�s expected utility from observing the signal is equal to a weightedaverage of the utility in each state of the world � 2 fL;Hg : Hence, there exist weights w (q)such that the utility from observing a signal correlated with the DM�s characteristics is

U (�) = w (q)� uH + [1� w (q)]� uL;

where � is the lottery that has outcome � = H with probability q and � = L with probability(1� q) : Clearly, the decision maker is an expected utility maximizer if w (q) = q. If w (q) < q;the agent is ambiguity averse.21See Piccione and Rubinstein (1997) for a discussion of decision problems with imperfect recall. In the present

model, because all nodes are reached with positive probability, the two equilibrium concepts proposed there(multiself consistency and modi�ed multiself consistency) coincide.

15

Simple algebraic manipulations of equation (7) yield

w (q) = q �q r (r

�H) + (1� q) f (f�L)

�u: (8)

Therefore, the model admits a non-expected utility representation, where the probability weightsare given by equation (8).

Theorem 1 The agent�s expected utility of observing the signal � 2 fL;Hg can be representedby

U (�) = w (q)uH + [1� w (q)]uL; (9)

where u� �Ru (�) dF (�j�) and w is given by equation 8. Furthermore,

i. w (0) = 0;

ii. w (1) = 1;

iii. w (q) � q for all q 2 [0; 1] ; and

iv. if r�H = 0 or f�L = 0; then w (q) 2 [0; 1] 8q 2 [0; 1] :

Theorem 1 implies that the weights w (q) are between 0 and 1 in Examples 1 and 2 and,therefore, can be thought of as probabilities (i.e., we can write U (�) = Ew [u (�)] ; where w isa probability measure). The expected utility representation is a special case of our model whenthere is no memory manipulation (r�H = f�L = 0 for all q).

Whenever r�H > 0 or f�L > 0; the individual attributes a lower weight to the good statew (q) < q: Hence, our model predicts ambiguity aversion whenever the individual engages inself-deception.

In order to compare lotteries with di¤erent probabilities of obtaining a high signal, it isimportant to keep the agent�s prior distribution over characteristics �xed. More speci�cally,when comparing two lotteries with di¤erent probabilities q of observing � = H; we want thechoice to be driven by the di¤erent informational content of each lottery while keeping the DM�sprior distribution over characteristics f (�) �xed. In general, this implies that �u is itself afunction of q: Therefore, in this section, we write �u (q) in order to stress this dependence.22

Following Dow and Werlang (1992), de�ne the degree of ambiguity aversion by

c (q;u) � 1� w (q)� w (1� q) (10)

=q r (r

�H (q)) + (1� q) f (f�L (q))

�u (q)

+(1� q) r (r�H (1� q)) + q f (f�L (1� q))

�u (1� q) :

Our model predicts that the most ambiguous events are those that lead to greatest amount ofself-deception e¤ort. The following proposition relates the amount of ambiguity aversion withthe net marginal bene�t (�u) and the marginal cost of self-deception:

22See Appendix B for an example.

16

Proposition 5 Under Assumption 2a; c (q;u) > 0: Under Assumption 2b; there exist � and �with � � � > 0 such that, in any PBE,

c (q;u) > 0 if max f�u (q) ;�u (1� q)g > �;

c (q;u) = 0 if max f�u (q) ;�u (1� q)g < �:

Furthermore, when � � max f�u (q) ;�u (1� q)g � �; there is a PBE where c (q;u) > 0 and aPBE where c (q;u) = 0:

Proof. The �rst part follows from Proposition 1 whereas the second part follows from Lemma1.

2.5 Information Acquisition

In this subsection, I analyze the e¤ects of memory manipulation for information acquisition whenindividuals have preferences over their characteristics (i.e., they have �ego utility�). The moststandard model of ego utility one could formulate consists of a basic application of expectedutility theory. As before, let the space of possible characteristics � be a non-empty subset ofR and let F (:) denote the agent�s prior distribution of �: The DM has preferences that arerepresented by a strictly increasing von Neumann-Morgenstern utility function u : �! R.

If the individual does not observe the signal, her utility isRu (�) dF (�) : If she observes a

signal �, the utility conditional on � isRu (�) dF (�j�) : Hence, the expected utility of observing

the signal isR�

R� u (�) dF (�j�) dG (�) ; where G is the distribution of signals �: By the law of

iterated expectations, we haveZu (�) dF (�) =

Z�

Z�u (�) dF (�j�) dG (�) ;

so that an individual with perfect memory and who behaves as an expected utility maximizer isalways indi¤erent between observing the signal or not. In other words, in this standard model ofego utility, the fact that an individual has preferences over her expected skills does not in�uenceher decision of acquiring information.

Note that the result above holds regardless of the shape of the utility function u: In order toa¤ect the decision of whether to acquire information, the utility function must be a non-linearfunction of probabilities. Several studies of information acquisition have, thus, assumed thatutility functions were non-linear in probabilities [for example, Philipson and Posner (1995) andCaplin and Eliaz (2003) analyze the case of testing for sexually transmitted diseases, K½oszegi(2003) considers a model of patient decision-making, K½oszegi (2006) studies information acqui-sition and �nancial decisions, and Caplin and Leahy (2004) study strategic information trans-mission]. With the exception of Philipson and Posner (1995), all these papers depart from thestandard expected utility model by adopting the psychological expected utility model (PEU) ofCaplin and Leahy (2001).23

As we have seen in the previous subsection, our model also leads to a utility function which isnon-linear in probabilities. However, in our case, the non-linearity arises endogenously throughmemory manipulation. Therefore, our model can be seen as providing a cognitive foundationfor a model of information acquisition.

23Philipson and Posner do not provide a justi�cation for the assumption of a utility function that is non-linearin probabilities.

17

More precisely, suppose the DM has the choice of collecting information about her charac-teristics before performing a task, which gives a random payo¤. Knowing one�s skills lead to amore favorable distribution over payo¤s.24 When will the decision maker prefer not to collectinformation about her characteristics?

As shown in Proposition 4, the DM requires a premium of q r (r�H)+(1� q)� f (f�L) in order

to observe a signal. Hence, the individual is only willing to observe the signal if the expectedcosts of making an uninformed decision are greater than this participation premium. Otherwise,the agent prefers to a obtain lower expected payo¤ and avoid the costs of self-deception. Thisintuition will play a key role in the applications presented in Subsections (4.3), (4.4), and (4.5).

An immediate consequence of avoiding information correlated with one�s skills is the emer-gence of �self-handicapping� strategies such as under-preparing for an examination or gettingtoo little sleep before a physical exercise [Berglas and Baumeister (1993)]. Self-handicappingstrategies reduce the informational content of the signal and, therefore, the model above predictsthat a person may engage in such strategies if the expected costs are not too high.

3 Dynamic Model

Consider an in�nitely repeated version of the game described previously. In each period n 2f1; 2; 3; :::; Ng, an independent draw of the signal �n 2 fH;Lg is made. Each �n has distributiongiven by Pr (�n = Hj�) and Pr (�n = Lj�), where � is the agent�s �true� characteristics. Theparameter � is not known. Instead, the decision-maker has a prior F (�) about its distribution.Hence, the prior over the distribution of the signal �n is

Pr (�n = i) =

ZPr (�nj� = i) dF (�) ;

where the conditional probability Pr (� = Hj�) is strictly increasing in �:After observing �n 2 fH;Lg ; the decision-maker engages in memory manipulation. She

recollects a signal �n 2 f?; L;Hg : As in the static game, this is modeled through a di¤erentself acting each time an information is forgotten.

A history at time n is a sequence of recollections hn�1 = (�1; �2; :::; �n�1) 2 f?; L;Hgn�1 : Ineach period, the stage-1 self is a short-run player that chooses actions (rH ; fL) : f?; L;Hgn�1 ![0; 1] � [0; 1] to maximize the discounted sum of all future stage-game payo¤s. The discountrate is � 2 (0; 1) : The stage-2 self always applies Bayes�rule. Denote the sequence of (history-dependent) manipulations by r � frH (hn) ;8hn; ng and f � ffL (hn) ;8hn; ng :

De�nition 2 A perfect Bayesian equilibrium (PBE) of the repeated game is a strategy pro�le(r; f) and posterior beliefs F (:j�) such that for all hn and all n = 1; 2; :::; N :

1. fL�L; hn�1

�2 argmax

fL(�L � fL)

�Ru (�) dF

��jL; hn�1

�+ �V

�L; hn�1

��+(1� �L + fL)

�Ru (�) dF

��j?; hn�1

�+ �V

�?; hn�1

��� f (fL)

2. rH�H;hn�1

�2 argmax

rH(�H + rH)

�Ru (�) dF

��jH;hn�1

�+ �V

�H;hn�1

��+(1� �H � rH)

�Ru (�) dF

��j?; hn�1

�+ �V

�?; hn�1

��� r (rH)

3. F (�jhn) is obtained by Bayes�rule if Pr (hnjf; r) > 0,24For simplicity, assume that, if the agent chooses not to collect information, the payo¤s are uninformative

about her skills.

18

4. The continuation payo¤ V satis�es25

V (hn) =1� �N�n

1� � �Zu (�) dF (�jhn)�

N�nXs=0

�sEhn+sjhn

�Pr (�t+s = Hjhn)� r (rH (hn+s))+Pr (�t+s = Ljhn)� f (fL (hn+s))

�:

We are interested in the PBE of the game when N is large. Let un (hn) denote the Bayesestimator of u (�) given history hn:

un (hn) �

Zu (�) dF (�jhn) :

Note that F (�jhn) is a function of r and f .I assume that �H > 0 and that there exists some �f < �L with F

��f�� sup� fu (�)g �

inf� fu (�)g : These assumptions ensure that the DM never forgets a signal �n 2 fL;Hg withprobability 1:26 The �rst issue is whether the Bayes estimator of u (�) is consistent. In otherwords, does the DM eventually learn her true type after observing a su¢ ciently large numberof signals?

If memory manipulation were exogenous (or, at least, constant), the answer would be imme-diate. Because, in this case, the recollections would be i.i.d., Doob�s Consistency theorem wouldimply that un (hn) converges to u (�) : This is formally stated in the following lemma:

Lemma 2 Suppose rn�hn�1

�= ~r and fn

�hn�1

�= ~f for all hn�1; n and let N ! 1: Then

un ! u (�) for almost all histories.

When memory manipulation is endogenous, however, it is not immediate that the DM even-tually learns her true type. Although observed signals �n are i.i.d., memory manipulation leadsto non-independent and non-identically distributed recollected signals �n. However, becausethe agent knows the equilibrium strategies, she knows the probability of each signal conditionalon the recollection. Therefore, intuitively, the agent correctly updates the recollections andeventually learns her true type regardless of how much manipulation e¤ort she exerts.

In order to show that this intuition is correct, I will use the following result:

Lemma 3 For any �xed history hn; F (�jhn; r; f) is increasing in r and in f:

The Lemma above implies that, conditional on reaching each history, the agent always prefersthat she had not engaged in memory manipulation. Because the agent is ultimately concernedabout �n, F (�jhn; r; f) is not a function of r and f in all histories that do not contain any�n = ?. However, whenever the agent recollects �n = ?; she is always better o¤ if she had notmanipulated memory (since memory manipulation increases the relative probability of arrivingat �n = ? after a low signal �n = L). Hence, F (�jhn; 0; 0) �rst-order stochastically dominatesF (�jhn; r; f) for all r; f:27

25From the law of iterated expectations, E�E�u (�) jhn+s

�jhn�= E [u (�) jhn] for all histories hn+s that follow

history hn:26Either one of these conditions are needed to ensure identi�cation. If �H = 0 and fL

�ht�= �L; then

rH�ht�= 0 for all ht implies that � = ? so that the bayesian posterior is equal to the prior. Therefore, there is

no hope for the Bayes estimator to be consistent.This assumption is not satis�ed in the model of Example 2.2.1 (�H > 0 is violated). However, it is straightfor-

ward to adjust the arguments from this section to establish the same results for that model.27The �rst-order dominance (FOSD) is for �xed ht: Since the probability of each history is itself a function of

r and f , it does not follow that there is unconditional FOSD.

19

A straightforward implication of Lemma 3 is that:

E�u (�) jhn; 1� �H ; �f

�� E [u (�) jhn; r; f ] � E [u (�) jhn; 0; 0] ; (11)

for all r and f and all histories ht: But, because Lemma 2 implies that both extremes in theinequality above converge to u (�), it thus follows that the term in the middle converges and haslimit u (�) : This result is formally stated in the following Proposition:28

Proposition 6 Let N !1: Then, un ! u (�) for almost all histories.

Since the expected ability eventually converges to the truth, the e¤ect of an additional signalconverges to zero. Hence, the bene�t of memory manipulation converges to zero and the agentdoes not engage in self deception after a su¢ ciently large amount of observations.

Proposition 7 Let N !1: Then, rH ! 0 and fL ! 0 for almost all histories. Furthermore,under assumption 2b; there exists a (�nite) �N 2 N such that n � �N implies rH (hn) = fL (h

n) =0 for almost all hn:

Denote by w�q;hn�1

�the weighting function associated with the utility of observing a signal

in period n given history hn�1 and denote the coe¢ cient of ambiguity aversion by c�q;u; hn�1

�:

Corollary 1 Let N !1: w (q; :)! q and c (q;u; :)! 0 for almost all histories. Furthermore,under assumption 2b; there exists a (�nite) �N 2 N such that n � �N implies w (q;hn) = q andc (q;u; hn) = 0 for almost all hn.

Therefore, when signals are observed frequently enough, agents will not engage in self-deception and will not exhibit ambiguity aversion. This is consistent with the usual intuitionthat people do not exhibit ambiguity aversion over frequently observed events or that expertsare subject to much less biases (e.g. List, 2003).

4 Lotteries over money

�We propose that the consequences of each bet include, besides monetary payo¤s,the credit or blame associated with the outcome. Psychic payo¤s of satisfaction orembarrassment can result from self-evaluation or from an evaluation by others. Ineither case, the credit and the blame associated with an outcome depend, we suggest,on the attributions for success and failure. In the domain of chance, both successand failure are attributed primarily to luck. The situation is di¤erent when a personbets on his or her judgement. If the decision maker has limited understanding of theproblem at hand, failure will be attributed to ignorance, whereas success is likely tobe attributed to chance. In contrast, if the decision maker is an "expert," success isattributable to knowledge, whereas failure can sometimes be attributed to chance.�

Heath and Tversky (1991, pp.7-8)

28Note that the probability of occurence of a history Pr�ht�is a function of the sequence of memory manip-

ulations r and f: The previous assumption implies that the sets of histories with zero measuse is the same forall relevant manipulation e¤orts: rH (hn) 2 [0; 1� �H ] and fL (h

n) 2�0; �f

�: Therefore, we ommit any explicit

reference to r and f when considering almost sure convergence of un (hn).

20

Figure 7: Extensive Form Representation

In this Section, the basic framework is generalized to allow for monetary lotteries. Consider adecision-maker (DM) with preferences over ability � 2 � and money t 2 R where, as before, theagent�s prior distribution of � over the set � � R has density F (:) : Preferences are representedby a von Neumann-Morgenstern utility function u : � � R ! R which is strictly increasing inboth arguments.

The informational structure of the model is the following. Upon observing a signal � = s;the DM receives a monetary payment s 2 fL;Hg ; L < H. A high signal is assumed to be morefavorable than a low signal in the sense of �rst-order stochastic dominance. The individual canmanipulate her recollections � by engaging in e¤orts rs and fs; where rs 2 [0; 1� �s] denotesthe e¤ort to remember and fs 2 [0; �s + rs] denotes the e¤ort to forget signal s. Memorymanipulation rs and fs involves a cost of r (rs) � 0 and f (fs) � 0: Figure 7 depicts theextensive form representation of the game.

An important assumption is that the agent does not remember exactly how much moneyt0 2 R she had before observing the signal. Therefore, she cannot fully infer which signal wasobserved from the amount of money she has in period 2.29 Her beliefs about t0 are given by the

29Alternatively, we could assume that, by forgetting the outcome of the lottery, the agent also forgets the

21

absolutely continuous c.d.f. G with strictly positive pdf g: Thus, the �nal monetary outcome is

t = t0 + s;

where s 2 fL;Hg the outcome of the lottery in period 1.

De�nition 3 A perfect Bayesian equilibrium (PBE) of the game is a strategy pro�le fr�H (t) ; f�L (t)gt2Rand posterior beliefs � (:j�; t) such that:

1. f�L (t) 2 argmaxfL

(�L � fL)Ru (�; t) d� (�j� = L; t)+(1� �L + fL)

Ru (�; t) d� (�j� = ?; t)�

f (fL)

2. r�H (t) 2 argmaxrH(�H + rH)

Ru (�; t) d� (�j� = H; t)+(1� �H � rH)

Ru (�; t) d� (�j� = ?; t)�

r (rH)

3. � (�j� = x; t) is obtained by Bayes�rule if Pr (� = xjf�L; r�H ; t) > 0.

A recollection � 2 fL;Hg can only occur if the agent observed a signal � = � in date 1:Therefore, Bayesian updating implies that � (�j�; t) = F (�j� = �) for � 2 fL;Hg and payo¤sare given by

uL (t) =

Zu (�; t) dF (�j� = L) ; and uH (t) =

Zu (�; t) dF (�j� = H) :

When the DM forgets which signal was observed, there are 2 possible nodes: (i) s = H andt0 = t�H; and (ii) s = L and t0 = t� L: Bayesian updating gives

Pr (� = Hj� = ?; t) = � (rH ; fL; t) ;

where � (rH ; fL; t) � q(1�rH��H)g(t�H)q(1�rH��H)g(t�H)+(1�q)(1�rL��L)g(t�L)

: Hence, the expected utility afterforgetting a signal and having a monetary outcome t is:

u? (t) = � (r�H (t) ; f�L (t) ; t)uH (t) + [1� � (r�H (t) ; f�L (t) ; t)]uL (t) ;

where r�H (t) and f�L (t) are the amount of memory manipulation that the date-2 self believes

that was employed in period 1.Proceeding as in Section 2, we obtain the following:

Proposition 8 There exists a PBE. Suppose that i is strictly convex, i 2 fL;Hg : Then, inany PBE we have:

1. If Assumption 2a holds, then f�L (t) > 0; r�H (t) > 0 8t,

2. If Assumption 2b holds, then

0f (0) � �u (t) =) f�L (t) = 0; and

0r (0) � �u (t) =) r�H (t) = 0:

amount of money she had before (so that she cannot infer the lottery outcome from the �nal amount of money).This assumption would not change the results of the model.

22

Proof. Analogous to Proposition 1.Denote the distribution of t by h (t) :30 Then, we have the following representation result:

Proposition 9 The agent�s (ex-ante) expected utility from the lottery is

U I �MC + �;

where:

� U I �Rt

R� u (�; t) f (�)h (t) dtd� (� and t independent),

� MC �Rt

�q r (r

�H (t)) + (1� q) f (f�L (t))

�h (t) dt; and

� � �Rt � (t)�u (t)h (t) dt;

� � (t) � [g (t�H)� g (t� L)] z (t) ; for some z (t) > 0:

The utility of a skill-dependent lottery can be decomposed in three terms. First, the utilityU I obtained from a lottery with the same distribution over monetary outcomes but whoseoutcomes are independent of �. Second, the expected manipulation costs MC: And, third, thedegree of complementarity � between characteristics and the monetary outcomes. If the lotteryis more likely to pay in states where the payo¤ gain from characteristics �u (t) is high, then �is positive so that the lottery is more valuable. Otherwise, � is negative and the lottery is lessvaluable.

Denote the expected utility of receiving a signal s 2 fL;Hg by vs �Rui (t)h (t) dt: The

following theorem presents the nonexpected utility representation of the model.

Theorem 2 The agent�s expected utility from the lottery can be represented by

U (L) = w (q) vH + [1� w (q)] vL;

where

w (q) = q +��MCR�u (t)h (t) dt

:

Furthermore,

i. w (0) = 0; and

ii. w (1) = 1.

Example 4 (Exogenous Memory) In the exogenous model of Example 3, we have w (q) =q+ �

E[�u] : Therefore, the DM prefers a skill-dependent lottery to a lottery with the same distribu-tion over monetary outcomes but whose monetary outcomes are uncorrelated with characteristicsif and only if � � Et [� ��u] > 0:30 It can be shown that h (t) = qg (t�H) + (1� q) g (t� L) ; t 2 R:

23

4.1 Discussion

The model presented here suggests that ambiguity aversion is a consequence of the lotteryoutcomes being informative about the decision-maker�s characteristics. When the decision-makerhas an imperfect memory, she may exhibit ambiguity aversion/lovingness. Several experimentalpapers have argued that ambiguity aversion may be related to an agent�s skill or knowledge.31

First, some experiments have contradicted the idea that ambiguity aversion is related to theimprecision of the probability distribution of the events as is usually argued. Budescu, Wein-berg, and Wallsten (1988), for example, compared decisions based on numerically, graphically(the shaded area in a circle), and verbally expressed probabilities. Numerical descriptions of aprobability are less vague than graphic descriptions which, in turn, are less vague than verbaldescriptions. Thus, if agents had a preference for more precise distributions, they should rankevents whose probabilities have a numerical description �rst, those with graphic descriptions sec-ond, and those with verbal descriptions last. However, unlike ambiguity aversion would predict,subjects were indi¤erent between these lotteries. Indeed, the authors could not reject that theagents behaved according to subjective expected utility theory and weighted events linearly.32

The idea that deviations from expected utility occur when lotteries are a¤ected by thedecision-maker�s skills dates back to Cohen and Hansel (1959) and Howell (1971). Howell con-sidered composite gambles involving throwing a dart and spinning a roulette wheel. They variedthe skill (dart) and chance (wheel) components of the lotteries while keeping the probability ofwinning constant. In general, people preferred lotteries with higher component of skill overchance. Cohen and Hansel presented a similar experiment where agents faced composite lotter-ies involving a mix of skill and chance components. The agents also tended to prefer lotterieswith a higher component of skill over chance.

Heath and Tversky (1991) proposed the "competence hypothesis", according to which peo-ple�s preferences over ambiguous events arise from the anticipation of feeling knowledgeable orcompetent.33 Their interpretation of the Ellsberg paradox is as follows:

"People do not like to bet on the unknown box, we suggest, because there is infor-mation,namely the proportion of red and green balls in the box, that is knowable inprinciple but unknown to them. The presence of such data makes people feel lessknowledgeable and less competent and reduces the attractiveness of the correspond-ing bet."

Fox and Tversky (1995) proposed that ambiguity is caused by comparative ignorance. Theyhave argued that "ambiguity aversion is produced by a comparison with less ambiguous events orwith more knowledgeable individuals." As in Heath and Tversky�s (1991) competence hypothesis,this "comparative ignorance hypothesis" argues that ambiguity aversion is driven by the feelingof incompetence. Fox and Weber (2002) replicated their results and extended them to the casesof familiar versus unfamiliar situations. Chow and Sarin (2001) obtained similar results.34

Similarly, Goodie (2003) proposed the perceived control hypothesis, according to which am-biguity aversion is generated by the agent�s belief that the distribution of outcomes is in�uenced

31See Goodie and Young (2007) for a more detailed discussion of this litterature.32See also Budescu et al. (2002).33Taylor (1995) and Keppe and Weber (1995) replicated their results. Grieco and Hogarth (2004) presented

experimental evidence suggesting that the high failure rate of new entrepreneurial ventures are associated withthe competence hypothesis.34 In Chow and Sarin (2001), however, preference for risk decreased but did not completely disappear in non-

comparative situations.

24

by characteristics such as knowledge or skill.35

4.2 First Order Risk Aversion

In this subsection, it is shown that memory imperfection may lead the DM to exhibit �rst-orderrisk aversion [Segal and Spivak (1990)]. First-order risk aversion has important economic im-plications. An individual with second-order risk aversion always accepts small gambles withpositive expected value.36 On the other hand, an individual with �rst-order risk aversion alwaysrejects small gambles as long as the positive expected value is su¢ ciently small. As a conse-quence, someone with �rst-order risk aversion may choose to be fully insured even if prices arenot actuarially fair.

Consider a lottery over money whose outcomes are informative about the decision maker�scharacteristics as described previously. The certainty equivalent of the lottery is de�ned bythe monetary amount CE 2 R that makes the agent indi¤erent between participating in thelottery or receiving CE for sure. In this case, receiving CE for sure implies consuming t0+CE;which gives utility

R� u (�; t0 + CE) dF (�) for each realization t0: Since t0 is a random variable

distributed according to a pdf g, the certainty equivalent is de�ned by the amount CE such that

U (L) =Zt0

Z�u (�; t0 + CE) dF (�) g (t0) dt0; (12)

where U (L) is the expected utility from the lottery as described in Theorem 2. The riskpremium associated to a lottery L is de�ned as the di¤erence between the expected paymentand the certainty equivalent: � � E [t]� CE:

Let ~" be a binary random variable such that E [~"] = 0: Consider a lottery that pays t = �~";where � > 0:We say that a decision maker has second-order risk aversion if lim�!0+ � (�) =� = 0;and we say that she has �rst-order risk aversion if lim�!0+ � (�) =� > 0.

First, let us consider the Exogenous Memory model of Example 3. In this case, there areno manipulation costs (f�L = r�H = 0) and the �rst-order risk attitude is determined solely bywhether characteristics � and money are complements or substitutes:

Lemma 4 In the Exogenous Memory model:

� the DM is �rst-order risk-averse ifRt�u (t) g

0 (t) dt > 0;

� the DM is �rst-order risk-loving ifRt�u (t) g

0 (t) dt < 0;

� the DM has risk preference of second-order ifRt�u (t) g

0 (t) dt = 0:

The intuition follows from the representation of Proposition 9. In that proposition, we havedecomposed the utility of a skill-dependent lottery in three terms: (i) the utility U I from a lotterywith the same distribution over monetary outcomes but whose outcomes are independent of �,(ii) the expected manipulation costsMC; and (iii) the degree of complementarity � between thecharacteristics � and the monetary outcomes. Because an expected utility maximizer has riskpreferences of second-order, the �rst term (U I) does not in�uence her �rst-order risk preferences.Moreover, in the case of exogenous memory, the second term (MC) vanishes since there are nomanipulation costs.

35There is a large experimental litterature on the e¤ect of perceived control on risk-taking [e.g. Chau andPhillips (1995) and Horswill and McKenna (1999)].36See Samuelson (1963), Segal and Spivak (1990), or Rabin (2000).

25

Therefore, in the exogenous memory model, �rst-order risk preferences are driven solely bythe third term: the degree of complementarity between the agent�s characteristics � and the mon-etary outcomes. Taking the limit whenH�L! 0 implies that � > 0 ()

Rt�u (t) g

0 (t) dt < 0.Hence, a DM with exogenous memory is �rst-order risk-averse if characteristics and the mon-etary outcomes are substitutes (g0 < 0) and is �rst-order risk-loving if they are complements(g0 > 0) :

When memory is endogenous, there is an additional term due to the costs of memory ma-nipulation which reduces the utility of the lottery. Denote the equilibrium amounts of memorymanipulation as a function of � by r�H (t; �) and f

�L (t; �) : It is useful to distinguish between two

cases:

i. Pr (lim�!0 r�H (t; �) > 0) > 0 or Pr (lim�!0 f

�L (t; �)) > 0; and

ii. Pr (lim�!0 r�H (t; �) > 0) = Pr (lim�!0 f

�L (t; �) > 0) = 0;

where the probability is with respect to the distribution of t: In case (i), the DM engages inmemory manipulation in some set with positive measure even when the monetary outcomes fromthe lottery are arbitrarily small. This would always be the case if the informational content ofthe lottery �u (t) is large enough (in some set with positive measure) or if the marginal cost ofmanipulation 0f (0) is small enough. In case (ii), the amount of memory manipulation convergesto zero almost surely.

In case (i), the DM would demand a participation premium in order to observe the signalwhen � = 0. Therefore, the certainty equivalent of the lottery converges to CE (0) < 0 and

lim�!0

� (�)

�= � lim

�!0

CE (�)

�= +1:

Thus, when manipulation costs do not converge to zero, they dominate the complementaritye¤ect and the DM always exhibits �rst-order risk aversion.

In case (ii), the answer depends on which e¤ect dominates. If characteristics � and themonetary outcomes are complements (g0 > 0) ; then both push towards risk-aversion and theDM exhibits �rst-order risk aversion. If � and the monetary outcomes are substitutes (g0 < 0) ;then the behavior of DM depends on the rate of convergence of manipulation costs compared tothe complementarity between � and the monetary outcomes.

These results are formally stated in the lemma below:

Lemma 5 Suppose Pr (lim�!0 r�H (t; �) > 0) > 0 or Pr (lim�!0 f

�L (t; �)) > 0: Then, CE (0) < 0

and DM displays �rst-order risk aversion. Suppose Pr (lim�!0 r�H (t; �) > 0) = Pr (lim�!0 f

�L (t; �)) =

0: Then, there exists some strictly increasing $ (q) > 0 such that

1. DM is �rst-order risk averse if

$ (q)

Zt�u (t) g0 (t) dt+

Zt

�q 0r (rH (t; 0)) r

0H (0) + (1� q) 0f (fL (t; �)) f 0L (0)

�g (t) dt > 0;

(13)and

2. DM is �rst-order risk loving if

$ (q)

Zt�u (t) g0 (t) dt+

Zt

�q 0r (rH (t; 0)) r

0H (0) + (1� q) 0f (fL (t; �)) f 0L (0)

�g (t) dt < 0:

26

Of course, when there are no complementarities between skill and money (e.g., when gis uniform) and when Pr (lim�!0 r

�H (t; �) > 0) = Pr (lim�!0 f

�L (t; �)) = 0; then DM exhibits

second-order risk preferences.Lemma 5 relies on assumptions made on the endogenous variables r�H and f�L: Next, we

present some results based on properties of the fundamentals i and �u (t):

Proposition 10 Under Assumption 2a; DM exhibits �rst-order risk aversion. Under Assump-tion 2b; there exist � and � with � � � > 0 such that, in any PBE,

� DM is �rst-order risk averse if Pr��u (t) > �

�> 0,

� DM is �rst-order risk loving if Pr (�u (t) > �) = 0 andRt�u (t) g

0 (t) dt < 0.

Proof. Follows from Lemmas 1 and 5.

Remark 2 Consider a repeated version of the game above. Proceeding as in Section 3, it followsthat �un (t) ! 0 for almost all histories. Therefore, the DM�s attitude towards risk convergesto second-order risk aversion as N !1 (for almost all histories).

4.3 The Endowment E¤ect

An individual that satis�es the axioms of expected utility theory does not display a di¤erencebetween the maximum willingness to pay for a good and the minimum compensation demandedto sell the same good (willingness to accept) when income e¤ects are small. However, severalempirical works have documented a discrepancy between these values. An individual tends tovalue one good more when the good becomes part of that person�s endowment. Thaler (1980)labeled this phenomenon an "endowment e¤ect".

Kahneman, Knetsch, and Thaler (1990) argued that the endowment e¤ect was caused by lossaversion.37 In this subsection, I provide an alternative explanation for the endowment e¤ect.The main idea is that, in most markets, trading requires certain skills or knowledge. At thevery least, the parties must form an expectation of how much each good is worth. In morecomplex markets, they must also estimate the future prices of the goods (which determine theopportunity cost of trading). Therefore, the outcome of trading reveals information about howskillful the person is.

As we have seen previously, an individual that cares about her self-image and is subjectto imperfect memory requires a participation premium in order to engage in an activity thatreveals information about her skills. Therefore, she may prefer not to trade if the price is onlyslightly above the expected value of the good.

The model presented below establishes this result formally. It consists of a special case ofthe model from Section 4. The main di¤erence from the original model is in the particularspeci�cation of the signal � observed in period 1:

A decision-maker owns good A and must choose whether or not to trade it for good B. GoodA has a known monetary value of vA 2 R. The value of good B is unknown. It may take valuesvB 2

�vL; vH

; where vL < vA < vH .

The DMmust evaluate whether trade is pro�table. This is formalized by a signal s 2 fT;NTgdenoting whether the agent should or should not trade. This signal is informative about the

37According to loss aversion, losses are weighed substantially more than gains. Then, the cost of losing a goodthat is much higher than the bene�t of winning a good.

27

probability of trade being pro�table. Predicting the realization of B correctly requires skills.Therefore, a correct prediction is good news about DM�s skills. However, s is only informativeabout the DM�s skills through the realization of each state (vL and vH). More speci�cally, sis uninformative about � if vB is not observed. This information structure is formally statedbelow:

Assumptions Let (�; vB; s) 2 � ��vL; vH

� fT;NTg be distributed according to the c.d.f.

F (�; vB; s) :

2.1 � and s are independent:

F (�; s = s) = F (�)� Pr (s = s) ; for s 2 fT;NTg :

2.2 The signal s 2 fT;NTg is informative about the value of good B, vB :

qH � Pr�vB = vH js = T

�> Pr

�vB = vH

�� �;

qL � Pr�vB = vLjs = NT

�> 1� �:

2.3 The signal s 2 fT;NTg is informative about � when the realization of vB is observed:

F��jvB = vH ; s = T

�� F

��jvB = vH ; s = NT

�;

F��jvB = vL; s = NT

�� F

��jvB = vH ; s = T

�;

which strict inequality over a set with non-zero measure.

2.4 It is pro�table to trade if and only if s = T:

qHvH +�1� qH

�vL > vA > �vH + (1� �) vL:

Note that Assumption 2.1 implies that the signal s is only informative about � if the valueof good B is also observed:

F (�js = NT ) = F (�js = T ) = F (�) : (14)

Assumption 2.4 states that it is pro�table to trade conditional on an s = T signal and thatit is not pro�table to trade if no signal is observed. It implies that it is also not pro�table totrade conditional on an s = NT signal (since s = NT is bad news about the pro�tability oftrade).

The timing of the model is as follows:

1. Nature determines �; the trading skills (or knowledge) of the DM. However, this is notobserved by the agent.

2. The DM observes a signal s, which is correlated with the bene�ts from trade but, by itself,uninformative about her skills. Then, she chooses whether to trade good A for good B:

3. If the DM chose to trade, she observes vB; the value of good B. The value of good B alongwith signal s is informative about her skills �:

4. The DM exerts e¤orts rH and fL in order to forget unsuccessful and remember successfuloutcomes. The e¤ort costs are given by L (fL) and H (rH) ; respectively.

28

For simplicity, I assume that t0 is uniformly distributed over a large interval so that there areno complementarities between ability and money (� = 0).38 Therefore, Proposition 9 impliesthat the agent always prefers a lottery whose outcomes are uncorrelated with skills to one withthe same distribution over monetary outcomes but where outcomes are correlated with her skills.Furthermore, I also assume that she is risk-neutral so that the von Neumann-Morgenstern utilityfunction can be written as u (�; t) = � (�) � t for some strictly increasing function � such thatR� (�) f (�) d� = 1:39

Upon observing a no-trade signal, trading leads to a lottery whose monetary outcomes arecorrelated with skills. Since � = 0, the expected utility of the lottery is equal to the expectedmonetary payo¤s qLvL+

�1� qL

�vH�vA minus the expected manipulation costs. Manipulation

costs are always weakly negative and, when s = NT; the expected monetary payo¤s are strictlynegative. Therefore, the agent prefers not to trade when she observes s = NT:

Consider the case where the DM observes a trade signal and suppose she chooses to tradethe good. If good B turns out to have a high value vB = vH so that the prediction was correct,then her posterior over � becomes F

��jvB = vH ; s = T

�: Letting t = t0 + vH where t0 is the

(unknown) initial amount of money as in Section 4, the expected payo¤ conditional on vB = vH

becomes

uH (t) = t�Z� (�) dF

��jvB = vH ; s = T

�:

If good B turns out to have a low value, her expected payo¤ becomes

uL (t) = t�Z� (�) dF

��jvB = vH ; s = T

�:

Let �� �R� (�) dF

��jvB = vH ; s = T

��R� (�) dF

��jvB = vH ; s = T

�> 0: Then, �u (t) =

t���:If the DM forgets the signal, there are 2 possibilities. Trade may have been successful but

she had a low initial endowment�vB = vH ; t0 = t�H

�, or trade may have been unsuccessful

but she had a high initial endowment�vB = vL; t0 = t� L

�: Bayesian updating yields

u? (t) = � (r�H (t) ; f�L (t))uH (t) + [1� � (r�H (t) ; f�L (t))]uL (t) ;

where � (rH ; fL) =qH�(1�rH��H)

q(1�rH��H)+(1�q)(1�rL��L)is the Bayesian weight.40

From Proposition 9, the utility from trading conditional on observing a signal s = T is

qHvH +�1� qH

�vL| {z }

Expected gains from trade

� MC|{z}Expected cost of self-deception

; (15)

whereas the utility from not-trading is vA: Since qHvH +�1� qH

�vL > vA; DM prefers to trade

if and only if manipulation costs are "not too high". If DM could commit not to engage in self-deception, she would always trade. Hence, self-deception implies that some trades with positiveexpected monetary gains do not occur. The agent balances a positive expected gain from tradewith the costs of self-deception.

38Let t0 be uniformly distributed over [0;K] for K large. When t 2 [L;H); DM infers that s = L regardless ofher recollections � and, therefore, does not engage in memory manipulation. Similarly, when t 2 (K +L;K +H];she infers that s = H and does not exert memory manipulation. When K is large, however, these intervals happenwith small probability. When t 2 [H;K + L); the solution is obtained as in Section 4.39The normalization

R� (�) f (�) d� = 1 implies that the expected utility of a lottery with expected monetary

payo¤ equal to �t and whose monetary outcomes are independent of skills is equal to �t:40Note that � is not a function of t because we have assumed that t0 is uniformly distributed.

29

We say that the agent displays an endowment e¤ect if she requires an expected value of tradestrictly higher than her valuation of the good in order to trade. In this model, the agent displaysan endowment e¤ect wheneverMC > 0 since she is only willing to trade if the expected value ofB is strictly greater than the value of A. The result is summarized in the following proposition:

Proposition 11 The agent will refuse trades with positive expected gains if

� self-deception is su¢ ciently costly (i.e. MC is high enough), or

� the expected value of trading is su¢ ciently small (i.e.��qHvH + �1� qH� vL � vA�� is small

enough).

Furthermore, under Assumption 2a; there is an endowment e¤ect.

Proof. Analogous to Propositions 8 and 10. Note that ��� and

��� are such that, for any �nite

�; Pr�t > �

��

�> 0.

Remark 3 Consider a repeated setting as in Section 3. Because memory manipulation con-verges to zero after the individual has played the lottery a su¢ ciently large number of times, themodel implies that people who trade often enough should not exhibit an endowment e¤ect. Thisresult is consistent with evidence from List (2003).

4.4 The Uncertainty E¤ect

In a set of experiments with over 1000 participants, Gneezy, List, and Wu (2006) found thata signi�cant proportion of participants seemed to value participating in a lottery less thanobtaining the lowest possible outcome for sure. For example, they were willing to pay $38 fora $50 gift certi�cate but only $28 for a lottery ticket that provided an equal chance of winningthe same $50 certi�cate and a $100 certi�cate with the same conditions. As Gneezy, List, andWu argue, �[t]his behavioral result, which we term the uncertainty e¤ect, not only contradictsexpected utility and prospect theory but is inconsistent with virtually all models of risky choice.�

The present model provides a memory manipulation explanation for the uncertainty e¤ect.If the lottery outcomes lead the decision-maker to exert memory manipulation, she may preferto obtain the worst monetary outcome for sure and avoid the additional memory costs.

If DM receives the lowest possible monetary outcome (L) for sure, the lottery is not infor-mative about her characteristics. Thus, she obtains an expected payo¤ equal toZ

t0

Z�u (�; t0 + L) g (t0) dF (�) dt0:

However, by participating in the lottery, the individual may infer something about her charac-teristics from the outcome. Therefore, from Proposition 9, her expected utility from the lotteryis Z

t

Z�u (�; t)h (t) dF (�) dt�MC + �:

Whether obtaining tL for sure is preferred to participating in the lottery depends on theexpected monetary gain from the possibility of winning a higher payment under the lottery,the expected self-deception costs, and the complementarity between characteristics and money.Whenever the expected monetary gains and the complementarity � between � and money arelower than the expected memory costs MC; she will prefer to obtain L for sure:

30

Lemma 6 A decision-maker prefers to obtain the lowest monetary outcome for sure instead ofparticipating in the lottery if and only ifZ

t0

Z�u (�; t0 + L) g (t0) dF (�) dt0 �

Zt

Z�u (�; t)h (t) dF (�) dt�MC + �: (16)

Suppose the DM is risk neutral so that u (�; t) = � (�) � t for some function � withR� (�) f (�) d� = 1: The, the expected monetary gain of participating in the lottery is q�t;

where �t � H � L:

Corollary 2 Suppose the DM is risk neutral. Then,she prefers to obtain the lowest monetaryoutcome for sure instead of participating in a lottery whose outcomes are informative about � ifand only if the expected monetary gains plus the gains from complementarity are greater thanthe expected memory manipulation costs:

MC � q�t+ �: (17)

We say that DM displays an uncertainty e¤ect if she prefers receiving L for sure instead ofparticipating in a lottery that pays L+�t with probability q and L with probability 1� q forsome �t > 0: As in Subsection 4.3, let t0 be uniformly distributed over a large interval so that� = 0. Condition (17) implies that, whenever DM engages in memory manipulation, there willbe an interval (0;��t) such that receiving L for sure is preferred to participating in the lottery:

Proposition 12 Suppose DM is risk neutral and let t0 be uniformly distributed over a largeinterval. Under Assumption 2a; DM displays an uncertainty e¤ect. Furthermore, there exists a��t > 0 such that the lowest monetary outcome is preferred to the lottery if �t < ��t:

Because the DM requires a participation premium in order to accept to participate in alottery that is informative about �, she will prefer to receive the lowest possible monetary payo¤for sure if the expected payments from the lottery are low enough.

4.5 The Sunk Cost Fallacy

�The consequences of any single decision (...) can have implications about theutility of previous choices as well as determine future events or outcomes. Thismeans that sunk costs may not be sunk psychologically but may enter into futuredecisions.�

Staw (1981, pp. 578)

Standard decision theory shows that only incremental costs and bene�ts should in�uence de-cisions. Historical costs, which have already been sunk, should be irrelevant. However, evidencesuggests that people often take sunk costs into account when making decisions.41

In a �eld experiment, Arkes and Blumer (1985) randomly selected sixty people to buy seasontickets to the Ohio University Theater and divided them in three groups of twenty. Patrons inthe �rst group paid the full price ($15). Those in the second group received a $2 discount, andpeople in the last group received a $7 discount. Patrons in the �rst group attended signi�cantlymore than those in the discount groups.

41Sunk costs e¤ects are also called �irrational escalation of commitment�, the �entrapment e¤ect�, or �toomuch invested to quit�.

31

Camerer and Weber (1999) analyzed the market for professional basketball players. Playerswho are drafted earlier represent larger sunk costs. They show that, conditional on performance,players who are drafted earlier get more playing time, which suggests the presence of sunk coste¤ects.

In this subsection, I present a self-deception model of sunk cost e¤ects. Psychologists havelong argued that self-deception may be an important cause of why sunk costs a¤ect choice. Forexample, Staw (1976) has shown that being personally responsible for an ine¢ cient investmentis an important factor in choosing to persist on it. Brockner et al. (1986) have documentedthat persisting on an ine¢ cient allocation of resources is increased when subjects are told thatoutcomes re�ected their �perceptual abilities and mathematical reasoning�.42

Whether previous investments succeed or fail has important e¤ects on the decision maker�sself-views. Then, as the opening quote suggests, a past choice may be associated to sunk mone-tary costs but real psychological costs. Abandoning a project usually involves admitting that awrong decision was made. Therefore, revising one�s position in the project reveals informationabout her skills or knowledge. As shown in Subsection 2.5, the DM requires a participationpremium in order to engage in an activity that reveals information about her characteristics.Hence, some projects with negative expected value will not be terminated. This subsectionincorporates the basic model os Section 4 into a reversible investment model.

Consider a decision-maker that has an opportunity invest in a project, which involves a sunkcost equal to K > 0. The project will give a monetary payo¤ equal to t 2

�t1; t2; :::; tN

;

t1 < t2 < ::: < tN ; N � 3; which is not known ex-ante. The monetary outcome of the projectis informative about the DM�s skills. Higher monetary payo¤s are better news about the DM�sskills:

F��jt = tn+1

�� F (�jt = tn) for all � 2 �;

with strict inequality in some set with positive measure for all n 2 f1; :::; N � 1g.After the sunk investment was made, the decision-maker can reevaluate the value of the

project. More precisely, she can choose whether to observe the value of the project t (at zerocost). She also decides whether to terminate it. Subsequently, she can manipulate her recollec-tions of the pro�tability of the project. If the project was not terminated, the DM receives amonetary payment of t:

As in Subsection 4.3, I will assume that t0 is uniformly distributed over a large interval(� = 0). Furthermore, the DM is assumed to be risk-neutral so that u (�; t) = � (�)� t for somefunction � with

R� (�) f (�) d� = 1:

The project is ex-ante e¢ cient but fails in some cases

E [t] � K; and (18)

t2 < 0 < t3:

Therefore, the project fails if t 2�t1; t2

: Denote the expected payo¤ from skills conditional on

ti by �i �R� (�) dF

��jt = ti

�dt: Then, the agent�s utility given t = ti is �i �

�ti + E (t0)

�:

The timing of the model is as follows:

1. The DM chooses whether or not to invest in the project. Investing costs K > 0:

2. If the investment was made, she decides whether or not to reassess the pro�tability of theproject, which takes value ti with probability qi 2 (0; 1) ; i = 1; :::; N .

42See Brockner (1992) for a review of the literature.

32

3. The DM chooses whether or not to abort the project.

4. If the project is aborted (A), the DM remembers the signal with probability �i + ri � fi;where �i is the natural rate of remembering signal i, ri is the amount of e¤ort to rememberand fi is the e¤ort to forget signal i: If it is not aborted (NA), the agent observes ti andobtains utility �i �

�ti + E (t0)

�:

The main idea is that abandoning the project is informative about the agent�s skills, whichleads to self-deception. In this model, memory manipulation is not e¤ective when the project isnot aborted since the agent will observe of the project pro�tability in the next period anyway.However, the DM may want to manipulate her memory when the project is aborted in order toconvince herself that her skills or knowledge are not extremely low even though the project wasabandoned. By not reassessing the project after the sunk investment was made, the DM avoidsthe self-deception costs that follow termination. Therefore, she would prefer not to reassess theproject if the value of information is lower than the costs of memory manipulation. In that case,however, ine¢ cient projects t 2 ft1; t2g are not aborted.

Suppose the DM chooses to observe the pro�tability of the project. Note that she will neverterminate the project if she observes tN since continuing gives a payo¤ of �N �

�tN + E (t0)

�;

which is strictly greater than any payo¤ she could get under termination (both �i and ti are

increasing in i and manipulation costs are nonnegative). Applying this reasoning inductively, itfollows that the agent will never terminate a project if she observes ti; i � 3 :

Lemma 7 Suppose Terminate 6= ?. Then, in any PBE, the decision-maker never terminatesa project after observing ti such that i � 3:

Now consider a project i 2 f1; 2g : Terminating yields a payo¤ of

[(�i + ri � fi)�i + (1� �i � ri + fi)E (�jTerminate)]� E (t0)�MCi;

where Terminate � f1; 2g is the set of projects that the agent terminates and MCi = L (fi)+ H (ri) is the manipulation cost after observing t

i. Continuing the project yields �i��ti + E (t0)

�:

The agent terminates a project after observing ti if

��iti � (1� � � ri + fi) [�i � E (�jTerminate)]� E (t0) +MCi: (19)

Hence, the DM continues project i if and only if the monetary loss from continuing the ine¢ cientproject

���iti > 0

�is lower than the gain from remembering that the project was not as bad as

the other projects which are interrupted (1� � � ri + fi) [�i � E (�jTerminate)] � E (t0) plusthe costs of memory manipulation MCi that would be exerted if the project was interrupted.

Proposition 13 Suppose Assumption 2a holds. There exist �t1 < 0 and �t2 < 0 such that ifeither t1 > �t1 or t2 > �t2; then the decision-maker does not terminate some ine¢ cient projectsin any PBE.

Thus, when the monetary losses (t1 and t2) from continuing an ine¢ cient project are "nottoo large", the DM will not terminate it. Although the investment involves sunk monetarycosts, abandoning ine¢ cient projects involve admitting one�s failure which leads to additionalpsychological costs.

33

5 Conclusion

This paper proposed a non-expected utility model based on self-deception. The model assumesthat people have preferences over their perceived characteristics and that they can, to someextend, manipulate their memories. However, because self-deception is costly, an individualmay prefer to avoid signals that are informative about her skills.

The predicted behavior is similar to that of a decision-maker with preferences over ambiguousevents. However, since the non-linearity arises from the anticipation of self-deception costs, itpredicts that individuals will avoid the events that lead to more self-deception. The model isconsistent with recent experimental evidence that suggests that ambiguity aversion/lovingnessis related to the decision-maker�s skills or knowledge. It also leads to a theory of ego utility andinformation acquisition.

It was shown that self-deception provides a uni�ed explanation for several biases in decision-making. I have argued that the endowment e¤ect can arise from an agent�s fear of learning thatthe trade turned out to be unpro�table. Similarly, sunk costs may in�uence behavior becausea decision-maker may not want to admit that she made an unpro�table investment. In caseswhere self-deception is severe or the potential monetary gains are small, the decision-maker mayeven prefer to obtain the smallest monetary outcome for sure instead of participating in a lottery(i.e., she exhibits an uncertainty e¤ect).

Because the decision-maker is assumed to be a Bayesian, she eventually learns the truth inan in�nitely repeated environment. Hence, the model implies that the her behavior convergesto that implied by expected utility theory as the decision-maker gains experience.

Appendix

A Proofs

Proof of the claim in Remark 1:

Let � and � denote the cumulative distribution functions of �� and ��; respectively. ��second-order stochastically dominates �� if and only if, for any concave function g : �! R;Z

g���

�d����

��Zg (��) d� (��) : (20)

But Zg (��) d� (��) = qg (�H) + (1� q) g (�L) ; andZ

g���

�d����

�= q (rH + �H) g (�H)+[q (1� rH � �H) + (1� q) (1 + fL � �L)] g

��?

�+(1� q) (�L � fL) g (�L) :

Substituting in inequality (20), yields

[q (1� rH � �H) + (1� q) (1 + fL � �L)] g��?

�> q (1� rH � �H) g (�H)+(1� q) (1� �L + fL) g (�L) :

34

Dividing both sides by q (1� rH � �H) + (1� q) (1 + fL � �L), we obtain:

g (� (fL; rH) �H + [1� � (fL; rH)] �L) � � (fL; rH) g (�H) + [1� � (fL; rH)] g (�L) ;

which is true because g is concave. �

Proof of Proposition 1:

De�ne fL (fL; rH) and rH (fL; rH) as the set of maxima of (2) and (3), respectively (i.e., theseare the best-response correspondences of the date-1 selves). Since (2) and (3) are continuous andconcave functions de�ned over a compact set, fL and rH are non-empty, convex, and compactsets. De�ne the transformation T : [0; �L] � [0; 1� �H ] ! [0; �L] � [0; 1� �H ] by T (fL; rH) =�fL (fL; rH) ; rH (fL; rH)

�: Then, Kakutani�s theorem establishes that there exists a �xed-point

of T , which is a PBE.The other claims follow from the Kuhn-Tucker conditions.1. For any fL 2 fL (f�L; r�H) and rH 2 rH (f�L; r�H) ;

� (f�L; r�H)�u > 0 =) fL > 0; and (21)

[1� � (f�L; r�H)]�u > 0 =) rH > 0:

Suppose f�L = 0: Then, we must have � (0; r�H)�u � 0 = 0f (0) : If � (0; r�H) > 0; we already

have a contradiction. Therefore, suppose � (0; r�H) = 0: But � (0; r�H) = 0 () 1 � �H = rH ;

which is strictly positive since �H < 1:Now suppose r�H = 0: Then, we must have [1� � (f�L; r�H)]�u � 0: As shown above fL >

0: But, fL > 0 implies 1 � � (fL; 0) > 0, which establishes that [1� � (f�L; r�H)]�u > 0 so thatr�H > 0.

2. For any fL 2 fL (f�L; r�H) and rH 2 rH (f�L; r�H) ;

� (f�L; r�H)�u � 0f (0) =) fL = 0; and

[1� � (f�L; r�H)]�u � 0r (0) =) rH = 0:

Since � (f�L; r�H) 2 [0; 1], it follows that fL (f�L; r�H) = rH (f

�L; r

�H) = f0g : �

Proof of Lemma 1:

There exists a PBE with f�L = r�H = 0 if

[1� � (0; 0)]�u � 0r (0) ; and

� (0; 0)�u � 0f (0) :

Because � (0; 0) = q(1��H)q(1��H)+(1�q)(1��L)

; letting

� = min

� 0r (0)

�1 +

q (1� �H)(1� q) (1� �L)

�; 0f (0)

�1 +

(1� q) (1� �L)q (1� �H)

��establishes that �u � � implies that there exists a PBE with f�L = r�H = 0.

Next, we need to show that there is no PBE with max ff�L; r�Hg > 0 when �u < �: The proofis obtained by contradiction. Suppose that we have a PBE such that max ff�L; r�Hg > 0. Thereare 8 possible cases:

35

Case 1) f�L = �L; r�H = 1� �H :

From the Kuhn-Tucker conditions, this is an equilibrium if

0 � 0f (�L) ; and �u � 0r (1� �H) :

Hence, 0f (�L) > 0 implies that we cannot have a PBE with f�L = �L; r

�H = 1� �H :

Case 2) f�L = 0; r�H = 1� �H :

This is a PBE if the following conditions hold:

� (0; 1� �H)�u � 0f (0) ;

[1� � (0; 1� �H)]�u � 0r (1� �H) :

Using the fact that � (0; 1� �H) = 0, it follows that the �rst inequality always hold (fromassumption 2b). Thus, we have a PBE with f�L = 0; r

�H = 1� �H if �u � 0r (1� �H) :

Case 3) f�L 2 (0; �L) ; r�H = 1� �H :Using Kuhn-Tucker�s conditions and the fact that � (fL; 1� �H) = 0, it follows that 0f (f�L) =

0; which is a contradiction. Hence, we cannot have a PBE with f�L 2 (0; �L) ; r�H = 1� �H :

Case 4) f�L = �L; r�H = 0 :

Kuhn-Tucker�s conditions give

� (f�L; r�H)�u � 0f (�L) ;

[1� � (f�L; r�H)]�u � 0r (0) :

Because � (�L; rH) =q(1��H�rH)

q(1��H�rH)+1�q; it follows that this is a PBE if:�

1 +q

1� q (1� �H)� 0r (0) � �u �

�1 +

1� qq (1� �H)

� 0f (�L) :

Case 5) f�L = �L; r�H 2 (0; 1� �H) :

Kuhn-Tucker�s conditions imply that this is a PBE if�1 +

q (1� �H � r�H)1� q

� 0r (r

�H) = �u �

�1 +

1� qq (1� �H � rH)

� 0f (�L) ;

for some r�H 2 (0; 1� �H) :

Case 6) f�L 2 (0; �L) ; r�H 2 (0; 1� �H) :This is a PBE if

�u = 0f (fL)

�1 +

(1� q) (1� �L + fL)q (1� �H � rH)

�= 0r (r

�H)

�1 +

q (1� �H � rH)(1� q) (1� �L + fL)

�;

for some rH 2 (0; 1� �H) ; and some fL 2 (0; �L) :

Case 7) f�L = 0; r�H 2 (0; 1� �H) :

Kuhn-Tucker�s conditions yield

0r (rH)

�1 +

q (1� �H � rH)(1� q) (1� �L)

�= �u � 0f (0)

�1 +

(1� q) (1� �L)q (1� �H � rH)

�;

36

for some rH 2 (0; 1� �H) :

Case 8) f�L 2 (0; �L) ; r�H = 0 :This is a PBE if

0f (fL)

�1 +

(1� q) (1� �L + fL)q (1� �H)

�= �u � 0r (0)

�1 +

q (1� �H)(1� q) (1� �L + fL)

�;

for some fL 2 (0; �L) :

It is immediate that we cannot have a PBE in cases 1 and 3. De�ne �1 as

min

� 0r (1� �H) ;

�1 +

1� qq (1� �H)

� 0f (�L)

�:

Then, when �u < �1; there are no PBE in cases 2 and 4.

Note that infrHnh1 + 1�q

q(1��H�rH)

i 0f (�L)

o> 0; infrH ;fL

n 0r (rH)

h1 + q(1��H�rH)

(1�q)(1��L+fL)

io>

0; and infrH ;fLn 0f (fL)

h1 + (1�q)(1��L+fL)

q(1��H�rH)

io> 0: De�ne �2 > 0 as the minimum of these

three terms. Then, when �u < �2; it follows that the conditions to Cases 5, 6, 7, and 8cannot be satis�ed. Hence, letting � = min f�1; �2g establishes that we cannot have a PBE withmax ff�L; r�Hg > 0 when �u < �; which concludes the proof. �

Proof of Proposition 2:

First, notice that existence follows from Proposition 1. The characterization follows fromKuhn-Tucker�s conditions:

�u � 0f (0) =) fL = 0;

�u � 0f (1)

q=) fL = 1;�

1 +1� qq

fL

� 0f (0) < �u <

�1 +

1� qq

fL

� 0f (1) =) � (fL)��u = 0f (fL) :

Suppose there exist at least two equilibria. There are 2 possibilities: f1L = 0; f2L > 0; f1L = 1;f2L < 1 (we cannot have 2 interior equilibria).

(i) f1L = 0; f2L 2 (0; 1) : We must have:

�u � 0f (0) ;

�u >

�1 +

1� qq

f2L

� 0f (0) :

Hence, we must have �1 +

1� qq

f2L

� 0f (0) < �u � 0f (0) :

This cannot happen if 0f (0) = 0 (since �u > 0). But if 0f (0) > 0; this is also impossible since

1 + 1�qq f2L > 1:

37

(ii) f1L = 1; f2L < 1: Then,

0f (1)

q� �u <

�1 +

1� qq

f2L

� 0f (1) ;

which is only possible if 1q � 1 +1�qq f2L: Rearranging yields

1� q � (1� q) f2L () 1 � f2L;

which is again a contradiction. This establishes uniqueness.Claim 1 follows by inspection. Let the cost of manipulation be f (fL; �) ; where � para-

metrizes the marginal cost of memory manipulation: @2 @fL@�

> 0: Therefore, higher ��s lead toa higher marginal cost of memory manipulation. Then, a simple computation shows that f�L is

decreasing in �: Condition (5) gives df�Ldq > 0: Thus, for interior solutions f�L is increasing in q:

Note that the range of parameters where f�L = 0 is constant in q whereas the range of parameterswhere f�L = 1 is increasing in q: This establishes Claim 3. �

Proof of Theorem 1:

Claims (i) and (ii) follow straight from equation (8) and the fact that

q = 0 =) f�L = 0 ) w (0) = 0� f (0)

�u= 0;

q = 1 =) r�H = 0 ) w (1) = 1� r (0)

�u= 1:

Equation (8) implies that, w (q) � q � 1 for all q; which establishes (iii).

It remains to be shown that r�H = 0 or f�L = 0 imply that w (q) � 0: First, consider the case

where f�L = 0: Then, the weighting function given by equation (8) is

w (q) = q � q r (r�H)

�u;

which is positive if and only if �u � r (r�H) : The argument follows by revealed preference on

r�H : Recall that it solves

maxrH

uL + f(�H + rH) [1� � (0; r�H)] + � (0; r�H)g�u� r (rH) :

In particular, this expression must be greater than when it is evaluated at rH = 0 :

uL + f�H [1� � (0; r�H)] + � (0; r�H)g�u� r (0) :

Hence,rH [1� � (0; r�H)]�u � r (rH)� r (0) :

But, since � (0; r�H) 2 [0; 1], r�H 2 [0; 1] ; and r (0) = 0; we obtain �u � r (r�H) ; which shows

that w (q) � 0:

Next, consider the case where r�H = 0: Note that w (q) � 0 if and only ifq1�q�u � F (f

�L) :

The argument also follows by revealed preference. Recall that f�L solves

maxfL

uL + � (f�L; 0) (1� �L + fL)�u� f (fL) :

38

This expression evaluated at f�L must be greater than when it is evaluated at fL = 0 :

uL + � (f�L; 0) (1� �L + f�L)�u� f (f�L) � uL + � (f

�L; 0) (1� �L)�u� f (0) :

Rearranging, yieldsqf�L

q + (1� q) f�L�u � f (f

�L) ;

where we have used the fact that � (f�L; 0) =q

q+(1�q)f�L: But qf�L

q+(1�q)f�L� q

1�q implies that

q

1� q�u � f (f�L) ;

which shows that w (q) � 0: �

Proof of Lemma 2:

Note that in this case recollections are i.i.d. Then, in order to apply Doob�s consistencytheorem, we need to check that there exists a set A 2 f?; L;Hg such that �1 6= �2 =)Pr�1 (A) 6= Pr�2 (A) : In each period, the probability of each recollection � (which are i.i.d.) is

Pr (� = Hj�) = Pr (� = Hj�)� �H ;Pr (� = Lj�) = [1� Pr (� = Hj�)] �L;Pr (� = ?j�) = Pr (� = Hj�) (1� �H) + [1� Pr (� = Hj�)] (1� �L) :

Since Pr (� = Hj�) is strictly increasing in �; it follows that �1 > �2 implies Pr�1 (� = H) >Pr�2 (� = H) and Pr�1 (� = L) < Pr�2 (� = L) ; which veri�es the condition. �

Proof of Lemma 3:

To simplify the notation, consider the distribution q instead of the distribution of �: This iswithout loss of generality since q = Pr (� = Hj�) is strictly increasing in �: With some abuse ofnotation, I will write F (qjhn) for the c.d.f. of q given history hn:

Denote by hnnk the history f�1; :::; �k�1; �k+1; :::; �ng : I will use the following result:

Claim 1 For any history hn; we have:

F�qjhnnk; �k = H

�� F

�qjhnnk; �k = L

�:

This claim states that, for any history, a high signal is good news about q and a low signalis bad news about q in terms of �rst-order stochastic dominance.

Note that the p.d.f. conditional on hn is

f (qjhn) =

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]

�Y

t:�t=H

q (�H + rt)�Y

t:�t=L

(1� q) (�L � ft)� f (q)

R 8>><>>:Y

t:�t=?[q (1� �H � rt) + (1� q) (1� �L + ft)]

�Y

t:�t=H

q (�H + rt)�Y

t:�t=L

(1� q) (�L � ft)� f (q)

9>>=>>; dq

:

39

Denote #H for # ft : �t = Hg and #L for # ft : �t = Lg : Then,

f (qjhn) =

(�H + rt)#H (�L � ft)#L � q#H � (1� q)#L � f (q)

�Y

t:�t=?[q (1� �H � rt) + (1� q) (1� �L + ft)]

(�H + rt)#H (�L � ft)#L �

R 8<: q#H � (1� q)#L � f (q)�

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]

9=; dq

=

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q)R Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dq:

Note that this is not a function of rt and ft for t such that �t 6= ?: Integrating, we obtain

F (xjhn) =

R x0

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dqR 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dq:

We are now ready to prove the Claim above.(Proof of the Claim). We have to show thatR x

0

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dqR 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dq�

R x0

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H�1 � (1� q)#L+1 � f (q) dqR 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H�1 � (1� q)#L+1 � f (q) dq:

When x = 0; both sides become 0 and, when x = 1; both sides are equal to 1:The derivative of the LHS with respect to x isY

t:�t=?[x (1� �H � rt) + (1� x) (1� �L + ft)]� x#H � (1� x)#L � f (x)R 1

0

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dq;

and the derivative of the RHS with respect to x isYt:�t=?

[x (1� �H � rt) + (1� x) (1� �L + ft)]� x#H�1 � (1� x)#L+1 � f (x)R 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H�1 � (1� q)#L+1 � f (q) dq:

40

Note that dRHSdq > dLHS

dq if and only ifYt:�t=?

[x (1� �H � rt) + (1� x) (1� �L + ft)]� x#H � (1� x)#L � f (x)R 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dq

�

Yt:�t=?

[x (1� �H � rt) + (1� x) (1� �L + ft)]� x#H�1 � (1� x)#L+1 � f (x)R 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H�1 � (1� q)#L+1 � f (q) dq

Rearranging, we obtain:

xR 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dq

� (1� x)R 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H�1 � (1� q)#L+1 � f (q) dq:

Thus, dRHSdq > dLHSdq if and only if

� (x) >

R 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dqR 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H�1 � (1� q)#L+1 � f (q) dq;

where � (x) = x1�x : Since � (0) = 0; � (1) = +1; � (x) is strictly increasing in x; and the term

on the right is a positive constant, there exists a unique �x such that

� (x) >

R 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dqR 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H�1 � (1� q)#L+1 � f (q) dqif x < �x;

� (x) <

R 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H � (1� q)#L � f (q) dqR 10

Yt:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)]� q#H�1 � (1� q)#L+1 � f (q) dqif x > �x:

Therefore, we have thatdRHSdq > dLHSdq if x < �x and dRHS

dq < dLHSdq if x > �x: Thus, the

inequality is satis�ed for all q (it is satis�ed with strict inequality whenever q 2 (0; 1) and withequality at q 2 f0; 1g.

Now, we are ready to prove the lemma:Proof of Lemma 3. As shown previously, F (xjhn) is not a function of rk and fk for k suchthat �k 6= ?: Therefore, we only need to establish the results for k such that �k = ?: Consider

41

an arbitrary k such that �k = ?: Then, F (xjhn) is equal to

(1� �H � rk)R x0 q

#H+1 � (1� q)#LY

t6=k:�t=?[q (1� �H � rt) + (1� q) (1� �L + ft)] f (q) dq

+(1� �L + fk)R x0 q

#H � (1� q)#L+1Y

t6=k:�t=?[q (1� �H � rt) + (1� q) (1� �L + ft)] f (q) dq

(1� �H � rk)R 10 q

#H+1 (1� q)#LY

t6=k:�t=?[q (1� �H � rt) + (1� q) (1� �L + ft)] f (q) dq

+(1� �L + fk)R 10 q

#H � (1� q)#L+1Y

t6=k:�t=?[q (1� �H � rt) + (1� q) (1� �L + ft)] f (q) dq

:

With some algebraic manipulations, it follows that dFdrk(xjhn) > 0 if and only ifR x

0 q#H � (1� q)#L+1

Yt6=k:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)] f (q) dqR 10 q

#H � (1� q)#L+1Y

t6=k:�t=?[q (1� �H � rt) + (1� q) (1� �L + ft)] f (q) dq

>

R x0 q

#H+1 � (1� q)#LY

t6=k:�t=?[q (1� �H � rt) + (1� q) (1� �L + ft)] f (q) dqR 1

0 q#H+1 � (1� q)#L

Yt6=k:�t=?

[q (1� �H � rt) + (1� q) (1� �L + ft)] f (q) dq

Note that the left-hand side is equal to F (xjhn; �n+1 = L) ; whereas the right-hand side isequal to F (xjhn; �n+1 = H) : From the previous claim, it follows that F (xjhn; �n+1 = L) �F (xjhn; �n+1 = H) ; which proves that the condition above is satis�ed.

Proof of Proposition 6:

The result is immediate from inequality 11, Lemma 2, and the fact that the sets of historieswith zero measure is the same for all relevant manipulation e¤orts. �

Proof of Proposition 7:

In period N; conditions 1 and 2 from the de�nition of a PBE imply that fL�L; hN�1

�maximizes

(�L � fL)Zu (�) dF

��jL; hN�1

�+ (1� �L + fL)

Zu (�) dF

��j?; hN�1

�� f (fL) ;

and rH�H;hn�1

�maximizes

(�H + rH)

�Zu (�) dF

��jH;hN�1

��+ (1� �H � rH)

�Zu (�) dF

��j?; hN�1

��� r (rH) :

From Proposition 6, it follows thatRu (�) dF

��jhN

�converges to u (�) for almost all histo-

ries. But, whenRu (�) dF

��jhN

�= u (�) ; it follows that fL

�L; hN�1

�maximizes

(�L � fL)u (�) + (1� �L + fL)u (�)� f (fL) = u (�)� f (fL) ;

which is strictly decreasing in fL: Hence, by continuity, it follows that fL�L; hN�1

�! 0 (a.s.).

Similarly, rH�H;hn�1

�maximizes u (�)� r (rH) ; which implies that rH

�H;hn�1

�! 0 (a.s.).

42

Under assumption 2b; it follows that there exists a ��u such that ifRu (�) dF

��jH;hN�1

��R

u (�) dF��jL; hN�1

�< ��u; then rH

�H;hN

�= fL

�L; hN�1

�= 0: Because

Ru (�) dF

��jH;hN�1

��R

u (�) dF��jL; hN�1

�! 0 (a.s.), it follows that there exists an �N such that N > �N implies

that rH�H;hN�1

�= fL

�L; hN�1

�= 0 for almost all histories. The continuation payo¤ at N�1

is then equal to V�hN�1

�= u (�) : Plugging back in the payo¤ at period N � 1 it follows that,

for large enough N; fL�L; hN�2

�= rH

�H;hn�1

�= 0 (a.s.). By induction, it follows that, when

N is large enough, there exists an �N such that rH (hn) = fL (hn) = 0 for almost all hn: �

Proof of Corollary 1:

Immediate from equation (10) and Proposition 6. �

Proof of Proposition 9:

The ex-ante utility net of manipulation costs isZt

�(1� q) [�L � fL (t)]uL (t) + q [�H + rH (t)]uH (t)

+ f(1� q) [1� �L + fL (t)] + q [1� �H � rH (t)]gu? (t)

�dH (t) :

Substituting u? (t) = �� uH (t) + (1� �)� uL (t) and rearranging, we obtainZt

�[1� q (rH + �H)]uL + q (�H + rH)uH

+� [(1� q) (1� �L + fL) + q (1� �H � rH)]�u

�dH (t)

where � = q(1�rH��H)g(t�H)q(1�rH��H)g(t�H)+(1�q)(1�rL��L)g(t�L)

(and we omit the dependence of �, u; rH ; andfL on t for notational simplicity).

The expected utility from a lottery whose monetary outcomes are independent of � is U I =Rt [quH (t) + (1� q)uL (t)] dH (t) : After some algebraic manipulations, we can express the ex-ante utility net of manipulation costs as

U I +

Z �(gH � gL)

q (1� q) (1� rH � �H) (1� �L + fL)q (1� rH � �H) gH + (1� q) (1� rL � �L) gL

�u

�dH:

De�ne the functions � (t) ; z (t) as

z (t) =q (1� q) (1� rH � �H) (1� �L + fL)

q (1� rH � �H) gH + (1� q) (1� rL � �L) gL> 0;

� (t) � [g (t�H)� g (t� L)] z (t) :

Then, the ex-ante utility incorporating the expected manipulation costs is equal to

U I + ��MC;

where � �Rt � (t)�u (t)h (t) dt and MC �

Rt

�q r (r

�H (t)) + (1� q) f (f�L (t))

�h (t) dt: �

Proof of Theorem 2:

From Proposition, U (L) = qRt uH (t) dH (t) + (1� q)

Rt uL (t) dH (t) + � �MC: Using the

de�nition of vs; we obtain

U (L) = qvH + (1� q) vL + ��MC:

43

Letting U (L) = w (q) vH + [1� w (q)] vL yields

w (q) = q +��MC

vH � vL:

Using the fact that vH � vL =R�u (t)h (t) dt concludes the �rst part of the proof.

Note also that when q = 0 or q = 1; it follows that MC = 0 and � = 0 (since z (t) = 0 forall t). Thus, U (L) = qvH + (1� q) vL so that w (q) = q: �

Proof of Lemma 4:

Special case of Lemma 5. �

Proof of Lemma 5:

We will use the following result:

Claim 2 If H = L = 0; thenRt [quH (t) + (1� q)uL (t)] [qg (t�H) + (1� q) g (t� L)] dt =R

t

R� u (�; t) dF (�) g (t) dt

Proof. Recall that uH (t) =Ru (�; t) dF (�j� = H) and uL (t) =

Ru (�; t) dF (�j� = L) : Fur-

thermore, qg (t�H) + (1� q) g (t� L) = g (t) : Then,Zt[quH (t) + (1� q)uL (t)] g (t) dt =

Zt

�qR� u (�; t) dF (�j� = H)

+ (1� q)R� u (�; t) dF (�j� = L)

�g (t) dt:

Using Bayes�rule, we obtain:Zt

�q

Z�u (�; t) dF (�j� = H) + (1� q)

Z�u (�; t) dF (�j� = L)

�g (t) dt =

Zt

Z�u (�; t) dF (�) g (t) dt:

The certainty equivalent is de�ned asZt

�f[quH (t) + (1� q)uL (t)]� [q H (rH (t; �)) + (1� q) L (fL (t; �))] + � (t; �)�u (t)g

� [qg (t� �"1) + (1� q) g (t� �"2)]

�dt

=

Zt

Z�u (�; t+ CE (�)) dF (�) g (t) dt: (22)

Evaluating at � = 0 and using the result from Claim 2, we obtain:Ztf� [q H (rH (t; 0)) + (1� q) L (fL (t; 0))] + � (t; 0)�u (t)g g (t) dt

=

Zt

Z�[u (�; t+ CE (0))� u (�; t)] dF (�) g (t) dt:

Because � (t; 0) = 0; we have

�Zt[q H (rH (t; 0)) + (1� q) L (fL (t; 0))] g (t) dt

=

Zt

Z�[u (�; t+ CE (0))� u (�; t)] dF (�) g (t) dt:

Thus:

44

�Rt [q H (rH (t; 0)) + (1� q) L (fL (t; 0))] g (t) dt = 0 =) CE (0) = 0;

�Rt [q H (rH (t; 0)) + (1� q) L (fL (t; 0))] g (t) dt > 0 =) CE (0) < 0:

If Pr (rH (t; 0) > 0) > 0 or Pr (fL (t; 0) > 0) > 0; then CE (0) < 0 and � (0) > 0: Therefore,lim�!0

�(�)� = +1 and the agent exhibits �rst-order risk aversion.

If Pr (rH (t; 0) > 0) = Pr (fL (t; 0) > 0) = 0, then CE (0) = 0 and � (0) = 0:Using L�Hospital�srule, it follows that

lim�!0

� (�)

�= �0 (0) = �CE0 (0) :

But, di¤erentiating equation (22) gives:

CE0 (0) = �

Rt

8<:hq 0H (0)� @rH

@� (t; 0) + (1� q) 0L (0)�

@fL@� (t; 0)

i+g0(t)g(t)

"11�q

hq(1�q)(1��H)(1��L)q(1��H)+(1�q)(1��L)

i�u (t)

9=; g (t) dt

Rt

R�dudt (�; t) dF (�) g (t) dt

:

Thus,

�0 (0) =

Rt

hq 0H (0)� @rH

@� (t; 0) + (1� q) 0L (0)�

@fL@� (t; 0)

ig (t) dt+ � (q)

Rt g0 (t)�u (t) dtR

t

R�dudt (�; t) dF (�) g (t) dt

;

where � (q) = "1q(1��H)(1��L)

q(1��H)+(1�q)(1��L)> 0:

Because the denominator is positive, it follows that:

� �0 (0) = 0 ifRt

hq 0H (0)

@rH@� (t; 0) + (1� q)

0L (0)

@fL@� (t; 0)

ig (t) dt+� (q)

Rt g0 (t)�u (t) dt =

0;

� �0 (0) > 0 ifRt

hq 0H (0)

@rH@� (t; 0) + (1� q)

0L (0)

@fL@� (t; 0)

ig (t) dt+� (q)

Rt g0 (t)�u (t) dt >

0; (�rst-order risk-averse) and

� �0 (0) < 0 ifRt

hq 0H (0)

@rH@� (t; 0) + (1� q)

0L (0)

@fL@� (t; 0)

ig (t) dt+� (q)

Rt g0 (t)�u (t) dt <

0 (�rst-order risk-loving),

which concludes the proof. �

Proof of Corollary 2:

Recall that u (�; t) = � (�)� t andR� (�) f (�) d� = 1: Thus,Z

t0

Z�u (�; t0 + L) f (�) g (t0) d�dt0 = L+ E (t0) :

Moreover,R�

Rt u (�; t) f (�)h (t) dtd� =

Rt t� h (t) dt = E (t) : Since t = t0 + s; it follows that

E (t) = E (t0) + qH + (1� q)L:

Substituting in equation (16), we obtain

MC � q (H � L) + �;

45

which concludes the proof. �

Proof of Proposition 12:

From Proposition 8, it follows that MC > 0 for any �t � 0; which establishes the �rstclaim. Moreover, for �t = 0; we have that MC � q�t in any equilibrium. By the theory of themaximum, it follows that MC � q�t when �t > 0 is su¢ ciently small. �

Proof of Lemma 7:

Let �� denote E (�jTermination). Terminating a project after observing tN gives an expectedpayo¤ of �

(�N + rN � fN )�N + (1� �N � rN + fN ) ���� E (t0)�MCN ;

whereas continuing gives �N ��tN + E (t0)

�: Thus, the project is continued if

�N tN + (1� �N � rN + fN )E (t0)

��N � ��

�� �MCN ;

which is true because the left-hand side is positive whereas the right-hand side is negative. Thus,the project is not terminated.

Hence, in any PBE where Termination 6= ?; Bayes�rule implies that �� � �N�1: The resultthen follows by induction. �

Proof of Proposition 13:

Suppose that both ine¢ cient projects 1 and 2 are terminated. Then, equation (19) yields

��iti � (1� �i � ri + fi)��i �

q1�1 + q2�2q1 + q2

�� E (t0) +MCi;

for i 2 f1; 2g : Rearranging yields, for i 2 f1; 2g ;

ti � � (1� �i � ri + fi)�1� 1

�i

q1�1 + q2�2q1 + q2

�� E (t0)�

MCi�i

:

Under Assumption 2a; inf fMCi : t � 0g > 0: Therefore, whenever ti > � inffMCi:t�0g�i

for

some i 2 f1; 2g ; project i is not terminated. Thus, setting �ti = � inffMCi:t�0g�i

< 0 establishesthat terminating 1 and 2 cannot be a PBE. �

B Example

In this Section, I present an example of explicit distributions of � and �: Suppose � � U [0; 1]and

� =

�H if � � 1� qL if � < 1� q :

Then, F (�j� = H) = �q �� (� � 1� q) and F (�j� = L) = �

1�q �� (� < 1� q) ; where � denotesthe indicator function. It follows that

uH =1

q

Z 1

1�qu (�) d�; and

uL =1

1� q

Z 1�q

0u (�) d�:

46

Hence, �u (q) = 1q

R 11�q u (�) d� �

11�q

R 1�q0 u (�) d� which, as stated in Subsection 2.4, is in

general a function of q. �u (q) can be an increasing or decreasing function of q depending onthe utility function u (�) :

For example, when u (�) = �2; then�u (q) = 23�

13q; which is decreasing in q:When u (�) = �;

then �u (q) = 12 . And, when u (�) =

p�; then �u (q) = 2

3q

�1�

p1� q

�; which is increasing in

q:

C Non-Bayesian Framework

In this Section, I consider deviations from the assumption that the decision-maker evaluates herexpected characteristics when a signal is forgotten according to Bayes rule. Suppose, instead,that upon recollecting � = ?; the DM attributes weight

� (fL; rH) �q (1� �H � rH)

q (1� �H � rH) + � (1� q) (1� �L + fL)

to � = H and 1 � � (fL; rH) to � = L; where � 2 [0; 1] denotes the degree of pessimism inthe agent�s updating rule. When � = 0; the agent believes that any forgotten signal is a highsignal. Thus, she updates information optimistically. When � = +1; the agent believes thatforgotten signals are low signals and, therefore, updates information pessimistically. Finally,� = 1 captures the Bayesian case described in the text (realistic updating).

It is straightforward to show that the ex-ante expected utility from observing the signal isequal to

1� (1� q) (�L � fL) + (1� q) (�L � fL) (�H + rH) (1� �)� (�H + rH) + � (1� q) (�H + rH)q + � (1� q)� q (�H + rH)� � (1� q) (�L � fL)

q�u

+uL � q H (rH)� (1� q) L (fL) :

For � = 0; the expression above becomes uH�(1� q) (�L � fL)�u�q H (rH)�(1� q) L (fL) :In that case, the DM would always choose r�H = 0 (since there is no loss from forgetting a highsignal when � = ? are interpreted as � = H). Thus, the ex-ante expected utility is

uH � (1� q) (�L � f�L)�u� (1� q) L (f�L)

The agent prefers to observe the signal if and only if

(1� �L + f�L)�u � L (f�L) :

Hence, if the manipulation costs are not too high, the DM may prefer to observe a signal whenshe has biased recollections in the sense of being more optimistic than implies by Bayes�rule.

When � = +1; the ex-ante expected utility from observing the signal is

uL + q (�H + rH)�u� q H (rH)� (1� q) L (fL) :

Since this expression is always smaller than quH+(1� q)uL; the DM will never prefer to observethe signal when � = +1: Indeed, this result is true for any agent with recollections that aremore pessimistic than implied by Bayes�rule (� > 1).

Extending this approach to the case of lotteries over money, it can be shown that deviationsfrom Bayes�rule introduce an additional term in the representation of Proposition 9. In that

47

case, the DM may prefer a lottery whose outcomes are correlated with characteristics even ifthere are no complementarities between characteristics and money.

It is interesting to note that the results from Section 3, where it was shown that the DM�sbehavior converges to the one predicted by expected utility theory after observing a su¢ cientlylarge number of signals, do not necessarily require the agent to update according to Bayes ruleor even to eventually learn her true type �.

Consider, for example, the case of extreme optimism: � = 0: In this case, the agent interprets� = ? as � = H: Hence, even though she does not update recollections according to Bayes�rule, the agent�s inference problem can be reinterpreted as if she observed signals � = L withprobability (1� q) (�L � fL) and � = H with probability 1 � (1� q) (�L � fL) and appliedBayes�rule.

Following the same arguments as in Section 3, we can show that the Bayes estimator unconverges to some function ~u (�) for almost every history when N ! +1. In general, we wouldhave ~u (�) 6= u (�) so that an extremely optimistic agent would not eventually learn her truetype. Nevertheless, because un converges, the bene�t of memory manipulation converges tozero. Therefore, the agent�s behavior also converges to the one of an expected utility maximizerdespite the fact that she does not update according to Bayes rule and does not eventually learnher true type �.

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