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MEMORY AND ANTICIPATION

B. Douglas Bernheim and Raphael Thomadsen

The introduction of memory imperfections into models of economic decision making creates anatural role for anticipatory emotions. Their combination has striking behavioural implica-tions. The paper first shows that agents can rationally select apparently dominated strategies.

We consider Newcombs Paradox and the Prisoners Dilemma. We provide a resolution forNewcombs Paradox and argue it requires the decision maker to ascribe only a tiny weight toanticipatory emotions. For some ranges of parameters, it is possible to obtain cooperation inthe Prisoners Dilemma with probability arbitrarily close to unity. The second half of the paperprovides a theory of reminders.

This paper studies decision problems with two twists: first, memory is imperfect,and second, the decision maker cares about anticipatory emotions. Previous workin this area has treated these phenomena separately. Recent analyses of decisions

with imperfect memory include Piccione and Rubinstein (1994, 1997a), Benabouand Tirole (2002) and Mullainathan (2002).1 Separately, the notion of anticipa-tory emotions has been studied by Loewenstein (1987), Elster and Loewenstein(1992), Caplin and Leahy (2001, 2004), Koszegi (2002, 2004) and others.

Why combine these two disparate lines of work? After all, many behaviouralalternatives to the standard model of decision making have been proposed and the

number of potential combinations is enormous. Slogging through every permu-tation seems at best a tedious task with uncertain prospects for useful insights.

Our decision to focus on this particular combination imperfect memory andanticipatory emotions is, however, not an arbitrary one. As we argue below,imperfect memory creates a natural role in decision making for anticipatoryemotions, one that does not exist when memory is perfect. In addition, we arguethat the combination of imperfect memory and anticipatory emotions yields somestriking and surprising implications for behaviour. The first half of the paper showsthat agents can rationally select apparently dominated strategies. We consider twoapplications: Newcombs Paradox and the Prisoners Dilemma. We provide aresolution for Newcombs Paradox and argue that it requires the decision maker toascribe only a tiny weight to anticipatory emotions. We also demonstrate that,under relatively weak conditions, it is possible to obtain cooperation in the Pris-oners Dilemma with probability arbitrarily close to unity. The second half of thepaper provides a theory of reminders. It shows that people may prefer to beuninformed, or to have coarse information, in situations where, eliminating eithermemory imperfections or anticipatory emotions, this would not be the case. Weexhibit a mechanism whereby the opportunity to leave a reminder can improve a

1 See also the symposium in Games and Economic Behaviordedicated to discussions of the issues raisedby Piccione and Rubinstein, including Aumann et al. (1997a,b), Battigalli (1997), Gilboa (1997), Groveand Halpern (1997), Halpern (1997), Lipman (1997) and Piccione and Rubinstein (1997b), as well as alater paper by Segal (2000).

The Economic Journal, 115 (April), 271304. Royal Economic Society 2005. Published by BlackwellPublishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

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concurrent decision, even though the reminder does not change the informationavailable when the decision is made. We also provide endogenous explanations foras-if overoptimism and behaviour associated with cognitive dissonance.

There are, of course, other theories purporting to explain a number of the

phenomena studied in this paper. Philosophers have proposed a variety of possibleresolutions for Newcombs Paradox (see e.g. the anthology edited by Campbelland Sowden (1985)), but the puzzle has received little attention among econo-mists (with a few exceptions such as Geanakoplos (1996)). Cooperation in the one-shot Prisoners Dilemma has been attributed to a variety of factors, such as altruismand concerns for fairness, social image, and self-image. A preference for ignorancearises in Carillo and Mariotti (2000), Benabou and Tirole (2002) and Koszegi(2002). Explanations for overoptimism and overconfidence appear in Rabin andSchrag (1999), Koszegi (2000), Hvide (2002), Benabou and Tirole (2002), Post-lewaite and Compte (2003) and Van den Steen (2003). Sources of cognitive dis-sonance have been studied by Akerlof and Dickens (1982) and others. To ourknowledge, the mechanisms proposed here are, however, novel.

The remainder of the paper is organised as follows. Section 1 discusses someissues concerning the modelling of imperfect memory. Section 2 explains howmemory imperfections create a natural role in decision-making for anticipatoryemotions. Section 3 explains how a rational player with imperfect memory andanticipatory emotions can rationally justify the selection of an apparently domin-ated strategy. Section 4 examines the role of reminders. Section 5 concludes.

1. Modelling Imperfect Memory

The literature on decision making with imperfect memory encompasses two fun-damentally different approaches. In one strand, the decision maker is naive, andalways acts as if he has forgotten nothing (Mullainathan, 2002). In the otherstrand, the decision maker is sophisticated, and draws rational inferences con-cerning things he may have forgotten, given his memory technology (Piccione andRubinstein, 1997a, Benabou and Tirole, 2002). In practice, behaviour is probablymarked both by naivete and sophistication. In this paper, we explore the behaviour

of sophisticated decision makers.As emphasised by Piccione and Rubinstein, various issues that are immaterialwhen modelling decisions with perfect recall emerge as important when analysingproblems with imperfect recall. These issues include the role of an initial planningstage and the decision makers ability to change a strategy during its execution.There are several coherent ways to resolve these issues and no single correctmodel of imperfect recall.

In this paper, we model imperfect recall using an approach proposed by Picci-one and Rubinstein (and adopted implicitly by Benabou and Tirole (2002)), whichthey call modified multiself consistency. The decision problem is viewed as a game

played by multiple incarnations of the decision maker, where a new incarnationtakes over whenever his memory fails. Behaviour corresponds to an equilibrium ofthis game. Each self has the ability to deviate from its prescribed equilibriumstrategy but it cannot choose to deviate for other incarnations.

272 [A P R I LT H E E C O N O M I C J O U R N A L

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For a simple illustration of this concept, see Figure 1, taken from Piccione andRubinstein (1994). This is a one player decision problem with imperfect recall.The individual starts at node A, and must choose either Left (L), placing him atnode B, or Right (R), placing him at node C. He immediately forgets this choice,

so Band Clie in the same information set. He must then choose eitherleft

(l) or

right (r). Payoffs are determined as shown in the Figure.With perfect recall, the individual would choose (R, r). To find modified

multiself consistent outcomes with imperfect recall, we imagine that there are twoseparate players, both of whom receive the same payoff; one makes the choice atnode A, and the other makes the choice at information setI containing B and C.One equilibrium of this two-player game involves the choice RatAand the choiceratI. Since the individual knows he will choose ratI, it is in his interests to chooseR at A. Similarly, even though he forgets his actual choice at A once he reaches I,he remembers that his strategy is to select R, so he infers that he is at C, in whichcase r is optimal. This is not, however, the only equilibrium. Another involves thechoice LatAand the choice latI. Since the individual knows he will choose latI,it is in his interests to choose L at A. Similarly, even though he forgets his actualchoice at A once he reaches I, he remembers that his strategy is to select L, so heinfers that he is atB, in which case lis optimal. Both (R, r) and (L, l) are multiselfconsistent outcomes.

A

BC

LR

l lr r

1 0 02

I

Fig. 1. A Simple Decision with Imperfect Recall

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This example is instructive in part because it highlights some key assumptions.The outcome (L, l) survives because, at node A, the individual is unable to change hisstrategy for the entire game. If there was an initial planning period in which he couldselect his strategy, he would clearly benefit from picking (R, r) rather than (L, l).

We interpret the Piccione-Rubinstein solution concept as follows. The equilib-rium decision strategy is a norm, in the sense that it describes how the individualnormally handles a certain type of decision problem. When he confronts one ofthese decision problems, he can choose to deviate from the norm. A deviation mayinclude the selection of an action other than the one prescribed by the norm, as

well as the adoption of a plan of contingent actions (a continuation strategy) thatdiffers from the norm. However, any time he experiences a memory lapse, heforgets any decision he may have made to follow a continuation plan that departsfrom the norm. He remembers the norm, and assumes he has always intended tofollow it.

An alternative approach would be to assume the individual can reformulate hisstrategy at any point during the decision process, and remember the reformulationat later stages even if he has forgotten actions and events. In this case, the outcome(L, l) would not survive. At node A, the individual would deviate not only to R, butalso to a continuation plan prescribing the choice ratI. Upon reaching I, he wouldforget his actual choice, but would remember his decision to adopt the strategy(R, r) rather than the strategy (L, l). From this he would infer that he is at node C,in which case ris optimal. Anticipating this later response makes the deviation atAattractive.

As we said, there is no right or wrong way to model imperfect recall. The Pic-cione-Rubinstein solution concept is one plausible approach. We find it appealingbecause it describes circumstances in which, upon experiencing a memory lapse,the decision maker asks himself what he usually does in similar circumstances.

2. Breaking the Law of Iterated Expectations

To clarify the connection between imperfect memory and anticipatory emotions,we consider a simple decision setting. A decision maker (abbreviated throughout

this article as DM) makes a choice in period 0 based on his available information,I0. He then receives or, in the case of imperfect recall, loses information. He waitsin period 1, forming a new expectation about his ultimate payoff based on hisavailable information, I1. Finally, in period 2, he receives his payoff, U.

In period 2, the individuals emotional well-being depends only on the payoffreceived, U. In period 1, his emotional well-being depends on his anticipatedperiod 2 payoff, V E(U j I1). Following Caplin and Leahy (2001), we assumethat, in period 0, he cares about his future emotional well-being both in period 1and in period 2. We summarise his preferences over his future emotional states bythe function W(V, U).

For the moment, let us suppose that W, is linear: W(V, U) V dU. Here andelsewhere in this paper, d is simply the weight attached to the ultimate payoffrelative to anticipatory emotions, rather than a discount factor. From the per-spective of period 0, his expected utility is




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EWV;U j I0 EEU j I1 j I0 dEU j I0:For the moment, let us also assume the individual has perfect recall (so that I1includes all of the information in I0). In that case, the law of iterated expectations

applies, andEEU j I1 j I0 EU j I0: 1

In that case,

EWV;U j I0 1 dEU j I0: 2

The implications of the preceding observation are important. We have formu-lated a model in which the individual cares about anticipatory emotions in period1. However, he acts in period 0 exactly as he would if he ignored anticipatory

emotions entirely and cared only about the final period 2 outcome. In this setting,the individual may care about future anticipatory emotions but this plays no role indecision making.

As Caplin and Leahy point out, anticipatory emotions can affect behaviour whenWis nonlinear. Consider, for example, the case where W(V, U) w(V) dU,

where w is strictly concave. In this case, the individual is averse to variation infuture anticipatory emotions (Caplin and Leahy call this anxiety). Thus, anyinformation received between periods 0 and 1 makes him worse off ex ante (fromthe perspective of period 0). Similarly, he prefers not to receive information priorto making a period 0 decision unless this allows him to select an action thatsufficiently improves his outcome.

With imperfect memory, anticipatory emotions matter, even if the function Wislinear, because the law of iterated expectations breaks down. For (1) to hold, I1must contain at least all of the information in I0. If it omits some information, thechain of logic leading to (2) does not hold.

Consider the following illustration. Suppose the state of the world is either bad,in which case the individuals payoff is 0, or good, in which case his payoff is 10.The two states occur with equal probability. The individual observes the state inperiod 0, but forgets it before period 1. In that case,

EU j I1 5and

EEU j I1 j I0 5:

However, E(U j I0 bad) 0, and E(U j I0 good) 10. Clearly, the law ofiterated expectations fails.

To see how this can affect decisions, suppose that, in the bad state only, theindividual has the option to take an action that cuts his losses, in which case hispayoff is 1 instead of 0. Imagine also that he remembers his action in period 1,even though he forgets the state.

Ignoring anticipatory emotions, the best choice is obviously to cut losses whenthe state is bad. However, with anticipatory emotions and d 4, he chooses not to




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cut losses. The reason is that he does not want to remind himself that the outcomewill be bad. If he chooses not to cut losses upon observing the bad state, his payoff(from the perspective of period 0) is 5 d 0 5. If he cuts losses, his payoff(from the perspective of period 0) is 1 d 1 1 d. So he is better off notcutting losses when d 4.If the weight on the final outcome (relative to anticipatory emotions) is suffi-ciently large (d ! 9), then there is a modified multiself consistent outcome in

which the DM chooses to cut losses. With a decision to cut losses, his payoff (fromthe perspective of period 0) is again 1 d 1 1 d. If he chooses not to cutlosses, his payoff (from the perspective of period 0) is 10 d 0 10 (in this case,he falsely infers from his equilibrium strategy that the state must have been good).Thus, he cuts losses ifd ! 9.

What happens when d 2 (4, 9)? It turns out that outcomes necessarily involverandomisations between cutting and not cutting losses. Suppose that, uponobserving the bad state, he cuts losses with probabilityk. When he chooses to cutlosses in the bad state, his period 0 expected payoff is again 1 d 1 1 d.

When he chooses not to cut losses in the bad state, his period 0 expected payoff is[1/(2 k)]10 d 0 10/(2 k). He is indifferent between these choices, and

willing to randomise, when 10/(2 k) 1 d, or

k 2 101 d :

When d 9, k 1 (he cuts losses with certainty). When d 4, k 0 (he nevercuts losses). As d declines from 9 to 4, k declines monotonically from 1 to 0. Thus,for all d 2 (4, 9), there is a mixed strategy outcome, involving some loss cutting. Atthe ends of this interval, the mixed strategy outcome converges to a pure strategyoutcome (not cutting losses for d # 4, and cutting losses for d " 9), so the outcomeis continuous in d.

3. Playing Dominated Strategies

The logic that prescribes avoidance of strictly dominated strategies rests on a

simple and appealing assumption: a players opponents

strategies are causallyindependent of his own strategy. Regardless of which strategies a playersopponents have chosen, these remain fixed when he alters his own choice. If, forsome reason, he does not believe in causal independence, then he could justifyplaying a dominated strategy. For example, if in the Prisoners Dilemma hebelieves that, by cooperating, he somehow makes it more likely that his oppo-nent cooperates, he might be able to justify behaving cooperatively rather thanopportunistically.

The general idea pursued in this Section is that players may induce ex postsubjective correlation between their choices by conditioning on something thatis commonly observed but then forgotten. Conceptually, conditioning onsomething that is commonly observed leads in the direction of correlatedequilibria but, of course, correlated equilibria never involve dominated strat-egies. Here, the added twist is that each player cares about the payoff he




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expects to receive after he has forgotten the signal but before the payoff isrealised. Though my decision does not causally affect the decision of myopponent, in this setting it can causally affect the inference that I make in thefuture about my opponents decision, thereby altering my anticipatory payoffs,

potentially for the better.There are at least two different ways to introduce the correlation. One is toassume that a parameter of the game is realised randomly, observed by all players,and then forgotten. Another is to introduce commonly observed but irrelevantsignals (sunspots). We use the first approach in the context of Newcombs Para-dox, and the second in the context of the Prisoners Dilemma. We begin withNewcombs Paradox because the analysis is more straightforward.

3.1. Newcombs Paradox

Newcombs Paradox is due to physicist William Newcomb, and was popularised bythe philosopher Robert Nozick (1969). The paradox involves interaction betweena human and a superior being. In some variants of the paradox, we are invited tothink of the superior being as God, but for reasons discussed below we prefer to

view this player as a psychic. The superior being asks the human to pick betweentwo boxes. One is open, and the other is closed. The open box contains onethousand dollars. The closed box contains either one million dollars or nothing.The subject has two choices: (1) take only the closed box or (2) take both boxes.So far, it seems as though it is clearly better to take both boxes. But there is a twist.

Before presenting the subject with this problem, the being predicts the subjectschoice. If it predicts the subject will choose only the closed box, it puts $1 millioninside. If it predicts the subject will choose both boxes, it puts nothing inside theclosed box. Moreover, the being has presented this same choice to hundreds ofthousands of humans. Some have chosen both boxes, and some have chosen theclosed box but the being has always predicted the choice correctly. What shouldthe subject do?

Choosing both boxes is the right answer from the perspective of dominance. Yetmany people say they would choose only the closed box. Their logic: if I were to

choose both boxes, then the being would know I am the kind of person that woulddo this and would put no money in the closed box. Yet this seems to suffer from amistaken view of causation. Suppose I am the kind of person who would tend notto choose the closed box and that the being somehow knows this. Then it will put$1 million in the closed box. When I make my decision, the beings decision is a

fait accomplit. It is clearly in my interests to resist my natural tendencies and chooseboth boxes, regardless of what the being has done. How then can we rationalisethe dominated choices that many people seem to favour?

Here we attempt to provide an explanation that neither endows the superiorbeing with the ability to defy causality, nor attributes to the human a belief that the

superior being possesses this ability. We assume that the being is superior only inthe sense that it is extremely knowledgeable and endowed with an ability to discernnuances of anothers preferences hence we prefer to think of it as a humanpsychic rather than as a divinity.




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The flavour of our explanation is as follows. There is a preference characteristicwhich, in equilibrium, is related to my choice, and which the being observes (this iswhy he can predict my choice). Once I make my choice, I recall my choice but donot recall the characteristic on which my choice was conditioned. From the choice

I made, I can use my equilibrium strategy to infer my characteristic and therebyinfer the beings choice. Thus, if I have chosen both boxes, Ill think it more likelythat my type is such that the being has put nothing in the closed box. If I havechosen only the closed box, Ill think it more likely that my type is such that thebeing has put $1 million in the closed box. When I make my choice, I recognisethat the beings decision is a fait accomplit, and does not depend on what I do.However, I care about my anticipatory state of mind once I have made my choiceand before I learn the ultimate outcome. I would rather anticipate receiving themillion dollars, so I choose only the closed box. As we will see, for this argument tobe valid, I need only place a very small amount of weight on my anticipatoryemotions.

3.1.1. The gameConsider a game played by two agents, a human (H) and a superior being (S).Choices and events unfold as follows.

1. Nature randomly selects a preference parameter, d 2 R (with CDF F), forH, which will govern the weight placed on the actual outcome relative toanticipatory emotions (see below). The value of this parameter is observedboth by

Hand by

S.

2. S chooses one of two actions: Z (for zero) and M (for million).3. Without having observed Ss choice, H selects one of two actions: C (for

closed) and B (for both). We will use x to denote Hs choice.4. H forgets the value ofd but recalls his action.5. H waits to learn the outcome and forms an expectation of what it will be.6. Payoffs are realised.

Payoffs are determined according to the matrix illustrated in Figure 2. In eachcell of this matrix, the first entry is Hs payoff; v is a Von Neumann-Morgenstern

utility function, and its argument is the monetary reward. When describing theproblem, we imagined that a 1,000,000, and b 1,000. The second entry ineach cell is Ss payoff. Notice thatSs payoffs imply that it wants to put one milliondollars in the box when H chooses only the closed box, and it wants to leave thebox empty when H chooses both boxes; S cares only about making the correctprediction and not about the monetary payment.

In stage 3, when making his decision, H thinks about how he will feel at bothstage 5 and stage 6. In stage 3, his expected stage 6 utility is E[v(y) j x, d]. In stage5, his expected stage 6 utility is E[v(y) j x], and this determines his stage 5 utility,

which he takes into account at stage 3. He maximises a weighted average of thesetwo terms: E[v(y) j x]dE[v(y) j x, d].

Suppose for the moment that H has a perfect memory. In that case, he alwayschooses x to maximise E[v(y) j x, d] (since this represents expected utility, as ofstage 3, for both stage 5 and stage 6). The standard dominance argument applies.




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The only subgame perfect equilibrium involves Hchoosing Band Schoosing Z. Inequilibrium, the superior being does predict the humans behaviour with perfectaccuracy. However, the human never picks only the closed box. This cannot ex-plain Newcombs Paradox because, as part of the scenario, we are asked to imaginethat the being has correctly predicted both choices in the past.

Now we look for the equilibria of the game with imperfect memory. We use thePiccione-Rubinstein multiself approach discussed in Section 2, and studysequential equilibria.

3.1.2. The main resultWhen H forgets d, there is always an equilibrium in which H chooses B and Schooses Zwith probability one. To verify this, we have to describe Hs beliefs when,in period 5, he recalls that he has chosen C(which occurs with probability zero onthe equilibrium path). We posit that H thinks S has chosen Z in this case. Hs

choice of B is then clearly optimal regardless of d.This is not, however, the only equilibrium. In stating our main result, we will

specialise to the case of linear utility, where v(y) y. After the theorem, we explainhow the result is modified in the presence of concave utility.

v(a), 1 v(0), 0

v(a+b), 0 v(b), 1

C

B

M Z

Hs

choice

Ss choice

Fig. 2. Newcombs Paradox




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Theorem 1: Letd (a b)/b, and suppose thatd lies on the interior of the supportof F. There exists an equilibrium in which

(i) ford

!d, H chooses B and S correctly predicts this (choosing Z), and

(ii) ford < d, H chooses C and S correctly predicts this (choosing M).

Proof. First imagine thatd! d. Hs prescribed choice is B, so Ss prescribed choiceof Z is optimal. Given Ss prescribed choices, is B optimal for H?

If H chooses B, his payoff is

Ey j B dEy j B; d:Let us start with E(y j B, d). Knowing d > d, H can infer from Ss equilibriumstrategy that S has chosen Z, so E(y

jB, d)

b. Now consider E(y

jB). Recalling

that he has chosen B, Hcan infer from his own equilibrium strategy thatd > d, inwhich case he concludes that S has chosen Z, so E(y j B) b. Thus,

Ey j B dEy j B; d 1 db:If H chooses C, his payoff is

Ey j C dEy j C; d:Let us start with E(y j C, d). Knowing d ! d, H can infer from Ss equilibriumstrategy that S has chosen Z, so E(y

jC, d)

0. Now consider E(y

jC). Recalling

that he has chosen C, H will infer from his own equilibrium strategy (incorrectly)thatd < d, in which case he concludes thatShas chosen M, so E(y j C) a. Thus,

Ey j C dEy j C; d a:Comparing these two payoffs, we see that it is indeed optimal for Hto choose B

provided that

1 db ! aor

d ! ab

1 a bb

d

But this is the case we are examining.Now imagine thatd < d. Hs prescribed choice is C, so Ss prescribed choice of

M is optimal. Given Ss prescribed choices, is C optimal for H?If H chooses C, his payoff is

Ey j C dEy j C; d:

Let us start with E(y j C, d). Knowing d < d, H can infer from Ss equilibriumstrategy that S has chosen M, so E(y j C, d) a. Now consider E(y j C). Recallingthat he has chosen C, Hwill infer from his own equilibrium strategy thatd < d, in

which case he concludes that S has chosen M, so E(y j C) a. Thus,




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Ey j C dEy j C; d 1 da:If H chooses B, his payoff is

Ey j B dEy j B; d:

Let us start with E(y j B, d). Knowing d < d, H can infer from Ss equilibriumstrategy that S has chosen M, s o E(y j B, d) a b. Now consider E(y j B).Recalling that he has chosen B, H will infer from his own equilibrium strategy(incorrectly) that d > d, in which case he concludes that S has chosen Z, soE(y j B) b. Thus,

Ey j B dEy j B; d b da b:Comparing these two payoffs, we see that it is indeed optimal for Hto choose C

provided that

1 da! b da bor

d a bb

d:

But this holds (with strict inequality) in the case we are examining.

The equilibrium mentioned in the theorem has the desired features. Thehuman subject makes both choices the closed box and both boxes with positiveprobability. The superior being always predicts this choice correctly, putting the

million dollars in the closed box when the human selects it alone, and leaving theclosed box empty when the human picks both boxes.

What is the intuition for this result? In equilibrium, the parties create correla-tion between their choices by conditioning on something commonly observed.The thing that is commonly observed is subsequently forgotten byH. Choosing theclosed box only does not change whats in the closed box. But, because of theequilibrium correlation, it does cause the subject to subsequently infer that S haspredicted closed box only, and consequently that the money is in the box. Cre-ating this inference is valuable because it improves the subjects anticipatory

feelings, and is more valuable when d is smaller.This analysis illustrates the general principle that, with imperfect memory andanticipatory emotions, people can rationally choose apparently dominated strat-egies. Is it also a good resolution of Newcombs Paradox? Perhaps. Think of d as acharacteristic about which the individual is only dimly (or possibly even subcon-sciously) aware to begin with. After making his choice, his reasons are at leastpartially obscure to himself. However, by hypothesis, he thinks the superior beingunderstands these reasons perfectly. It is therefore reasonable for him to concludethat the choice he made is correlated with the beings decision.

3.1.3. The weight attached to anticipatory emotionsDoes this theory require H to put an implausibly large amount of weight onanticipatory emotions relative to the actual outcome, to rationalise choosing only




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the closed box? Some simple calculations shed light on this issue. For a 1,000,000 and b 1,000, we have d 999. Thus, even an H who places, say, 998times as much weight on actual outcomes as on anticipatory emotions chooses onlythe closed box.

Notice that d rises with a and falls with b. This makes intuitive sense. Thesubject will be less willing to risk losing a (by choosing both boxes) when a islarger, and will be less willing to pass on b (by choosing only the open box) whenb is larger.

How does this result extend to cases with a strictly concave utility function, v?Precisely the same reasoning identifies two thresholds, d1 and d2, defined as fol-lows:

d1 va vbvb v0

and

d2 va vbva b va :

With v strictly concave, v(b) v(0) > v(a b) v(a), so d1 < d2. We constructequilibria as follows. For d < d1, we have H choose C and Schoose M. For d > d2,

we have H choose B and S choose Z. For any d 2 [d1, d2], we can either have Hchoose Cand Schoose M, or have Hchoose Band Schoose Z. This indeterminacygives rise to a class of equilibria. Resolving choice for all d

2[d1, d2] in favour of

C results in C being chosen for a wider range of d than for v linear. Likewise,resolving choice for all d in this range of indeterminacy in favour of B results inCbeing chosen for a smaller range ofd than for v linear. To illustrate, image thatvy ffiffiyp, a 1,000,000, and b 1,000. Then d1 30.6 and d2 1937. Resol-

ving the indeterminacy in favour of C wherever possible, H does not choose bothboxes unless he places nearly two thousand times as much weight on the actualoutcome as on the anticipation.

This last conclusion becomes even more striking as we increase the curvature.For example, when v(y) y2, d2 5.01 108. So there is an equilibrium in

which putting 500 million times as much weight on the actual outcome as on theanticipatory emotion is still consistent with choosing the closed box only.

3.2. The Prisoners Dilemma

In the previous subsection, we justified the selection of an apparently dominatedstrategy by a rational agent (with imperfect memory and anticipatory emotions)in a game where players could condition choices on a randomly drawn andcommonly observed feature of the game. It is also possible to do somethingsimilar by allowing players to condition choices on a randomly drawn andcommonly observed but otherwise irrelevant signal. We illustrate this possibilityin the context of the one-shot Prisoners Dilemma. As we show, under relatively

weak conditions it is possible to obtain cooperation with probability arbitrarilyclose to unity.




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3.2.1. The gameConsider a game played by two agents, A and B. Choices and events unfold asfollows.

1. Nature randomly selects a signal x, distributed uniformly over the interval

[0, 1]. (We choose the uniform distribution here for notational simplicity.Any continuous distribution will clearly suffice since we can transform thevariable to make its distribution uniform. Specifically, if F is the CDF ofx,then the realised value of F(x) has a uniform distribution.) The value ofthis parameter is observed both by A and by B.

2. Aand Bsimultaneously choose one of two actions: C(for cooperate) and N(for not cooperate).

3. A and B both forget the value ofx.4. Aand Bwait to learn the outcome, and form expectations of what it will be.

5. Payoffs are realised.

Payoffs are determined according to the matrix illustrated in Figure 3. In eachcell of this matrix, the first entry is As payoff and the second is Bs. We will refer toplayer is decision as xi, and his payoff as ui. We impose two restrictions on theparameters:

a, a b, c

c, b d, d

C

N

C N

As

choice

Bs choice

Fig. 3. The Prisoners Dilemma




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Assumption P1: c> a> d> b

Assumption P2: d b> c a

Assumption P1 is what makes this game a Prisoners Dilemma. It implies thatNisa dominant strategy for each player but both would do better if both played C.

Assumption P2 says that the gain from playing N rather than C is greater whenones opponent plays N rather than C; in combination with Assumption P1, itimplies c b< 2a, which means that mutual cooperation produces the greatestaggregate payoff. As we will see, this assumption identifies circumstances in whichcooperation is achievable.

When making a decision in stage 2, a player thinks about how he will feel both atstage 4 and stage 5. In stage 2, his expected stage 5 utility is E(ui

jxi, x). In stage 4,

his expected stage 5 utility is E(ui j xi), and this determines his stage 4 utility, whichhe anticipates at stage 2. He maximises a weighted average of these two terms:E(ui j xi) dE(ui j xi, x).

Suppose for the moment that A and B have perfect memory. In that case, theyalways choose xi to maximise E(ui j xi, x) (since this represents expected utility, asof stage 2, both for stage 4 and stage 5 utility). The standard dominance argumentapplies. The only subgame perfect equilibrium involves A and B choosing N. Nocooperation is observed.

Now we look for the equilibria of the game with imperfect memory. Again we

use the Piccione-Rubinstein multiself approach discussed in Section 1 and studysequential equilibria.

3.2.2. The main resultWhen Aand B forgetx, there is always an equilibrium in which they both chooseN. To verify this, we have to describe a players beliefs when, in period 4, he recallsthat he has chosen C(which occurs with probability zero on the equilibrium path).

We posit that i thinks j has chosen N in this case. Choosing N is then clearlyoptimal regardless ofx.

This is not, however, the only equilibrium. As the following result shows, pro-vided d is not too large, there exists an equilibrium for which cooperation occurswith probability arbitrarily close to unity.

Theorem 2: Suppose thatd < (a d)/(c a). Then, for all e > 0, there exists anequilibrium for which the players cooperate (play C) with probability greater than 1 e.

The proof of Theorem 2 appears in the Appendix available on the Journals

website http://www.res.org.uk. For d 2 a dd b; a dc a (a non-empty interval byAssumption P2), there is a simpler proof, which we offer here in the text to helpbuild intuition.

Consider strategies of the following form: for some b 2 (0, 1),




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If x b; play CIf x > b; play N:

Is it an equilibrium for both players to use this strategy? We verify that the

prescribed choices are optimal for all values ofx.First suppose first thatx b is observed. Ifichooses Cas prescribed, his payoff

is

Eui j C dEui j C;x:Let us start with E(ui j C, x). Knowing x b, i can infer from js equilibriumstrategy that j has chosen C, so

Eui j C;x a:

Now consider E(ui j C). Recalling that he has chosen C, i can infer from his ownequilibrium strategy thatx b, in which case he concludes thatjhas chosen C, so

Eui j C a:Thus,

Eui j C dEui j C;x 1 da: 3If i instead chooses N, his payoff is

E

ui

jN

dE

ui

jN;x

:

Let us start with E(ui j N, x). Knowing x b, i can infer from js equilibriumstrategy that j has chosen C, so

Eui j N;x c:Now consider E(ui j N). In stage 4, i will recall that he has chosen N. Given hisequilibrium strategy, he will conclude (mistakenly) thatx > b; from js equilibri-ums strategy, he then infers that j has chosen N, so

Eui j N d:Thus,

Eui j N dEui j N;x d dc: 4Combining (3) and (4), we have

Eui j C dEui j C;x Eui j N dEui j N;x 1 da d dc 5 a d dc a ! 0;

where the last inequality follows from the assumption that d (a d)/(c a).Thus, upon observing x b, it is in is interest to play C, as prescribed.Now suppose first thatx > b is observed. Ifichooses Nas prescribed, his payoff is

Eui j N dEui j N;x




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Let us start with E(ui j N, x). Knowing x > b, i can infer from js equilibriumstrategy that j has chosen N, so

Eui j N;x d:

Now consider E(ui j N). In stage 4, i will recall that he has chosen N. Given hisequilibrium strategy, he will conclude thatx > b; from js equilibrium strategy, hethen infers that j has chosen N, so

Eui j N d:Thus,

Eui j N dEui j N;x 1 dd: 6If i instead chooses C, his payoff is

Eui j C dEui j C;x:Let us start with E(ui j C, x). Knowing x > b, i can infer from js equilibriumstrategy that j has chosen N, so

Eui j C;x b:Now consider E(ui j C). Recalling that he has chosen C, i will infer (mistakenly)from his own equilibrium strategy thatx b, in which case he concludes that jhas chosen C, so

E

ui j

C

a:

Thus,

Eui j C dEui j C;x a db: 7Combining (7) and (6), we have

Eui j N dEui j N;x Eui j C dEui j C;x 1 dd a db dd b a d ! 0;

where the last inequality follows from the assumption that d ! (a d)/(d b).Thus, upon observing x > b, it is in is interest to play N, as prescribed.

Notice that this equilibrium arises for all values of b. In particular, by taking bcloser and closer to unity, we can construct an equilibrium where cooperationemerges with probability arbitrarily close to unity.

There is also a Perfect Bayesian equilibrium where both players select C withprobability one, but it is somewhat problematic. For this equilibrium, there is zeroprobability that player i learns in stage 4 that he has previously chosen N. Byassigning to this out-of-equilibrium event the belief that j has chosen N withprobability one, we ensure that C is is best choice for all x. However, if we selectany sequence of completely mixed strategies converging uniformly to the equi-librium strategies, the implied posteriors when i chooses N will, as we pass to thelimit, place nearly unitary probability on the event that j has chosen C. Thus, the




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equilibrium strategies and beliefs are not consistent in the sense of Kreps andWilson (1982), at least when the strategy space is endowed with the topology ofuniform convergence.2

Theorem 2 also encompasses cases where d < (a d)/(d b). For the equi-librium described above, we impose the condition d ! (a d)/(d b) to makesure that a player does not place so much weight on anticipatory emotions that heis tempted to select C (thereby producing the subsequent inference that hisopponent has also chosen C) even when he is supposed to play N. Without thiscondition, a more subtle argument is required; the proof involves mixed strategiesinstead of pure strategies, and uses a more complex limiting argument to establishthat one can find an equilibrium with cooperation probabilities arbitrarily close tounity (see the Appendix for details).

Theorem 2 may seem counterintuitive. After all, since N is a dominant strategy,and since the signal is pure noise, one would not ordinarily expect the players tochoose C. The intuition for the theorem has to do with the failure of the law ofiterated expectations. In equilibrium, the parties create correlation between theirchoices by conditioning on the commonly observed signal that both subsequentlyforget. While playing cooperatively does not cause ones opponent to playcooperatively, it does cause the player to subsequently infer that his opponent hasplayed cooperatively. Creating this inference is more valuable when the playersattach more importance to anticipatory emotions (that is, d must be sufficientlysmall).

3.2.3. Relation to observed behaviourLaboratory experiments consistently show that subjects cooperate in the one-shotPrisoners Dilemma game with non-trivial frequency. Formal game theory lacks anexplanation of cooperation in this setting, though various explanations have beenoffered involving other-regarding preferences.

On occasion, one hears informal explanations for cooperation in the one-shotPrisoners Dilemma such as the following. Each player believes the other player is

like

himself. He expects the other player to go through the same thoughtprocess when choosing a strategy. Thus, if a player concludes that he ought toplay C, then he thinks it is likely that the opponent also reaches the same con-clusion. Similarly, if a player concludes that he ought to play N, he thinks it islikely the opponent also settles on N. Under these conditions, it is argued, C isthe better choice.

We have always regarded the logic of this argument as highly suspect. After all,the opponent will choose what the opponent will choose; by changing his choice

2 This conclusion is sensitive to the choice of topology. To illustrate, choose some sequence of scalars

bj 2 (0, 1) converging to unity. We construct a sequence of strictly mixed strategy profiles as follows: foreach j, each player selects N with probability (1 bj)2 (and C otherwise) ifx j2 (bj, 1), and selects Nwith probabilitybj (and C otherwise) ifx 2 (bj, 1). Upon observing that one player has selected N, theconditional probability that the other has played N converges to unity as j 1, which corresponds tothe beliefs used to construct the Perfect Bayesian Equilibrium. While this strategy profile does notconverge uniformly to the equilibrium profile, it does converge pointwise.




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from N to C, or from C to N, player i cannot affect js choice. To assume he canascribes causality in a setting where no causal link can possibly exist (as in New-combs paradox). And yet, Theorem 2 captures some of this intuition. The signalinduces correlation between the choices of the players. Ex ante, iunderstands that,

given any signal, js choice will not change just because i

s choice changes. How-ever, ialso understands that, at a later date prior to observing the outcome, he will

have forgotten his signal, and that, from this intermediate perspective, it willappear that js choice is correlated with his. In particular, if i has chosen C, he

will ascribe greater likelihood to the possibility that his opponent has chosen C; if ihas chosen N, he will ascribe greater likelihood to the possibility that his opponenthas chosen N.

We remain agnostic indeed, we are at least somewhat sceptical about theextent to which our theory accounts for experimental results. On the favourableside, work by Shafir and Tversky (1992) suggests that, as predicted by our model,uncertainty about an opponents choice plays a positive and significant role inproducing unselfish choices.3 We acknowledge, however, that this phenomenonmay have other causes. Irrespective of its applicability to specific laboratoryexperiments, our analysis provides a potentially reasonable explanation for theselection of apparently dominated cooperative strategies in situations wherethe elapsed time between a decision and an outcome is substantial, and where theoutcome is sufficiently significant to generate anticipation. It also raises the pos-sibility that, in behaving unselfishly, experimental subjects may follow rules ofthumb that have rational origins.

4. Reminders

When a decision maker suffers from imperfect recall, it is natural to think that heor she may attempt to improve or supplement memory. Some strategies forimproving memory are internal (e.g. rehearsal) while others are external(reminders). Here we focus on external reminders, though one could modelinternal mechanisms similarly.

With standard preferences, the analysis of reminders is relatively straight-

forward. The decision maker wishes to be as well-informed in the future aspossible, and trades off the gains from better information against the cost

3 In standard treatments, these authors found that subjects cooperate in the one-shot prisoners dilemma game in roughly 37% of trials. In other treatments, subjects were informed of their opponents choices before they made their own decisions. When informed that the opponent had played selfishly,only 3% played unselfishly. When informed that the opponent had played unselfishly, only 16% playedunselfishly. Note that both of these figures are significantly lower than the 37% figure obtained whenthe opponents choice was not revealed. Thus, uncertainty about the opponents choice plays a positiveand important role in producing unselfish play. Our analysis roughly fits this pattern; it predictscooperation only if the opponents choice is not known when a player makes his or her choice. It doesnot, however, explain the fact that 3% and 16% (respectively), rather than 0%, play unselfishly when the

opponents choice is known. In contrast, theories of reciprocity would predict counterfactually that thefrequency of unselfish play should be highest when the subject knows the opponent has playedunselfishly. Reciprocity may nevertheless help to explain the experimental results as a contributoryfactor, in as much as the frequency of unselfish play is strictly positive when the DM knows the opponenthas played unselfishly, and significantly higher than when the DM knows the opponent has playedselfishly.




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of providing it. When we add anticipatory emotions to the mix, thingsbecome considerably more interesting. The decision maker must also consi-der the effects of reminders on future emotional states. As we will see,this leads to a variety of striking and, in some cases surprising, behavioural

implications.

4.1. Types of Reminders

How do reminders work? We consider two different reminder technologies: hardreminders and soft reminders. A hard reminder consists of hard, unequivocalinformation. A soft reminder consists of a pure message.

To make this distinction more concrete, Let us consider a simple example.4

Once a year, you set aside time to take stock of your personal finances. You keep

pertinent information in a file, and you always start by reviewing the files contents.In making your plans for each year, you want take into account the performance ofyour portfolio over the past year. This information appears on monthly statementsbut you tend to glance at these quickly and forget most of the details. Conse-quently, you make a habit of including this information in your personal financefile. One possibility is to place asset statements in the file. These are hardreminders. Another possibility is to write yourself notes and leave them in the file.These are soft reminders.

We model hard reminders as hard information, much as in the literature ondisclosure; see e.g. Grossman and Hart (1980), Grossman (1981), Milgrom(1981) and Dye (1985). We model soft reminders as cheap talk, as in the lit-erature on pure communication (Crawford and Sobel, 1982). We acknowledge,however, that our treatment of soft reminders is potentially controversial, andmay be inappropriate in some circumstances. Our analysis implicitly assumes thatthe individual can attempt to lie to himself through soft reminders, and that heinterprets all soft reminders in light of his incentives to do so. Whether this isplausible depends on the technology of memory. If a soft reminder triggers aspecific memory of hard evidence, then there is no difference between a softreminder and a hard reminder. Alternatively, if an individual is not in the habit

of lying to himself, then deceptive soft reminders may be self-defeating. Whenleaving an inaccurate soft reminder is a significant departure from the individ-uals normal practice, he may, upon receiving the reminder, jog a specificmemory of his intent to deceive himself. In that case, soft reminders would againfunction much like hard information. Our treatment of soft reminders isappropriate in situations where the individual can conjure up no specificmemory either of the original information, or of his thought process in leavingthe reminder. We believe there are situations that fit this description, as well assituations that do not.

4 Another concrete and highly vivid illustration appears in the motion picture Memento(NewmarketFilms, 2001). After incurring an injury that destroys his ability to form new memories, the main char-acter leaves himself hard reminders by taking photographs, and soft reminders by annotating thephotographs and tatooing messages on his body. As in our analysis, he sometimes seeks to manipulatehis subsequent beliefs and behaviour by crafting potentially misleading reminders.




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4.2. Framework

Events and choices unfold as follows.

1. A payoff-relevant state of nature, x 2 [0, 1], is realised.2. The DM observes information s. Unless specified otherwise, s x (theDM observes the state of nature). He then makes two decisions. First, he

chooses reminders. For the hard reminder technology, he selectsh 2 f0, 1g, where h 1 causes him to recall s at a later date, whereash 0 does not. For the soft reminder technology, he selects a message m.

We could in principle allow the message space to be arbitrarily complex,but in this setting all that will matter is that it has at least two elements, so

we suppose that m 2 f0, 1g. Second, he chooses an initial actionx1 2 f0, 1g.

3. If the DM has not left a hard reminder, he forgets s. He recalls x1, m, andalso s if he has left a hard reminder.

4. The DM selects a second action, x2 2 f0, 1g. Time passes.5. The outcome is realised.

The setting described above is extremely simple. This is intentional, since ourobject is to illustrate basic ideas as transparently as possible. The analysis extendsdirectly to cases in which recall is probabilistic, where information is observedprobabilistically rather than with certainty, where reminders are probabilisticallymisplaced (and therefore have no effect), and where the sets of potential actionsare continuous rather than dichotomous.

The DMs state of emotional well-being in Stage 5, U, is given by

U u1x1;x u2x2;x ch;where c represents the cost of leaving a hard reminder. We assume that softreminders are costless.

Assumption R: ui is differentiable and strictly increasing in x withui(0, 0) > ui(1, 0) and ui(1, 1) > ui(0, 1) and oui(0, x)/ox < oui(1, x)/ox.

Assumption Rtells that there is some cutoff state, x1, such that the first-best rule(from the perspective of Stage 5 well-being) involves choosing xi 0 whenx < x1, and xi 1 when x > x1. At this cutoff, ui0; x1 ui1; x1.

The DMs state of emotional well-being in Stage 4, V, is given by his expectedwell-being in stage 5, conditional on his information:

V EU j m; x1 without a hard reminder

V EU j m; x1; s with a hard reminder

In Stage 2, his utility is given byW(U, V). We will focus on cases where W(U, V) dU w(V). Unless specified otherwise, we assume that w is the identity function,so W(U, V) dU V.




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In some cases, we will also add a decision in Stage 1 (whether to acquireinformation). In these cases, we assume the DM acts to maximise the expected

value of W(U, V).This model has a number of components: an initial (Stage 2) action, a delayed

(Stage 4) action, hard reminders and soft reminders. We will focus on one or twocomponents at a time, shutting the others off in each case. In one case we will alsovary the informativeness of the signal, s. To provide baseline results, we start byanalysing the role of reminders without actions. Next we will focus on the initialaction, first examining choice without reminders, and then indicating how thischoice changes in the presence of hard and soft reminders. Finally, we will focuson the delayed action, once again examining choice without reminders, and thenindicating how reminders affect this decision.

For concreteness, we suggest the following life-cycle planning problem as anapplication. Suppose the DM switches jobs and must relocate to a new city. The stateof nature determines the generosity of the new employers defined benefit pensionplan and/or post-retirement medical benefits. The DM receives information aboutthese resources upon accepting the position. Though he processes this information,it is relatively complicated and the details are easily forgotten. To jog his memory, hecan leave himself reminders by creating an easily accessible file containing plandescriptions, notes, correspondence with his employers human resources depart-ment, and other materials, or he can throw these materials out. Upon relocating, hebuys a new house, and may also invest in other durable goods such as automobiles(the initial action, x1). He makes other consumption decisions (the second action,

x2) after his detailed recollection of the retirement plan fades but before reachingretirement. Though his memory is imperfect, he continues to observe x1 (he lives inthe house and drives the cars) and he occasionally consults any materials he mayhave filed. During the pre-retirement period, his well-being depends on con-sumption (both x1 and x2), and on his anticipated happiness in retirement, which inturn reflects his unspent income and the generosity of the retirement plan.

4.3. Reminders Without Actions

Suppose there is no initial or delayed action (equivalently, x1 and x2 are bothdegenerate). The payoff in Stage 6 is simply given by u(x) ch. Without antici-patory emotions, reminders serve no purpose. With anticipatory emotions,reminders may be useful because they can influence the DMs anticipated emo-tional state in Stage 4.

In this setting, however, soft reminders are useless. Irrespective of which x isrealised, the DM would like to convince his Stage 4 incarnation thatx is as high aspossible. If two distinct messages lead to different inferences about x, he willalways choose the message leading to a higher inferred value. Thus, no degree ofseparation is sustainable. The only equilibrium outcome involves babbling the

message is uninformative.The case of hard reminders is more interesting. An equilibrium consists of a

mapping from states of the world to a binary choice set, rh : [0, 1] f0, 1g,where rh 1 (rh 0) indicates that he leaves (does not leave) a hard reminder,




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along with a mapping from choices to beliefs about types, where the second isderived from the first where possible, and where the first is optimal given thesecond for all states of nature.

The following result tells us several things. The DM leaves hard reminders in

good states and does not leave hard reminders in bad states. The dividing linebetween good and bad states depends on the cost of leaving a hard reminder.When the cost is zero, the DM leaves hard reminders in all states. When the cost issufficiently high, he leaves no reminders. The theorem also provides an expressionfor equilibrium payoffs.

Theorem 3: Suppose hard reminders are available, and that there are no actions(initial or delayed). An equilibrium exists. Moreover, in any equilibrium, there existsbxhc 2 0; 1 with rh(x) 0 for x < bxhc, and rh(x) 1 for x > bxhc. Fur-thermore,

bxh0 0; limc#0

bxhc 0, and

bxhc 1 for c sufficiently large. The DMs

expected equilibrium payoff, from the perspective of Stage 1, isEux Prw> bxhccf g1 d:

Proof. First we show that, in any equilibrium, rh is weakly increasing in the stateof nature. Suppose the DM has observed x. Let V0 denote the Stage 4 emotionalstate experienced in equilibrium when he does not leave a reminder (clearly, thiscannot depend on x). From the perspective of Stage 2, the net gain from leaving areminder is

u

x

c

1

d

V0

du

x

u

x

V0

1

d

c:

8

Since this expression is strictly increasing in x, the desired conclusion followsimmediately. Thus, we look for a cutoff value bxhc such that rh(x) 0 forx < bxhc, and rh(x) 1 for x > bxhc.

Suppose c 0. We claim that there exists an equilibrium in which the DM alwaysleaves a hard reminder, and where he believes that, in the absence of a reminder,the state is x 0 (which implies V0 u(0)). Using (8), we see that the gain fromleaving a reminder in any state x0 is u(x0) u(0), which is strictly positive for allx > 0 and zero for x 0, as required. To see that there is no equilibrium with acutoff value bx > 0 (with the DM leaving a reminder for x0 > bx and no reminderfor x0 < bx), note that this would imply V0 Eux j x bx. Using (8), we seethat the gain from leaving a reminder in any state x0 would then beux0 Eux j x bx. Notice that this expression is strictly positive for x0slightly below bx, which implies that the DM would leave a reminder upon obser-

ving x0, a contradiction.Now suppose

c ! u1 u01 d c

: 9

We claim that there exists an equilibrium in which the DM never leaves a hardreminder (which implies V0 E[u(x)]). Using (8), we see that the gain fromleaving a reminder in any state x0 is




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ux0 Eux 1 dc ux0 u1 u0 Euxf g;which is strictly negative as required. To see that there is no equilibrium with acutoff value bx < 1 (with the DM leaving a reminder for x0 > bx and no reminderfor x0 < bx), note that this would imply V0 Eux j x bx. Using (8), we seethat the gain from leaving a reminder in any state x0 would then be

ux0 Eux j x bx 1 dc ux0 u1 u0 Eux j x bxf g;which is strictly negative for all x0 2 bx; 1, a contradiction.

Now suppose that c 2 (0, c). We claim that an equilibrium exists, and that thecutoff, bxhc, lies in (0, 1]. Using (8), we see that, with a cutoff bx (which impliesV0 Eux j x

bx), the gain from leaving a reminder in state x0 is

ux0; bx; c ux0 Eux j x bx 1 dc:If u(1, 1, c) 0, then there is plainly an equilibrium for which the DM neverleaves a reminder. Suppose instead that u(1, 1, c) > 0. Trivially, u(0, 0, c) < 0.Since u is continuous, there exists bxh 2 0; 1 for which ubxh; bxh; c 0. Sinceux0; bxh; c > 0 for x0 > bxh and ux0; bxh; c < 0 for x0 < bxh, there is clearly anequilibrium for which the DM leaves a reminder in states x0 > bxh and does notleave a reminder in states x0 < bxh.

Next we show that limc#0

bxhc 0. We know thatu

bxhc;

bxhc; c 0 (with

equality when

bxhc < 1). Thus, as c # 0, u

bxhc Eux j x

bxhc 0.

But this can occur only if bxhc 0.In any of the equilibria described above, the DMs expected payoff from the

perspective of Stage 2 upon observing state x0 is [u(x0) c](1 d) ifx0 > bxhc,and Eux j x bxhc dux0 for x0 < bxhc. Thus, his expected payoff priorto observing the state is

Prx0 < bxhcE Eux j x bxhc dux0 j x0 < bxhcf g Prx0 > bxhcE ux0 c1 d j x0 > bxhcf g

Prx0 bxhc cf g1 d 1 d Eux0 Prx0 > bxhccf g;

as claimed.

Given the close parallel to standard disclosure problems (Grossman and Hart,1980; Grossman, 1981; Milgrom, 1981; Dye, 1985), nothing about this theorem isparticularly surprising. However, in this context, it has three important implications.

First, when reminders are costless, the DM ends up with full information inStage 4. This observation will be relevant when we discuss the effects of reminders

on actions.Second, when reminders are costly, the DM reminds himself of favourable

states of nature, and does not remind himself of unfavourable states. Thus, ourmodel endogenously produces a systematic memory bias: people tend to recall




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favourable information (e.g., in our proposed application, about the generosityof an employers pension plan), and forget unfavourable information.5 With thisobservation in mind, a small extension of our model endogenously produces aphenomenon associated with cognitive dissonance: the tendency to pay attention to

information that confirms beliefs supporting prior decisions, and to ignorecontradictory information (Akerlof and Dickens, 1982). In particular, imaginethat in some initial stage, the DM must choose either left, in which case hispayoffs are given by u(x) ch, or right, in which case his payoffs are given byu(1 x) ch. In other words, the choice of right, rather than left, simplyreverses which states are favourable and which are unfavourable. Given eachchoice, the solutions to the continuation problems are symmetric and describedby Theorem 3. Thus, when the DM has chosen left, he endogenously forgetsinformation when it tells him that x is low and that right would have been abetter choice. Similarly, when he has chosen right, he endogenously forgetsinformation when it tells him that x is high and that left would have been abetter choice.

Third, equilibrium payoffs are non-monotonic in the cost of reminders. Whenreminders are free, the ex ante expected equilibrium payoff is E[u(x)]. Likewise,

when reminders are sufficiently costly, the DM does not use them in any state ofnature and again the ex ante expected equilibrium payoff is E[u(x)]. For inter-mediate values of c, the DM uses reminders in some states of the world and hisexpected equilibrium payoff is less than E[u(x)]. To understand why, note thatleaving a free collection of state-specific reminders is a wash from an ex ante per-

spective losses in some states exactly offset gains in others. Accordingly, ex ante,the DMs expected payoff falls by the expected cost of the reminders he leaves afterlearning x. In this setting, he is better off not receiving the information to begin

with; he would certainly not invest to acquire it and would even pay to avoid it. Ifhe nevertheless receives information, he is better off without a reminder tech-nology.

This third implication is reminiscent of an existing result due to Caplin andLeahy (2001), who point out that a decision maker may prefer ignorance when heis averse to variation in future emotional states. To illustrate the implications of

this point in our setting, imagine that w is strictly concave and that c 0(reminders are costless). Then once again the DM will leave reminders in all states(this is just the standard disclosure result). His expected equilibrium payoff (fromthe perspective of Stage 2) isZ1

0wuxfxdx d

Z10

uxfxdx

< w

Z10

uxfxdx

dZ1

0

uxfxdx:

5 In a model where an excessively positive view of ones ability improves delayed choices from anex ante perspective (by offsetting a distortion arising from present-biased preferences), Benabou andTirole (2002) show that a decision maker will tend to repress unfavourable information. The mech-anism is related to ours in that the decision maker attempts to forget less favourable information so hissubsequent beliefs will be more positive.




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The right hand side of this expression would be his expected equilibrium payoff(from the perspective of Stage 2) if he either received no information or wasable to refrain from leaving reminders. Once again, leaving reminders makeshim worse off, but he does it anyway. This extends Caplin and Leahys result by

showing that ignorance is preferable with w strictly concave, even though thedecision maker could in principle disregard the information by leaving noreminders. Theorem 3 goes beyond this result and identifies conditions under

which an agent may prefer ignorance even when he is not averse to variation infuture emotional states.

The third implication sounds like an example of dynamically inconsistentpreferences in Stage 1, the DM would like to avoid leaving reminders in Stage 2,but is unable to follow through once Stage 2 arrives. However, it is a differentphenomenon. Given the equilibrium inferences hell make in Stage 4, the DMsStage 1 self concurs with his Stage 2s decision in each state of nature. He wishes toconstrain his future actions not merely to change the actions themselves (as wouldbe the case with dynamically inconsistent preferences), but also to change infer-ences (which is what makes the change in actions desirable).

Theorem 3 is somewhat related to a result in Caplin and Leahys (2004) analysisof information transmission between doctors and patients, where doctors areempathetic and patients experience anticipatory emotions. In stage 2 of ourmodel, the DM is in the position of informing his later self concerning likelyoutcomes, much as an empathetic doctor would inform a patient concerningdiagnosis, necessary procedures, and prognosis. Caplin and Leahy consider doctor-

patient communication through costless hard information (a verifiable diagnosis).In a model with binary information, they demonstrate that, if anxiety primarilydepends on pessimism rather than on the degree of uncertainty, informationrevelation is complete (see their Proposition 2).6 Specialising to the case wherec 0, Theorem 3 provides the same result in a setting with continuous informa-tion.

4.4. The Initial Action

Now let us suppose there is an initial action (but no delayed action). In formu-lating our model, we assumed that the initial action is recalled even though thesignal is forgotten. Alternatively, one could imagine that the initial action is for-gotten as well. In that case, anticipatory emotions from Stage 4 would not affect thedecision, and the DM would simply make the first-best choice.

The problem becomes more interesting when the initial action is recalled. Inthat case, the action also serves the role of a reminder. An equilibrium consists of amapping from states of the world to choices, r1 : [0, 1] f0, 1g (not to beconfused with rh from the previous Section), along with a mapping from choicesto beliefs about types, where the second is derived from the first where possible,

and where the first is optimal given the second for all states of nature. Since the

6 In their model, patients also provide doctors with information concerning their susceptibilities todifferent sources of anxiety but this aspect of their analysis has no parallel in the current paper becausehere the DM knows his true preferences.




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problem involves signalling, we also rule out some implausible outcomes byimposing the intuitive criterion (Cho and Kreps, 1987).

Theorem 4: Suppose there is an initial action, no delayed action and no reminders. Anequilibrium exists. Moreover, in any equilibrium, there existsbx1 2 0; x

1

withr1(x)

0

forx < bx1, andr1(x) 1 forx > bx1.Proof. It is easy to check that, under Assumption R, the choice must be weakly

increasing in the state of nature. Thus, we look for a cutoff value bx1 such thatr1(x) 0 for x < bx1, and r1(x) 1 for x > bx1. If bx1 2 0; 1, then the DMmust, upon observing bx1, be indifferent between the two choices, so

Eu10;x j x bx1 du10; bx1 Eu11;x j x ! bx1 du11; bx1: 10Since u1(1, x)

u1(0, x) is strictly increasing in x, (10) implies that the DM

strictly prefers x1 0 when x < bx1 and x1 1 when x > bx1, exactly as requiredin an equilibrium. This configuration trivially satisfies the intuitive criterion sinceno action is chosen with zero probability in equilibrium.

Consider the question of existence. Setting bx1 1, it is clear that the right-hand side of (10) exceeds the left (from this observation it is easy to checkthat applying the intuitive criterion rules out the possibility that bx1 1 withr1(1) 0). If

u10; 0 du10; 0 > Eu11;x du11; 0 11

then there is clearly an intermediate solution to (10) on (0,1). If (11) does nothold, then there is an equilibrium where r1(x) 1 for all x (and, upon recallingx1 0, the DM infers thatx 0) that is, bx1 0. In this case, indifference maynot hold for x bx1, so we set r1(0) 1. If, upon observing x1 0 (a zero-probability event), the DM infers that the state is x 0, the intuitive criterion issatisfied (this is easy to check).

Now we show that bx1 < x1. For all x0 ! x1, we know that u1(0, x

0) u1(1, x0), and thereforeEu10;x j x x0 < u10;x0

u11;x0 Eu11;x j x ! x0:

It follows that (10) cannot hold for any such x0. Combining this observation withexistence establishes the claim.

The fact that bx1 < x1 is no great surprise. Self-signalling distorts choices infavour of alternatives that lead to more favourable inferences and therefore morepositive Stage 4 anticipatory emotions. However, the result has two importantimplications.

First, without reminders, the DM acts as if he is excessively optimistic. Observingonly his choices (e.g. concerning housing and durable purchases in our proposedapplication), one could offer a rationalisation based on the assumption that he isoverly optimistic (concerning the generosity of his employers retirement plan in




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the application), attaching too great a likelihood to favourable states of the world(since bx1 < x1). This is significant because the phenomenon of excessive opti-mism is reasonably well-documented, and has recently received a significantamount of attention (Rabin and Schrag, 1999; Koszegi, 2000; Hvide, 2002; Bena-

bou and Tirole, 2002; Postlewaite and Compte, 2003; Van den Steen, 2003). Ourmodel produces excessively optimistic behaviour endogenously. Though the indi-viduals expectations are, on average, correct, he acts as if he is excessively opti-mistic in an attempt to fool himself.

Second, without reminders, equilibrium choices are not first-best. As with thecase of a delayed action, there is a potential role for reminders.

Both of these conclusions extend in a straightforward way to efficient separatingequilibria in environments where the set of actions is continuous. They reflect asimple property of signalling equilibria: the sender distorts choices in the direc-tion of types he is trying to imitate (here, those with more favourable information).

How do reminders affect decisions? The most obvious mechanism which westudy in the next subsection is to equip the decision maker with more infor-mation at the point in time when he makes a choice. However, this is not the onlymechanism. Here, we examine the influence of reminders on the initial action,assuming there is no delayed actions. In our model, initial actions are taken inStage 2 along with decisions to leave reminders. Consequently, reminders do nothelp inform these decisions. Nevertheless, as we show, reminders can influenceconcurrent decisions.

The mechanism studied here is intuitive. Theorem 4 tells us that, when a

reminder technology is not available, initial choices serve dual functions as payoff-relevant actions and reminders. From the perspective of maximising ultimatepayoffs, the choice of an action is distorted by concerns about the effects of choices(as reminders) on anticipatory well-being in Stage 4. When reminders are avail-able, the decision maker no longer needs to use actions to serve two objectives. Inprinciple, he can address concerns about anticipatory well-being in Stage 4through reminders, leaving actions undistorted. Here we ask whether things workout this way in equilibrium.

Soft reminders are unhelpful in this context. Since the DM wishes to induce the

same favourable inferences regardless of the state of nature, onlybabbling

emerges as an equilibrium. Adding soft reminders has no effect on initial actions.Hard reminders have more interesting effects. For simplicity, we will focus on

the case where c 0 (reminders are costless).Theorem 5: With an initial action, no delayed action, and costless hard reminders, the

DM chooses the first-best action in every state (that is, x1 0 whenx < x1, and x1 1whenx > x1) and is perfectly informed about the state in Stage 4.

Proof. First we verify that there is an equilibrium with the properties describedin the theorem. Suppose the DM leaves hard reminders in all states, chooses x1 0 when x < x1, and chooses x1 1 when x > x1. To complete the descriptionof an equilibrium, we need to supplement this description of equilibrium actions

with out-of-equilibrium beliefs. If in Stage 4 the DM does not receive a reminder,




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he infers thatx 0. To see that prescribed choices are optimal given these beliefs,first notice that the DM cannot improve his payoff for anyx by continuing to leavea reminder but choosing a different action. The only remaining question is

whether he can improve his payoff by failing to leave a reminder. In that case, his

payoff (from the perspective of Stage 2) isu1x1; 0 du1x1;x u1x1;x1 d

u1r1x;x1 d(where r1 assigns x1 0 when x < x1, and x1 1 when x > x1) Since the lastexpression is the DMs equilibrium payoff in state x, the deviation is not beneficial.

Next we demonstrate that this is the only possible equilibrium outcome.The first step is to show that the DM is perfectly informed about the state in

Stage 4. Assume not. Then there is a non-empty set of states X in which the DM

does not leave a hard reminder. LetX0 X be the set of states in X such that theDM chooses x1 0 , and letX1 X be the set of states in X such that the DM

chooses x1 1. Since the DM is, by assumption, not perfectly informed in Stage 4,either X1 or X0 must contain at least two states. Assume Xi contains at least twostates. Then there exists x0 2 Xi such that u(i, x0) > E[u(i, x) j x 2 Xi] (thisfollows because Xi contains at least two distinct states and u(i, x) is strictlyincreasing in x). The DM could do better in state x0 by leaving a hard reminderand choosing r1x0, a contradiction.

Now assume there is some state x00 for which r1x00 6 r1x00. The DMsequilibrium payoff in state x00 (from the perspective of Stage 2) is1 du1r1x00; x00 < 1 du1r1x00; x00, which immediately implies thathe would be better off in state x0 0 by choosing r1x00 and leaving a hard reminder.From this contradiction, we infer thatr1x r1x for all x.

Though simple, Theorem 5 has a striking implication: the ability to leave hardreminders completely restores the DMs ability to make first-best decisions (in ourproposed application, concerning housing and other durable consumption). Thisoccurs even though the reminders do not improve the quality of information on

which decisions are based. Instead, the presence of reminders removes thetemptation to influence anticipatory well-being in Stage 4 by distorting the choiceof an action.

4.5. The Delayed Action

Now let us suppose there is a delayed action (but no initial action). Withoutreminders, the DM takes this action without information. Consequently, hechooses x2 to maximise E[u2(x2, x)]. This outcome is clearly not first-best. Inprinciple, reminders can improve decision making by improving the DMs infor-

mation at the point in time when he makes the decision.With costless hard reminders, the DM leaves a reminder in all states (except

possibly the lowest, when he is indifferent), and always selects the first-best actionin Stage 4. This is in the spirit of Theorems 3 and 5, and the formal proof (omitted




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to conserve space) invokes the same full disclosure logic. Thus, with costless hardreminders, the first-best outcome is always achieved, and it makes no difference

whether an action is immediate or delayed.7 Of course, this equivalence dependson the assumption that reminders are perfectly effective.

The case of soft reminders (e.g. filed notes reflecting the DMs own interpret-ation of his retirement plans generosity) is more interesting. In this setting, cheaptalk can successfully convey information, at least for some parameter values. In aninformative equilibrium, the DM partitions the state space into two non-emptysegments, 0; bx2 and bx2; 1. One can place the boundary point, bx2, in eithersegment; here we place it in the higher segment by convention but the choice isimmaterial. When x 2 0; bx2, he chooses a message, m0, that induces him to pickx2 0 in Stage 4. When x 2 bx2; 1, he chooses a message, m1, that induces him topick x2 1 in Stage 4.

There are three requirements for this to be an equilibrium. First, in Stage 4,having received m0, he must prefer to pick x2 0:

Eu20;x j x < bx2 ! Eu21;x j x < bx2: 12Second, in Stage 4, having received m1, he must prefer to pick x2 1:

Eu21;x j x > bx2 ! Eu20;x j x > bx2: 13Third, given his subsequent responses, in Stage 2 he must prefer to send themessage m0 (leading to x2 0) when x

bx2. This requirement is satisfied

provided that he is indifferent between these choices for x bx2:Eu20;x j x bx2 du20; bx2 14

Eu21;x j x ! bx2 du21; bx2:Notice that (14) is identical to (10), except that the subscripts are 2s instead of

1s. Thus, when the same payoff function is used for initial actions and delayedactions (u1 u2), the correspondence between states and choices is the same withan initial action and no reminders, and with a delayed action and soft reminders(

bx1

bx2) provided of course that an informative equilibrium exists in the latter

case (which depends on conditions (12) and (13)). It is worth mentioning that thisequivalence would not hold with continuous choices (with no reminders, there willtypically be an equilibrium with full separation, and this is impossible to achievethrough cheap talk).

Given the preceding observation and Theorem 4, we know that bx2 < x2 (thecutoff for first-best decisions, defined analogously to x1). Consequently, when(14) is satisfied, so is (12). An informative equilibrium therefore exists whenthe value of bx2 defined implicitly in (14) is strictly positive and also satisfies(13).

These informative equilibria are always inefficient. Indeed, as the next resultdemonstrates, the DM always strictly prefers (from the perspective of Stage 1) to

7 With costly hard reminders, the DM behaves as in Theorem 3, perfectly informing delayed choiceswhen the state is sufficiently good and leaving uncertainty when the state is sufficiently bad.




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receive categorical information instead of continuous information. That is, hewould strictly prefer to contract with an information provider in Stage 1 to tell himwhether the state of nature is above or below some threshold (e.g. whether hisretirement plan is either generous or miserly). Our model therefore gives rise

(endogenously) to a taste for categorical information. (For this interpretation ofthe result to hold, we must assume the DM would remember whether he pur-chased categorical or continuous information.)

Theorem 6: Suppose there is a delayed action and no initial action and that only soft

reminders are available. Suppose also that there exists an informative equilibrium. There exists

a dichotomous signal function, s:[0, 1] f0, 1g, such that, given the choice in Stage 1between this dichotomous signal and a fully informative signal, the DM will choose the

dichotomous signal, and thereby achieve a higher level of well-being (evaluated in Stage 1).

Proof. Without worrying about incentive compatibility, let us assume the DMmechanically adheres to the following rule: for some arbitrarily selected x2, sendm0 when x < x2, send m1 when x > x2, choose x2 0 upon receiving m0, andchoose x2 1 upon receiving m1. In that case, his expected payoff (from theperspective of Stage 1) isZx2

0

Eu20;x0 j x0 < x2 du20;xf gfxdx

Z

1

x2

Eu21;x0 j x0 > x2 du21;xf gfxdx

Eu20;x0 j x0 < x2Fx2 dZx2

0

u20;xfxdx

Eu21;x0 j x0 > x21 Fx2 dZ1x2

u21;xfxdx

1 dZx2

0

u20;xfxdx Z1x2

u21;xfxdx:

Taking the derivative of this final expression with respect to x2 gives us

1 du20;x2 u21;x2fx2:Since bx2 < x2, we know that this term is strictly positive evaluated at x2 bx2.Thus, a small increase in the cutoff state from its equilibrium value would improvethe DMs equilibrium payoff. The question is: how do we make this incentivecompatible?

Suppose the signal s takes on only one of two values: s s0 when x < x2, ands s1 when x > x2. Let us attempt to construct an equilibrium with the followingproperties: in Stage 2, the DM sends one message, m0, upon receiving signal s0, anda different message, m1, upon receiving signal s1; in Stage 4, he chooses x2

0

upon receiving m0, and chooses x2 1 upon receiving m1. There are fourrequirements for this to be an equilibrium. First, in Stage 4, having received m0, hemust prefer to pick x2 0:




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Eu20;x j x < x2 ! Eu21;x j x < x2: 15Second, in Stage 4, having received m1, he must prefer to pick x2 1:

Eu21;x j x > x2 ! Eu20;x j x > x2: 16Third, given his subsequent responses, in Stage 2 he must prefer to send themessage m0 (leading to x2 0) when learning thatx < x2:

Eu20;x j x < x2 dEu20;x2 j x < x2 17! Eu21;x j x > x2 dEu21;x2 j x < x2:

Fourth, he must prefer to send the message m1 (leading to x2 1) when learningthatx > x2:

Eu21;x j x > x2 dEu21;x2 j x > x2 18! Eu20;x j x < x2 dEu20;x2 j x > x2:

Let us evaluate each of these constraints at x2 bx2. When (14) holds withequality, both (17) and (18) hold with strict inequality. Therefore, (17) and (18)continue to hold for x2 slightly larger than bx2. So does (15), providedx2 2 bx2; x2. Finally, if (16) holds for x2 bx2, it also holds for slightly larger

values ofx2. To see this, note that

d

dx2

Z1x2

u21;x u20;xfxdx

u20;x2 u21;x2fx2;

which is strictly positive for x2 bx2 < x2. Thus, we have an equilibrium, and theDM is strictly better off.

Intuitively, why does this result hold? When the DM chooses a cheap-talk mes-sage, he is concerned both with the quality of the subsequent decision and withinducing a favourable inference in Stage 4. Creating a favourable inference mayhelp in one particular state, but it cannot change the overall ex ante expectationconcerning the state what he gains in one state, he loses in another. Conse-quently, it cannot increase his expected payoff prior to making his decision. Thus,

the decision is distorted with no offsetting gain from an ex ante perspective.Reversing this distortion therefore improves the ex ante expected payoff.Theorem 6 bears some resemblance to a result by Fischer and Stocken (2001),

who show in a special parametric case of Crawford and Sobels (1982) cheap talkmodel that reducing the quality of information received can increase the amountof information communicated (as measured by the number of distinct inferencesdraw from all messages in equilibrium). Our focus here is not on the amount ofinformation communicated, but on the quality of the decision made, and wedemonstrate that it is possible to improve the quality of the decision withoutincreasing the informativeness of the equilibrium, in the sense of Fisher andStocken. In addition, our result does not appear to require special parametricassumptions, even when a continuum of actions is available.

Some of the analysis in this section is also related to Koszegis (2004) analysis ofinformation transmission between doctors and patients. Like Caplin and Leahy




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(2004) (discussed in Section 5.3), Koszegi assumes that doctors are empathetic andpatients experience anticipatory emotions; in addition, he also assumes that patientstake actions (treatment) after receiving information from doctors. The doctorsdecision to advise the patient is analogo

bernheim thomadsen

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