Confidence-Enhanced Performance

By OLIVIER COMPTE AND ANDREW POSTLEWAITE

There is ample evidence that emotions affect performance Positive emotions canimprove performance while negative ones can diminish it For example the fearsinduced by the possibility of failure or of negative evaluations have physiologicalconsequences (shaking loss of concentration) that may impair performance insports on stage or at school There is also ample evidence that individuals havedistorted recollection of past events and distorted attributions of the causes ofsuccess or failure Recollection of good events or successes is typically easier thanrecollection of bad ones or failures Successes tend to be attributed to intrinsicaptitudes or effort while failures are attributed to bad luck In addition theseattributions are often reversed when judging the performance of others Theobjective of this paper is to incorporate the phenomenon that emotions affectperformance into an otherwise standard decision theoretic model and show that ina world where performance depends on emotions biases in information processingenhance welfare (JEL D8)

A personrsquos psychological state can affect per-formance One common form in which this ismanifested is ldquochokingrdquo which is a physicalresponse to a perceived psychological situationusually fear of not performing well or failingThe fear response physically affects the individ-ual breathing gets short and muscles tightenThis may be the response to fear of performingbadly on tests on the golf course or whenspeaking in public in other words it may occurany time that an individualrsquos performance atsome activity matters The fear of performingbadly is widespread it is said that in rankingtheir fears people put public speaking abovedeath1 The response is ironic in that whenperformance matters the physiological re-sponse may compromise performance Whether

itrsquos a lawyer presenting a case to the SupremeCourt a professional golfer approaching agame-winning putt or an academic giving a jobmarket seminar the probability of success di-minishes if legs are trembling hands are shak-ing or breathing is difficult2

The fear that triggers the physiological re-sponse in a person faced with a particular ac-tivity is often related to experience and is likelyto be especially strong if the person performedless than optimally at those times Knowing thatone has failed frequently at a task may makesucceeding at the task in the future difficultStated succinctly the likelihood of an individ-ualrsquos succeeding at a task may not be indepen-dent of his or her beliefs about the likelihood ofsuccess

Neoclassical decision theory does not accom-modate this possibility individuals choosewhether or not to undertake an activity depend-ing on the expected utility (or some other crite-rion) of doing so where it is assumed that theprobability distribution over the outcomes isexogenously given We depart from neoclassi-cal decision theory by incorporating a ldquoperfor-

Compte CERAS-ENPC 48 Boulevard Jourdan Paris75007 France (e-mail compteenpcfr) Postlewaite De-partment of Economics University of Pennsylvania 3718Locust Walk Philadelphia PA 19104 (e-mail apostleweconupennedu) Much of this work was done while Postle-waite was a visitor at CERAS Their support is gratefullyacknowledged as is the support of the National ScienceFoundation We thank Douglas Bernheim and two anony-mous referees for suggestions that greatly improved thepaper

1 From the Speech Anxiety Website at Rochester Com-munity and Technical College (httpwwwrochedudeptspchcomanxiety_introductionhtm)

2 There may of course be physiological responses thatare beneficial such as a surge of adrenaline that enhancessuccess in some athletic pursuits Our focus in this paper ison those responses that diminish performance

1536

mance technologyrdquo whereby a personrsquos historyof successes and failures at an activity affectsthe probability of success in future attempts Weshow that the presence of this type of interactionmay induce biases in information processingsimilar to biases that have been identified bypsychologists These biases are sometimes seenby economists as anomalies that are difficult toreconcile with rational decision making Ratherthan being a liability in our model these biasesincrease an individualrsquos welfare

In Section I we review related literature de-scribing the effect of psychological state onperformance and systematic biases in peoplersquosdecision making In Section II we present thebasic model including the possibility that anindividualrsquos perceptions can be biased and showin Section III the optimality of biased perceptionsWe conclude with a discussion section

I Related Work in Psychology

We will review briefly some work done bypsychologists that supports the view that anindividualrsquos psychological state affects his orher performance as well as the evidence con-cerning biased perceptions

A Psychological State Affecting Performance

Psychological state can affect performance indifferent ways Claude Steele and Joshua Aron-son (1995) provide evidence that stress canimpair performance In Graduate Record Exam-inations blacks and whites were given a seriesof difficult verbal problems to solve under twoconditions the first was described as diagnosticto be interpreted as evaluating the individualtaking the test while the second was describedas determining how individuals solved prob-lems Steele and Aronson interpret the diagnos-tic condition as putting black subjects at risk offulfilling the racial stereotype about their intel-lectual ability and causing self-doubt and anxi-ety about living up to this stereotype Blacksperformed more poorly when stress was in-duced (the diagnostic condition) than under theneutral condition3

Aronson et al (1999) demonstrate a similareffect in white males When they were told priorto a test that there was a stereotype that Asianstudents outperform Caucasian students in math-ematics these math-proficient white males didmeasurably worse than students to whom thisstatement was not made

Shelley E Taylor and Jonathan D Brown(1988) survey a considerable body of researchthat suggests that overly positive self-evalua-tion exaggerated perceptions of mastery andunrealistic optimism are characteristic of nor-mal individuals and that moreover these illu-sions appear to promote productive and creativework

There are other psychological states that canaffect performance Henry C Ellis et al (1997)show that mood affects subjectsrsquo ability to de-tect contradictory statements in written pas-sages They induced either a neutral or adepressed mood by having participants readaloud a sequence of 25 statements An exampleof a depressed statement was ldquoI feel a littledown todayrdquo and an example of a neutral state-ment was ldquoSante Fe is the capital of New Mex-icordquo4 They found that those individuals inwhom a depressed mood was induced wereconsistently impaired in detecting contradic-tions in prose passages Frank McKenna and CLewis (1994) similarly demonstrated that in-duced mood affects articulation Depressed orelated moods were induced in participants whowere then asked to count aloud to 50 as quicklyas possible Performance was retarded in thedepressed group

Nico Brand et al (1997) demonstrate thatphysical reaction time is affected by inducedmood Subjects were randomly assigned to twogroups with positive mood induced in onegroup and negative mood in the other Positivemood was induced by showing the Dutchsubjects a seven-minute videotape consisting offragments of the 1988 European soccer cham-pionship in which the Dutch team won the titlefollowed by a two-minute clip of the movieldquoBeethovenrdquo including a few scenes of apuppy Negative mood was induced by showingsubjects an eight-minute clip of the film ldquoFaces

3 See also Steele and Aronson (1998) for a closely re-lated experiment

4 See John D Teasdale and M Russell (1983) for a fullerdescription and discussion of this type of mood induction

1537VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

of Deathrdquo consisting of a live recorded execu-tion (by electric chair) of a criminal5 Subjectswith positive induced mood showed faster re-sponse times than did subjects with negativeinduced mood

Simon Baker et al (1997) also induce elatedand depressed moods in subjects6 and show thatinduced mood affects subjectsrsquo performance ona verbal fluency task In addition they measureregional cerebral blood flow using PositronEmission Tomography (PET) They find thatinduced mood is associated with activation ofareas of the brain associated with the experienceof emotion This finding is of particular interestin that it points to demonstrable physiologicaleffects of mood

In our formal model below we assume thatthe psychological state that affects performanceof a particular task is associated with relatedtasks done in the past We should point out thatonly the first half (approximately) of the worksurveyed above can be considered as supportingour finding We include the remaining workbecause it provides evidence of the broaderpoint that we think importantmdashthat the proba-bility that an individual will be successful at atask should not always be taken as being inde-pendent of psychological factors

B Biased Perception

Psychologists have compiled ample evidencethat people have biased perceptions and indi-viduals systematically evaluate themselvesmore highly than others do Chris Guthrie et al(2001) distributed a questionnaire to 168 federalmagistrate judges as part of the Federal JudicialCenterrsquos Workshop for Magistrate Judges II inNew Orleans in November 1999 The respon-dents were assured that they could not be iden-tified from their questionnaires and were toldthat they could indicate if they preferred their

answers not be included in the research projectOnly one of the 168 judges chose not to havethe answers included

To test the relationship of the judgesrsquo esti-mates of their ability relative to that of otherjudges they were asked the following questionldquoUnited States magistrate judges are rarelyoverturned on appeal but it does occur If wewere to rank all of the magistrate judges cur-rently in this room according to the rate atwhich their decisions have been overturned dur-ing their careers [what] would your rate berdquoThe judges were then asked to place themselvesin the quartile corresponding to their respectivereversal rates highest (75 percent) secondhighest (50 percent) third highest (25 per-cent) or lowest

The judgesrsquo answers are very interesting561 percent put themselves into the lowestquartile 316 percent into the second-lowestquartile 77 percent in the second-highest quar-tile and 45 percent in the highest quartile Inother words nearly 90 percent thought them-selves above average

These judges are not alone in overestimatingtheir abilities People routinely overestimatethemselves relative to others in driving (OlaSvenson 1981) the likelihood their marriagewill succeed (Lynn Baker and Robert Emery1993) and the likelihood that they will havehigher salaries and fewer health problems thanothers (Neil D Weinstein 1980) Michael Rossand Fione Sicoly (1979) report that when mar-ried couples are asked to estimate the percent-age of household tasks they perform theirestimates typically add up to far more than 100percent

People overestimate not only their likelihoodof success relative to others but also the likeli-hood of success in situations not involving oth-ers When subjects are asked questions andasked the likelihood that their answers are cor-rect their ldquohit raterdquo is typically 60 percent whenthey state that they are 90 percent certain (seeeg Baruch Fischoff et al 1977 and SarahLichtenstein et al 1982)

Biases are common when in judging the rel-ative likelihood of particular events and manyscholars have attempted to trace the source ofthese biases One often-mentioned source is theoverutilization of simple heuristics such asthe availability heuristic (Amos Tversky and

5 Subjects in this group were told in advance that theywere free to stop the video if the emotional impact becametoo strong Seven of the 20 subjects did so about halfwaythrough the video

6 Depressed neutral and elated states are induced byhaving subjects listen to extracts of Prokofievrsquos ldquoRussiaunder the Mongolian Yokerdquo at half speed in the depressedstate ldquoStressbustersrdquo a recording of popular classics in theneutral condition and Delibesrsquo ldquoCoppeliardquo in the elatedcondition

1538 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Daniel Kahneman 1973) in assessing the like-lihood of particular events people are ofteninfluenced by their past experiences For exam-ple a worker who is often in contact with job-less individuals because he is jobless himselfwould typically overestimate the rate of unem-ployment similarly an employed worker wouldtypically underestimate the rate of unemploy-ment (see Richard E Nisbett and Ross 1980)In the same vein biases in self-evaluations maybe the result of particular experiences beingmore readily available than others if good ex-periences are more easily remembered than badones or if failures are to be disregarded orattributed to atypical circumstances people tendto have overly optimistic self-evaluations (seeMartin Seligman 1990 for more details on theconnection between attributional styles andoptimism)

Economists often see these biases asldquoshortcomingsrdquo of judgement or pathologiesthat can only lower the welfare of an individ-ual As mentioned in the introduction in ourmodel such biases will emerge as welfareenhancing

II Basic Model

We consider an agent who faces a sequenceof decisions on whether to undertake a riskyactivity This activity may be either a success ora failure Consider a lawyer who is faced with anumber of cases that he may accept or declineAccepting a case is a risky prospect as he maywin or lose the case The lawyer will receive apayoff of 1 in case of success and a payoffnormalized to 0 in case of failure

Our primary departure from conventional de-cision theory is that we assume that the lawyerrsquosprobability of winning the case depends on hisconfidence7 if he is unsure about his abilities oranxious he will have less chance of convincingthe jury To capture the idea that confidenceaffects performance in the simplest way weidentify these feelings of anxiety or self assur-ance with the lawyerrsquos perception of success inprevious cases A lawyer who is more confident

about his chances of success will be assumed tohave greater chances of succeeding than a law-yer who is not confident

In what follows we present two main build-ing blocks of our model First we describe therisky activity the true prospects of the agent ifhe undertakes it and the effect of confidenceSecond we describe the decision process theagent follows

A The Risky Activity

We start by modelling confidence and itseffect on performance We make two assump-tions First higher confidence will translate intohigher probability of success Second confi-dence will depend on the agentrsquos perception ofhow successful he has been in the past

Formally we denote the probability of suc-cess and we take the parameter (0 1] as ameasure of the agentrsquos confidence We assumethat

0

where 0 (0 1) The probability 0 dependssolely on the characteristics of the project beingundertaken Absent any psychological consider-ations the agent would succeed with probability0 Lack of confidence reduces performance8

Note that we do not model the process bywhich performance is reduced It is standard toassume that performance should depend onlyon the agentrsquos actions while undertaking theproject Hence one might object that we need toexplain how confidence alters the decisionsmade by the agent while undertaking theproject We do not however model these deci-sions We assume the agent does not have fullconscious control over all the actions that needto be undertaken and that his psychological statehas an effect on these actions that is beyond hisconscious control

We now turn to modelling how confidence(or lack of confidence) arises Various factorsmay contribute to decreasing confidence suchas the remembrance of past failures or the

7 We use the term ldquoconfidencerdquo in a broad sense toinclude feelings of assuredness and lack of anxiety

8 Note that we do not assume that the agent knows (nor or 0) We discuss this issue in Section III B when wedescribe how the agent forms beliefs about his chance ofsuccess

1539VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

agentrsquos perception of how likely he is to suc-ceed In the basic version of our model weassume confidence depends solely on theagentrsquos perception of the empirical frequency ofpast success

Formally we denote by s (respectively f ) thenumber of successes (respectively failures) theagent recalls We explicitly allow this numberto deviate from the actual number9 We willexplain below how these data are generated Wedefine the perceived empirical frequency of suc-cess as

s

s f

and we assume that confidence is a smooth andincreasing function of that is

where 0 (0) 0 and without loss ofgenerality (1) 110

Combining the two assumptions above wemay view the probability of success as a func-tion of the agentrsquos perception and define theperformance function as follows

0

Last we assume that the effect of confidenceon performance is not too strong that is that

1

This assumption is made primarily for technicalconvenience It will be discussed below and inthe Appendix We draw the function in Figure1 The positive slope captures the proposal thathigh (respectively low) perceptions affect per-formance positively (respectively negatively)

Among the possible perceptions that anagent may have about the frequency of success

there is one that will play a focal role Considerthe (unique) perception such that11

Whenever the agent has a perception that isbelow the objective probability of success() exceeds his perception Similarly when-ever the agent has the perception that is above the objective probability of success is belowhis perception At the agentrsquos perception isequal to the objective probability of success Anagent who holds this perception would not findhis average experience at variance with hisperception

We now turn to how the numbers s and f aregenerated Obviously there should be someclose relationship between true past outcomesand what the agent perceives or recalls In manycontexts however the psychology literature hasidentified various factors such as generation ofattributions or inferences memorability per-ceptual salience and vividness that may affectwhat the agent perceives records and recallshence the data that are available to the agent

To illustrate for example the role of attribu-tions consider the case of our lawyer When heloses he may think that this was due to the factthat the defendant was a member of a minoritygroup while the jury was all white the activitywas a failure but the reasons for the failure areseen as atypical and not likely to arise in thefuture He may then disregard the event when

9 Our modelling here is similar to that of Matthew Rabinand Joel Schrag (1999) we discuss the relationship betweenthe models in Section IV

10 When no data are available we set 1 Our resultshowever will not depend on that assumption

11 There is a unique satisfying this equation due toour assumption that [0 1)

FIGURE 1

1540 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

evaluating his failure In extreme cases he mayeven convince himself that had the circum-stances been ldquonormalrdquo he would have won thecase and mentally record the event as a success

This basic notion is not new there is evi-dence in the psychology literature that peopletend to attribute positive experiences to thingsthat are permanent and to attribute negativeexperiences to transient effects12 If one gets apaper accepted by a journal one attributes thisto being a good economist while rejections areattributed to the referee not understanding theobviously important point being made Withsuch attribution successes are likely to be per-ceived as predicting further successes whilefailures have no predictive effect

We now formalize these ideas We assumethat after undertaking the activity if the out-come is a failure the agent with probability [0 1) attributes the outcome to atypicalcircumstances Events that have been labelledatypical are not recorded When 0 percep-tions are correct and the true and perceivedfrequencies of success coincide When 0perceptions are biased the agent overestimatesthe frequency of past success

To assess precisely how the perception bias affects the agentrsquos estimate of his past successrate define the function [0 1] 3 [0 1] as

1 1

() is essentially the proportion of recordedevents that are recorded as success when thetrue frequency of success is and is theperception bias True and perceived frequenciesof success are random variables Yet in the longrun as will be shown in the Appendix theirdistributions will be concentrated around a sin-gle value and a true frequency of success willtranslate into a perceived frequency of success

Equivalently for the agent to perceive a successrate of in the long run the true long-run fre-

quency of success has to be equal to ()1()Figure 2 illustrates such a function with 0Note that with our assumptions perceptions areoptimistic they always exceed the true frequencyof past success that is ()

B Decision Process

As described in the outline of our modelabove the agent faces a sequence of decisionson whether or not to undertake a risky activityWe assume that undertaking the activity entailsa cost c possibly because of a lost opportunityFor example in the example above there mightalways be available some riskless alternative totaking on a new case such as drawing up a willWe assume the cost c is stochastic and takesvalues in [0 1] We further assume that therandom variables ctt1

the costs at each timet are independent that the support of the ran-dom variables is [0 1] and that at the time theagent chooses whether to undertake the activityat t he knows the realization of ct

To evaluate whether undertaking the projectis worthwhile the agent forms a belief p aboutwhether he will succeed and we assume thisbelief is based on the data he recollects Thiscan be viewed as a formalization of the avail-ability heuristic only the outcomes that arerecollected are available to the agent We do notmodel however the fine details of how recol-lections of past successes and failures aremapped into the agentrsquos belief about the prob-ability of success on the next try rather wemodel this process as a function

p s f12 See eg Seligman (1990)

FIGURE 2

1541VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

One can think of this function as capturing thedynamic process by which an agent who isinitially unsure about 0 would update his initialbeliefs as he accumulates experience13 Alter-natively one might imagine an agent who fol-lows a simple rule of thumb in forming beliefsThe only restrictions we place on the function are the following14

(i) s f 0 0 (s f ) 1(ii) There exists A 0 such that s f 0

(s f ) s(s f ) 13 A(s f )

The first assumption is a statement that be-liefs must lie between 0 and 1 The secondassumption is an ldquoasymptotic consistencyrdquo con-dition that rules out belief formation processesfor which there is a permanent divergence be-tween the agentrsquos perceived successes and fail-ures and his beliefs when the number ofrecalled past outcomes is large the belief of theagent must approach the perceived empiricalfrequency of past successes

Under these conditions when the data arenot biased beliefs are correct in the longrun We insist on the same conditions holdingeven when the data are biased because ourview is that the agent is ignorant of the factthat the process by which these data are gen-erated might be biased and lead him to biasedbeliefs15

Having formed a belief pt about his chance ofsuccess at date t if he undertakes the activitythe agent compares the expected payoff fromundertaking the activity to the cost of undertak-

ing it That is the agent undertakes the activityif and only if 16

pt ct

Finally we define the expected payoff v(p )to the agent who has a belief p and perception and who does not yet know the realization of thecost of the activity17

v p Prp cE cp c13

This completes the description of the modelThe key parameters of the model are the tech-nology the belief function and the per-ception bias Each triplet ( ) induces aprobability distribution over beliefs percep-tions decisions and outcomes18 We are inter-ested in the limit distribution and the agentrsquosexpected gain under that limit distribution whenthe perception bias is We will verify in theAppendix that this limit distribution is well-defined Formally let

Vt Evpt t

We are interested in the long-run payoff definedby

V limt3

Vt

We will show that V(0) 0 that is that

13 Depending on the priors of the agent and on whatthe agent is trying to estimate Bayesian learning expla-nations for the formation of beliefs might depend on thesequence of successes and failures To facilitate ex-position we have chosen to describe the function as a function of the aggregate numbers of recalledsuccesses and failures only One could alternativelyconsider more complicated functions determining theagentrsquos belief about the probability of success that de-pend on the particular sequencing of recalled successesand failures

14 These restrictions are in addition to the assumptionthat it is only the aggregate numbers of recalled successesand failures that matter

15 We will return to this issue in Section IV and examinethe case where the agent is aware that his data may be biasedand attempts to correct his beliefs accordingly

16 As mentioned above we have not assumed that theagent knows 0 hence he might wish to experiment Wecould allow for more sophisticated rules of behavior ac-counting for the fact that the agent might want to experi-ment for a while and undertake the activity when pt tct

with t 13 1 Our results would easily carry over to this caseas long as with probability 1 there is no experimentation inthe long run t 3 1

17 Note that this expected utility is from the perspectiveof an outside observer since it is calculated with the trueprobability of success (p) From the agentrsquos point of viewthe expected value is

Prp cEp cp c13

18 The distribution at date t also depends on the initialcondition that is the value of confidence when no data areavailable (which has been set equal to 1 see footnote 10)Under the assumption that is not too steep however thelimit distribution is independent of initial condition

1542 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

according to this criterion some bias in percep-tions will make the decision maker better off

Note that our focus on long-run payoffs im-plies that we do not analyze what effect biasedbeliefs have on the decision makerrsquos welfare inthe transition period leading up to steady stateThat biases in perceptions might be welfareenhancing in transition would not be surprisinghowever in our model the decision makertypically starts with erroneous beliefs19 andhence make poor decisions in transition20

Biases in perceptions could induce better de-cisions because they may induce a quickermove toward beliefs that would be closer totrue probability of success We discuss thisfurther in Section IV C

III The Optimality of Biased Perceptions

A A Benchmark The Cost of BiasedPerceptions

We begin with the standard case where con-fidence does not affect performance that is thecase in which 1 independently of Ourobjective is to highlight the potential cost asso-ciated with biased perceptions

The probability of success is equal to 0 Bythe law of large numbers as the number ofinstances where the activity is undertaken in-creases the frequency of success gets close to0 (with probability close to 1) When percep-tions are correct ( 0) the true and perceivedfrequencies of past success coincide Hencegiven our assumption (iii) concerning theagentrsquos belief must converge to 0 as well Itfollows that

V 0 0c

0 cgc dc

where g is the density function for the randomcost

How do biased perceptions affect payoffs

With biased perceptions ( 0) the true fre-quency of success still gets close to 0 (withprobability close to 1) The perceived frequencyof success however will get close to (0) aswill his belief about his chance of success Theagent will thus decide to undertake the projectwhenever c 13 (0) It follows that

V 0 c

0 cgc dc

The only effect of the perception bias is tochange the circumstances under which the ac-tivity is undertaken When 0 there areevents for which

0 c 0

In these events the agent undertakes the activitywhen he should not do so (since they havenegative expected payoff with respect to thetrue probability of success) and in the otherevents he is taking the correct decision So theagentrsquos (true) welfare would be higher if he hadcorrect perceptions This is essentially the argu-ment in classical decision theory that biasingperceptions can only harm agents

B Confidence-Enhanced Performance

When confidence affects performance it isno longer true that correct perceptions maxi-mize long-term payoffs It will still be the casethat agents with biased perceptions will haveoverly optimistic beliefs and consequently beinduced to undertake the activity when theyshould not have On those projects they under-take however their optimism leads to higherperformance that is they have higher probabil-ity of success We will compare these two ef-fects and show that having some degree ofoptimism is preferable to correct perceptions

The key observation (see the Appendix) isthat in the long run the perceived frequency ofsuccess tends to be concentrated around a singlevalue say and the possible values of aretherefore easy to characterize The realized fre-quency of success t (t) is with high prob-ability near () True and perceivedfrequencies of success are thus concentrated

19 Our model puts few constraints on beliefs except inthe long run In the long run if his perceptions are unbiasedour decision maker will asymptotically have correct beliefs

20 For example an initially pessimistic agent undertakesthe project too infrequently

1543VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

around and () respectively The onlypossible candidates for must therefore sat-isfy

(31)

Below we will restrict our attention to thecase where there is a unique such value Underthe assumption 1 this is necessarily thecase when is not too large21 We will discussin the Appendix the more general case whereequation (31) may have several solutions

Figure 3 illustrates geometrically for anagent with biased perceptions (and conse-quently optimistic beliefs)

Note that in the case that perceptions arecorrect coincides with the rational belief defined earlier but that for an agent for whom 0

In the long run the agent with positive per-ception bias thus has better performance than anagent with correct perceptions (() ())but he is overly optimistic about his chances ofsuccess ( ())

Turning to the long-run payoff we obtain

V c

cgc dc

As a benchmark the long-run payoff whenperceptions are correct is equal to

V 0 c

cgc dc

To understand how V() compares to V(0)we write V() V(0) as the sum of threeterms

(32) V V 0

c13

gc dc

cgc dc

cgc dc

The first term is positive it corresponds to theincrease in performance that arises due to opti-mism regarding the activities that would havebeen undertaken even if perceptions had beencorrect The second term is positive it corre-sponds to the realizations of costs for which theactivity is profitable to the agent only becausehe is optimistic Finally the third term is neg-ative it corresponds to the realizations of costsfor which the agent should not have undertakenthe activity and undertakes it because he isoptimistic The shaded regions in Figure 4 rep-resent these three terms when c is uniformlydistributed

One implication of equation (32) is thatwhen confidence positively affects performance( 0) some degree of optimism alwaysgenerates higher long-run payoff There are twoforces operating distorting perceptions distortbeliefsmdashhence decisionsmdashbut they also distortthe technology that maps past outcomes intofuture probabilities of success The first is badthe second is good and there is a tradeoffStarting at the point where perceptions are cor-rect ( 0) the distortion in decisions has a21 This is because for not too large ( ) 1

FIGURE 3

1544 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second order on welfare the distortion appliesto few realizations of costs and for these real-izations the loss is small22 In contrast theimprovement in the technology has a first-ordereffect on welfare Formally

dV

d

0

d

d

0

Pr c

Since (dd)0 0 we obtain

PROPOSITION If 0 there exists a bi-ased perception 0 such that V() V(0)

C Robustness to Alternative Assumptions

The Genesis of ConfidencemdashIn our modelconfidence depends on the agentrsquos perception ofthe empirical frequency of past success Onemotivation for this assumption is the idea thatperceived bad average past performance wouldtrigger anxiety hence a decrease in perfor-mance There are plausible alternative assump-tions concerning the genesis of confidence (orlack of confidence) or more generally how pastperceptions affect current performance

One plausible alternative assumption is thatupon undertaking the project the agent recalls a(possibly small) number of past experiences

and that recollections of failures among thesepast experiences decrease confidence or perfor-mance for example because they generate un-productive intrusive thoughts One could alsoimagine that recent past experiences are morelikely to be recalled

Let us give two more specific examples(i) Suppose that upon undertaking the project

the agent recalls one experience among all re-corded past experiences and that confidence is equal to 1 if that experience was a successand equal to 1 (with (0 1)) if thatexperience was a failure Under that specifica-tion confidence is thus a random variable thatmay take two values (1 or 1 ) with thehigher value having probability ss f

(ii) Suppose that confidence depends on (andis increasing in) the fraction of successes ob-tained in the last T recorded experiences Underthat specification confidence is a random vari-able that may take T 1 values

The first example falls within the scope of ourmodel because it yields an expected perfor-mance equal to ()0 where () 1 We provide in the Appendix an alternativeform of biased perception that encompasses thesecond example where current performance de-pends on the particular history of past percep-tions rather than just the summary statistic Afeature of that model is that a change in date tperceived outcome from failure to success canonly increase later expected confidence Weshow that our result holds with this alternativespecification

The Welfare CriterionmdashAs discussed abovewe evaluate the agentrsquos welfare using long-runor steady-state payoffs only One alternativeassumption is to evaluate welfare using dis-counted payoffs With a discount factor suffi-ciently close to one our insight would clearlyremain valid

For smaller discount factors we would haveto analyze how biased perceptions affect wel-fare in the transition to steady state In transi-tion however our argument does not applySince beliefs may not be correct in transitionthe distortion in beliefs induced by biased per-ceptions may not be second order depending onwhether the agent is currently too optimistic ortoo pessimistic the bias may either increase orreduce the period during which beliefs remain

22 We emphasize that the welfare reduction stemmingfrom the distortion in decisions is a second-order effect onlywhen there is initially no distortion There are clearly casesin which nonmarginal increases in confidence can be verycostly

FIGURE 4

1545VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

of Deathrdquo consisting of a live recorded execu-tion (by electric chair) of a criminal5 Subjectswith positive induced mood showed faster re-sponse times than did subjects with negativeinduced mood

Simon Baker et al (1997) also induce elatedand depressed moods in subjects6 and show thatinduced mood affects subjectsrsquo performance ona verbal fluency task In addition they measureregional cerebral blood flow using PositronEmission Tomography (PET) They find thatinduced mood is associated with activation ofareas of the brain associated with the experienceof emotion This finding is of particular interestin that it points to demonstrable physiologicaleffects of mood

In our formal model below we assume thatthe psychological state that affects performanceof a particular task is associated with relatedtasks done in the past We should point out thatonly the first half (approximately) of the worksurveyed above can be considered as supportingour finding We include the remaining workbecause it provides evidence of the broaderpoint that we think importantmdashthat the proba-bility that an individual will be successful at atask should not always be taken as being inde-pendent of psychological factors

B Biased Perception

Psychologists have compiled ample evidencethat people have biased perceptions and indi-viduals systematically evaluate themselvesmore highly than others do Chris Guthrie et al(2001) distributed a questionnaire to 168 federalmagistrate judges as part of the Federal JudicialCenterrsquos Workshop for Magistrate Judges II inNew Orleans in November 1999 The respon-dents were assured that they could not be iden-tified from their questionnaires and were toldthat they could indicate if they preferred their

answers not be included in the research projectOnly one of the 168 judges chose not to havethe answers included

To test the relationship of the judgesrsquo esti-mates of their ability relative to that of otherjudges they were asked the following questionldquoUnited States magistrate judges are rarelyoverturned on appeal but it does occur If wewere to rank all of the magistrate judges cur-rently in this room according to the rate atwhich their decisions have been overturned dur-ing their careers [what] would your rate berdquoThe judges were then asked to place themselvesin the quartile corresponding to their respectivereversal rates highest (75 percent) secondhighest (50 percent) third highest (25 per-cent) or lowest

The judgesrsquo answers are very interesting561 percent put themselves into the lowestquartile 316 percent into the second-lowestquartile 77 percent in the second-highest quar-tile and 45 percent in the highest quartile Inother words nearly 90 percent thought them-selves above average

These judges are not alone in overestimatingtheir abilities People routinely overestimatethemselves relative to others in driving (OlaSvenson 1981) the likelihood their marriagewill succeed (Lynn Baker and Robert Emery1993) and the likelihood that they will havehigher salaries and fewer health problems thanothers (Neil D Weinstein 1980) Michael Rossand Fione Sicoly (1979) report that when mar-ried couples are asked to estimate the percent-age of household tasks they perform theirestimates typically add up to far more than 100percent

People overestimate not only their likelihoodof success relative to others but also the likeli-hood of success in situations not involving oth-ers When subjects are asked questions andasked the likelihood that their answers are cor-rect their ldquohit raterdquo is typically 60 percent whenthey state that they are 90 percent certain (seeeg Baruch Fischoff et al 1977 and SarahLichtenstein et al 1982)

Biases are common when in judging the rel-ative likelihood of particular events and manyscholars have attempted to trace the source ofthese biases One often-mentioned source is theoverutilization of simple heuristics such asthe availability heuristic (Amos Tversky and

5 Subjects in this group were told in advance that theywere free to stop the video if the emotional impact becametoo strong Seven of the 20 subjects did so about halfwaythrough the video

6 Depressed neutral and elated states are induced byhaving subjects listen to extracts of Prokofievrsquos ldquoRussiaunder the Mongolian Yokerdquo at half speed in the depressedstate ldquoStressbustersrdquo a recording of popular classics in theneutral condition and Delibesrsquo ldquoCoppeliardquo in the elatedcondition

1538 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Daniel Kahneman 1973) in assessing the like-lihood of particular events people are ofteninfluenced by their past experiences For exam-ple a worker who is often in contact with job-less individuals because he is jobless himselfwould typically overestimate the rate of unem-ployment similarly an employed worker wouldtypically underestimate the rate of unemploy-ment (see Richard E Nisbett and Ross 1980)In the same vein biases in self-evaluations maybe the result of particular experiences beingmore readily available than others if good ex-periences are more easily remembered than badones or if failures are to be disregarded orattributed to atypical circumstances people tendto have overly optimistic self-evaluations (seeMartin Seligman 1990 for more details on theconnection between attributional styles andoptimism)

Economists often see these biases asldquoshortcomingsrdquo of judgement or pathologiesthat can only lower the welfare of an individ-ual As mentioned in the introduction in ourmodel such biases will emerge as welfareenhancing

II Basic Model

We consider an agent who faces a sequenceof decisions on whether to undertake a riskyactivity This activity may be either a success ora failure Consider a lawyer who is faced with anumber of cases that he may accept or declineAccepting a case is a risky prospect as he maywin or lose the case The lawyer will receive apayoff of 1 in case of success and a payoffnormalized to 0 in case of failure

Our primary departure from conventional de-cision theory is that we assume that the lawyerrsquosprobability of winning the case depends on hisconfidence7 if he is unsure about his abilities oranxious he will have less chance of convincingthe jury To capture the idea that confidenceaffects performance in the simplest way weidentify these feelings of anxiety or self assur-ance with the lawyerrsquos perception of success inprevious cases A lawyer who is more confident

about his chances of success will be assumed tohave greater chances of succeeding than a law-yer who is not confident

In what follows we present two main build-ing blocks of our model First we describe therisky activity the true prospects of the agent ifhe undertakes it and the effect of confidenceSecond we describe the decision process theagent follows

A The Risky Activity

We start by modelling confidence and itseffect on performance We make two assump-tions First higher confidence will translate intohigher probability of success Second confi-dence will depend on the agentrsquos perception ofhow successful he has been in the past

Formally we denote the probability of suc-cess and we take the parameter (0 1] as ameasure of the agentrsquos confidence We assumethat

0

where 0 (0 1) The probability 0 dependssolely on the characteristics of the project beingundertaken Absent any psychological consider-ations the agent would succeed with probability0 Lack of confidence reduces performance8

Note that we do not model the process bywhich performance is reduced It is standard toassume that performance should depend onlyon the agentrsquos actions while undertaking theproject Hence one might object that we need toexplain how confidence alters the decisionsmade by the agent while undertaking theproject We do not however model these deci-sions We assume the agent does not have fullconscious control over all the actions that needto be undertaken and that his psychological statehas an effect on these actions that is beyond hisconscious control

We now turn to modelling how confidence(or lack of confidence) arises Various factorsmay contribute to decreasing confidence suchas the remembrance of past failures or the

7 We use the term ldquoconfidencerdquo in a broad sense toinclude feelings of assuredness and lack of anxiety

8 Note that we do not assume that the agent knows (nor or 0) We discuss this issue in Section III B when wedescribe how the agent forms beliefs about his chance ofsuccess

1539VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

agentrsquos perception of how likely he is to suc-ceed In the basic version of our model weassume confidence depends solely on theagentrsquos perception of the empirical frequency ofpast success

Formally we denote by s (respectively f ) thenumber of successes (respectively failures) theagent recalls We explicitly allow this numberto deviate from the actual number9 We willexplain below how these data are generated Wedefine the perceived empirical frequency of suc-cess as

s

s f

and we assume that confidence is a smooth andincreasing function of that is

where 0 (0) 0 and without loss ofgenerality (1) 110

Combining the two assumptions above wemay view the probability of success as a func-tion of the agentrsquos perception and define theperformance function as follows

0

Last we assume that the effect of confidenceon performance is not too strong that is that

1

This assumption is made primarily for technicalconvenience It will be discussed below and inthe Appendix We draw the function in Figure1 The positive slope captures the proposal thathigh (respectively low) perceptions affect per-formance positively (respectively negatively)

Among the possible perceptions that anagent may have about the frequency of success

there is one that will play a focal role Considerthe (unique) perception such that11

Whenever the agent has a perception that isbelow the objective probability of success() exceeds his perception Similarly when-ever the agent has the perception that is above the objective probability of success is belowhis perception At the agentrsquos perception isequal to the objective probability of success Anagent who holds this perception would not findhis average experience at variance with hisperception

We now turn to how the numbers s and f aregenerated Obviously there should be someclose relationship between true past outcomesand what the agent perceives or recalls In manycontexts however the psychology literature hasidentified various factors such as generation ofattributions or inferences memorability per-ceptual salience and vividness that may affectwhat the agent perceives records and recallshence the data that are available to the agent

To illustrate for example the role of attribu-tions consider the case of our lawyer When heloses he may think that this was due to the factthat the defendant was a member of a minoritygroup while the jury was all white the activitywas a failure but the reasons for the failure areseen as atypical and not likely to arise in thefuture He may then disregard the event when

9 Our modelling here is similar to that of Matthew Rabinand Joel Schrag (1999) we discuss the relationship betweenthe models in Section IV

10 When no data are available we set 1 Our resultshowever will not depend on that assumption

11 There is a unique satisfying this equation due toour assumption that [0 1)

FIGURE 1

1540 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

evaluating his failure In extreme cases he mayeven convince himself that had the circum-stances been ldquonormalrdquo he would have won thecase and mentally record the event as a success

This basic notion is not new there is evi-dence in the psychology literature that peopletend to attribute positive experiences to thingsthat are permanent and to attribute negativeexperiences to transient effects12 If one gets apaper accepted by a journal one attributes thisto being a good economist while rejections areattributed to the referee not understanding theobviously important point being made Withsuch attribution successes are likely to be per-ceived as predicting further successes whilefailures have no predictive effect

We now formalize these ideas We assumethat after undertaking the activity if the out-come is a failure the agent with probability [0 1) attributes the outcome to atypicalcircumstances Events that have been labelledatypical are not recorded When 0 percep-tions are correct and the true and perceivedfrequencies of success coincide When 0perceptions are biased the agent overestimatesthe frequency of past success

To assess precisely how the perception bias affects the agentrsquos estimate of his past successrate define the function [0 1] 3 [0 1] as

1 1

() is essentially the proportion of recordedevents that are recorded as success when thetrue frequency of success is and is theperception bias True and perceived frequenciesof success are random variables Yet in the longrun as will be shown in the Appendix theirdistributions will be concentrated around a sin-gle value and a true frequency of success willtranslate into a perceived frequency of success

Equivalently for the agent to perceive a successrate of in the long run the true long-run fre-

quency of success has to be equal to ()1()Figure 2 illustrates such a function with 0Note that with our assumptions perceptions areoptimistic they always exceed the true frequencyof past success that is ()

B Decision Process

As described in the outline of our modelabove the agent faces a sequence of decisionson whether or not to undertake a risky activityWe assume that undertaking the activity entailsa cost c possibly because of a lost opportunityFor example in the example above there mightalways be available some riskless alternative totaking on a new case such as drawing up a willWe assume the cost c is stochastic and takesvalues in [0 1] We further assume that therandom variables ctt1

the costs at each timet are independent that the support of the ran-dom variables is [0 1] and that at the time theagent chooses whether to undertake the activityat t he knows the realization of ct

To evaluate whether undertaking the projectis worthwhile the agent forms a belief p aboutwhether he will succeed and we assume thisbelief is based on the data he recollects Thiscan be viewed as a formalization of the avail-ability heuristic only the outcomes that arerecollected are available to the agent We do notmodel however the fine details of how recol-lections of past successes and failures aremapped into the agentrsquos belief about the prob-ability of success on the next try rather wemodel this process as a function

p s f12 See eg Seligman (1990)

FIGURE 2

1541VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

One can think of this function as capturing thedynamic process by which an agent who isinitially unsure about 0 would update his initialbeliefs as he accumulates experience13 Alter-natively one might imagine an agent who fol-lows a simple rule of thumb in forming beliefsThe only restrictions we place on the function are the following14

(i) s f 0 0 (s f ) 1(ii) There exists A 0 such that s f 0

(s f ) s(s f ) 13 A(s f )

The first assumption is a statement that be-liefs must lie between 0 and 1 The secondassumption is an ldquoasymptotic consistencyrdquo con-dition that rules out belief formation processesfor which there is a permanent divergence be-tween the agentrsquos perceived successes and fail-ures and his beliefs when the number ofrecalled past outcomes is large the belief of theagent must approach the perceived empiricalfrequency of past successes

Under these conditions when the data arenot biased beliefs are correct in the longrun We insist on the same conditions holdingeven when the data are biased because ourview is that the agent is ignorant of the factthat the process by which these data are gen-erated might be biased and lead him to biasedbeliefs15

Having formed a belief pt about his chance ofsuccess at date t if he undertakes the activitythe agent compares the expected payoff fromundertaking the activity to the cost of undertak-

ing it That is the agent undertakes the activityif and only if 16

pt ct

Finally we define the expected payoff v(p )to the agent who has a belief p and perception and who does not yet know the realization of thecost of the activity17

v p Prp cE cp c13

This completes the description of the modelThe key parameters of the model are the tech-nology the belief function and the per-ception bias Each triplet ( ) induces aprobability distribution over beliefs percep-tions decisions and outcomes18 We are inter-ested in the limit distribution and the agentrsquosexpected gain under that limit distribution whenthe perception bias is We will verify in theAppendix that this limit distribution is well-defined Formally let

Vt Evpt t

We are interested in the long-run payoff definedby

V limt3

Vt

We will show that V(0) 0 that is that

13 Depending on the priors of the agent and on whatthe agent is trying to estimate Bayesian learning expla-nations for the formation of beliefs might depend on thesequence of successes and failures To facilitate ex-position we have chosen to describe the function as a function of the aggregate numbers of recalledsuccesses and failures only One could alternativelyconsider more complicated functions determining theagentrsquos belief about the probability of success that de-pend on the particular sequencing of recalled successesand failures

14 These restrictions are in addition to the assumptionthat it is only the aggregate numbers of recalled successesand failures that matter

15 We will return to this issue in Section IV and examinethe case where the agent is aware that his data may be biasedand attempts to correct his beliefs accordingly

16 As mentioned above we have not assumed that theagent knows 0 hence he might wish to experiment Wecould allow for more sophisticated rules of behavior ac-counting for the fact that the agent might want to experi-ment for a while and undertake the activity when pt tct

with t 13 1 Our results would easily carry over to this caseas long as with probability 1 there is no experimentation inthe long run t 3 1

17 Note that this expected utility is from the perspectiveof an outside observer since it is calculated with the trueprobability of success (p) From the agentrsquos point of viewthe expected value is

Prp cEp cp c13

18 The distribution at date t also depends on the initialcondition that is the value of confidence when no data areavailable (which has been set equal to 1 see footnote 10)Under the assumption that is not too steep however thelimit distribution is independent of initial condition

1542 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

according to this criterion some bias in percep-tions will make the decision maker better off

Note that our focus on long-run payoffs im-plies that we do not analyze what effect biasedbeliefs have on the decision makerrsquos welfare inthe transition period leading up to steady stateThat biases in perceptions might be welfareenhancing in transition would not be surprisinghowever in our model the decision makertypically starts with erroneous beliefs19 andhence make poor decisions in transition20

Biases in perceptions could induce better de-cisions because they may induce a quickermove toward beliefs that would be closer totrue probability of success We discuss thisfurther in Section IV C

III The Optimality of Biased Perceptions

A A Benchmark The Cost of BiasedPerceptions

We begin with the standard case where con-fidence does not affect performance that is thecase in which 1 independently of Ourobjective is to highlight the potential cost asso-ciated with biased perceptions

The probability of success is equal to 0 Bythe law of large numbers as the number ofinstances where the activity is undertaken in-creases the frequency of success gets close to0 (with probability close to 1) When percep-tions are correct ( 0) the true and perceivedfrequencies of past success coincide Hencegiven our assumption (iii) concerning theagentrsquos belief must converge to 0 as well Itfollows that

V 0 0c

0 cgc dc

where g is the density function for the randomcost

How do biased perceptions affect payoffs

With biased perceptions ( 0) the true fre-quency of success still gets close to 0 (withprobability close to 1) The perceived frequencyof success however will get close to (0) aswill his belief about his chance of success Theagent will thus decide to undertake the projectwhenever c 13 (0) It follows that

V 0 c

0 cgc dc

The only effect of the perception bias is tochange the circumstances under which the ac-tivity is undertaken When 0 there areevents for which

0 c 0

In these events the agent undertakes the activitywhen he should not do so (since they havenegative expected payoff with respect to thetrue probability of success) and in the otherevents he is taking the correct decision So theagentrsquos (true) welfare would be higher if he hadcorrect perceptions This is essentially the argu-ment in classical decision theory that biasingperceptions can only harm agents

B Confidence-Enhanced Performance

When confidence affects performance it isno longer true that correct perceptions maxi-mize long-term payoffs It will still be the casethat agents with biased perceptions will haveoverly optimistic beliefs and consequently beinduced to undertake the activity when theyshould not have On those projects they under-take however their optimism leads to higherperformance that is they have higher probabil-ity of success We will compare these two ef-fects and show that having some degree ofoptimism is preferable to correct perceptions

The key observation (see the Appendix) isthat in the long run the perceived frequency ofsuccess tends to be concentrated around a singlevalue say and the possible values of aretherefore easy to characterize The realized fre-quency of success t (t) is with high prob-ability near () True and perceivedfrequencies of success are thus concentrated

19 Our model puts few constraints on beliefs except inthe long run In the long run if his perceptions are unbiasedour decision maker will asymptotically have correct beliefs

20 For example an initially pessimistic agent undertakesthe project too infrequently

1543VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

around and () respectively The onlypossible candidates for must therefore sat-isfy

(31)

Below we will restrict our attention to thecase where there is a unique such value Underthe assumption 1 this is necessarily thecase when is not too large21 We will discussin the Appendix the more general case whereequation (31) may have several solutions

Figure 3 illustrates geometrically for anagent with biased perceptions (and conse-quently optimistic beliefs)

Note that in the case that perceptions arecorrect coincides with the rational belief defined earlier but that for an agent for whom 0

In the long run the agent with positive per-ception bias thus has better performance than anagent with correct perceptions (() ())but he is overly optimistic about his chances ofsuccess ( ())

Turning to the long-run payoff we obtain

V c

cgc dc

As a benchmark the long-run payoff whenperceptions are correct is equal to

V 0 c

cgc dc

To understand how V() compares to V(0)we write V() V(0) as the sum of threeterms

(32) V V 0

c13

gc dc

cgc dc

cgc dc

The first term is positive it corresponds to theincrease in performance that arises due to opti-mism regarding the activities that would havebeen undertaken even if perceptions had beencorrect The second term is positive it corre-sponds to the realizations of costs for which theactivity is profitable to the agent only becausehe is optimistic Finally the third term is neg-ative it corresponds to the realizations of costsfor which the agent should not have undertakenthe activity and undertakes it because he isoptimistic The shaded regions in Figure 4 rep-resent these three terms when c is uniformlydistributed

One implication of equation (32) is thatwhen confidence positively affects performance( 0) some degree of optimism alwaysgenerates higher long-run payoff There are twoforces operating distorting perceptions distortbeliefsmdashhence decisionsmdashbut they also distortthe technology that maps past outcomes intofuture probabilities of success The first is badthe second is good and there is a tradeoffStarting at the point where perceptions are cor-rect ( 0) the distortion in decisions has a21 This is because for not too large ( ) 1

FIGURE 3

1544 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second order on welfare the distortion appliesto few realizations of costs and for these real-izations the loss is small22 In contrast theimprovement in the technology has a first-ordereffect on welfare Formally

dV

d

0

d

d

0

Pr c

Since (dd)0 0 we obtain

PROPOSITION If 0 there exists a bi-ased perception 0 such that V() V(0)

C Robustness to Alternative Assumptions

The Genesis of ConfidencemdashIn our modelconfidence depends on the agentrsquos perception ofthe empirical frequency of past success Onemotivation for this assumption is the idea thatperceived bad average past performance wouldtrigger anxiety hence a decrease in perfor-mance There are plausible alternative assump-tions concerning the genesis of confidence (orlack of confidence) or more generally how pastperceptions affect current performance

One plausible alternative assumption is thatupon undertaking the project the agent recalls a(possibly small) number of past experiences

and that recollections of failures among thesepast experiences decrease confidence or perfor-mance for example because they generate un-productive intrusive thoughts One could alsoimagine that recent past experiences are morelikely to be recalled

Let us give two more specific examples(i) Suppose that upon undertaking the project

the agent recalls one experience among all re-corded past experiences and that confidence is equal to 1 if that experience was a successand equal to 1 (with (0 1)) if thatexperience was a failure Under that specifica-tion confidence is thus a random variable thatmay take two values (1 or 1 ) with thehigher value having probability ss f

(ii) Suppose that confidence depends on (andis increasing in) the fraction of successes ob-tained in the last T recorded experiences Underthat specification confidence is a random vari-able that may take T 1 values

The first example falls within the scope of ourmodel because it yields an expected perfor-mance equal to ()0 where () 1 We provide in the Appendix an alternativeform of biased perception that encompasses thesecond example where current performance de-pends on the particular history of past percep-tions rather than just the summary statistic Afeature of that model is that a change in date tperceived outcome from failure to success canonly increase later expected confidence Weshow that our result holds with this alternativespecification

The Welfare CriterionmdashAs discussed abovewe evaluate the agentrsquos welfare using long-runor steady-state payoffs only One alternativeassumption is to evaluate welfare using dis-counted payoffs With a discount factor suffi-ciently close to one our insight would clearlyremain valid

For smaller discount factors we would haveto analyze how biased perceptions affect wel-fare in the transition to steady state In transi-tion however our argument does not applySince beliefs may not be correct in transitionthe distortion in beliefs induced by biased per-ceptions may not be second order depending onwhether the agent is currently too optimistic ortoo pessimistic the bias may either increase orreduce the period during which beliefs remain

22 We emphasize that the welfare reduction stemmingfrom the distortion in decisions is a second-order effect onlywhen there is initially no distortion There are clearly casesin which nonmarginal increases in confidence can be verycostly

FIGURE 4

1545VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Daniel Kahneman 1973) in assessing the like-lihood of particular events people are ofteninfluenced by their past experiences For exam-ple a worker who is often in contact with job-less individuals because he is jobless himselfwould typically overestimate the rate of unem-ployment similarly an employed worker wouldtypically underestimate the rate of unemploy-ment (see Richard E Nisbett and Ross 1980)In the same vein biases in self-evaluations maybe the result of particular experiences beingmore readily available than others if good ex-periences are more easily remembered than badones or if failures are to be disregarded orattributed to atypical circumstances people tendto have overly optimistic self-evaluations (seeMartin Seligman 1990 for more details on theconnection between attributional styles andoptimism)

Economists often see these biases asldquoshortcomingsrdquo of judgement or pathologiesthat can only lower the welfare of an individ-ual As mentioned in the introduction in ourmodel such biases will emerge as welfareenhancing

II Basic Model

We consider an agent who faces a sequenceof decisions on whether to undertake a riskyactivity This activity may be either a success ora failure Consider a lawyer who is faced with anumber of cases that he may accept or declineAccepting a case is a risky prospect as he maywin or lose the case The lawyer will receive apayoff of 1 in case of success and a payoffnormalized to 0 in case of failure

Our primary departure from conventional de-cision theory is that we assume that the lawyerrsquosprobability of winning the case depends on hisconfidence7 if he is unsure about his abilities oranxious he will have less chance of convincingthe jury To capture the idea that confidenceaffects performance in the simplest way weidentify these feelings of anxiety or self assur-ance with the lawyerrsquos perception of success inprevious cases A lawyer who is more confident

about his chances of success will be assumed tohave greater chances of succeeding than a law-yer who is not confident

In what follows we present two main build-ing blocks of our model First we describe therisky activity the true prospects of the agent ifhe undertakes it and the effect of confidenceSecond we describe the decision process theagent follows

A The Risky Activity

We start by modelling confidence and itseffect on performance We make two assump-tions First higher confidence will translate intohigher probability of success Second confi-dence will depend on the agentrsquos perception ofhow successful he has been in the past

Formally we denote the probability of suc-cess and we take the parameter (0 1] as ameasure of the agentrsquos confidence We assumethat

0

where 0 (0 1) The probability 0 dependssolely on the characteristics of the project beingundertaken Absent any psychological consider-ations the agent would succeed with probability0 Lack of confidence reduces performance8

Note that we do not model the process bywhich performance is reduced It is standard toassume that performance should depend onlyon the agentrsquos actions while undertaking theproject Hence one might object that we need toexplain how confidence alters the decisionsmade by the agent while undertaking theproject We do not however model these deci-sions We assume the agent does not have fullconscious control over all the actions that needto be undertaken and that his psychological statehas an effect on these actions that is beyond hisconscious control

We now turn to modelling how confidence(or lack of confidence) arises Various factorsmay contribute to decreasing confidence suchas the remembrance of past failures or the

7 We use the term ldquoconfidencerdquo in a broad sense toinclude feelings of assuredness and lack of anxiety

8 Note that we do not assume that the agent knows (nor or 0) We discuss this issue in Section III B when wedescribe how the agent forms beliefs about his chance ofsuccess

1539VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

agentrsquos perception of how likely he is to suc-ceed In the basic version of our model weassume confidence depends solely on theagentrsquos perception of the empirical frequency ofpast success

Formally we denote by s (respectively f ) thenumber of successes (respectively failures) theagent recalls We explicitly allow this numberto deviate from the actual number9 We willexplain below how these data are generated Wedefine the perceived empirical frequency of suc-cess as

s

s f

and we assume that confidence is a smooth andincreasing function of that is

where 0 (0) 0 and without loss ofgenerality (1) 110

Combining the two assumptions above wemay view the probability of success as a func-tion of the agentrsquos perception and define theperformance function as follows

0

Last we assume that the effect of confidenceon performance is not too strong that is that

1

This assumption is made primarily for technicalconvenience It will be discussed below and inthe Appendix We draw the function in Figure1 The positive slope captures the proposal thathigh (respectively low) perceptions affect per-formance positively (respectively negatively)

Among the possible perceptions that anagent may have about the frequency of success

there is one that will play a focal role Considerthe (unique) perception such that11

Whenever the agent has a perception that isbelow the objective probability of success() exceeds his perception Similarly when-ever the agent has the perception that is above the objective probability of success is belowhis perception At the agentrsquos perception isequal to the objective probability of success Anagent who holds this perception would not findhis average experience at variance with hisperception

We now turn to how the numbers s and f aregenerated Obviously there should be someclose relationship between true past outcomesand what the agent perceives or recalls In manycontexts however the psychology literature hasidentified various factors such as generation ofattributions or inferences memorability per-ceptual salience and vividness that may affectwhat the agent perceives records and recallshence the data that are available to the agent

To illustrate for example the role of attribu-tions consider the case of our lawyer When heloses he may think that this was due to the factthat the defendant was a member of a minoritygroup while the jury was all white the activitywas a failure but the reasons for the failure areseen as atypical and not likely to arise in thefuture He may then disregard the event when

9 Our modelling here is similar to that of Matthew Rabinand Joel Schrag (1999) we discuss the relationship betweenthe models in Section IV

10 When no data are available we set 1 Our resultshowever will not depend on that assumption

11 There is a unique satisfying this equation due toour assumption that [0 1)

FIGURE 1

1540 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

evaluating his failure In extreme cases he mayeven convince himself that had the circum-stances been ldquonormalrdquo he would have won thecase and mentally record the event as a success

This basic notion is not new there is evi-dence in the psychology literature that peopletend to attribute positive experiences to thingsthat are permanent and to attribute negativeexperiences to transient effects12 If one gets apaper accepted by a journal one attributes thisto being a good economist while rejections areattributed to the referee not understanding theobviously important point being made Withsuch attribution successes are likely to be per-ceived as predicting further successes whilefailures have no predictive effect

We now formalize these ideas We assumethat after undertaking the activity if the out-come is a failure the agent with probability [0 1) attributes the outcome to atypicalcircumstances Events that have been labelledatypical are not recorded When 0 percep-tions are correct and the true and perceivedfrequencies of success coincide When 0perceptions are biased the agent overestimatesthe frequency of past success

To assess precisely how the perception bias affects the agentrsquos estimate of his past successrate define the function [0 1] 3 [0 1] as

1 1

() is essentially the proportion of recordedevents that are recorded as success when thetrue frequency of success is and is theperception bias True and perceived frequenciesof success are random variables Yet in the longrun as will be shown in the Appendix theirdistributions will be concentrated around a sin-gle value and a true frequency of success willtranslate into a perceived frequency of success

Equivalently for the agent to perceive a successrate of in the long run the true long-run fre-

quency of success has to be equal to ()1()Figure 2 illustrates such a function with 0Note that with our assumptions perceptions areoptimistic they always exceed the true frequencyof past success that is ()

B Decision Process

As described in the outline of our modelabove the agent faces a sequence of decisionson whether or not to undertake a risky activityWe assume that undertaking the activity entailsa cost c possibly because of a lost opportunityFor example in the example above there mightalways be available some riskless alternative totaking on a new case such as drawing up a willWe assume the cost c is stochastic and takesvalues in [0 1] We further assume that therandom variables ctt1

the costs at each timet are independent that the support of the ran-dom variables is [0 1] and that at the time theagent chooses whether to undertake the activityat t he knows the realization of ct

To evaluate whether undertaking the projectis worthwhile the agent forms a belief p aboutwhether he will succeed and we assume thisbelief is based on the data he recollects Thiscan be viewed as a formalization of the avail-ability heuristic only the outcomes that arerecollected are available to the agent We do notmodel however the fine details of how recol-lections of past successes and failures aremapped into the agentrsquos belief about the prob-ability of success on the next try rather wemodel this process as a function

p s f12 See eg Seligman (1990)

FIGURE 2

1541VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

One can think of this function as capturing thedynamic process by which an agent who isinitially unsure about 0 would update his initialbeliefs as he accumulates experience13 Alter-natively one might imagine an agent who fol-lows a simple rule of thumb in forming beliefsThe only restrictions we place on the function are the following14

(i) s f 0 0 (s f ) 1(ii) There exists A 0 such that s f 0

(s f ) s(s f ) 13 A(s f )

The first assumption is a statement that be-liefs must lie between 0 and 1 The secondassumption is an ldquoasymptotic consistencyrdquo con-dition that rules out belief formation processesfor which there is a permanent divergence be-tween the agentrsquos perceived successes and fail-ures and his beliefs when the number ofrecalled past outcomes is large the belief of theagent must approach the perceived empiricalfrequency of past successes

Under these conditions when the data arenot biased beliefs are correct in the longrun We insist on the same conditions holdingeven when the data are biased because ourview is that the agent is ignorant of the factthat the process by which these data are gen-erated might be biased and lead him to biasedbeliefs15

Having formed a belief pt about his chance ofsuccess at date t if he undertakes the activitythe agent compares the expected payoff fromundertaking the activity to the cost of undertak-

ing it That is the agent undertakes the activityif and only if 16

pt ct

Finally we define the expected payoff v(p )to the agent who has a belief p and perception and who does not yet know the realization of thecost of the activity17

v p Prp cE cp c13

This completes the description of the modelThe key parameters of the model are the tech-nology the belief function and the per-ception bias Each triplet ( ) induces aprobability distribution over beliefs percep-tions decisions and outcomes18 We are inter-ested in the limit distribution and the agentrsquosexpected gain under that limit distribution whenthe perception bias is We will verify in theAppendix that this limit distribution is well-defined Formally let

Vt Evpt t

We are interested in the long-run payoff definedby

V limt3

Vt

We will show that V(0) 0 that is that

13 Depending on the priors of the agent and on whatthe agent is trying to estimate Bayesian learning expla-nations for the formation of beliefs might depend on thesequence of successes and failures To facilitate ex-position we have chosen to describe the function as a function of the aggregate numbers of recalledsuccesses and failures only One could alternativelyconsider more complicated functions determining theagentrsquos belief about the probability of success that de-pend on the particular sequencing of recalled successesand failures

14 These restrictions are in addition to the assumptionthat it is only the aggregate numbers of recalled successesand failures that matter

15 We will return to this issue in Section IV and examinethe case where the agent is aware that his data may be biasedand attempts to correct his beliefs accordingly

16 As mentioned above we have not assumed that theagent knows 0 hence he might wish to experiment Wecould allow for more sophisticated rules of behavior ac-counting for the fact that the agent might want to experi-ment for a while and undertake the activity when pt tct

with t 13 1 Our results would easily carry over to this caseas long as with probability 1 there is no experimentation inthe long run t 3 1

17 Note that this expected utility is from the perspectiveof an outside observer since it is calculated with the trueprobability of success (p) From the agentrsquos point of viewthe expected value is

Prp cEp cp c13

18 The distribution at date t also depends on the initialcondition that is the value of confidence when no data areavailable (which has been set equal to 1 see footnote 10)Under the assumption that is not too steep however thelimit distribution is independent of initial condition

1542 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

according to this criterion some bias in percep-tions will make the decision maker better off

Note that our focus on long-run payoffs im-plies that we do not analyze what effect biasedbeliefs have on the decision makerrsquos welfare inthe transition period leading up to steady stateThat biases in perceptions might be welfareenhancing in transition would not be surprisinghowever in our model the decision makertypically starts with erroneous beliefs19 andhence make poor decisions in transition20

Biases in perceptions could induce better de-cisions because they may induce a quickermove toward beliefs that would be closer totrue probability of success We discuss thisfurther in Section IV C

III The Optimality of Biased Perceptions

A A Benchmark The Cost of BiasedPerceptions

We begin with the standard case where con-fidence does not affect performance that is thecase in which 1 independently of Ourobjective is to highlight the potential cost asso-ciated with biased perceptions

The probability of success is equal to 0 Bythe law of large numbers as the number ofinstances where the activity is undertaken in-creases the frequency of success gets close to0 (with probability close to 1) When percep-tions are correct ( 0) the true and perceivedfrequencies of past success coincide Hencegiven our assumption (iii) concerning theagentrsquos belief must converge to 0 as well Itfollows that

V 0 0c

0 cgc dc

where g is the density function for the randomcost

How do biased perceptions affect payoffs

With biased perceptions ( 0) the true fre-quency of success still gets close to 0 (withprobability close to 1) The perceived frequencyof success however will get close to (0) aswill his belief about his chance of success Theagent will thus decide to undertake the projectwhenever c 13 (0) It follows that

V 0 c

0 cgc dc

The only effect of the perception bias is tochange the circumstances under which the ac-tivity is undertaken When 0 there areevents for which

0 c 0

In these events the agent undertakes the activitywhen he should not do so (since they havenegative expected payoff with respect to thetrue probability of success) and in the otherevents he is taking the correct decision So theagentrsquos (true) welfare would be higher if he hadcorrect perceptions This is essentially the argu-ment in classical decision theory that biasingperceptions can only harm agents

B Confidence-Enhanced Performance

When confidence affects performance it isno longer true that correct perceptions maxi-mize long-term payoffs It will still be the casethat agents with biased perceptions will haveoverly optimistic beliefs and consequently beinduced to undertake the activity when theyshould not have On those projects they under-take however their optimism leads to higherperformance that is they have higher probabil-ity of success We will compare these two ef-fects and show that having some degree ofoptimism is preferable to correct perceptions

The key observation (see the Appendix) isthat in the long run the perceived frequency ofsuccess tends to be concentrated around a singlevalue say and the possible values of aretherefore easy to characterize The realized fre-quency of success t (t) is with high prob-ability near () True and perceivedfrequencies of success are thus concentrated

19 Our model puts few constraints on beliefs except inthe long run In the long run if his perceptions are unbiasedour decision maker will asymptotically have correct beliefs

20 For example an initially pessimistic agent undertakesthe project too infrequently

1543VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

around and () respectively The onlypossible candidates for must therefore sat-isfy

(31)

Below we will restrict our attention to thecase where there is a unique such value Underthe assumption 1 this is necessarily thecase when is not too large21 We will discussin the Appendix the more general case whereequation (31) may have several solutions

Figure 3 illustrates geometrically for anagent with biased perceptions (and conse-quently optimistic beliefs)

Note that in the case that perceptions arecorrect coincides with the rational belief defined earlier but that for an agent for whom 0

In the long run the agent with positive per-ception bias thus has better performance than anagent with correct perceptions (() ())but he is overly optimistic about his chances ofsuccess ( ())

Turning to the long-run payoff we obtain

V c

cgc dc

As a benchmark the long-run payoff whenperceptions are correct is equal to

V 0 c

cgc dc

To understand how V() compares to V(0)we write V() V(0) as the sum of threeterms

(32) V V 0

c13

gc dc

cgc dc

cgc dc

The first term is positive it corresponds to theincrease in performance that arises due to opti-mism regarding the activities that would havebeen undertaken even if perceptions had beencorrect The second term is positive it corre-sponds to the realizations of costs for which theactivity is profitable to the agent only becausehe is optimistic Finally the third term is neg-ative it corresponds to the realizations of costsfor which the agent should not have undertakenthe activity and undertakes it because he isoptimistic The shaded regions in Figure 4 rep-resent these three terms when c is uniformlydistributed

One implication of equation (32) is thatwhen confidence positively affects performance( 0) some degree of optimism alwaysgenerates higher long-run payoff There are twoforces operating distorting perceptions distortbeliefsmdashhence decisionsmdashbut they also distortthe technology that maps past outcomes intofuture probabilities of success The first is badthe second is good and there is a tradeoffStarting at the point where perceptions are cor-rect ( 0) the distortion in decisions has a21 This is because for not too large ( ) 1

FIGURE 3

1544 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second order on welfare the distortion appliesto few realizations of costs and for these real-izations the loss is small22 In contrast theimprovement in the technology has a first-ordereffect on welfare Formally

dV

d

0

d

d

0

Pr c

Since (dd)0 0 we obtain

PROPOSITION If 0 there exists a bi-ased perception 0 such that V() V(0)

C Robustness to Alternative Assumptions

The Genesis of ConfidencemdashIn our modelconfidence depends on the agentrsquos perception ofthe empirical frequency of past success Onemotivation for this assumption is the idea thatperceived bad average past performance wouldtrigger anxiety hence a decrease in perfor-mance There are plausible alternative assump-tions concerning the genesis of confidence (orlack of confidence) or more generally how pastperceptions affect current performance

One plausible alternative assumption is thatupon undertaking the project the agent recalls a(possibly small) number of past experiences

and that recollections of failures among thesepast experiences decrease confidence or perfor-mance for example because they generate un-productive intrusive thoughts One could alsoimagine that recent past experiences are morelikely to be recalled

Let us give two more specific examples(i) Suppose that upon undertaking the project

the agent recalls one experience among all re-corded past experiences and that confidence is equal to 1 if that experience was a successand equal to 1 (with (0 1)) if thatexperience was a failure Under that specifica-tion confidence is thus a random variable thatmay take two values (1 or 1 ) with thehigher value having probability ss f

(ii) Suppose that confidence depends on (andis increasing in) the fraction of successes ob-tained in the last T recorded experiences Underthat specification confidence is a random vari-able that may take T 1 values

The first example falls within the scope of ourmodel because it yields an expected perfor-mance equal to ()0 where () 1 We provide in the Appendix an alternativeform of biased perception that encompasses thesecond example where current performance de-pends on the particular history of past percep-tions rather than just the summary statistic Afeature of that model is that a change in date tperceived outcome from failure to success canonly increase later expected confidence Weshow that our result holds with this alternativespecification

The Welfare CriterionmdashAs discussed abovewe evaluate the agentrsquos welfare using long-runor steady-state payoffs only One alternativeassumption is to evaluate welfare using dis-counted payoffs With a discount factor suffi-ciently close to one our insight would clearlyremain valid

For smaller discount factors we would haveto analyze how biased perceptions affect wel-fare in the transition to steady state In transi-tion however our argument does not applySince beliefs may not be correct in transitionthe distortion in beliefs induced by biased per-ceptions may not be second order depending onwhether the agent is currently too optimistic ortoo pessimistic the bias may either increase orreduce the period during which beliefs remain

22 We emphasize that the welfare reduction stemmingfrom the distortion in decisions is a second-order effect onlywhen there is initially no distortion There are clearly casesin which nonmarginal increases in confidence can be verycostly

FIGURE 4

1545VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

agentrsquos perception of how likely he is to suc-ceed In the basic version of our model weassume confidence depends solely on theagentrsquos perception of the empirical frequency ofpast success

Formally we denote by s (respectively f ) thenumber of successes (respectively failures) theagent recalls We explicitly allow this numberto deviate from the actual number9 We willexplain below how these data are generated Wedefine the perceived empirical frequency of suc-cess as

s

s f

and we assume that confidence is a smooth andincreasing function of that is

where 0 (0) 0 and without loss ofgenerality (1) 110

Combining the two assumptions above wemay view the probability of success as a func-tion of the agentrsquos perception and define theperformance function as follows

0

Last we assume that the effect of confidenceon performance is not too strong that is that

1

This assumption is made primarily for technicalconvenience It will be discussed below and inthe Appendix We draw the function in Figure1 The positive slope captures the proposal thathigh (respectively low) perceptions affect per-formance positively (respectively negatively)

Among the possible perceptions that anagent may have about the frequency of success

there is one that will play a focal role Considerthe (unique) perception such that11

Whenever the agent has a perception that isbelow the objective probability of success() exceeds his perception Similarly when-ever the agent has the perception that is above the objective probability of success is belowhis perception At the agentrsquos perception isequal to the objective probability of success Anagent who holds this perception would not findhis average experience at variance with hisperception

We now turn to how the numbers s and f aregenerated Obviously there should be someclose relationship between true past outcomesand what the agent perceives or recalls In manycontexts however the psychology literature hasidentified various factors such as generation ofattributions or inferences memorability per-ceptual salience and vividness that may affectwhat the agent perceives records and recallshence the data that are available to the agent

To illustrate for example the role of attribu-tions consider the case of our lawyer When heloses he may think that this was due to the factthat the defendant was a member of a minoritygroup while the jury was all white the activitywas a failure but the reasons for the failure areseen as atypical and not likely to arise in thefuture He may then disregard the event when

9 Our modelling here is similar to that of Matthew Rabinand Joel Schrag (1999) we discuss the relationship betweenthe models in Section IV

10 When no data are available we set 1 Our resultshowever will not depend on that assumption

11 There is a unique satisfying this equation due toour assumption that [0 1)

FIGURE 1

1540 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

evaluating his failure In extreme cases he mayeven convince himself that had the circum-stances been ldquonormalrdquo he would have won thecase and mentally record the event as a success

This basic notion is not new there is evi-dence in the psychology literature that peopletend to attribute positive experiences to thingsthat are permanent and to attribute negativeexperiences to transient effects12 If one gets apaper accepted by a journal one attributes thisto being a good economist while rejections areattributed to the referee not understanding theobviously important point being made Withsuch attribution successes are likely to be per-ceived as predicting further successes whilefailures have no predictive effect

We now formalize these ideas We assumethat after undertaking the activity if the out-come is a failure the agent with probability [0 1) attributes the outcome to atypicalcircumstances Events that have been labelledatypical are not recorded When 0 percep-tions are correct and the true and perceivedfrequencies of success coincide When 0perceptions are biased the agent overestimatesthe frequency of past success

To assess precisely how the perception bias affects the agentrsquos estimate of his past successrate define the function [0 1] 3 [0 1] as

1 1

() is essentially the proportion of recordedevents that are recorded as success when thetrue frequency of success is and is theperception bias True and perceived frequenciesof success are random variables Yet in the longrun as will be shown in the Appendix theirdistributions will be concentrated around a sin-gle value and a true frequency of success willtranslate into a perceived frequency of success

Equivalently for the agent to perceive a successrate of in the long run the true long-run fre-

quency of success has to be equal to ()1()Figure 2 illustrates such a function with 0Note that with our assumptions perceptions areoptimistic they always exceed the true frequencyof past success that is ()

B Decision Process

As described in the outline of our modelabove the agent faces a sequence of decisionson whether or not to undertake a risky activityWe assume that undertaking the activity entailsa cost c possibly because of a lost opportunityFor example in the example above there mightalways be available some riskless alternative totaking on a new case such as drawing up a willWe assume the cost c is stochastic and takesvalues in [0 1] We further assume that therandom variables ctt1

the costs at each timet are independent that the support of the ran-dom variables is [0 1] and that at the time theagent chooses whether to undertake the activityat t he knows the realization of ct

To evaluate whether undertaking the projectis worthwhile the agent forms a belief p aboutwhether he will succeed and we assume thisbelief is based on the data he recollects Thiscan be viewed as a formalization of the avail-ability heuristic only the outcomes that arerecollected are available to the agent We do notmodel however the fine details of how recol-lections of past successes and failures aremapped into the agentrsquos belief about the prob-ability of success on the next try rather wemodel this process as a function

p s f12 See eg Seligman (1990)

FIGURE 2

1541VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

One can think of this function as capturing thedynamic process by which an agent who isinitially unsure about 0 would update his initialbeliefs as he accumulates experience13 Alter-natively one might imagine an agent who fol-lows a simple rule of thumb in forming beliefsThe only restrictions we place on the function are the following14

(i) s f 0 0 (s f ) 1(ii) There exists A 0 such that s f 0

(s f ) s(s f ) 13 A(s f )

The first assumption is a statement that be-liefs must lie between 0 and 1 The secondassumption is an ldquoasymptotic consistencyrdquo con-dition that rules out belief formation processesfor which there is a permanent divergence be-tween the agentrsquos perceived successes and fail-ures and his beliefs when the number ofrecalled past outcomes is large the belief of theagent must approach the perceived empiricalfrequency of past successes

Under these conditions when the data arenot biased beliefs are correct in the longrun We insist on the same conditions holdingeven when the data are biased because ourview is that the agent is ignorant of the factthat the process by which these data are gen-erated might be biased and lead him to biasedbeliefs15

Having formed a belief pt about his chance ofsuccess at date t if he undertakes the activitythe agent compares the expected payoff fromundertaking the activity to the cost of undertak-

ing it That is the agent undertakes the activityif and only if 16

pt ct

Finally we define the expected payoff v(p )to the agent who has a belief p and perception and who does not yet know the realization of thecost of the activity17

v p Prp cE cp c13

This completes the description of the modelThe key parameters of the model are the tech-nology the belief function and the per-ception bias Each triplet ( ) induces aprobability distribution over beliefs percep-tions decisions and outcomes18 We are inter-ested in the limit distribution and the agentrsquosexpected gain under that limit distribution whenthe perception bias is We will verify in theAppendix that this limit distribution is well-defined Formally let

Vt Evpt t

We are interested in the long-run payoff definedby

V limt3

Vt

We will show that V(0) 0 that is that

13 Depending on the priors of the agent and on whatthe agent is trying to estimate Bayesian learning expla-nations for the formation of beliefs might depend on thesequence of successes and failures To facilitate ex-position we have chosen to describe the function as a function of the aggregate numbers of recalledsuccesses and failures only One could alternativelyconsider more complicated functions determining theagentrsquos belief about the probability of success that de-pend on the particular sequencing of recalled successesand failures

14 These restrictions are in addition to the assumptionthat it is only the aggregate numbers of recalled successesand failures that matter

15 We will return to this issue in Section IV and examinethe case where the agent is aware that his data may be biasedand attempts to correct his beliefs accordingly

16 As mentioned above we have not assumed that theagent knows 0 hence he might wish to experiment Wecould allow for more sophisticated rules of behavior ac-counting for the fact that the agent might want to experi-ment for a while and undertake the activity when pt tct

with t 13 1 Our results would easily carry over to this caseas long as with probability 1 there is no experimentation inthe long run t 3 1

17 Note that this expected utility is from the perspectiveof an outside observer since it is calculated with the trueprobability of success (p) From the agentrsquos point of viewthe expected value is

Prp cEp cp c13

18 The distribution at date t also depends on the initialcondition that is the value of confidence when no data areavailable (which has been set equal to 1 see footnote 10)Under the assumption that is not too steep however thelimit distribution is independent of initial condition

1542 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

according to this criterion some bias in percep-tions will make the decision maker better off

Note that our focus on long-run payoffs im-plies that we do not analyze what effect biasedbeliefs have on the decision makerrsquos welfare inthe transition period leading up to steady stateThat biases in perceptions might be welfareenhancing in transition would not be surprisinghowever in our model the decision makertypically starts with erroneous beliefs19 andhence make poor decisions in transition20

Biases in perceptions could induce better de-cisions because they may induce a quickermove toward beliefs that would be closer totrue probability of success We discuss thisfurther in Section IV C

III The Optimality of Biased Perceptions

A A Benchmark The Cost of BiasedPerceptions

We begin with the standard case where con-fidence does not affect performance that is thecase in which 1 independently of Ourobjective is to highlight the potential cost asso-ciated with biased perceptions

The probability of success is equal to 0 Bythe law of large numbers as the number ofinstances where the activity is undertaken in-creases the frequency of success gets close to0 (with probability close to 1) When percep-tions are correct ( 0) the true and perceivedfrequencies of past success coincide Hencegiven our assumption (iii) concerning theagentrsquos belief must converge to 0 as well Itfollows that

V 0 0c

0 cgc dc

where g is the density function for the randomcost

How do biased perceptions affect payoffs

With biased perceptions ( 0) the true fre-quency of success still gets close to 0 (withprobability close to 1) The perceived frequencyof success however will get close to (0) aswill his belief about his chance of success Theagent will thus decide to undertake the projectwhenever c 13 (0) It follows that

V 0 c

0 cgc dc

The only effect of the perception bias is tochange the circumstances under which the ac-tivity is undertaken When 0 there areevents for which

0 c 0

In these events the agent undertakes the activitywhen he should not do so (since they havenegative expected payoff with respect to thetrue probability of success) and in the otherevents he is taking the correct decision So theagentrsquos (true) welfare would be higher if he hadcorrect perceptions This is essentially the argu-ment in classical decision theory that biasingperceptions can only harm agents

B Confidence-Enhanced Performance

When confidence affects performance it isno longer true that correct perceptions maxi-mize long-term payoffs It will still be the casethat agents with biased perceptions will haveoverly optimistic beliefs and consequently beinduced to undertake the activity when theyshould not have On those projects they under-take however their optimism leads to higherperformance that is they have higher probabil-ity of success We will compare these two ef-fects and show that having some degree ofoptimism is preferable to correct perceptions

The key observation (see the Appendix) isthat in the long run the perceived frequency ofsuccess tends to be concentrated around a singlevalue say and the possible values of aretherefore easy to characterize The realized fre-quency of success t (t) is with high prob-ability near () True and perceivedfrequencies of success are thus concentrated

19 Our model puts few constraints on beliefs except inthe long run In the long run if his perceptions are unbiasedour decision maker will asymptotically have correct beliefs

20 For example an initially pessimistic agent undertakesthe project too infrequently

1543VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

around and () respectively The onlypossible candidates for must therefore sat-isfy

(31)

Below we will restrict our attention to thecase where there is a unique such value Underthe assumption 1 this is necessarily thecase when is not too large21 We will discussin the Appendix the more general case whereequation (31) may have several solutions

Figure 3 illustrates geometrically for anagent with biased perceptions (and conse-quently optimistic beliefs)

Note that in the case that perceptions arecorrect coincides with the rational belief defined earlier but that for an agent for whom 0

In the long run the agent with positive per-ception bias thus has better performance than anagent with correct perceptions (() ())but he is overly optimistic about his chances ofsuccess ( ())

Turning to the long-run payoff we obtain

V c

cgc dc

As a benchmark the long-run payoff whenperceptions are correct is equal to

V 0 c

cgc dc

To understand how V() compares to V(0)we write V() V(0) as the sum of threeterms

(32) V V 0

c13

gc dc

cgc dc

cgc dc

The first term is positive it corresponds to theincrease in performance that arises due to opti-mism regarding the activities that would havebeen undertaken even if perceptions had beencorrect The second term is positive it corre-sponds to the realizations of costs for which theactivity is profitable to the agent only becausehe is optimistic Finally the third term is neg-ative it corresponds to the realizations of costsfor which the agent should not have undertakenthe activity and undertakes it because he isoptimistic The shaded regions in Figure 4 rep-resent these three terms when c is uniformlydistributed

One implication of equation (32) is thatwhen confidence positively affects performance( 0) some degree of optimism alwaysgenerates higher long-run payoff There are twoforces operating distorting perceptions distortbeliefsmdashhence decisionsmdashbut they also distortthe technology that maps past outcomes intofuture probabilities of success The first is badthe second is good and there is a tradeoffStarting at the point where perceptions are cor-rect ( 0) the distortion in decisions has a21 This is because for not too large ( ) 1

FIGURE 3

1544 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second order on welfare the distortion appliesto few realizations of costs and for these real-izations the loss is small22 In contrast theimprovement in the technology has a first-ordereffect on welfare Formally

dV

d

0

d

d

0

Pr c

Since (dd)0 0 we obtain

PROPOSITION If 0 there exists a bi-ased perception 0 such that V() V(0)

C Robustness to Alternative Assumptions

The Genesis of ConfidencemdashIn our modelconfidence depends on the agentrsquos perception ofthe empirical frequency of past success Onemotivation for this assumption is the idea thatperceived bad average past performance wouldtrigger anxiety hence a decrease in perfor-mance There are plausible alternative assump-tions concerning the genesis of confidence (orlack of confidence) or more generally how pastperceptions affect current performance

One plausible alternative assumption is thatupon undertaking the project the agent recalls a(possibly small) number of past experiences

and that recollections of failures among thesepast experiences decrease confidence or perfor-mance for example because they generate un-productive intrusive thoughts One could alsoimagine that recent past experiences are morelikely to be recalled

Let us give two more specific examples(i) Suppose that upon undertaking the project

the agent recalls one experience among all re-corded past experiences and that confidence is equal to 1 if that experience was a successand equal to 1 (with (0 1)) if thatexperience was a failure Under that specifica-tion confidence is thus a random variable thatmay take two values (1 or 1 ) with thehigher value having probability ss f

(ii) Suppose that confidence depends on (andis increasing in) the fraction of successes ob-tained in the last T recorded experiences Underthat specification confidence is a random vari-able that may take T 1 values

The first example falls within the scope of ourmodel because it yields an expected perfor-mance equal to ()0 where () 1 We provide in the Appendix an alternativeform of biased perception that encompasses thesecond example where current performance de-pends on the particular history of past percep-tions rather than just the summary statistic Afeature of that model is that a change in date tperceived outcome from failure to success canonly increase later expected confidence Weshow that our result holds with this alternativespecification

The Welfare CriterionmdashAs discussed abovewe evaluate the agentrsquos welfare using long-runor steady-state payoffs only One alternativeassumption is to evaluate welfare using dis-counted payoffs With a discount factor suffi-ciently close to one our insight would clearlyremain valid

For smaller discount factors we would haveto analyze how biased perceptions affect wel-fare in the transition to steady state In transi-tion however our argument does not applySince beliefs may not be correct in transitionthe distortion in beliefs induced by biased per-ceptions may not be second order depending onwhether the agent is currently too optimistic ortoo pessimistic the bias may either increase orreduce the period during which beliefs remain

22 We emphasize that the welfare reduction stemmingfrom the distortion in decisions is a second-order effect onlywhen there is initially no distortion There are clearly casesin which nonmarginal increases in confidence can be verycostly

FIGURE 4

1545VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

evaluating his failure In extreme cases he mayeven convince himself that had the circum-stances been ldquonormalrdquo he would have won thecase and mentally record the event as a success

This basic notion is not new there is evi-dence in the psychology literature that peopletend to attribute positive experiences to thingsthat are permanent and to attribute negativeexperiences to transient effects12 If one gets apaper accepted by a journal one attributes thisto being a good economist while rejections areattributed to the referee not understanding theobviously important point being made Withsuch attribution successes are likely to be per-ceived as predicting further successes whilefailures have no predictive effect

We now formalize these ideas We assumethat after undertaking the activity if the out-come is a failure the agent with probability [0 1) attributes the outcome to atypicalcircumstances Events that have been labelledatypical are not recorded When 0 percep-tions are correct and the true and perceivedfrequencies of success coincide When 0perceptions are biased the agent overestimatesthe frequency of past success

To assess precisely how the perception bias affects the agentrsquos estimate of his past successrate define the function [0 1] 3 [0 1] as

1 1

() is essentially the proportion of recordedevents that are recorded as success when thetrue frequency of success is and is theperception bias True and perceived frequenciesof success are random variables Yet in the longrun as will be shown in the Appendix theirdistributions will be concentrated around a sin-gle value and a true frequency of success willtranslate into a perceived frequency of success

Equivalently for the agent to perceive a successrate of in the long run the true long-run fre-

quency of success has to be equal to ()1()Figure 2 illustrates such a function with 0Note that with our assumptions perceptions areoptimistic they always exceed the true frequencyof past success that is ()

B Decision Process

As described in the outline of our modelabove the agent faces a sequence of decisionson whether or not to undertake a risky activityWe assume that undertaking the activity entailsa cost c possibly because of a lost opportunityFor example in the example above there mightalways be available some riskless alternative totaking on a new case such as drawing up a willWe assume the cost c is stochastic and takesvalues in [0 1] We further assume that therandom variables ctt1

the costs at each timet are independent that the support of the ran-dom variables is [0 1] and that at the time theagent chooses whether to undertake the activityat t he knows the realization of ct

To evaluate whether undertaking the projectis worthwhile the agent forms a belief p aboutwhether he will succeed and we assume thisbelief is based on the data he recollects Thiscan be viewed as a formalization of the avail-ability heuristic only the outcomes that arerecollected are available to the agent We do notmodel however the fine details of how recol-lections of past successes and failures aremapped into the agentrsquos belief about the prob-ability of success on the next try rather wemodel this process as a function

p s f12 See eg Seligman (1990)

FIGURE 2

1541VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

One can think of this function as capturing thedynamic process by which an agent who isinitially unsure about 0 would update his initialbeliefs as he accumulates experience13 Alter-natively one might imagine an agent who fol-lows a simple rule of thumb in forming beliefsThe only restrictions we place on the function are the following14

(i) s f 0 0 (s f ) 1(ii) There exists A 0 such that s f 0

(s f ) s(s f ) 13 A(s f )

The first assumption is a statement that be-liefs must lie between 0 and 1 The secondassumption is an ldquoasymptotic consistencyrdquo con-dition that rules out belief formation processesfor which there is a permanent divergence be-tween the agentrsquos perceived successes and fail-ures and his beliefs when the number ofrecalled past outcomes is large the belief of theagent must approach the perceived empiricalfrequency of past successes

Under these conditions when the data arenot biased beliefs are correct in the longrun We insist on the same conditions holdingeven when the data are biased because ourview is that the agent is ignorant of the factthat the process by which these data are gen-erated might be biased and lead him to biasedbeliefs15

Having formed a belief pt about his chance ofsuccess at date t if he undertakes the activitythe agent compares the expected payoff fromundertaking the activity to the cost of undertak-

ing it That is the agent undertakes the activityif and only if 16

pt ct

Finally we define the expected payoff v(p )to the agent who has a belief p and perception and who does not yet know the realization of thecost of the activity17

v p Prp cE cp c13

This completes the description of the modelThe key parameters of the model are the tech-nology the belief function and the per-ception bias Each triplet ( ) induces aprobability distribution over beliefs percep-tions decisions and outcomes18 We are inter-ested in the limit distribution and the agentrsquosexpected gain under that limit distribution whenthe perception bias is We will verify in theAppendix that this limit distribution is well-defined Formally let

Vt Evpt t

We are interested in the long-run payoff definedby

V limt3

Vt

We will show that V(0) 0 that is that

13 Depending on the priors of the agent and on whatthe agent is trying to estimate Bayesian learning expla-nations for the formation of beliefs might depend on thesequence of successes and failures To facilitate ex-position we have chosen to describe the function as a function of the aggregate numbers of recalledsuccesses and failures only One could alternativelyconsider more complicated functions determining theagentrsquos belief about the probability of success that de-pend on the particular sequencing of recalled successesand failures

14 These restrictions are in addition to the assumptionthat it is only the aggregate numbers of recalled successesand failures that matter

15 We will return to this issue in Section IV and examinethe case where the agent is aware that his data may be biasedand attempts to correct his beliefs accordingly

16 As mentioned above we have not assumed that theagent knows 0 hence he might wish to experiment Wecould allow for more sophisticated rules of behavior ac-counting for the fact that the agent might want to experi-ment for a while and undertake the activity when pt tct

with t 13 1 Our results would easily carry over to this caseas long as with probability 1 there is no experimentation inthe long run t 3 1

17 Note that this expected utility is from the perspectiveof an outside observer since it is calculated with the trueprobability of success (p) From the agentrsquos point of viewthe expected value is

Prp cEp cp c13

18 The distribution at date t also depends on the initialcondition that is the value of confidence when no data areavailable (which has been set equal to 1 see footnote 10)Under the assumption that is not too steep however thelimit distribution is independent of initial condition

1542 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

according to this criterion some bias in percep-tions will make the decision maker better off

Note that our focus on long-run payoffs im-plies that we do not analyze what effect biasedbeliefs have on the decision makerrsquos welfare inthe transition period leading up to steady stateThat biases in perceptions might be welfareenhancing in transition would not be surprisinghowever in our model the decision makertypically starts with erroneous beliefs19 andhence make poor decisions in transition20

Biases in perceptions could induce better de-cisions because they may induce a quickermove toward beliefs that would be closer totrue probability of success We discuss thisfurther in Section IV C

III The Optimality of Biased Perceptions

A A Benchmark The Cost of BiasedPerceptions

We begin with the standard case where con-fidence does not affect performance that is thecase in which 1 independently of Ourobjective is to highlight the potential cost asso-ciated with biased perceptions

The probability of success is equal to 0 Bythe law of large numbers as the number ofinstances where the activity is undertaken in-creases the frequency of success gets close to0 (with probability close to 1) When percep-tions are correct ( 0) the true and perceivedfrequencies of past success coincide Hencegiven our assumption (iii) concerning theagentrsquos belief must converge to 0 as well Itfollows that

V 0 0c

0 cgc dc

where g is the density function for the randomcost

How do biased perceptions affect payoffs

With biased perceptions ( 0) the true fre-quency of success still gets close to 0 (withprobability close to 1) The perceived frequencyof success however will get close to (0) aswill his belief about his chance of success Theagent will thus decide to undertake the projectwhenever c 13 (0) It follows that

V 0 c

0 cgc dc

The only effect of the perception bias is tochange the circumstances under which the ac-tivity is undertaken When 0 there areevents for which

0 c 0

In these events the agent undertakes the activitywhen he should not do so (since they havenegative expected payoff with respect to thetrue probability of success) and in the otherevents he is taking the correct decision So theagentrsquos (true) welfare would be higher if he hadcorrect perceptions This is essentially the argu-ment in classical decision theory that biasingperceptions can only harm agents

B Confidence-Enhanced Performance

When confidence affects performance it isno longer true that correct perceptions maxi-mize long-term payoffs It will still be the casethat agents with biased perceptions will haveoverly optimistic beliefs and consequently beinduced to undertake the activity when theyshould not have On those projects they under-take however their optimism leads to higherperformance that is they have higher probabil-ity of success We will compare these two ef-fects and show that having some degree ofoptimism is preferable to correct perceptions

The key observation (see the Appendix) isthat in the long run the perceived frequency ofsuccess tends to be concentrated around a singlevalue say and the possible values of aretherefore easy to characterize The realized fre-quency of success t (t) is with high prob-ability near () True and perceivedfrequencies of success are thus concentrated

19 Our model puts few constraints on beliefs except inthe long run In the long run if his perceptions are unbiasedour decision maker will asymptotically have correct beliefs

20 For example an initially pessimistic agent undertakesthe project too infrequently

1543VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

around and () respectively The onlypossible candidates for must therefore sat-isfy

(31)

Below we will restrict our attention to thecase where there is a unique such value Underthe assumption 1 this is necessarily thecase when is not too large21 We will discussin the Appendix the more general case whereequation (31) may have several solutions

Figure 3 illustrates geometrically for anagent with biased perceptions (and conse-quently optimistic beliefs)

Note that in the case that perceptions arecorrect coincides with the rational belief defined earlier but that for an agent for whom 0

In the long run the agent with positive per-ception bias thus has better performance than anagent with correct perceptions (() ())but he is overly optimistic about his chances ofsuccess ( ())

Turning to the long-run payoff we obtain

V c

cgc dc

As a benchmark the long-run payoff whenperceptions are correct is equal to

V 0 c

cgc dc

To understand how V() compares to V(0)we write V() V(0) as the sum of threeterms

(32) V V 0

c13

gc dc

cgc dc

cgc dc

The first term is positive it corresponds to theincrease in performance that arises due to opti-mism regarding the activities that would havebeen undertaken even if perceptions had beencorrect The second term is positive it corre-sponds to the realizations of costs for which theactivity is profitable to the agent only becausehe is optimistic Finally the third term is neg-ative it corresponds to the realizations of costsfor which the agent should not have undertakenthe activity and undertakes it because he isoptimistic The shaded regions in Figure 4 rep-resent these three terms when c is uniformlydistributed

One implication of equation (32) is thatwhen confidence positively affects performance( 0) some degree of optimism alwaysgenerates higher long-run payoff There are twoforces operating distorting perceptions distortbeliefsmdashhence decisionsmdashbut they also distortthe technology that maps past outcomes intofuture probabilities of success The first is badthe second is good and there is a tradeoffStarting at the point where perceptions are cor-rect ( 0) the distortion in decisions has a21 This is because for not too large ( ) 1

FIGURE 3

1544 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second order on welfare the distortion appliesto few realizations of costs and for these real-izations the loss is small22 In contrast theimprovement in the technology has a first-ordereffect on welfare Formally

dV

d

0

d

d

0

Pr c

Since (dd)0 0 we obtain

PROPOSITION If 0 there exists a bi-ased perception 0 such that V() V(0)

C Robustness to Alternative Assumptions

The Genesis of ConfidencemdashIn our modelconfidence depends on the agentrsquos perception ofthe empirical frequency of past success Onemotivation for this assumption is the idea thatperceived bad average past performance wouldtrigger anxiety hence a decrease in perfor-mance There are plausible alternative assump-tions concerning the genesis of confidence (orlack of confidence) or more generally how pastperceptions affect current performance

One plausible alternative assumption is thatupon undertaking the project the agent recalls a(possibly small) number of past experiences

and that recollections of failures among thesepast experiences decrease confidence or perfor-mance for example because they generate un-productive intrusive thoughts One could alsoimagine that recent past experiences are morelikely to be recalled

Let us give two more specific examples(i) Suppose that upon undertaking the project

the agent recalls one experience among all re-corded past experiences and that confidence is equal to 1 if that experience was a successand equal to 1 (with (0 1)) if thatexperience was a failure Under that specifica-tion confidence is thus a random variable thatmay take two values (1 or 1 ) with thehigher value having probability ss f

(ii) Suppose that confidence depends on (andis increasing in) the fraction of successes ob-tained in the last T recorded experiences Underthat specification confidence is a random vari-able that may take T 1 values

The first example falls within the scope of ourmodel because it yields an expected perfor-mance equal to ()0 where () 1 We provide in the Appendix an alternativeform of biased perception that encompasses thesecond example where current performance de-pends on the particular history of past percep-tions rather than just the summary statistic Afeature of that model is that a change in date tperceived outcome from failure to success canonly increase later expected confidence Weshow that our result holds with this alternativespecification

The Welfare CriterionmdashAs discussed abovewe evaluate the agentrsquos welfare using long-runor steady-state payoffs only One alternativeassumption is to evaluate welfare using dis-counted payoffs With a discount factor suffi-ciently close to one our insight would clearlyremain valid

For smaller discount factors we would haveto analyze how biased perceptions affect wel-fare in the transition to steady state In transi-tion however our argument does not applySince beliefs may not be correct in transitionthe distortion in beliefs induced by biased per-ceptions may not be second order depending onwhether the agent is currently too optimistic ortoo pessimistic the bias may either increase orreduce the period during which beliefs remain

22 We emphasize that the welfare reduction stemmingfrom the distortion in decisions is a second-order effect onlywhen there is initially no distortion There are clearly casesin which nonmarginal increases in confidence can be verycostly

FIGURE 4

1545VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One can think of this function as capturing thedynamic process by which an agent who isinitially unsure about 0 would update his initialbeliefs as he accumulates experience13 Alter-natively one might imagine an agent who fol-lows a simple rule of thumb in forming beliefsThe only restrictions we place on the function are the following14

(i) s f 0 0 (s f ) 1(ii) There exists A 0 such that s f 0

(s f ) s(s f ) 13 A(s f )

The first assumption is a statement that be-liefs must lie between 0 and 1 The secondassumption is an ldquoasymptotic consistencyrdquo con-dition that rules out belief formation processesfor which there is a permanent divergence be-tween the agentrsquos perceived successes and fail-ures and his beliefs when the number ofrecalled past outcomes is large the belief of theagent must approach the perceived empiricalfrequency of past successes

Under these conditions when the data arenot biased beliefs are correct in the longrun We insist on the same conditions holdingeven when the data are biased because ourview is that the agent is ignorant of the factthat the process by which these data are gen-erated might be biased and lead him to biasedbeliefs15

Having formed a belief pt about his chance ofsuccess at date t if he undertakes the activitythe agent compares the expected payoff fromundertaking the activity to the cost of undertak-

ing it That is the agent undertakes the activityif and only if 16

pt ct

Finally we define the expected payoff v(p )to the agent who has a belief p and perception and who does not yet know the realization of thecost of the activity17

v p Prp cE cp c13

This completes the description of the modelThe key parameters of the model are the tech-nology the belief function and the per-ception bias Each triplet ( ) induces aprobability distribution over beliefs percep-tions decisions and outcomes18 We are inter-ested in the limit distribution and the agentrsquosexpected gain under that limit distribution whenthe perception bias is We will verify in theAppendix that this limit distribution is well-defined Formally let

Vt Evpt t

We are interested in the long-run payoff definedby

V limt3

Vt

We will show that V(0) 0 that is that

13 Depending on the priors of the agent and on whatthe agent is trying to estimate Bayesian learning expla-nations for the formation of beliefs might depend on thesequence of successes and failures To facilitate ex-position we have chosen to describe the function as a function of the aggregate numbers of recalledsuccesses and failures only One could alternativelyconsider more complicated functions determining theagentrsquos belief about the probability of success that de-pend on the particular sequencing of recalled successesand failures

14 These restrictions are in addition to the assumptionthat it is only the aggregate numbers of recalled successesand failures that matter

15 We will return to this issue in Section IV and examinethe case where the agent is aware that his data may be biasedand attempts to correct his beliefs accordingly

16 As mentioned above we have not assumed that theagent knows 0 hence he might wish to experiment Wecould allow for more sophisticated rules of behavior ac-counting for the fact that the agent might want to experi-ment for a while and undertake the activity when pt tct

with t 13 1 Our results would easily carry over to this caseas long as with probability 1 there is no experimentation inthe long run t 3 1

17 Note that this expected utility is from the perspectiveof an outside observer since it is calculated with the trueprobability of success (p) From the agentrsquos point of viewthe expected value is

Prp cEp cp c13

18 The distribution at date t also depends on the initialcondition that is the value of confidence when no data areavailable (which has been set equal to 1 see footnote 10)Under the assumption that is not too steep however thelimit distribution is independent of initial condition

1542 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

according to this criterion some bias in percep-tions will make the decision maker better off

Note that our focus on long-run payoffs im-plies that we do not analyze what effect biasedbeliefs have on the decision makerrsquos welfare inthe transition period leading up to steady stateThat biases in perceptions might be welfareenhancing in transition would not be surprisinghowever in our model the decision makertypically starts with erroneous beliefs19 andhence make poor decisions in transition20

Biases in perceptions could induce better de-cisions because they may induce a quickermove toward beliefs that would be closer totrue probability of success We discuss thisfurther in Section IV C

III The Optimality of Biased Perceptions

A A Benchmark The Cost of BiasedPerceptions

We begin with the standard case where con-fidence does not affect performance that is thecase in which 1 independently of Ourobjective is to highlight the potential cost asso-ciated with biased perceptions

The probability of success is equal to 0 Bythe law of large numbers as the number ofinstances where the activity is undertaken in-creases the frequency of success gets close to0 (with probability close to 1) When percep-tions are correct ( 0) the true and perceivedfrequencies of past success coincide Hencegiven our assumption (iii) concerning theagentrsquos belief must converge to 0 as well Itfollows that

V 0 0c

0 cgc dc

where g is the density function for the randomcost

How do biased perceptions affect payoffs

With biased perceptions ( 0) the true fre-quency of success still gets close to 0 (withprobability close to 1) The perceived frequencyof success however will get close to (0) aswill his belief about his chance of success Theagent will thus decide to undertake the projectwhenever c 13 (0) It follows that

V 0 c

0 cgc dc

The only effect of the perception bias is tochange the circumstances under which the ac-tivity is undertaken When 0 there areevents for which

0 c 0

In these events the agent undertakes the activitywhen he should not do so (since they havenegative expected payoff with respect to thetrue probability of success) and in the otherevents he is taking the correct decision So theagentrsquos (true) welfare would be higher if he hadcorrect perceptions This is essentially the argu-ment in classical decision theory that biasingperceptions can only harm agents

B Confidence-Enhanced Performance

When confidence affects performance it isno longer true that correct perceptions maxi-mize long-term payoffs It will still be the casethat agents with biased perceptions will haveoverly optimistic beliefs and consequently beinduced to undertake the activity when theyshould not have On those projects they under-take however their optimism leads to higherperformance that is they have higher probabil-ity of success We will compare these two ef-fects and show that having some degree ofoptimism is preferable to correct perceptions

The key observation (see the Appendix) isthat in the long run the perceived frequency ofsuccess tends to be concentrated around a singlevalue say and the possible values of aretherefore easy to characterize The realized fre-quency of success t (t) is with high prob-ability near () True and perceivedfrequencies of success are thus concentrated

19 Our model puts few constraints on beliefs except inthe long run In the long run if his perceptions are unbiasedour decision maker will asymptotically have correct beliefs

20 For example an initially pessimistic agent undertakesthe project too infrequently

1543VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

around and () respectively The onlypossible candidates for must therefore sat-isfy

(31)

Below we will restrict our attention to thecase where there is a unique such value Underthe assumption 1 this is necessarily thecase when is not too large21 We will discussin the Appendix the more general case whereequation (31) may have several solutions

Figure 3 illustrates geometrically for anagent with biased perceptions (and conse-quently optimistic beliefs)

Note that in the case that perceptions arecorrect coincides with the rational belief defined earlier but that for an agent for whom 0

In the long run the agent with positive per-ception bias thus has better performance than anagent with correct perceptions (() ())but he is overly optimistic about his chances ofsuccess ( ())

Turning to the long-run payoff we obtain

V c

cgc dc

As a benchmark the long-run payoff whenperceptions are correct is equal to

V 0 c

cgc dc

To understand how V() compares to V(0)we write V() V(0) as the sum of threeterms

(32) V V 0

c13

gc dc

cgc dc

cgc dc

The first term is positive it corresponds to theincrease in performance that arises due to opti-mism regarding the activities that would havebeen undertaken even if perceptions had beencorrect The second term is positive it corre-sponds to the realizations of costs for which theactivity is profitable to the agent only becausehe is optimistic Finally the third term is neg-ative it corresponds to the realizations of costsfor which the agent should not have undertakenthe activity and undertakes it because he isoptimistic The shaded regions in Figure 4 rep-resent these three terms when c is uniformlydistributed

One implication of equation (32) is thatwhen confidence positively affects performance( 0) some degree of optimism alwaysgenerates higher long-run payoff There are twoforces operating distorting perceptions distortbeliefsmdashhence decisionsmdashbut they also distortthe technology that maps past outcomes intofuture probabilities of success The first is badthe second is good and there is a tradeoffStarting at the point where perceptions are cor-rect ( 0) the distortion in decisions has a21 This is because for not too large ( ) 1

FIGURE 3

1544 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second order on welfare the distortion appliesto few realizations of costs and for these real-izations the loss is small22 In contrast theimprovement in the technology has a first-ordereffect on welfare Formally

dV

d

0

d

d

0

Pr c

Since (dd)0 0 we obtain

PROPOSITION If 0 there exists a bi-ased perception 0 such that V() V(0)

C Robustness to Alternative Assumptions

The Genesis of ConfidencemdashIn our modelconfidence depends on the agentrsquos perception ofthe empirical frequency of past success Onemotivation for this assumption is the idea thatperceived bad average past performance wouldtrigger anxiety hence a decrease in perfor-mance There are plausible alternative assump-tions concerning the genesis of confidence (orlack of confidence) or more generally how pastperceptions affect current performance

One plausible alternative assumption is thatupon undertaking the project the agent recalls a(possibly small) number of past experiences

and that recollections of failures among thesepast experiences decrease confidence or perfor-mance for example because they generate un-productive intrusive thoughts One could alsoimagine that recent past experiences are morelikely to be recalled

Let us give two more specific examples(i) Suppose that upon undertaking the project

the agent recalls one experience among all re-corded past experiences and that confidence is equal to 1 if that experience was a successand equal to 1 (with (0 1)) if thatexperience was a failure Under that specifica-tion confidence is thus a random variable thatmay take two values (1 or 1 ) with thehigher value having probability ss f

(ii) Suppose that confidence depends on (andis increasing in) the fraction of successes ob-tained in the last T recorded experiences Underthat specification confidence is a random vari-able that may take T 1 values

The first example falls within the scope of ourmodel because it yields an expected perfor-mance equal to ()0 where () 1 We provide in the Appendix an alternativeform of biased perception that encompasses thesecond example where current performance de-pends on the particular history of past percep-tions rather than just the summary statistic Afeature of that model is that a change in date tperceived outcome from failure to success canonly increase later expected confidence Weshow that our result holds with this alternativespecification

The Welfare CriterionmdashAs discussed abovewe evaluate the agentrsquos welfare using long-runor steady-state payoffs only One alternativeassumption is to evaluate welfare using dis-counted payoffs With a discount factor suffi-ciently close to one our insight would clearlyremain valid

For smaller discount factors we would haveto analyze how biased perceptions affect wel-fare in the transition to steady state In transi-tion however our argument does not applySince beliefs may not be correct in transitionthe distortion in beliefs induced by biased per-ceptions may not be second order depending onwhether the agent is currently too optimistic ortoo pessimistic the bias may either increase orreduce the period during which beliefs remain

22 We emphasize that the welfare reduction stemmingfrom the distortion in decisions is a second-order effect onlywhen there is initially no distortion There are clearly casesin which nonmarginal increases in confidence can be verycostly

FIGURE 4

1545VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

according to this criterion some bias in percep-tions will make the decision maker better off

Note that our focus on long-run payoffs im-plies that we do not analyze what effect biasedbeliefs have on the decision makerrsquos welfare inthe transition period leading up to steady stateThat biases in perceptions might be welfareenhancing in transition would not be surprisinghowever in our model the decision makertypically starts with erroneous beliefs19 andhence make poor decisions in transition20

Biases in perceptions could induce better de-cisions because they may induce a quickermove toward beliefs that would be closer totrue probability of success We discuss thisfurther in Section IV C

III The Optimality of Biased Perceptions

A A Benchmark The Cost of BiasedPerceptions

We begin with the standard case where con-fidence does not affect performance that is thecase in which 1 independently of Ourobjective is to highlight the potential cost asso-ciated with biased perceptions

The probability of success is equal to 0 Bythe law of large numbers as the number ofinstances where the activity is undertaken in-creases the frequency of success gets close to0 (with probability close to 1) When percep-tions are correct ( 0) the true and perceivedfrequencies of past success coincide Hencegiven our assumption (iii) concerning theagentrsquos belief must converge to 0 as well Itfollows that

V 0 0c

0 cgc dc

where g is the density function for the randomcost

How do biased perceptions affect payoffs

With biased perceptions ( 0) the true fre-quency of success still gets close to 0 (withprobability close to 1) The perceived frequencyof success however will get close to (0) aswill his belief about his chance of success Theagent will thus decide to undertake the projectwhenever c 13 (0) It follows that

V 0 c

0 cgc dc

The only effect of the perception bias is tochange the circumstances under which the ac-tivity is undertaken When 0 there areevents for which

0 c 0

In these events the agent undertakes the activitywhen he should not do so (since they havenegative expected payoff with respect to thetrue probability of success) and in the otherevents he is taking the correct decision So theagentrsquos (true) welfare would be higher if he hadcorrect perceptions This is essentially the argu-ment in classical decision theory that biasingperceptions can only harm agents

B Confidence-Enhanced Performance

When confidence affects performance it isno longer true that correct perceptions maxi-mize long-term payoffs It will still be the casethat agents with biased perceptions will haveoverly optimistic beliefs and consequently beinduced to undertake the activity when theyshould not have On those projects they under-take however their optimism leads to higherperformance that is they have higher probabil-ity of success We will compare these two ef-fects and show that having some degree ofoptimism is preferable to correct perceptions

The key observation (see the Appendix) isthat in the long run the perceived frequency ofsuccess tends to be concentrated around a singlevalue say and the possible values of aretherefore easy to characterize The realized fre-quency of success t (t) is with high prob-ability near () True and perceivedfrequencies of success are thus concentrated

19 Our model puts few constraints on beliefs except inthe long run In the long run if his perceptions are unbiasedour decision maker will asymptotically have correct beliefs

20 For example an initially pessimistic agent undertakesthe project too infrequently

1543VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

around and () respectively The onlypossible candidates for must therefore sat-isfy

(31)

Below we will restrict our attention to thecase where there is a unique such value Underthe assumption 1 this is necessarily thecase when is not too large21 We will discussin the Appendix the more general case whereequation (31) may have several solutions

Figure 3 illustrates geometrically for anagent with biased perceptions (and conse-quently optimistic beliefs)

Note that in the case that perceptions arecorrect coincides with the rational belief defined earlier but that for an agent for whom 0

In the long run the agent with positive per-ception bias thus has better performance than anagent with correct perceptions (() ())but he is overly optimistic about his chances ofsuccess ( ())

Turning to the long-run payoff we obtain

V c

cgc dc

As a benchmark the long-run payoff whenperceptions are correct is equal to

V 0 c

cgc dc

To understand how V() compares to V(0)we write V() V(0) as the sum of threeterms

(32) V V 0

c13

gc dc

cgc dc

cgc dc

The first term is positive it corresponds to theincrease in performance that arises due to opti-mism regarding the activities that would havebeen undertaken even if perceptions had beencorrect The second term is positive it corre-sponds to the realizations of costs for which theactivity is profitable to the agent only becausehe is optimistic Finally the third term is neg-ative it corresponds to the realizations of costsfor which the agent should not have undertakenthe activity and undertakes it because he isoptimistic The shaded regions in Figure 4 rep-resent these three terms when c is uniformlydistributed

One implication of equation (32) is thatwhen confidence positively affects performance( 0) some degree of optimism alwaysgenerates higher long-run payoff There are twoforces operating distorting perceptions distortbeliefsmdashhence decisionsmdashbut they also distortthe technology that maps past outcomes intofuture probabilities of success The first is badthe second is good and there is a tradeoffStarting at the point where perceptions are cor-rect ( 0) the distortion in decisions has a21 This is because for not too large ( ) 1

FIGURE 3

1544 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second order on welfare the distortion appliesto few realizations of costs and for these real-izations the loss is small22 In contrast theimprovement in the technology has a first-ordereffect on welfare Formally

dV

d

0

d

d

0

Pr c

Since (dd)0 0 we obtain

PROPOSITION If 0 there exists a bi-ased perception 0 such that V() V(0)

C Robustness to Alternative Assumptions

The Genesis of ConfidencemdashIn our modelconfidence depends on the agentrsquos perception ofthe empirical frequency of past success Onemotivation for this assumption is the idea thatperceived bad average past performance wouldtrigger anxiety hence a decrease in perfor-mance There are plausible alternative assump-tions concerning the genesis of confidence (orlack of confidence) or more generally how pastperceptions affect current performance

One plausible alternative assumption is thatupon undertaking the project the agent recalls a(possibly small) number of past experiences

and that recollections of failures among thesepast experiences decrease confidence or perfor-mance for example because they generate un-productive intrusive thoughts One could alsoimagine that recent past experiences are morelikely to be recalled

Let us give two more specific examples(i) Suppose that upon undertaking the project

the agent recalls one experience among all re-corded past experiences and that confidence is equal to 1 if that experience was a successand equal to 1 (with (0 1)) if thatexperience was a failure Under that specifica-tion confidence is thus a random variable thatmay take two values (1 or 1 ) with thehigher value having probability ss f

(ii) Suppose that confidence depends on (andis increasing in) the fraction of successes ob-tained in the last T recorded experiences Underthat specification confidence is a random vari-able that may take T 1 values

The first example falls within the scope of ourmodel because it yields an expected perfor-mance equal to ()0 where () 1 We provide in the Appendix an alternativeform of biased perception that encompasses thesecond example where current performance de-pends on the particular history of past percep-tions rather than just the summary statistic Afeature of that model is that a change in date tperceived outcome from failure to success canonly increase later expected confidence Weshow that our result holds with this alternativespecification

The Welfare CriterionmdashAs discussed abovewe evaluate the agentrsquos welfare using long-runor steady-state payoffs only One alternativeassumption is to evaluate welfare using dis-counted payoffs With a discount factor suffi-ciently close to one our insight would clearlyremain valid

For smaller discount factors we would haveto analyze how biased perceptions affect wel-fare in the transition to steady state In transi-tion however our argument does not applySince beliefs may not be correct in transitionthe distortion in beliefs induced by biased per-ceptions may not be second order depending onwhether the agent is currently too optimistic ortoo pessimistic the bias may either increase orreduce the period during which beliefs remain

22 We emphasize that the welfare reduction stemmingfrom the distortion in decisions is a second-order effect onlywhen there is initially no distortion There are clearly casesin which nonmarginal increases in confidence can be verycostly

FIGURE 4

1545VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

around and () respectively The onlypossible candidates for must therefore sat-isfy

(31)

Below we will restrict our attention to thecase where there is a unique such value Underthe assumption 1 this is necessarily thecase when is not too large21 We will discussin the Appendix the more general case whereequation (31) may have several solutions

Figure 3 illustrates geometrically for anagent with biased perceptions (and conse-quently optimistic beliefs)

Note that in the case that perceptions arecorrect coincides with the rational belief defined earlier but that for an agent for whom 0

In the long run the agent with positive per-ception bias thus has better performance than anagent with correct perceptions (() ())but he is overly optimistic about his chances ofsuccess ( ())

Turning to the long-run payoff we obtain

V c

cgc dc

As a benchmark the long-run payoff whenperceptions are correct is equal to

V 0 c

cgc dc

To understand how V() compares to V(0)we write V() V(0) as the sum of threeterms

(32) V V 0

c13

gc dc

cgc dc

cgc dc

The first term is positive it corresponds to theincrease in performance that arises due to opti-mism regarding the activities that would havebeen undertaken even if perceptions had beencorrect The second term is positive it corre-sponds to the realizations of costs for which theactivity is profitable to the agent only becausehe is optimistic Finally the third term is neg-ative it corresponds to the realizations of costsfor which the agent should not have undertakenthe activity and undertakes it because he isoptimistic The shaded regions in Figure 4 rep-resent these three terms when c is uniformlydistributed

One implication of equation (32) is thatwhen confidence positively affects performance( 0) some degree of optimism alwaysgenerates higher long-run payoff There are twoforces operating distorting perceptions distortbeliefsmdashhence decisionsmdashbut they also distortthe technology that maps past outcomes intofuture probabilities of success The first is badthe second is good and there is a tradeoffStarting at the point where perceptions are cor-rect ( 0) the distortion in decisions has a21 This is because for not too large ( ) 1

FIGURE 3

1544 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second order on welfare the distortion appliesto few realizations of costs and for these real-izations the loss is small22 In contrast theimprovement in the technology has a first-ordereffect on welfare Formally

dV

d

0

d

d

0

Pr c

Since (dd)0 0 we obtain

PROPOSITION If 0 there exists a bi-ased perception 0 such that V() V(0)

C Robustness to Alternative Assumptions

The Genesis of ConfidencemdashIn our modelconfidence depends on the agentrsquos perception ofthe empirical frequency of past success Onemotivation for this assumption is the idea thatperceived bad average past performance wouldtrigger anxiety hence a decrease in perfor-mance There are plausible alternative assump-tions concerning the genesis of confidence (orlack of confidence) or more generally how pastperceptions affect current performance

One plausible alternative assumption is thatupon undertaking the project the agent recalls a(possibly small) number of past experiences

and that recollections of failures among thesepast experiences decrease confidence or perfor-mance for example because they generate un-productive intrusive thoughts One could alsoimagine that recent past experiences are morelikely to be recalled

Let us give two more specific examples(i) Suppose that upon undertaking the project

the agent recalls one experience among all re-corded past experiences and that confidence is equal to 1 if that experience was a successand equal to 1 (with (0 1)) if thatexperience was a failure Under that specifica-tion confidence is thus a random variable thatmay take two values (1 or 1 ) with thehigher value having probability ss f

(ii) Suppose that confidence depends on (andis increasing in) the fraction of successes ob-tained in the last T recorded experiences Underthat specification confidence is a random vari-able that may take T 1 values

The first example falls within the scope of ourmodel because it yields an expected perfor-mance equal to ()0 where () 1 We provide in the Appendix an alternativeform of biased perception that encompasses thesecond example where current performance de-pends on the particular history of past percep-tions rather than just the summary statistic Afeature of that model is that a change in date tperceived outcome from failure to success canonly increase later expected confidence Weshow that our result holds with this alternativespecification

The Welfare CriterionmdashAs discussed abovewe evaluate the agentrsquos welfare using long-runor steady-state payoffs only One alternativeassumption is to evaluate welfare using dis-counted payoffs With a discount factor suffi-ciently close to one our insight would clearlyremain valid

For smaller discount factors we would haveto analyze how biased perceptions affect wel-fare in the transition to steady state In transi-tion however our argument does not applySince beliefs may not be correct in transitionthe distortion in beliefs induced by biased per-ceptions may not be second order depending onwhether the agent is currently too optimistic ortoo pessimistic the bias may either increase orreduce the period during which beliefs remain

22 We emphasize that the welfare reduction stemmingfrom the distortion in decisions is a second-order effect onlywhen there is initially no distortion There are clearly casesin which nonmarginal increases in confidence can be verycostly

FIGURE 4

1545VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second order on welfare the distortion appliesto few realizations of costs and for these real-izations the loss is small22 In contrast theimprovement in the technology has a first-ordereffect on welfare Formally

dV

d

0

d

d

0

Pr c

Since (dd)0 0 we obtain

PROPOSITION If 0 there exists a bi-ased perception 0 such that V() V(0)

C Robustness to Alternative Assumptions

The Genesis of ConfidencemdashIn our modelconfidence depends on the agentrsquos perception ofthe empirical frequency of past success Onemotivation for this assumption is the idea thatperceived bad average past performance wouldtrigger anxiety hence a decrease in perfor-mance There are plausible alternative assump-tions concerning the genesis of confidence (orlack of confidence) or more generally how pastperceptions affect current performance

One plausible alternative assumption is thatupon undertaking the project the agent recalls a(possibly small) number of past experiences

and that recollections of failures among thesepast experiences decrease confidence or perfor-mance for example because they generate un-productive intrusive thoughts One could alsoimagine that recent past experiences are morelikely to be recalled

Let us give two more specific examples(i) Suppose that upon undertaking the project

the agent recalls one experience among all re-corded past experiences and that confidence is equal to 1 if that experience was a successand equal to 1 (with (0 1)) if thatexperience was a failure Under that specifica-tion confidence is thus a random variable thatmay take two values (1 or 1 ) with thehigher value having probability ss f

(ii) Suppose that confidence depends on (andis increasing in) the fraction of successes ob-tained in the last T recorded experiences Underthat specification confidence is a random vari-able that may take T 1 values

The first example falls within the scope of ourmodel because it yields an expected perfor-mance equal to ()0 where () 1 We provide in the Appendix an alternativeform of biased perception that encompasses thesecond example where current performance de-pends on the particular history of past percep-tions rather than just the summary statistic Afeature of that model is that a change in date tperceived outcome from failure to success canonly increase later expected confidence Weshow that our result holds with this alternativespecification

The Welfare CriterionmdashAs discussed abovewe evaluate the agentrsquos welfare using long-runor steady-state payoffs only One alternativeassumption is to evaluate welfare using dis-counted payoffs With a discount factor suffi-ciently close to one our insight would clearlyremain valid

For smaller discount factors we would haveto analyze how biased perceptions affect wel-fare in the transition to steady state In transi-tion however our argument does not applySince beliefs may not be correct in transitionthe distortion in beliefs induced by biased per-ceptions may not be second order depending onwhether the agent is currently too optimistic ortoo pessimistic the bias may either increase orreduce the period during which beliefs remain

22 We emphasize that the welfare reduction stemmingfrom the distortion in decisions is a second-order effect onlywhen there is initially no distortion There are clearly casesin which nonmarginal increases in confidence can be verycostly

FIGURE 4

1545VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

far from the truth Thus we cannot be assuredthat the improvement in technology dominatesthe distortion of decisions as a result the effectof the bias is indeterminate

Nevertheless if one were to evaluate welfareusing discounted payoffs it might be legitimateto allow for a bias in perception that would betime dependent In that case our model suggeststhat welfare is increased when is positive inthe long run

IV Discussion

We have identified circumstances in whichhaving biased perceptions increases welfareHow should one interpret this result Our viewis that it is reasonable to expect that in suchcircumstances agents will end up holding opti-mistic beliefs and the welfare that they derivewhen they are optimistic is greater than whentheir beliefs are correct

We do not however conceive optimism asbeing the outcome of a deliberate choice by theagent In our model agents do not choose per-ceptions or beliefs Rather these are the productof recalled past experiences which are them-selves based on the types of attributions agentsmake upon failing for example Our view isthat there are undoubtedly limits to the extent towhich perceptions can be biased but that it isreasonable to expect that the agentsrsquo attributionsadjust some so as to generate some bias inperceptions

We have described earlier a particularly sim-ple form of biased attribution We do nothowever suggest that precisely this type ofattributions should arise Other types of attribu-tions as well as other types of information-processing biases may lead to optimistic beliefs

We start by providing some examples Wethen question the main predictions of our mod-el that rational agents would persistently havebiased perceptions and be persistently overcon-fident about their chances of success

A Alternative Information-Processing Biases

In the model analyzed above the agent some-times attributes the cause of failure to eventsthat are not likely to occur again in the futurewhile he never makes such attributions in case

of success (nonpermanence of failureperma-nence of success) An alternative way in whichan agentrsquos perceptions may be biased is thatrather than attributing failures to transient cir-cumstances he attributes successes to a broaderset of circumstances than is appropriate in thepsychology literature this is referred to as per-vasiveness of success bias (see eg Seligman1990)

To illustrate this idea of pervasiveness con-sider again the example of the lawyer Supposethat cases that are available to the lawyer can beof two types high-profile cases that will attractpopular attention and low-profile cases that willnot Suppose that in high-profile cases therewill be many more people in court includingreporters and photographers and consequentlymuch more stress which we assume impairsperformance Suppose that one day the lawyerwins a case when there was no such attentionAlthough the lawyer realizes that success wouldhave been less likely had there been attentionhe thinks ldquoThe arguments I made were so goodI would have won even if this had been ahigh-profile caserdquo In thinking this way the law-yer is effectively recording the experience as asuccess whether it had been high profile or not23

Another channel through which beliefs aboutchances of success may become biased is whenthe agent has a biased memory technology andfor example tends to remember more easilypast successes than past failures24 Optimismmay also stem from self-serving bias Such biasoccurs in settings where there is not a compel-ling unique way to measure success When

23 More generally this is an example of how inferencesmade at the time of the activity affect later recollection ofexperiences This inability to distinguish between outcomesand inferences when one attempts to recall events consti-tutes a well-documented source of bias A compelling ex-ample demonstrating this is given by Sharon L Hanniganand Mark T Reinitz (2001) Subjects were presented asequence of slides depicting ordinary routines and laterreceived a recognition test The sequence of slides wouldsometimes contain an effect scene (oranges on the floor) andnot a cause scene (woman taking oranges from the bottomof pile) and sometimes a cause scene and not an effectscene In the recognition test participants mistook newcause scenes as old when they had previously viewed theeffect

24 See William N Morris (1999) for a survey of theevidence that an individualrsquos mood affects his or her recol-lection process

1546 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

giving a lecture a speaker might naturally feel itwas a success if the audience enthusiasticallyapplauds If unfortunately that doesnrsquot occurbut an individual comes up to him following thelecture and tells him that it was very stimulat-ing he can choose that as a signal that thelecture was a ldquosuccessrdquo

B Can a Rational Agent Fool Himself

One prediction of our model is that whenperceptions affect performance rational agentsprefer that their perceptions are biased How-ever can a rational agent fool himself

Imagine for example an agent who sets hiswatch ahead in an attempt to ensure that he willbe on time for meetings Suppose that when helooks at his watch the agent takes the dataprovided by the watch at face value In thisway the agent fools himself about the correcttime and he easily arrives in time for eachmeeting One suspects however that thistrick may work once or twice but that even-tually the agent will become aware of hisattempt to fool himself and not take the watchtime at face value

There are two important differences betweenthis example and the problems we addressFirst in the example the agent actively makesthe decision to ldquofoolrdquo himself and sets hiswatch accordingly In our model one should notthink of the agent choosing to bias his percep-tions but rather the bias in perceptions is sub-conscious and arises to the extent that the biashas instrumental value that is increases thewelfare of people with the bias The seconddifference is that the individual who sets hiswatch ahead will get constant feedback aboutthe systematic difference between correct timeand the time shown on his watch Our model isaimed at problems in which this immediate andsharp feedback is absent An academic giving atalk a student writing a term paper and a quar-terback throwing a pass have substantial lati-tude in how they perceive a quiet reception totheir talk a poor grade on the paper or anincompletion Each can decide that his perfor-mance was actually reasonably good and theseeming lack of success was due to the fault ofothers or to exogenous circumstances Impor-tantly an individual with biased perceptionsmay never confront the gap in the way one who

sets his watch ahead would While it may behard for people actively to fool themselves itmay be relatively easy (and potentially benefi-cial) to be passively fooled25

How easy is it to be fooled This presumablydepends on the accuracy of the feedback theagent gets about his performance One canexpect that fooling oneself is easier when theoutcome is more ambiguous because it isthen easier to interpret the outcome as a suc-cess Along these lines putting agents inteams is likely to facilitate biased interpreta-tions of the outcomes and make it easier foreach agent to keep his confidence high onecan always put the blame on others in case offailure26 These observations suggest thatwhen confidence affects performance theremay be a value to designing activities in a waythat facilitates biases in information processing

C What If the Agent Is Aware ThatConfidence Matters

A related objection to our model might bethat when forming beliefs a more sophisticatedagent would take into account the fact that thedata may be biased This would be a reason-able assumption if the agent is aware thatconfidence matters and understands the pos-sible effect that this may have on his attribu-tions Would we still obtain optimistic beliefs(hence overconfidence) as a prediction of ourmodel if we allowed the agent to reassess hisbeliefs over chances of success to account fora possible overrepresentation of successes inthe data

The bias in perception clearly has instru-mental value but the resulting bias in belief(hence the agentrsquos overconfidence) has noinstrumental value The agent might try re-assessing his beliefs regarding chances ofsuccess to benefit from biased perceptions(through increased performance) without in-curring the cost of suboptimal decision mak-ing induced by overconfidence

25 This is of course not an argument that there are nolimits on the extent to which an agent can be fooled Weargue only that it is plausible that the limit is not zero

26 There may of course be a cost to agents in gettingless reliable feedback that offsets at least partially the gainfrom the induced high confidence

1547VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

These second thoughts in evaluating chancesof success could limit however the benefit thatstems from biased perception Reevaluating hischances of success may lead the agent to reas-sess his perceptions over past successes andfailures as well Introspection about his beliefsmay lead him to recall past failures otherwiseforgotten leading in turn to diminished futureperformance Thus even if the agent could learnto reevaluate his chance of success in a way thatmaximizes long-run welfare we should expecta reduced reevaluation that does not completelyoffset the bias in perception

More formally sophisticated agents who un-derstand that their data may be biased mightadjust their prediction of success on the next trydownward to say p (s f ) where 13 1captures the extent of the adjustment (Thestronger the bias as perceived by the agent thegreater the adjustment) As a result the perception would differ from the belief p in the long runand the agentrsquos overconfidence might be reducedThe comments above however suggest that forthese agents performance would depend not onlyon first-hand perception but also on the extentto which the agent thinks his data are biased

where is increasing in (a larger adjustmentdiminishes performance)27

We have examined the case where is setequal to 1 that is the agent is unaware that hisperceptions are biased and there is no adjust-ment We note that for any fixed 1 ouranalysis would carry over to this more generalsetting For any fixed 1 there exists a bias that exactly offsets the adjustment andleads to correct beliefs in the long run From () a small further increase in the biasmust be welfare increasing because the induceddistortion in decisions has only a second-ordereffect on welfare In other words if a sophisticatedagent moderates his prediction of success on thenext attempt to offset the bias in perception afurther bias in perception is welfare increasing

But in this more general setting our argu-ment applies even if the bias in perception isfixed or if no further bias is possible For anyfixed bias there exists an adjustment thatexactly offsets the bias From () a smallincrease in (that is less adjustment of theagentrsquos beliefs) is welfare increasing as wellThus if both the bias and the adjustment thatthe agent thinks appropriate tend to evolve in away that maximizes long-run welfare then weshould expect that the agent underestimates theextent to which the data are biased (hence theoverconfidence)28 Therefore while in our mainmodel the decision maker was unaware of thepossibility that his data were biased (ie 0but 1) in this extended model the agentwill be overconfident if he is simply unaware ofthe extent to which the data are biased ( ()) a much weaker assumption

Finally we wish to point out a simple exten-sion of our model that would generate overcon-fidence even if no relationship between p and were imposed Though we have assumed thatconfidence depends solely on perceptions aquite natural assumption would be that confi-dence also depends on the beliefs of the agentabout the likelihood of success that is

p

where both beliefs and perceptions may have apositive effect on confidence This is a simpleextension of our model that would confer in-strumental value to beliefs being biased29

D Related Literature

There is a growing body of literature in eco-nomics dealing with overconfidence or confi-dence management We now discuss several ofthese papers and the relation to our work

27 Whether an adjustment has a large or small effect onperformance could for example depend on whether theagent can simultaneously hold two somewhat contradictorybeliefsmdashbiased perceptions and unbiased or less biasedbeliefs This ability is clearly welfare enhancing

28 Depending on the shape of more sophistication (ie 1) need not be welfare increasing

29 More generally our insight could be applied to psy-chological games (see eg John Geanakoplos et al 1989and Rabin 1993) In these games beliefs affect utilities soif agents have some (possibly limited) control over theirattributional styles they should learn to bias their beliefs ina way that increases welfare and in the long run we shouldnot expect beliefs to be correct Our analysis thus questionswhether the standard assumption that beliefs should becorrect in equilibrium is a valid one

1548 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

One important modelling aspect concerns thetreatment of information In Roland Benabouand Jean Tirole (2002) and Botund Koszegi(2000) the agent tries to assess how able he isin a purely Bayesian way he may actively de-cide to forget selectively some negative experi-ences (Benabou and Tirole) or stop recordingany further signal (Koszegi) but he is perfectlyaware of the way information is processed andproperly takes it into account when formingbeliefs Thus beliefs are not biased on averageonly the distribution over posterior beliefs isaffected (which is why referring to confidencemanagement rather than overconfidence may bemore appropriate to describe this work) An-other notable difference is our focus on long-runbeliefs For agents who in the long run have aprecise estimate of their ability the Bayesianassumption implies that this estimate must bethe correct one (with probability close to 1) Soin this context confidence management essen-tially has only temporary effects

Concerning the treatment of information ourapproach is closer in spirit to that of Rabin andSchrag (1999) They analyze a model of ldquocon-firmatory biasrdquo in which agents form firstimpressions and bias future observations ac-cording to these first impressions In formingbeliefs agents take their ldquoobservationsrdquo at facevalue without realizing that their first impres-sions biased what they believed were their ldquoob-servationsrdquo Rabin and Schrag then focus on howthis bias affects beliefs in the long run and findthat on average the agent is overconfident that ison average he puts more weight than a Bayesianobserver would on the alternative he deems mostlikely A notable difference with our work is thatin their case the bias has no instrumental value

Another important aspect concerns the under-lying reason why information processing biasesmay be welfare enhancing In Michael Wald-man (1994) or Benabou and Tirole (2002) thebenefit stems from the fact that the agentrsquoscriterion for decision making does not coincidewith welfare maximization while in Koszegithe benefit stems from beliefs being directly partof the utility function30

Waldman (1994) analyzes an evolutionarymodel in which there may be a divergence be-tween private and social objectives He consid-ers a model of sexual inheritance of traits (nodisutility of effortdisutility of effort correctassessment of abilityoverconfidence) in whichfitness would be maximized with no disutility ofeffort and correct assessment of ability Disutil-ity of effort leads to a divergence between thesocial objective (offspring) and the private ob-jective (utility) As a result individuals whopossess the trait ldquodisutility of effortrdquo havehigher fitness if they are overconfident (becausethis induces greater effort)31

In Benabou and Tirole (2002) the criterionfor decision making also diverges from (exante) welfare maximization Benabou and Ti-role study a two-period decision problem inwhich the agent has time-inconsistent preferencesAs a consequence of this time inconsistency thedecision criterion in the second period does notcoincide with ex ante welfare maximization andbiased beliefs in this context may induce decisionsthat are better aligned with welfare maximization

To illustrate formally why a divergence be-tween private objectives and welfare gives in-strumental value to biased beliefs assume thatin our model performance is independent ofbeliefs but that the agent undertakes the activityonly when

(41) p c with 1

One interpretation is that the agent lacks the willto undertake the risky activity and that he un-dertakes it only when its expected return ex-ceeds 1 by a sufficient amount32

30 See Larry Samuelson (2004) for another example of amodel in which an incorrectly specified decision criterion leadsto welfare enhancing biases and Hanming Fang (2001) and

Aviad Heifetz and Yossi Spiegel (2001) for examples of mod-els where in a strategic context beliefs are assumed to beobservable hence directly affect opponentsrsquo reactions

31 Waldmanrsquos (1994) main insight is actually strongerhe shows that the trait ldquodisutility of effortrdquo and hence thedivergence between private and social objectives may ac-tually persist the combination of disutility of effort andoverconfidence in ability can be evolutionarily stable Thereason is that for individuals who possess the trait ldquoover-confidencerdquo disutility of effort is optimal so the combina-tion disutility of effort and overconfidence in ability is alocal maximum for fitness (only one trait may change at atime)

32 This formulation can actually be viewed as a reducedform of the agentrsquos decision problem in Benabou and Tirole

1549VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Given this criterion for making the decisionwhether to undertake the risky activity or notwe may write expected welfare as a function ofthe bias We obtain

w c

cgc dc

which implies

w

d

dg

Hence w() has the same sign as [()]and is positive at 0 (correct beliefs) since0() and 1 Intuitively biased beliefsallow the agent to make decisions that are betteraligned with welfare maximization

Finally Eric Van den Steen (2002) providesanother interesting model of overconfidencethat explains why most agents would in generalbelieve they have higher driving ability than theaverage population Unlike some of the workmentioned above this overconfidence does notstem from biases in information processing butfrom the fact that agents do not use the samecriterion to evaluate their strength

APPENDIX A CONVERGENCE RESULT

Recall that () 0() Define that is

1 1

We first assume that the equation

(42)

has a unique solution denoted We wish to show that in the long run the perception getsconcentrated around Formally we will show that for any 033

lim t Prt 13 1

Our proof builds on a large deviation result in statistics The proof of this result as well as that ofthe Corollary that builds on it are in Appendix B

LEMMA 1 (large deviations) Let X1 Xn be n independent random variables and let S yeni XiAssume Xi M for all i Define m (1n)E yeni Xi and h(x S) supt0 xt (1n) yeni ln E exp tXiFor any x m we have

Pr1

n i

Xi x 13 exp nhx S

(2001) where is interpreted as a salience for the presentTo see why assume that (i) there are three periods (ii) thedecision to undertake the activity is made in period 2 (iii)the benefits are enjoyed in period 3 and (iv) there is nodiscounting The salience for the present has the effect ofinflating the cost of undertaking the activity in period 2 henceinduces a divergence from ex ante welfare maximization (thatwould prescribe undertaking the activity whenever p c)

33 In what follows date t corresponds to the tth attempt to undertake the project That is we ignore dates at which theproject is not undertaken This can be done without loss of generality because at dates where the project is not undertakenthe perception does not change

1550 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Besides for any x m h(x S) h0 for some h0 0 that can be chosen independentlyof n

To assess how the perception evolves from date t to date t T we derive below bounds on thenumber of outcomes N and the number of successes S that are recorded during that period of timeThese bounds will be obtained as a Corollary of Lemma 1

Formally consider the random variable yt which takes value 0 or 1 depending on whether theoutcome is failure or a success and the random variable zt which takes value 1 or 0 depending onwhether the agent is (or is not) subject to a perception bias at t (the outcome is not recorded in theevent yt 0 zt 1) Now consider a sequence of T projects undertaken at date t t T 1and suppose that performance at any of these dates is bounded above by The numbers S ofrecorded success and N of recorded outcomes are respectively S yeni0

T1 yt i and N yeni0T1 yt i

(1 yt i)(1 zt i) We have

COROLLARY 2 Assume t i 13 for all i 0 T 1 Then for any small x 0 there existsh 0 that can be chosen independently of T such that

PrN

T 1 2 and

S

N

1 1 x 1 exp hT

Corollary 2 implies that when perceptions remain below so that performance remains boundedabove by () then the ratio of recorded successes SN cannot exceed () by much Convergenceto will then result from the fact that as long as exceeds () is below More preciselysince is smooth since (0) 0 and (1) 1 and since (42) has a unique solution we have

LEMMA 3 For any 0 there exists 0 such that for any ()

In words when current perceptions exceed by more than the adjusted probability of success is below current perceptions by more than Combining this with Corollary 2 we should expectperceptions to go down after a large enough number of periods The next Lemma makes thisstatement precise

LEMMA 4 Fix 0 and (0 ) as in Lemma 3 and let (2)((1 )2) There existsh 0 and t 0 such that for any date t0 t for any T 13 t0 and for any

Prt0T t0 T

t0 T

2t0 1 exp hT

PROOFLet nt0

st0 ft0

For t large enough and t0 t the event nt013 (1 )t02 has probability at most

equal to exp ht0 for some h independent of t0 Consider now the event nt0 (1 )t02 Let S

and N be the number of successes and recorded events during t0 t0 T 1 We have

(43) t0T st0 S

nt0 N

nt0

nt0 Nt0

N

nt0 N

S

N t0

N

nt0 N S

N t0

During this period of time t remains below t0 Tn0 hence below t0

2 It follows that tremains below (t0

2) We now apply Corollary 2 with x 4 and choose h accordingly Theevent (SN) (t0

2) 4 and N (1 )T2 thus has probability 1 exp hT Underthat event (and using Lemma 3) we have (SN) t0

4 which implies (using 43)

1551VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

t0T t0 1 T2

t0 1 T24

We can now conclude the proof We will show that for t0 large enough (i) for any t0

the perception eventually gets below with probability close to 1 and (ii) if t0

then the perception always remains below 2 with probability close to 1(i) This is obtained by iterative application of Lemma 4 We set tk (1 )tk1 So long as tk

we may apply Lemma 4 and get that tk1

tk with probability 1 exp htk where

2(2( 1)) We thus obtain (after at most K (1 ) iterations)

(44) Pr t tK t t0 1 exp ht0K

Since K and h are independent of t0 the right-hand side is arbitrarily close to 1 when t0 is large(ii) Consider the event As where gets above 2 for the first time (after t0) at date s For

any s t0 choose ts such that (1 )ts s Under the event nt (1 )t2 at least (1 )t2(t) periods are required for to get from to 2 so we must have ts

and we may apply Lemma 4 to obtain

PrAsnts 1 ts 2 13 exp h0s

for some h0 independent of s which further implies

(45) Pr st0

As 13 st0

Prnts 1 ts 2 PrAsn 1 ts 2 13 exp ht0

for some h independent of t0Combining (44) and (45) we finally obtain

lim Prt 1

Proving that for any 0 lim Prt 1 can be done similarly

Generalization

CONVERGENCEOur convergence result can be easily generalized to the case where equation (42) has a finite

number of solutions To fix ideas consider the case where there are three solutions denoted i i 1 2 3 with 1 2 3 While Lemma 3 no longer holds the following weaker statement does

For any 0 there exists 0 such that for any (1 2 ) and any 3 ()

Lemma 4 thus applies to any (1 2 2) or any 3 and convergence resultsas before from (iterative) application of Lemma 4 For example if t0

13 2 2 then withprobability 1 exp ht0 t will get below 1 and never get back above 1 2

MONOTONICITYTo prove our main result (Proposition 1) we used the fact that when increases the (unique)

solution to equation (42) increases as well When there are several solutions to (42) this propertydoes not hold for all solutions In particular it does not hold when for the solution i considered

1552 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(i) 1 as would typically be the case for i 2 Nevertheless these solutions are unstable evenif convergence to such unstable solutions occurs with positive probability for some values of 34 aslightly higher value of would make these trajectories converge to a higher (and stable) solution

Intuitively a higher bias has two effects (i) it reduces the number of failures that are recorded(failures that would have been recorded under are not recorded with probability ) hence itmechanically increases the perception of past success rate t (ii) because t increases there is alsoan indirect effect through increased confidence successes are more likely

Both these effects go in the direction of increasing t and the first effect alone increases t by apositive factor with probability close to 1 When 2() 13 2() trajectories leading to 2() whenthe bias is equal to must lead (with probability one) to trajectories leading to higher long-runperception when the bias is equal to The only possible candidate is then 3()

Robustness

We next examine the robustness of our result to alternative formulations where performance atdate t depends on the sequence of the most recent perceived outcomes rather than on the summarystatistic

Formally at any date t at which the project is undertaken the outcome yt may take two valuesand we set yt 0 or 1 depending on whether the outcome is failure or a success We consider thepossibility that the agentrsquos perception is biased as follows with probability a failure is actuallyperceived as a success We denote by Yt the history of outcomes by yt the perceived outcome at tand by Yt (respectively Yt

T) the history of perceived outcomes (respectively last T perceivedoutcomes)35 We assume that performance at t depends on the sequence Yt

T

t YtT

and that performance at t is increasing in YtT that is there exists a positive number a such that for

all histories YT1 (of length T 1)

YT 1 1 YT 0 a

As in our model the triplet ( ) induces a probability distribution over beliefs and histories ofperceptions Although performance no longer converges to a point distribution it is standard toshow that Yt

T converges to a unique invariant distribution It is not difficult then to show that the per-ceived frequency of success hence beliefs also converges to some p()36 Long-run expectedwelfare may thus be written as

V E c13p

YtT c13gc dc

As in our model it then follows that starting from 0 (where in the long run beliefs are correctand coincide with lim E0(Yt

T)) a positive bias 0 induces a distortion in decisions that has a

34 Convergence to such unstable solutions cannot be excluded for some values of 35 Note that we omit dates on which the project is not undertaken because performance will be assumed to depend only

on the sequence of perceived outcomes and not on the actual date at which these outcomes occurred Thus we have Yt (y1 yt1) Yt ( y1 yt1)Yt

T Yt for t 13 T and YtT ( ytT yt1) for t T

36 Alternatively we could have assumed that the agent attempts to estimate the probability of success at the next trialcontingent on the last T perceived outcomes (and that these conditional beliefs each get close to the corresponding empiricalfrequencies) Conditional on YT the agentrsquos belief would then converge to some p( YT) that coincides with (Yt

T) when thereis no bias ( 0)

1553VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

second-order effect on welfare To show that our result carries over to this setting we need to checkthat the increase in generates a first-order increase in long-run expected performance

To verify this we consider the two stochastic processes associated respectively with 0 and 0 0 Consider a sequence Z (zt)t1 where zt 0 1 for all t which captures whether the agentmay or may not be subject to a perception bias at t Formally we define the function yt b(yt zt)such that yt 1 if yt 1 or zt 1 and yt 0 otherwise37 The function Yt Bt(Yt Zt) where Ztdenotes the truncated sequence (z1 zt1) is defined accordingly The first stochastic process ( 0) corresponds to the case where zt 0 for all t The second stochastic process corresponds to thecase where (zt)t1 is a sequence of iid random variable that takes value 0 with probability 1 0and 1 with probability 0

Now denote by X ( xt)t1 a sequence of iid random variables uniformly distributed on [0 1]The outcome at date t can be viewed as a function of Yt and the realization xt where yt Rt(Yt xt) 1 if xt (Yt) and yt 0 otherwise So the outcome Yt1 can be viewed as a function of Zt and Xt1say Yt1 St1(Zt Xt1) Since Bt is non-decreasing in Yt and in Zt and since Rt is non-decreasingin Yt we obtain (by induction on t) that for all t St is non-decreasing in Zt

For any given realization X of X let Yt0 denote the sequence of outcomes that obtains up to date

t under the first stochastic process Yt0 St(0 Xt) For any given realizations (Xt Zt) the difference

between expected performance at t under the two stochastic processes can be written as

Bt St Zt 1 Xt Zt St 0 Xt

and since St is non-decreasing in Zt this difference is at least equal to

Bt Yt0 Zt Yt

0

Then a first-order increase in performance would obtain in the long run because at any date theevent (yt zt) (0 1) has positive probability38

APPENDIX B LARGE DEVIATIONS RESULTS

PROOF OF LEMMA 1For any z 0 we have

(46) PrS

n x PrzS

n x 0 Prexp zS

n x 1 13 E exp zS

n x

Since the random variables Xi are independent

E expz

n i

Xi i

E expz

nXi

We may thus write letting t (zn)

37 Note that b(yt zt) max(yt zt) so b( ) is non-decreasing in each of its arguments38 Note that the argument above is quite general and applies even when performance depends on the whole history as was

the case in the main body of the paper Indeed in that case when the number of trials becomes large the number ofrealizations for which (yt zt) (0 1) is (with positive probability) a positive (and non-vanishing) fraction of the number oftrials So (Bt(Yt

0 Zt)) (Yt0) is non-negative and bounded away from 0 with positive probability

1554 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

E exp zS

n exp ln i

E exp tXi exp i

ln E exp tXi

which implies using (46)

(47) PrS

n x 13 exp ntx

1

n i

ln exp tXi

Since (47) holds for any t 0 we obtain the desired inequality Besides since Xi M for all ifor t close to 0 xt (1n) yeni ln E exp tXi t(x m ) O(t2) hence for x m h(x S) is boundedbelow by some h0 0 that depends only on M and x m

PROOF OF COROLLARY 2For any small x 0 we set x0 such that

x0

x0 1 x0 1 x0

1 1 x

Choose x small enough so that x0 (1 )2 Consider first the hypothetical case whereperformance would be equal to at all dates Then S is the sum of T independent random variablesApplying Lemma 1 to the sum S we get that there exists h0 0 that can be chosen independentlyof T such that

(48) PrS

T x0 13 exp h0T

where the index indicates that outcomes are generated assuming performance is equal to Becauseperformance actually depends on past realizations S is not the sum of independent variables Underthe assumption that performance remains bounded above by however we have

PrS

T x0 13 PrS

T x0

Consider now realizations of ST for which 13 x0 Conditional on a realization of STyeni (1 zt i)(1 yt i) is the sum of T S T(1 ) independent and identical random variableseach having (1 ) as expected value Applying Lemma 1 we thus obtain

(49) PrN

T 1 1 x0S

T 13 exp h11 n

for some h1 0 that can be chosen independently of n Set h2 h1(1 ( x0)) Since (1 )(1 x0) (1 )2 and since for any 13 x0 (by definition of x0)

1 1 x

1 1 x0

and inequality (49) implies

1555VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

PrN

T 1 2 or

S

N

1 1 xS

T13 x0 13 exp h2T

Combining this with (48) we obtain the desired inequality

REFERENCES

Aronson Joshua Lustina Michael Good Cathe-rine and Keough Kelli ldquoWhen White MenCanrsquot Do Math Necessary and SufficientFactors in Stereotype Threatrdquo Journal of Ex-perimental Social Psychology 1999 35(1)pp 29ndash46

Baker Lynn and Emery Robert ldquoWhen EveryRelationship Is above Average Perceptionsand Expectations of Divorce at the Time ofMarriagerdquo Law and Human Behavior 1993(17) pp 439ndash50

Baker Simon C Frith Christopher and DolanRaymond ldquoThe Interaction between Moodand Cognitive Function Studied with PETrdquoPsychological Medicine 1997 27(3) pp565ndash78

Benabou Roland and Tirole Jean ldquoSelf-Confidence and Personal Motivationrdquo Quar-terly Journal of Economics 2002 117(3) pp871ndash915

Brand Nico Vierspui Lisalotte and Oving Ar-jen ldquoInduced Mood and Selective AttentionrdquoPerceptual and Motor Skills 1997 84(2) pp455ndash63

Ellis Henry C Ottaway Scott A Varner LarryJ Becker Andrew S and Moore Brent AldquoEmotion Motivation and Text Comprehen-sion The Detection of Contradictions in Pas-sagesrdquo Journal of Experimental Psychology1997 126(2) pp 131ndash46

Fang Hanming ldquoConfidence Management andthe Free-Riding Problemrdquo Unpublished Pa-per 2001

Fischoff Baruch Slovic Paul and LichtensteinSarah ldquoKnowing with Certainty The Appro-priateness of Extreme Confidencerdquo Journalof Experimental Psychology Human Percep-tion and Performance 1977 3(4) pp 552ndash64

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-iour 1989 1(1) pp 60ndash79

Guthrie Chris Rachlinski Jeffrey and WistrichAndrew ldquoInside the Judicial Mind Heuristics

and Biasesrdquo Cornell Law Review 2001 (56)pp 777ndash830

Hannigan Sharon L and Reinitz Mark T ldquoADemonstration and Comparison of TwoTypes of Inference-Based Memory ErrorsrdquoJournal of Experimental Psychology Learn-ing Memory and Cognition 2001 27(4) pp931ndash40

Heifetz Aviad and Spiegel Yossi ldquoThe Evolutionof Biased Perceptionsrdquo Unpublished Paper2001

Koszegi Botund ldquoEgo Utility Overconfidenceand Task Choicerdquo Unpublished Paper 2000

Lichtenstein Sarah Fischoff Baruch and Phil-lips Lawrence ldquoCalibration of ProbabilitiesThe State of the Art to 1980rdquo in D Kahne-man P Slovic and A Tversky eds Judge-ment under uncertainty Heuristics andbiases Cambridge Cambridge UniversityPress 1982 pp 306ndash34

McKenna Frank and Lewic C ldquoA Speech RateMeasure of Laboratory Induced Affect TheRole of Demand Characteristics RevisitedrdquoBritish Journal of Clinical Psychology 199433(3) pp 345ndash51

Morris William N ldquoThe Mood Systemrdquo in DKahneman E Diener and N Schwarts edsWell-being The foundations of hedonic psy-chology New York Russell Sage Founda-tion 1999

Nisbett Richard E and Ross Lee Human infer-ence Strategies and shortcomings of socialjudgement Englewood Cliffs Prentice-HallInc 1980

Rabin Matthew ldquoIncorporating Fairness intoGame Theory and Economicsrdquo AmericanEconomic Review 1993 83(5) pp 1281ndash302

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ross Michael and Sicoly Fiore ldquoEgocentrics Bi-ases in Availability and Attributionrdquo Journalof Personality and Social Psychology 1979(37) pp 322ndash36

Samuelson Larry ldquoInformation-Based Relative

1556 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

Consumption Effectsrdquo Econometrica 200472(1) pp 93ndash118

Seligman Martin Learned optimism NewYork Knopf 1990

Steel Claude M and Aronson Joshua ldquoStereo-type Threat and the Intellectual Test Perfor-mance of African-Americansrdquo Journal ofPersonality and Social Psychology 199569(5) pp 797ndash811

Steel Claude M and Aronson Joshua ldquoHowStereotypes Influence the Standardized TestPerformance of Talented African AmericanStudentsrdquo in C Jencks and M Phillips edsThe black-white test score gap WashingtonDC Brookings Institution 1998 pp 401ndash27

Svenson Ola ldquoAre We All Less Risky and MoreSkillful Than Our Fellow Driversrdquo ActaPsychologica 1981 47(2) pp 145ndash46

Taylor Shelley E and Brown Jonathan D ldquoIllu-sion and Well-Being A Social Psychological

Perspective on Mental Healthrdquo Psychologi-cal Bulletin 1988 103(3) pp 193ndash210

Teasdale John D and Russell M ldquoDifferentialEffects of Induced Mood on the Recall ofPositive Negative and Neutral Moodsrdquo Brit-ish Journal of Clinical Psychology 198322(3) pp 163ndash71

Tversky Amos and Kahneman Daniel ldquoAvail-ability A Heuristic for Judging Frequencyand Provabilityrdquo Cognitive Science 19735(2) pp 207ndash32

Van den Steen Eric ldquoSkill or Luck Biases ofRational Agentsrdquo Unpublished Paper2002

Waldman Michael ldquoSystematic Errors and theTheory of Natural Selectionrdquo American Eco-nomic Review 1994 84(3) pp 482ndash97

Weinstein Neil D ldquoUnrealistic Optimism aboutFuture Life Eventsrdquo Journal of Personalityand Social Psychology 1980 39(5) pp806ndash20

1557VOL 94 NO 5 COMPTE AND POSTLEWAITE CONFIDENCE-ENHANCED PERFORMANCE