self-confidence and personal motivation ...rbenabou/papers/qje2002.pdfself-confidence and personal...

SELF-CONFIDENCE AND PERSONAL MOTIVATION

ROLAND BENABOU AND JEAN TIROLE

We analyze the value placed by rational agents on self-condence and thestrategies employed in its pursuit Condence in onersquos abilities generally en-hances motivation making it a valuable asset for individuals with imperfectwillpower This demand for self-serving beliefs (which can also arise from hedonicor signaling motives) must be weighed against the risks of overcondence On thesupply side we develop a model of self-deception through endogenous memorythat reconciles the motivated and rational features of human cognition Theresulting intrapersonal game of strategic communication typically leads to mul-tiple equilibria While ldquopositive thinkingrdquo can improve welfare it can also beself-defeating (and nonetheless pursued)

Believe what is in the line of your needs for only by such belief is the needfullled Have faith that you can successfully make it and your feet arenerved to its accomplishment [William James Principles of Psychology]

I have done this says my memory I cannot have done that says my prideremaining inexorable Finallymdashmemory yields [Friedrich Nietzsche BeyondGood and Evil]

I had during many years followed the Golden Rule namely that whenever apublished fact a new observation or thought came across me which wasopposed to my general results to make a memorandum of it without fail andat once for I had found by experience that such (contrary and thus unwel-come) facts and thoughts were far more apt to escape from memory thanfavorable ones [Charles Darwin in The Life of Charles Darwin by FrancisDarwin]

INTRODUCTION

The maintenance and enhancement of self-esteem has al-ways been identied as a fundamental human impulse Philoso-phers writers educators and of course psychologists all haveemphasized the crucial role played by self-image in motivationaffect and social interactions The aim of this paper is to bring

This paper was previously titled ldquoSelf-Condence Intrapersonal Strategies[1999]rdquo For helpful comments and discussion we are grateful to Dilip AbreuOlivier Blanchard Isabelle Brocas Edward Glaeser Daniel Gilbert Ian JewittDavid Laibson George Loewenstein Andrew Postlewaite Marek Pycia MatthewRabin Julio Rotemberg and three anonymous referees We also thank confer-ence and seminar participants at the University of Chicago Columbia Uni-versity Cornell University the Massachusetts Institute of Technology theNational Bureau of Economic Research Northwestern University New YorkUniversity the Oxford Young Economistsrsquo Conference at Oxford University theUniversity of Pennsylvania Princeton University Stanford University and YaleUniversity Benabou gratefully acknowledges nancial support from the NationalScience Foundation (SES-0096431)

copy 2002 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics August 2002

871

these concerns into the realm of economic analysis and show thatthis has important implications for how agents process informa-tion and make decisions Conversely the tools of economic mod-eling can help shed light on a number of apparently irrationalbehaviors documented by psychologists

Indeed both the demand and the supply sides of self-con-dence appear at odds with economistsrsquo view of human behaviorand cognition Why should people prefer rosy views of themselvesto accurate ones or want to impart such beliefs to their childrenFrom car accidents failed dotcom rms and day trading to thespace shuttle disaster and lost wars the costs of overcondenceare plain for all to see Even granting that some ldquopositive illu-sionsrdquo could be desirable is it even possible for a rational Bayes-ian individual to deceive himself into holding them Finally thewelfare consequences of so-called self-serving beliefs are far fromclear while ldquothinking positiverdquo is often viewed as a good thingself-deception is not even though the former is only a particularform of the latter

To analyze these issues we develop a simple formal frame-work that unies a number of themes from the psychology liter-ature and brings to light some of their economic implications Werst consider the demand side of self-condence and identify inSection I three main reasons why people may prefer optimisticself-views to accurate ones a consumption value a signalingvalue and a motivation value First people may just deriveutility from thinking well of themselves and conversely nd apoor self-image painful Second believingmdashrightly or wronglymdashthat one possesses certain qualities may make it easier to con-vince others of it Finally condence in his abilities and efcacycan help the individual undertake more ambitious goals andpersist in the face of adversity While we shall mostly focus onthis last explanation all three should be seen as complementaryand for many purposes work equally well with the supply side ofour model (self-deception)

The main reason why we emphasize the motivation theory isits substantially broader explanatory power Indeed it yields anendogenous value of self-condence that responds to the situa-tions and incentives which the individual faces in a way that canaccount for both ldquocan-dordquo optimism and ldquodefensiverdquo pessimism Italso readily extends to economic and social interactions (altruisticor not) explaining why people generally prefer self-condent

872 QUARTERLY JOURNAL OF ECONOMICS

partners to self-doubting ones and invest both time and effort insupporting the latterrsquos morale

The rst premise of the motivation theory is that people haveimperfect knowledge of their own abilities or more generally ofthe eventual costs and payoffs of their actions1 The second one isthat ability and effort interact in determining performance inmost instances they are complements so that a higher self-con-dence enhances the motivation to act As demonstrated by theopening quote from James [1890] this complementarity has longbeen familiar in psychology2 It is also consistent with the stan-dard observation that morale plays a key role in difcult endeav-ors conversely when people expect to fail they fail quite effec-tively and failure leads to failure more readily for individualscharacterized with low self-esteem [Salancik 1977]

The fact that higher self-condence enhances the individualrsquosmotivation gives anyone with a vested interest in his performancean incentive to build up and maintain his self-esteem First themanipulator could be another person (parent teacher spousefriend colleague manager) who is eager to see him ldquoget his acttogetherrdquo or otherwise apply himself to the task at hand Suchinterpersonal strategies are studied in Benabou and Tirole [2001]Second for an individual suffering from time inconsistency (eghyperbolic discounting) the current self has a vested interest inthe self-condence of future selves as it helps counter theirnatural tendency to quit too easily It is in this context whichbuilds on Carrillo and Mariotti [2000] that we shall investigate avariety of intrapersonal strategies of self-esteem maintenanceWe shall thus see how and when people may choose to remainignorant about their own abilities and why they sometimes de-liberately impair their own performance or choose overambitioustasks in which they are sure to fail (self-handicapping)

Section II thus turns to the supply side of the self-condence

1 The psychology literature generally views introspection as quite inaccu-rate [Nisbett and Wilson 1977] and stresses that learning about oneself is anongoing process Furthermore the self is constantly changing (eg Rhodenwalt[1986]) personal characteristics evolve with age the goals pursued shift overonersquos career and life cycle (often as the result of interactions with others) and thepersonal or economic environment in which these objectives are rewarded istypically variable

2 Thus Gilbert and Cooper [1985] note that ldquothe classic attributional modelof the causes of behavior [is described by] the well-known conceptual equation(E 3 A) 6 TD 5 B in which effort times ability plus or minus task difcultyequals the behavioral outcomerdquo Additional references are given in Section I Notehowever that there are also instances where ability and effort are substitutes Asdiscussed below we shall consider this case as well

873SELF-CONFIDENCE AND PERSONAL MOTIVATION

problem and the ldquoreality constraintsrdquo that limit the extent towhich people can engage in wishful thinking In our model wemaintain the standard assumption of individuals as rational(Bayesian) information processors While almost universal in eco-nomics this view is more controversial in psychology On onehand a lot of the classical literature has emphasized rationalityand information-seeking in the process of self-identicationdocumenting the ways in which people update their beliefs accord-ing to broad Bayesian principles3 On the other hand the morerecent cognitive literature abundantly documents the less ratio-nal (or at least subjectively motivated) side of human inference

For instance a substantial body of evidence suggests thatpeople tend to recall their successes more than their failures andhave self-servingly biased recollections and interpretations oftheir past performances4 Similarly they tend to overestimatetheir abilities and other desirable traits as well as the extent towhich they have control over outcomes They also rate their ownprobabilities as above average for favorable future life eventsand below average for unfavorable ones the more controllablethese events through their future actions the more so5

We shall capture this class of self-deception phenomena witha simple game-theoretic model of endogenous memory or aware-ness-management which represents one of the main contribu-tions of this paper Drawing on evidence about the mechanics andlimitations of memory it shows how to reconcile the motivated(ldquohotrdquo) and rational (ldquocoldrdquo) features of human cognition andcould be used in any setting where a demand for motivated beliefsarises The basic idea is that the individual can within limits and

3 Thus attribution theory [Heider 1958] emphasizes the distinction betweentemporary (situational) and enduring (dispositional) characteristics In economicsparlance the individual lters out noise in order to extract information from pastevents In the social comparison process [Festinger 1954] individuals assess theirability by comparing their performance with that of people facing similar condi-tions (familial cultural educational etc) In other words they use ldquorelativeperformance evaluationrdquo or ldquobenchmarkingrdquo for self-evaluation A good perfor-mance by others in onersquos reference group is thus generally detrimental to self-esteem and conversely some comfort is derived when others experience adversity(Schadenfreude) Relatively sophisticated updating also applies to the interpre-tation of praise and criticism a person takes into account not only what others say(or do) but also their possible intentions

4 Why they would want to do so in a social context is obvious The interestingquestion is why they may bias their own inference process

5 See eg Taylor and Brown [1988] Weinstein [1980] Alloy and Abra-hamson [1979] and the many other references given in Section II For recentoverviews of the general phenomenon of self-deception see Gilbert and Cooper[1985] and especially Baumeister [1998] on the psychological evidence and Elster[1999] and Mele [1999] for the philosophical debates and implications


possibly at a cost affect the probability of remembering a givenpiece of data At the same time we maintain rational inferenceso people realize (at least to some extent) that they have a selec-tive memory or attention

The resulting structure is that of a game of strategic commu-nication between the individualrsquos temporal selves In decidingwhether to try to repress bad news the individual weighs thebenets from preserving his effort motivation against the risk ofbecoming overcondent Later on however he appropriately dis-counts the reliability of rosy recollections and rationalizationsThe implications of this game of asymmetric information arequite different from those of ex ante decisions about informationacquisition (eg self-handicapping or selective search) In particu-lar multiple intrapersonal equilibria (ldquoself-trapsrdquo) may ariseranging from systematic denial to complete self-honesty Moregenerally we characterize the set of perfect Bayesian equilibriaand its dependence on the individualrsquos degree of time inconsis-tency and repression costs (ldquodemand and supplyrdquo parameters)

The model also has interesting implications for the distribu-tion of optimism and pessimism across agents which we examinein Section III We show that when the costs of repression are lowenough most people typically believe themselves to be more ablethan they actually are as well as more able than both the averageand the median of the population A minority will have eitherrealistically low assessments or actually severely underestimatethemselves We also highlight the key role played by Bayesian-like introspection (understanding at least partially onersquos ownincentives for self-esteem maintenance) in the modelrsquos resultsand why incorporating this essential human trait is more fruitfulthan modeling agents as naively taking all recollections andself-justications at face value

Section IV examines the welfare impact of equilibrium self-deception Is a more active self-esteem maintenance strategywhen chosen always benecial How can people be ldquoin denialrdquo ifit does not serve their best interests We show that in addition tothe trade-off mentioned earlier between the condence-mainte-nance motive and the risks of overcondence ex ante welfarereects a third effect namely the spoiling of good news by self-doubt Intuitively when adverse signals about his ability aresystematically repressed the individual can never be sure thatonly positive ones were received even when this is actually trueWe characterize the conditions under which always ldquolooking at


the bright siderdquo pays off on average or conversely when it wouldbe better to always ldquobe honest with yourselfrdquo as Charles Darwinapparently concluded

In Section V we turn to the case where ability and effort aresubstitutes rather than complements This typically occurs whenthe payoff for success is of a ldquopass-failrdquo nature or characterized bysome other form of satiation Since a high perceived ability maynow increase the temptation to exert low effort (ldquocoastingrdquo) thiscase allows us to account for what psychologists refer to as ldquode-fensive pessimismrdquo the fact that people sometimes minimizerather than aggrandize their previous accomplishments and ex-pectations of future success Another variant of the model consid-ered in this section involves replacing the motivation value ofself-condence with a purely affective one Section VI concludesthe paper All proofs are gathered in the Appendix

This paper is related to several strands of the new literaturethat tries to better link economics and psychology A hedonicconcern for self-image in the form of preferences over beliefs wasrst explored in Akerlof and Dickensrsquo [1982] well-known model ofdissonance reduction and more recently in Rabin [1995] Wein-berg [1999] and Koszegi [1999] In emphasizing an endogenousvalue of self-condence and retaining the constraint of Bayesianrationality our paper is most closely related to the work of Car-rillo and Mariotti [2000] who rst showed how information ma-nipulation may serve as a commitment device for time-inconsis-tent individuals (see also Brocas and Carrillo [1999]) The centralrole played by memory also relates our model to Mullainathan[2002] and Laibson [2001] although one of its main features is tomake recall endogenous

I THE DEMAND FOR SELF-CONFIDENCE

In most societies self-condence is widely regarded as avaluable individual asset Going back at least to William Jamesan important strand in psychology has advocated ldquobelieving inoneself rdquo as a key to personal success Today an enormous ldquoself-helprdquo industry ourishes a sizable part of which purports to helppeople improve their self-esteem shed ldquolearned helplessnessrdquo andreap the benets of ldquolearned optimismrdquo6 American schools place

6 These last two terms are borrowed from Seligman [1975 1990]


such a strong emphasis on imbuing children with self-condence(ldquodoing a great jobrdquo) that they are often criticized for giving itpreeminence over the transmission of actual knowledge Hencethe general question why is a positive view of oneself as opposedto a fully accurate one seen as such a good thing to have

Consumption value A rst reason may be that thinking ofoneself favorably just makes a person happier self-image isthen simply another argument in the utility function Indeedpsychologists emphasize the affective benets of self-esteem aswell as the functional ones on which we shall focus One may alsohypothesize that such preferences over beliefs could have beenselected for through evolution the overcondence that typicallyresults may propel individuals to undertake activities (explora-tion foraging combat) which are more risky than warranted bytheir private material returns but confer important externalbenets on the species In subsection VB we shall explain how ahedonic self-image motive can readily be incorporated into ourgeneral framework

Signaling value A second explanation may be that believingoneself to be of high ability or morality makes it easier to convinceothers (rightly or wrongly) that one does have such qualitiesIndeed it is often said that to lie most convincingly a person mustbelieve his own lies While the idea that people are ldquotransparentrdquoand have trouble misrepresenting their private information mayseem unusual in economics one could easily obtain an instru-mental value of self-condence from a signaling game wherethose who truly believe in their own abilities face lower costs ofrepresenting themselves favorably to others

Motivation value The explanation that we emphasize mostis that self-condence is valuable because it improves the indi-vidualrsquos motivation to undertake projects and persevere in thepursuit of his goals in spite of the setbacks and temptations thatperiodically test his willpower Morale is universally recognizedas key to winning a medal performing on stage getting intocollege writing a great book doing innovative research settingup a rm losing weight nding a mate and so forth The linkbetween self-condence and motivation is also pervasive in thepsychology literature from early writers like James [1890] tocontemporary ones like Bandura [1977] according to whom ldquobe-


liefs of personal efcacy constitute the key factor of humanagencyrdquo (see also eg Deci [1975] or Seligman [1990]) The mo-tivation theory also readily extends to economic (nonaltruistic)interactions explaining why people typically prefer self-condentcoworkers managers employees teammates soldiers etc toself-doubting ones and why they spend substantial time andeffort supporting the morale of those with whom they end upbeing matched7

IA The Motivation Problem

Had I been less denitively determined to start working I might have madean effort to begin right away But because my resolve was absolute andwithin twenty-four hours in the empty frames of the next day where every-thing t so well since I was not yet there my good resolutions would easilybe accomplished it was better not to choose an evening where I was illdisposed for a beginning to which alas the following days would turn out tobe no more propitious [Marcel Proust Remembrance of Things Past]

Consider a risk-neutral individual with a relevant horizon ofthree periods t 5 012 At date 0 he selects an action thatpotentially affects both his ow payoff u0 and his date 1 informa-tion structure8 At date 1 he decides whether to undertake a taskor project (exert effort which has disutility cost c 0) or not(exert no effort) With some probability u which denes his abil-ity the project will succeed and yield a benet V at date 2 failuregenerates no benet The individualrsquos beliefs over u (dening hisself-condence or self-esteem) are described by distribution func-tions F(u ) at date 0 and F1(u ) at date 1 In the intervening periodnew information may be received or previous signals forgottenwe shall focus here on the rst more standard case and turn tomemory in Section II Note that with risk neutrality the meanu 1 [ 0

1 u dF1(u ) will be a sufcient statistic for F1 for brevitywe shall also refer to it as the agentrsquos date 1 self-condence

Finally we assume that the individualrsquos preferences exhibittime-inconsistency due to quasi-hyperbolic discounting There isindeed considerable experimental and everyday evidence thatintertemporal choices exhibit a ldquosalience of the presentrdquo in the

7 Note that this last observation cannot readily be accounted for by theldquosignalingrdquo theory of self-condence either

8 The simplest date 0 action is thus the choice of the amount of informationthat will be available at date 1 (eg soliciting feedback taking a test keeping ordestroying records) Alternatively this information may be derived from theoutcome of some activity pursued for its own sake at date 0 (learning by doingdrinking a lot of wine)


sense that discount rates are much lower at short horizons thanat more distant ones9 Denoting u t and Et[ z ] the ow payoffs andexpectations at t 5 012 the intertemporal utility perceived bythe individual as of date 1 is

(1) u1 1 bdE1u2 5 2c 1 bdu1V

when he undertakes the activity and 0 when he does not Bycontrast intertemporal utility conditional on the same informa-tion set at date 1 but evaluated from the point of view of date 0is

(2) u0 1 bE0du1 1 d2u2uu1 5 u0 1 bd2c 1 du 1V

if the activity is undertaken at date 1 and u0 otherwise10

Whereas d is a standard discount factor b reects the momentarysalience of the present When b 1 the individual at date 0 (ldquoSelf0rdquo) is concerned about his date 1 (ldquoSelf 1rsquosrdquo) excessive preferencefor the present or lack of willpower which leads to the under-provision of effort (procrastination) Indeed Self 1 only exertseffort in the events where u 1 cbdV whereas from the point ofview of Self 0 it should be undertaken whenever u 1 cdV Notethat while we focus here on the case where the individualrsquosintrinsic ability u is unknown it could equally be the expectedpayoff in case of success V the ldquosurvivalrdquo probability d or thetaskrsquos difculty measured by the cost of effort c All that mattersfor our theory is that the individual be uncertain of the long-termreturn to effort udVc which he faces

IB Condence Maintenance versus Overcondence

In an important paper Carrillo and Mariotti [2000] showedthat in the presence of time inconsistency (TI) Blackwell gar-blings of information may increase the current selfrsquos payoff Thisresult can be usefully applied and further developed in ourcontext

Suppose that at date 0 our individual can choose betweenjust two information structures for date 1 In the ner one Self 1learns his ability u exactly In the coarser one he learns nothing

9 See Ainslie [1992 2001] for the evidence and Strotz [1956] Phelps andPollack [1968] Loewenstein and Prelec [1992] Laibson [1997 2001] andOrsquoDonoghue and Rabin [1999] for formal models and economic implications

10 Note that the equality in (2) makes use of the identity E0[uuE1[u] 5 u1] 5u1 which holds whenmdashand only whenmdashthere is no information loss between dates0 and 1


that Self 0 did not know F1(u ) 5 F(u ) and hence u 1 5 01 u

dF1(u ) [ u F Let us rst assume that in the absence of infor-mation Self 1 will undertake the task u F cbdV The valueattached by Self 0 to Self 1rsquos learning the value of u is therefore bdtimes

(3) F EcbdV

1

~duV 2 c dF~u 2 ~du FV 2 c 5 F 2 F

where

(4) F E0

cdV

~c 2 duV dF~u

(5) F EcdV

cbdV

~duV 2 c dF~u

F stands for the gain from being informed which arises from thefact that better information reduces the risk of overcondence onthe part of Self 1 Overcondence occurs when the individualrsquosability is below cdV but he is unaware of it and thus inappro-priately undertakes or perseveres in the project F stands for theloss from being informed which may depress the individualrsquosself-condence if he learns that u is in some intermediate rangecdV u cbdV he will procrastinate at date 1 even though exante it was optimal to work Information is thus detrimental tothe extent that it creates a risk that the individual will fall intothis time-inconsistency (TI) region If this condence maintenancemotive is strong enough ( F F) the individual will prefer toremain uninformed F 0 More generally note that F is lowerthe lower is b By contrast in the absence of time inconsistency(b 5 1) we have F 5 0 and thus F $ 0 in classical decisiontheory information is always valuable

The overcondence effect calls for more information con-dence maintenance for less This trade-off has been noted byempirical researchers For instance Leary and Downs [1995]summarize the literature by noting that a) ldquopersons with highself-esteem perform better after an initial failure and are morelikely to persevere in the face of obstaclesrdquo b) ldquohigh self-esteem isnot always functional in promoting task achievement Peoplewith high self-esteem may demonstrate nonproductive persis-


tence at insoluble tasks thereby undermining their effectivenessThey may also take excessive and unrealistic risks when theirself-esteem is threatenedrdquo

To understand the last statement let us turn to the casewhere u F cbdV Since Self 1 now always exerts (weakly) lesseffort than Self 0 would like him to information can only help theindividual restore his decient motivation Indeed

(6) F 5 EcbdV

1

~duV 2 c dF~u 0

Moreover F is now higher the lower is b In such situations theindividual will avidly seek feedback on his ability and his choicesof tasks and social interactions will have the nature of ldquogamblesfor resurrectionrdquo of his self-esteem

Putting together the different cases we see that the value ofinformation is not monotonic with respect to initial self-con-dence Indeed for someone with condence so low that u F cbdV F is always positive and increasing with respect to (sto-chastic) increases in u11 For an individual with F(cdV) 5 0 butF(cbdV) 1 F is always negative Finally for a person soself-assured that F(cbdV) 5 0 motivation is not a concern (as ifb were equal to 1) but neither is overcondence F 5 F 5 F 50 Therefore there must exist some intermediate range where F

rst declines and becomes negative then increases back towardzero

IC What Types of People Are Most Eager to Maintain TheirSelf-Condence

Let us now consider two individuals with different degrees ofinitial self-condence and ask which one is least receptive toinformation We denote their prior distributions over abilities asF(u ) and G(u ) with densities f(u )g(u ) and means u F u G Tomake condence maintenance meaningful let u F u G cbdVFor comparing levels of self-condence however just looking atexpected abilities turns out not to be sufcient

DEFINITION 1 An individual with distribution F over ability u hashigher self-condence than another one with distribution G ifthe likelihood ratio f(u )g(u ) is increasing in u

11 Rewrite (6) as F 5 01 1u$ c b dV (duV 2 c) dF(u ) where 1 z denotes the

indicator function and note that the integrand is increasing in u


Abstracting for the moment from any cost attached to learn-ing or not learning the true ability it is easy to see from (3) that

F $ 0 if and only if

(7) E0

cbdV F~u

F~cbdVdu $ S 1 2 b

b D S cdVD

The monotone likelihood ratio property (MLRP) implies thatF(u )F(cbdV) G(u )G(cbdV) for all u cbdV Therefore theleft-hand side of (7) is smaller under F than under G meaningthat the person with the more positive self-assessment will acceptinformation about his ability for a smaller set of parametersIntuitively he has more to lose from information and is thereforemore insecure

PROPOSITION 1 If an individual prefers not to receive informationin order to preserve his self-condence so will anyone withhigher initial self-condence if G 0 for some distributionG over u then F 0 for any distribution F such that thelikelihood ratio fg is increasing

ID Self-Handicapping

A well-documented and puzzling phenomenon is that peoplesometimes create obstacles to their own performance12 In experi-ments subjects with fragile self-condence opt to take perfor-mance-impairing drugs before an intelligence test In real lifepeople withhold effort prepare themselves inadequately or drinkalcohol before undertaking a task They also set themselves over-ambitious goals where they are almost sure to fail Test anxietyand ldquochokingrdquo under pressure are yet other common examplesPsychologists have long suggested that self-handicapping is oftena self-esteem maintenance strategy (instinctive or deliberate)directed both at oneself and at others13

12 See eg Berglas and Jones [1978] Arkin and Baumgardner [1985]Fingarette [1985] or Gilovich [1991]

13 See eg Berglas and Baumeister [1993] Of course self-handicappinginvolves both intrapersonal (self-esteem maintenance) and intrapersonal (self-presentation) motives our model captures only the former As Baumeister [1998]notes ldquoby self-handicapping one can forestall the drawing of unattering attri-butions about oneself Self-handicapping makes failure meaningless and so ifpeople think you are intelligent the upcoming test cannot change this impressionrdquoIn particular people apparently self-handicap more in public situations [Kolditzand Arkin 1982] They then reap a double dividend as this provides an excuse forpoor performance both to themselves and to others


To examine this question consider an individual with priorbeliefs F(u ) faced at date zero with a choice between an efcientaction that (for simplicity) will fully reveal his ability and aninefcient ldquoself-handicappingrdquo one that entails an expected costh0(F) $ 0 but is totally uninformative about u Assuming thatu F cbdV as before equation (7) immediately generalizes toshow that he will self-handicap if and only if 2bd F $ h0(F) or

(8) S 1 2 b

b D c 2 dVF E0

cbdV F~u

F~cbdVduG $

h0~F

bdF~cbdV

Note rst that multiplying (8) by F(cbdV) yields a decreasingfunction of b on the left-hand side Therefore people who aremore concerned about sustaining motivation (more time-inconsis-tent) are more likely to self-handicap and will choose to do sowhen the short-run costs of doing so are not too large Next let uscompare individuals with different prior beliefs about them-selves As before those who are initially more self-condent havemore to lose from learning about their ability by the MLRP theleft-hand side of (8) is larger than if F were replaced with GHowever the more self-condent are also less likely to receive badnews and this reduces the return on ldquoinvesting in ignorancerdquo theMLRP implies that F(cbdV) G(cbdV) which tends to makethe right-hand side of (8) also larger under F than under G Thusin general one cannot conclude whether people with higher orlower self-condence are more likely to self-handicap14 This am-biguity is fundamentally linked to the nonmonotonicity notedearlier for the value of information while the MLRP ensures thatthe sign of F varies monotonically with initial beliefs the abso-lute amount does not When self-handicapping costs are relativelysmall howevermdashwhich is often the case in experimentsmdashtheldquomore to loserdquo effect identied in Proposition 1 will prevail

It is interesting in this respect to note that psychologistshave also not reached a rm conclusion on whether high or lowself-condence people are the most defensive of their egos al-though there does seem to be somewhat more evidence in favor ofthe rst hypothesis Thus Greenier Kernis and Wasschul [1995]contrast ldquohumanistically oriented theories according to

14 A third consideration is whether the expected cost of self-handicappingrises or falls with initial self-condence h0(F) h0(G) In Benabou and Tirole[2001] we provide examples of tasks that correspond to each case


which high self-esteem individualsrsquo feelings of self-worth are builton solid foundations that do not require continual validationrdquo withexperimental research showing that ldquohigh self-esteem individualsare the more likely to display self-serving attributions self-hand-icap to enhance the potentially positive implications of good per-formance set inappropriately risky goals when ego-threatenedand actively create less fortunate others with whom they cancompare favorablyrdquo

II THE PSYCHOLOGICAL IMMUNE SYSTEM

Just as there was in his study a chest of drawers which he managed never tolook at which he went out of his way not to encounter when walking in or outbecause in one drawer was held the chrysanthemum which she had givenhim on the rst evening so there was in him a place which he never lethis mind approach imposing on it if necessary the detour of a long reason-ing it was there that lived the memories of happier days [Marcel ProustRemembrance of Things Past]

We now turn to the supply side of the self-esteem problemGiven that a positive self-assessment may be desirable (whetherfor motivational signaling or hedonic reasons) what are themeans through which it can be achieved or at least pursued

Wired-in optimism A rst hypothesis could be that evolutionhas selected for a particular cognitive bias in humans causingthem to systematically and involuntarily underweigh adversesignals about themselves and overweigh positive ones This ex-planation is rather problematic the extent of overcondence oroveroptimism varies both over time and across tasks and a greatmany people actually suffer from undercondence (the extremecase being depression) Furthermore individuals often ldquoworkrdquoquite hard at defending their self-image when it is threatenedgoing though elaborate schemes of denial self-justication fur-niture-avoidance and the like

Blissful ignorance When self-condence is valuable capitalit may be preferable to remain uninformed than to put it at riskby exposing oneself to new information In particular as seen inthe previous section ex ante strategic ignorance [Carrillo andMariotti 2000] or even self-handicapping may help a time-incon-sistent individual safeguard his motivation In a context of he-donic beliefs workers in a hazardous job may not want to knowabout the exact risks involved [Akerlof and Dickens 1982]


Self-deception Most often the relevant issue is not whetherto seek or avoid information ex ante (before knowing what it willturn out to say) but how to deal with the good and especially thebad news concerning onersquos performances and abilities that lifeinevitably brings This is where the mechanisms of defensivedenial repression self-serving attributions and the like soprominently emphasized in psychology come into play We shallcapture this class of phenomena with a simple game-theoreticmodel of endogenously selective memory

IIA Managing Awareness The Role of Memory

Psychologists and before them writers and philosophershave long documented peoplersquos universal tendency to deny ex-plain away and selectively forget ego-threatening informationFreudian repression is the most obvious example but variousother forms of motivated cognition and self-deception gure promi-nently in contemporary psychology Thus a lot of research hasconrmed that people tend to recall their successes more thantheir failures (eg Korner [1950] and Mischel Ebbesen andZeiss [1976]) have self-servingly biased recollections of their pastperformances [Crary 1966] and readily nd ldquoevidencerdquo in theirpersonal histories that they possess characteristics which theyview (sometimes as the result of experimental manipulation) ascorrelated with success in professional or personal life [Kundaand Sanitioso 1989 Murray and Holmes 1993] Similarly theyoften engage in ldquobeneffactancerdquo viewing themselves as instru-mental for good but not bad outcomes [Zuckerman 1979] Whenthey commit a bad deed they reframe the facts to try and con-vince themselves that it was not so bad (ldquohe deserved itrdquo ldquothedamage was limitedrdquo) or attribute the responsibility to others[Snyder 1985]

At the same time the impossibility of simply choosing thebeliefs we like has always stood in the way of a fully consistenttheory of self-deception Sartre [1953] argued that the individualmust simultaneously know and not know the same informationGur and Sackeim [1979] dened self-deception as a situation inwhich a) the individual holds two contradictory beliefs b) he isnot aware of holding one of the beliefs c) this lack of awareness ismotivated

Our intertemporal model allows us to unbundle the ldquoself thatknowsrdquo from the ldquoself that doesnrsquot knowrdquo and thereby reconcilethe motivation (ldquohotrdquo) and cognition (ldquocoldrdquo) aspects of self-decep-


tion within a standard information-theoretic framework The ba-sic idea is that the individual can within limits affect the prob-ability of remembering a given piece of data Under timeinconsistency there is an incentive to try to recall signals thathelp sustain long-term goals and forget those that underminethem This is the motivation part15 On the other hand we main-tain the rational inference postulate so people realize (at least tosome extent) that they have a selective memory or attention Thisis the cognition part

ASSUMPTION 1 (MEMORY OR AWARENESS MANAGEMENT) The individ-ual can at a cost increase or decrease the probability ofremembering an event or its interpretation

Formally let l [ [01] denote the probability that giveninformation received at date 0 will be recalled or accessed at date1 We dene the natural rate of recall lN [ [01] as that whichmaximizes the date 0 ow payoff u0 Increasing or decreasing lthus involves a ldquomemory costrdquo M(l) ie a reduction in u0 withM(lN ) 5 0 M9(l) 0 for l lN and M9(l) $ 0 for l lN 16

Whether it refers to the subconscious or points to the differ-ential accessibility and decay of memories stored in specializedparts of the brain virtually all modern psychology recognizes thatonly part of the individualrsquos accumulated stock of information isreadily available for conscious purposive processing and deci-sion-making Furthermore the encoding and retrieval process issubject to systematic inuences both internal and external a)

15 Alternatively it could arise from an affective or signaling value ofself-esteem

16 By denition ldquoforgettingrdquo means that an actual loss of information (acoarsening of the informational partition) occurs Thus if at date 1 the individualdoes not remember a date 0 signal s he also does not recall any other piece ofinformation that is perfectly correlated with s such as the costs M(l) incurred inthe process of forgetting This complete forgetting is only a simplied represen-tation of a richer attribution (signal extraction) problem however Let d measurethe level of an action that affects recall but can also be undertaken for its ownsake amount of wine consumed time spent with friendly rather than criticalpeople attention paid to the details of competing information effort in makingsafekeeping or disposing of physical records spatial or mental detours aroundcertain potential cues etc Choosing d following an event s leads to a recallprobability l 5 L(d) and has a direct utility u0 (de) where e is a random tasteshock Later on the agent may recall the action d(se) [ arg maxdu0(de) 1bdE0[u1 1 du2 uds] that he took and possibly the associated consequencesu0(d(se)e) but not the particular realization of e that occurred (eg how muchdid I really want to drink wine or sit next to that person at dinner) Recallingd(se) is thus generally insufcient to fully reconstruct s or to separate outwithin realized utility the cost M(L(d(se)e)) [ maxd u0 (de) 2 u0(d(se)e)that was incurred purely for memory manipulation This problem remains whenthe functions u0 L or M or the distribution of e depend on s


information that is rehearsed often is better remembered (indeedthat is why we cram for an exam) conversely if one is preoccu-pied or distracted when an event unfolds one has greater dif-culty remembering the details b) direct behavioral experiencemakes the information more accessible in memory because lateron recall is more likely to be activated by situational cues17

Such mechanisms seem to be at work in experiments wheresubjects who are asked to behave in a self-deprecating mannerlater report lower self-esteem than earlier while persons who areasked to display self-enhancing behavior report higher self-es-teem [Jones et al 1981] This may be due to the fact that theywere led to rehearse unfavorable or favorable information aboutthemselves thus increasing the probability of remembering itlater on Similarly receiving positive feedback seems to trigger acue-based ldquowarm glowrdquo effect which automatically makes acces-sible to the individual other instances of himself in positive situa-tions [Greenwald 1980]

These frictions in the mechanics of memory give the individ-ual some discretion about what data he is more likely to con-sciously recall later onmdashthereby opening the door to motivatedcognition Thus a person who wishes to remember good news andforget the bad can linger over praise or positive feedback re-hearse it periodically and choose to be more frequently in envi-ronments or with people who will remind him of his past suc-cesses18 Conversely he can eschew situations that remind him ofbad news tear up the picture of a former lover or like thenarrator in Proustrsquos novel avoid passing by a chest of drawerswhich contains cues to painful memories He can work unusuallyhard to ldquoforgetrdquo (really not think about) a failed relationship orfamily problem or even use drugs and alcohol

The individual can also nd ways to discount self-threaten-ing news in the rst place A common such strategy is to seek outinformation that derogates the informativeness of the initial data[Frey 1981 Gilovich 1991] After being criticized in a seminar or

17 For evidence and discussions see eg Schacter [1996] and Fazio andZanna [1981] Mullainathan [2002] and Laibson [2001] provide models of cue-dependent consumption

18 Thus ldquowe can expect [an author in a meeting] to spend more timeconsidering the comments of the lone dissenter who praised the project (andconrmed his self-conception) than of the colleagues who disliked it thus merci-fully softening the cavalcade of criticismsrdquo [Gilbert and Cooper 1985] For adiscussion of self-presentation strategies and their link with self-enhancementsee Rhodewalt [1986]


referee report a researcher will look hard for reasons why thecommenter has poor taste a limited understanding of the issuesa vested interest in a competing theory or body of empiricalevidence and so forth Interpersonal strategies can also be calledupon thus a verbal ght with onersquos spouse or someone whocriticizes onersquos work may (consciously or not) serve the purpose ofcreating a distraction that will impair accurate recollection of thedetails of the criticism (of course it has costs as well )

As these examples make clear it is important to note that weneed not literally assume that the individual can directly andmechanically suppress memories Our model is equally consistentwith a Freudian view where memories get buried in the uncon-scious (with some probability of reappearance) and with morerecent cognitive psychology which holds that memory itself can-not be controlled but emphasizes the different ways in whichawareness can be affected the choice of attention when the infor-mation accrues the search for or avoidance of cues the process ofselective rehearsal afterwards and again the choice of attentionat the time the information is (voluntarily or accidentally) re-trieved19 We shall therefore use the terms ldquomemoryrdquo and ldquoaware-ness of past informationrdquo interchangeably

ASSUMPTION 2 (METACOGNITION) While the individual can manipu-late his conscious self-knowledge he is aware that incentivesexist that result in selective memory

As illustrated by the opening quotations from Nietzsche andDarwin if a person has a systematic tendency to forget distort orrepress certain types of information he will likely become (or bemade) aware of it and not blindly take at face value the fact thatmost of what comes to his mind when thinking about his pastperformances and the feedback he received is good news Insteadusing (some) rational inference he will realize that what he mayhave forgotten are not random events20 Formally this introspec-tion or skepticism with respect to the reliability of onersquos ownself-knowledge is represented by Bayesrsquo rule which implies thata person cannot consistently fool himself in the same direction

19 One could even adhere to a minimalist version of the model where theindividual can only improve his rate of recall (through rehearsal record-makingetc) but never lower it (M(l) 5 ` for all l lN ) All that matters is the potentialfor a differential rate of recall or awareness in response to desirable or undesirableinformation

20 As Gilbert and Cooper [1995] note ldquowe are all insightful naive psycholo-gists well aware of human tendencies to be self-servingrdquo


Less sophisticated inference processes lead to similar results solong as they are not excessively naive (see subsection IIIA)

IIB The Game of Self-Deception

Let the agent receive at date 0 a signal s about his ability uTo make things simple let s take only two values with proba-bility 1 2 q the agent receives bad news s 5 L and withprobability q he receives no news at all s 5 A In other words ldquononews is good newsrdquo Let

(9) uL Euus 5 L Euus 5 A uH

Since s is informative about the return to date 1 effort theagentrsquos Self 1 would benet from having this signal If it isego-threatening however Self 0 may have an interest in sup-pressing it The recollection at date 1 of the news s will bedenoted s [ AL We assume that memories can be lost but notmanufactured ex nihilo so s 5 A always leads to s 5 A A signals 5 L on the other hand may be forgotten due to naturalmemory decay or voluntary repression Let l denote the proba-bility that bad news will be remembered accurately

(10) l Prs 5 Lus 5 L

As explained earlier the agent can increase or decrease thisprobability with respect to its ldquonaturalrdquo value lN 1 choosing arecall probability l involves a ldquomemory costrdquo M(l) We shall nowanalyze the equilibrium in several stages

1 Inference problem of Self 1 Faced with a memory s [LA Self 1 must rst assess its credibility Given that memoriescannot be invented unfavorable ones are always credible WhenSelf 1 does not recall any adverse signals on the other hand hemust ask himself whether there was indeed no bad news at date0 or whether it may have been lost or censored If Self 1 thinksthat bad news are recalled with probability l he uses Bayesrsquorule to compute the reliability of a ldquono recollectionrdquo message as

(11) r Prs 5 Aus 5 Al 5q

q 1 ~1 2 q~1 2 l

His degree of self-condence is then

(12) u ~r ruH 1 ~1 2 ruL


2 Decisions and payoffs We normalize the payoff in case ofsuccess to V 5 1 and assume that the cost of date 1 effort isdrawn from an interval [cI c ] with probability distribution F(c)and density w(c) 0 We assume that c bduH bduL cI which means that at date 1 there is always a positive probabilityof no effort and a positive probability of effort21

Given a signal s at date 0 and a memory s at date 1 Selves0 and 1 respectively assess the productivity of date 1 effort asE[uus] and E[uus] Self 1 only works when the realization of theeffort cost is c bdE[uus] so Self 0rsquos payoff is

(13) bd E0

bdEuus

~dEuus 2 c dF~c

To build intuition suppose for a moment that Self 0 could freelyand costlessly manipulate Self 1rsquos expectation E[uus] What be-liefs would he choose for a naive Self 1 Maximizing (13) we ndthat Self 0 would like to set E[uus] equal to E[uus]b This makesclear how time consistency gives Self 0 an incentive to boost ormaintain Self 1rsquos self-condence the problem of course is thatSelf 1 is not so easily fooled These two effects are consistent withthe common view in psychology that a moderate amount of ldquoposi-tive illusionrdquo about oneself is optimal but that many people ndit quite difcult to strike this desirable balance

3 Costs and benets of selective memory or attention Focus-ing on the ldquobad newsrdquo case denote as UC (uL ur) the expectedutility of Self 0 (gross of memory-management costs) when theadverse information is successfully forgotten and as UT (uL ) thecorresponding value when it is accurately recalled The subscriptsC and T stand for ldquocensoredrdquo and ldquotruthrdquo respectively Hidingfrom Self 1 the signal s 5 L raises his self-condence from uL tou (r) leading him to exert effort in the additional states of theworld where bduL c bdu (r) As with ex ante ignorance thishas both costs and benets thus if r is high enough thatbu (r) uL the net gain or loss from self-deception is

21 In Section I we took the distribution of u to be continuous and c was xedIn this section c has a continuous distribution and u can take only two values Thetwo formulations are actually isomorphic (even if the latter happens to be moreconvenient here) all that really matters is the distribution of duV c


(14) UC~uLur 2 UT~uL

5 bdS EbduL

duL

~duL 2 c dF~c 2 EduL

bdu ~r

~c 2 duL dF~cD

The rst integral is decreasing in b becoming zero at b 5 1 itrepresents the gain from condence-building which alleviatesSelf 1rsquos motivation problem The second integral is increasing inb it reects the loss from overcondence which causes Self 1 toattempt the task in states of the world where even Self 0 wouldprefer that he abstain Note that these effects are now endoge-nous Thus the overcondence cost in (14) is larger the morereliable Self 1 considers the memory process to be ie the largerr Conversely if r is so low as to have bu (r) uL theovercondence effect disappears entirely but the condence-building effect is now limited by u (r)

4 Strategic memory or awareness management Faced witha signal s 5 L that is hurtful to his self-esteem Self 0 chooses therecall probability l so as to solve

(15) maxl

$lUT~uL 1 ~1 2 lUC~uLur 2 M~l

Given the convexity of M(l) the optimum is uniquely determined(given r) by the rst-order condition which involves comparingthe marginal benet from self-deception UC (uL ur) 2 UT (uL )with the marginal cost M9(l) Finally the Bayesian rationality ofSelf 1 means that he is aware of Self 0rsquos choosing the recall strategyl opportunistically according to (15) and uses this optimal l inhis assessment of the reliability of memories (or lack thereof)

DEFINITION 2 A Perfect Bayesian Equilibrium (PBE) of the mem-ory game is a pair (lr) [ [01] 3 [q1] that solves (11)and (15) meaning thati) The recall strategy of Self 0 is optimal given Self 1rsquos

assessment of the reliability of memoriesii) Self 1 assesses the reliability of memories using Bayesrsquo

rule and Self 0rsquos recall strategy

We shall be interested in two main issues

1 Nature and multiplicity of equilibria What modes of self-esteem management are sustainable (from ldquosystematic denialrdquo to


ldquocomplete self-acceptancerdquo) depending on a personrsquos characteris-tics such as his time-discounting prole or cost of memory ma-nipulation Can the same person or otherwise similar people beldquotrappedrdquo in different modes of cognition and behavior

2 Welfare analysis Is a more active self-esteem mainte-nance strategy always benecial or can it end up being self-defeating Would a person rather be free to manage his self-condence and awareness as he sees t or prefer a priori to ndmechanisms (friends mates environments occupations etc)that ensure that he will always be confronted with the truthabout himself no matter how unpleasant it turns out to be

Because PBErsquos are related to the solutions r [ [q1] to thenonlinear equation obtained by substituting (11) into the rst-order condition for (15) namely

(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0

we shall use a sequence of simpler cases to demonstrate the mainpoints that emerge from our model Note for further referencethat c(rb) represents Self 0rsquos (net) marginal incentive to forget

IIC Costless Memory or Awareness Management

Repression is automatic forgetting [Henry Laughlin The Ego and Its De-fenses 1979]

We rst solve the model in the case where the manipulationof memory is costless M [ 0 While it does not capture thepsychological costs of repression (as opposed to the informationalones) this case already yields several key insights and is verytractable22

PROPOSITION 2 When M [ 0 there exist b

and b in (01) b

b with the following properties For low degrees of time incon-sistency b b the unique equilibrium involves minimumrepression (l 5 1) for high degrees b b

it involves

22 While we assume here that l can be freely varied between 0 and 1 theresults would be identical if it were constrained to lie in some interval [l

l ] With

l

0 one can never forget (or avoid undesired cues) for sure


maximum repression (l 5 0) For intermediate degrees oftime inconsistency b [ [b

b ] there are three equilibria

including a partially repressive one l [ 0L(b)1 whereL(b) decreases from 1 to 0 as b rises from b

to b

These results are illustrated on Figure I The intuition issimple and apparent from (14) When b is high enough overcon-dence is the dominant concern therefore adverse signals aresystematically transmitted Conversely for low values of b thecondence-building motive dominates so ego-threatening signalsare systematically forgotten For intermediate values both effectsare relevant allowing multiple equilibria including one wherememory is partially selective What makes all three equilibriaself-fullling is precisely the introspection or ldquometacognitionrdquo ofthe Bayesian individual who understands that his self-knowl-edge is subject to opportunistic distortions The more censoringby Self 0 the more Self 1 discounts the ldquono bad news to reportrdquorecollection and therefore the lower the risk that he will beovercondent As a result the greater is Self 0rsquos incentive tocensor Conversely if Self 0 faithfully encodes all news into mem-ory Self 1 is more likely to be overcondent when he cannot recallany bad signals and this incites Self 0 to be truthful

Note that in the censoring equilibrium (l 5 0) none of Self0rsquos information is ever transmitted to Self 1 r 5 q In thelanguage of communication games this is a ldquobabbling equilib-riumrdquo The mechanism at work here is nonetheless very differentfrom the ex ante suppression of information considered earlierwhen analyzing self-handicapping or in Carrillo and Mariotti[2000] Self 0 does not want to suppress good news only bad

FIGURE I


news but in doing the latter he cannot help but also do theformer As we shall see later on this may end up doing him moreharm than good whereas the usual ldquostrategic ignorancerdquo is onlychosen when it improves ex ante welfare

The last observation to be drawn from Figure I is that as brises from 0 to 1 there is necessarily (ie for any equilibriumselection) at least one point where l has an upward discontinu-ity Small differences in the psychic or material costs of memorymanagement repression etc can thus imply large changes inthe selectivity of memory hence in the variability of self-con-dence and ultimately in performance

IID Costly Memory or Awareness Management

To break down the renewed assaults of my memory my imagination effec-tively labored in the opposite direction [Marcel Proust Remembrance ofThings Past]

In this subsection we use specic functional forms to studythe problem set up in subsection IIB The memory cost functionis

(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l

with a 0 and b $ 0 It is minimized at the ldquonaturalrdquo recall ratelN 5 a(a 1 b) and precludes complete repression When b 0perfect recall is also prohibitively costly and M is U-shaped As tothe distribution of effort costs we take it to be uniform w(c) 5 1con [0c ] with c bduH

With these assumptions the incentive to forget namelyUC (uL ur) 2 UT (uL) in (16) is proportional to a third degreepolynomial in r with either one or three roots in [q1] (see theAppendix) Therefore for any (ab) there are again either one orthree equilibria One can go further and obtain explicit compara-tive statics results by focusing on the simpler case where recall iscostless but repression is costly The following proposition is illus-trated in Figures IIa to IIb

PROPOSITION 3 Let b 5 0 A higher degree of time inconsistency ora lower cost of repression increases the scope for memorymanipulation generating partially repressive equilibria andpossibly even making perfect recall unsustainable Formally1) For any given b there exist thresholds aI and a with 0 aI

a and continuous functions l1(a)l2(a) respectively increas-ing and decreasing in a such that (i) for a [ (0aI ) the unique


equilibrium corresponds to l 5 l1(a) (ii) for a [ (aI a ) thereare three equilibria l [ l1(a)l2(a)1 (iii) for a [ (a 1`)the unique equilibrium corresponds to l 5 1

2) There exist critical values b1 b2 b3 such that (i) for b $b3 a 5 0 (ii) for b [ [b2b3) aI 5 0 a as in Figure IIa (iii)for b [ (b1b2) 0 aI a as in Figure IIb (iv) for b b1 0 aI 5 a as in Figure IIc

The most representative case is that of Figure IIb whereeach of the three ranges [0aI ] [aI a ] and [a 1 `) correspond-ing respectively to high repression multiplicity and truthful-ness is nonempty The effects of a are intuitive we just note that

FIGURE IIaCase b2 b b3

FIGURE IIbCase b1 b b2


small changes in awareness costs can induce large changes inself-esteem and behavior23 Interestingly a lower willpower b byshifting the equilibrium set toward lower lrsquos tends to make theindividual incur higher repression costs

III BELIEFS AND MAKE-BELIEFS24

As discussed earlier surveys experiments and daily obser-vation consistently suggest that most people overestimate theirpast achievements abilities and other desirable traits both inabsolute terms and relative to others (eg Weinstein [1980] andTaylor and Brown [1988]) Well-educated reective individualsseem to be no exception as Gilovich [1991] relates ldquoa survey ofcollege professors found that 94 thought they were better thantheir average colleaguerdquo These ndings are often put forward asevidence of pervasive irrationality in human inference25 It turns

23 It is also interesting to note that the specication with uniformly distrib-uted costs is formally equivalent to one where c is xed (say c [ 1) but effort isa continuous decision with net discounted payoff bdue 2 e2 2 for Self 1 andbd(due 2 e2 2) for Self 0 Thus in our model discontinuities in behavior are notpredicated on an indivisibility

24 The terminology of ldquobeliefs and make-beliefsrdquo is borrowed from Ainslie[2001]

25 Without denying the validity of this kind of evidence we would like toemphasize that it should be interpreted with caution Answers to surveys orexperimental questionnaires may reect self-presentation motives (for the bene-ts of the interviewer) or selective memory rehearsal strategies (for the individ-ualrsquos own benet as predicted by our model) Second for every person who isldquoovercondentrdquo about how great they are (professionally intellectually socially

FIGURE IIcCase b b1


out however that rational self-deception by Bayesian agents canhelp account for most people holding biased self-serving beliefswhich in turn have aggregate effects

IIIA Optimistic and Pessimistic Biases

Continuing to work with our awareness-management modellet us compare the cross-sectional distributions of true and self-perceived abilities26 In a large population a proportion 1 2 q ofindividuals are of low ability uL having received a negativesignal s 5 L The remaining q having received s 5 A have highability uH Average ability is quH 1 (1 2 q)uL 5 u (q) we assumethat q 1 2 so that median ability is uL Consider now thedistribution of self-evaluations Suppose for simplicity that whenfaced with ego-threatening information (s 5 L) everyone usesthe same censoring probability l [ (01)27 As before let rdenote the corresponding reliability of memory given by (11)Thus when individuals make decisions at date 1

mdash a fraction (1 2 q)(1 2 l) overestimate their ability byu (r) 2 uL 5 r(uH 2 uL )

mdash a fraction q underestimate it by uH 2 u (r) 5 (1 2r)(uH 2 uL )

If the costs of repression or forgetting are low enough (eg asmall a in Figure IIc) one can easily have (1 2 q)(1 2 l) 12and even (1 2 q)(1 2 l) rsquo

1 Thus most people believethemselves to be more able than they actually are more able thanaverage and more able than the majority of individuals28 Addingthose who had truly received good news (s 5 A) the fraction ofthe population who think they are better than average is evenlarger namely 1 2 l(1 2 q) The remaining minority think

maritally) another one may be found who is undercondent depressed paralyzedby guilt and self-doubt but unlikely to acknowledge this to anyone except hisclosest condant counselor or therapist These could even be the same people atdifferent points in time

26 The interesting distinction is between self-perceived ability E[uus] andobjectively assessed ability E[uus] rather than between E[uus] and the individ-ualrsquos true u which it may measure only imperfectly To simplify the exposition weshall therefore assume in this section that s is perfectly informative about u ieu 5 E[u us] [ uL uH Alternatively one could just read ldquoobjectively assessedabilityrdquo wherever ldquoabilityrdquo appears

27 Either this is the unique equilibrium as in Figure IIc or else we focus ona symmetric situation for simplicity

28 Note that these statements (like most experimental data) are about theagentrsquos perception of his rank in the distribution of true abilitiesmdashnot in thedistribution of self-assessments which as a Bayesian he realizes are generallyoveroptimistic


correctly that they are worse than average as a result they havelow motivation and are unlikely to undertake challenging tasksThey t the experimental ndings of depressed people as ldquosadderbut wiserrdquo realists compared with their nondepressed counter-parts who are much more likely to exhibit self-serving delusions[Alloy and Abrahamson 1979]

As seen above Bayesrsquo law does not constrain the skewness inthe distribution of biases29 it only requires that the average biasacross the (1 2 q)(1 2 l) optimists and the q pessimists be zeroindeed (1 2 q)(1 2 l)r 2 q(1 2 r) 5 0 by (11) In otherwords Bayesian rationality only imposes a trade-off between therelative proportions of overcondent versus undercondentagents in the population and their respective degrees of over- orundercondence Note however that a zero average bias in noway precludes self-esteem maintenance strategies from havingaggregate economic effects Clearly in our model they do affectaggregate effort output and welfare as none of these is a linearfunction of perceived ability

IIIB To Bayes or Not to Bayes

Having shown that even rational agents can deceive them-selves most of the time (albeit not all the time) we nonethelessrecognize that it may be more realistic to view people as imperfectBayesians who do not fully internalize the fact that their recol-lections may be self-serving At the other extreme taking beliefsas completely naive would be even more implausible As arguedin subsection IIA (also recall Nietzsche and Darwin) if a personconsistently destroys represses or manages not to think aboutnegative news he will likely become aware that he has thissystematic tendency and realize that the absence of adverseevidence or recollections should not be taken at face value Thisintrospection is the fundamental trait of the human mind whichthe Bayesian assumption captures in our model Without it self-delusion would be very easy and when practiced always optimal(ex ante) With even some of this metacognition self-deceptionbecomes a much more subtle and complex endeavor

In Benabou and Tirole [1999] we relax Bayesian rationalityand allow the agent at date 1 to remain unaware not just of what

29 This was rst pointed out by Carrillo and Mariotti [2000] for strategicignorance and is a feature that our model also shares with those of Brocas andCarrillo [1999] and Koszegi [1999]


he may have forgotten but also of the fact that he forgets Self 1rsquosassessment of the reliability of a recollection s 5 A is thusmodied to

(18) rp~l Prs 5 Aus 5 A l 5q

q 1 p~1 2 q~1 2 l

where l is the actual recall strategy and p [ [01] parameterizescognitive sophistication ranging from complete naivete (r0(l) [1) to full rationality (r1(l) [ r(l)) As long as p is above acritical threshold meaning that the individualrsquos self-conception isnot too unresponsive to his actual pattern of behavior all theBayesian modelrsquos results on multiplicity and welfare rankings ofpersonal equilibria go through Thus for the purpose of under-standing self-deception and overoptimism the explanatory powergained by departing from rational inference is rather limited(perception biases need no longer sum to zero) whereas much canbe lost if the departure is too drastic without sufcient introspec-tion one cannot account for ldquoself-trapsrdquo or self-doubt

IV WELFARE ANALYSIS OF SELF-DECEPTION

The art of being wise is the art of knowing what to overlook [William JamesPrinciples of Psychology 1890]

There is nothing worse than self-deceptionmdashwhen the deceiver is at homeand always with you [Plato quoted by Mele 1997]

Is a person ultimately better off in an equilibrium with astrategy of active self-esteem maintenance and ldquopositive think-ingrdquo (l 1) or when he always faces the truth Like Plato andWilliam James psychologists are divided between these two con-icting views of self-deception On one side are those who endorseand actively promote the self-efcacyself-esteem movement (egBandura [1977] and Seligman [1990]) pointing to studies whichtend to show that a moderate dose of ldquopositive illusionsrdquo hassignicant affective and functional benets30 On the other sideare skeptics and outright critics (eg Baumeister [1998] andSwann [1996]) who see instead a lack of convincing evidence and

30 There is of course a huge industry based on that premise with countlessweb sites devoted to ldquoself-esteemrdquo and hundreds of books with titles such as Howto Raise Your Self-Esteem 31 Days to High Self-Esteem How to Change Your LifeSo You Have Joy Bliss amp Abundance 365 Ways to Build Your Childrsquos Self-Esteem501 Ways to Boost Your Childrsquos Self-Esteem 611 Ways to Boost Your Self EsteemAccept Your Love Handles and Everything About Yourself ABC I Like Me etc


point to the dangers of overcondence as well as the loss ofstandards that results when negative feedback is systematicallywithheld in the name of self-esteem preservation Our model willprovide insights into the reasons for this ambiguity

Consider an equilibrium with recall probability l 1 andassociated credibility r (via (11)) With probability 1 2 q Self 0receives bad news which he then forgets with probability 1 2 lthe resulting expected payoff is lUT (uL ) 1 (1 2 l)UC (uL ur) 2M(l) With probability q the news is good which means that noadverse signal is received The problem is that the credibility of aldquono bad newsrdquo memory in the eyes of Self 1 may be quite low sothat he will not exert much effort even when it is actually optimalto do so Indeed the payoff to Self 0 following genuinely ldquogoodnewsrdquo is only

(19) UT~uHur 5 bd E0

bdu ~r

~duH 2 c dF~c

which is clearly less than UT (uH u1) whenever cost realizationsbetween u (r) and u(1) have positive probability In that casethere is a loss from self-distrust or self-doubt compared with asituation where Self 0 always truthfully records all events intomemory Like a ruler whose entourage dares not bring him badnews or a child whose parents praise him indiscriminately anindividual with some understanding of the self-serving tendencyin his attention or memory can never be sure that he really ldquodidgreatrdquo even in instances where this was actually true

Averaging over good and bad news the agentrsquos ex ante wel-fare in equilibrium equals

(20) W~lr qUT~uHur

1 ~1 2 qlUT~uL 1 ~1 2 lUC~uLur 2 M~l

Let us now assume that truth (perfect recall) is also an equilib-rium strategy with cost M(1) as we shall see a very similaranalysis applies if l 5 1 is achieved by using some a prioricommitment mechanism (chosen before s is observed) Denotingthe difference in welfare with this benchmark case asDW(lr) [ W(lr) 2 W(11) we have

DW~lr 5 ~1 2 q1 2 l~UC~uLur 2 UT~uL

2 M~l 1 M~1 2 qUT~uHu1 2 UT~uHur


or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c

2 M~l 1 M~1D 2 q Ebdu ~r

bduH

~duH 2 c dF~c

The rst expression describes the net gain from forgetting badnews the second one the loss from disbelieving good news Whilethe individual is better motivated or even overmotivated follow-ing a negative signal about his ability he may actually be under-motivated following a good signal A few general results can beimmediately observed

First if memory manipulation is costless (M [ 0) thena partial recall (mixed strategy) equilibrium can never bebetter than perfect recall Indeed in such an equilibrium thegain from hiding bad news is zero (UC (uL ur) 5 UT (uL )) be-cause the self-enhancement and overcondence effects just cancelout The cost from self-distrust on the other hand is alwayspresent

When repression is costly this reasoning no longer appliesas the term in large brackets in (21) becomes M(1) 2 M(l) 2(1 2 l) M9(l) 0 by the convexity of M Similarly whensystematic denial (l 5 0) is an equilibrium it generates a posi-tive ldquosurplusrdquo in the event of bad news UC (uL uq) 2 UT (uL ) M9(0) $ 0 How does this gain compare with the loss fromself-distrust in the good-news state As seen from (21) the keyintuition involves the likelihood of cost realizations sufcientlyhigh to discourage effort in the absence of adverse recollections(s 5 A) When such events are infrequent the self-distrust effectis small or even absent and on average self-deception pays offWhen they are relatively common the reverse is true

PROPOSITION 4 Let M [ 0 If the cost density w(c) decreases fastenough

2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c

then ex ante welfare is higher if all bad news is censored frommemory than if it is always recalled If the inequality is


reversed so is the welfare ranking For a given cost distri-bution w self-deception is thus more likely to be benecial fora less time-consistent individual

Note that even the second result was far from obvious apriori since both the gain and the loss in (21) decrease with b inequilibrium memory manipulation tends to alleviate procrasti-nation when s 5 L but worsen it when s 5 A31 To summarizewe have shown that

1) When the tasks one faces are very difcult and onersquoswillpower is not that strong a strategy of active self-esteem maintenance ldquolooking on the bright siderdquo avoid-ing ldquonegativerdquo thoughts and people etc as advocated innumerous ldquoself-helprdquo books can indeed pay off

2) When the typical task is likely to be only moderatelychallenging and time inconsistency is relatively mild onecan only lose by playing such games with oneself and itwould be better to always ldquobe honest with yourself rdquo andldquoaccept who you arerdquo

It is important to note that in the second case the individualmay still play such denial games even though self-honesty wouldbe better First he could be trapped in an inferior equilibriumSecond motivated cognition may be the only equilibrium yet stillresult in lower welfare than if the individual could commit tonever try to fool himself32 A couple of examples will help makethese results more concrete

a) With a uniform density on [0c ](c bduH ) self-deceptionis always harmful compared with truth-telling This ap-plies whether both l 5 0 and l 5 1 are in the equilib-rium set or only one of them (See Proposition 2 andFigure I) It also applies a fortiori when repression iscostly

b) Conversely self-deception is always benecial whenw(c) 5 gc2 n on [cI 1`) with 0 cI bduL g chosen so

31 The intuition is relatively simple however The net loss across statesfrom a ldquohear no evil-see no evilrdquo strategy l 5 0 namely 2DW(q0) is simply theex ante value of information (always recalling the true s rather than having onlythe uninformed prior u (0) 5 q) Only when time inconsistency is strong enoughcan this value can be negative

32 This case is also interesting because it involves two degrees of lack ofcommitment it is because the agent cannot commit to working at date 1 that hisinability to commit not to tamper with memory at date 0 becomes an issue whichmay end up hurting him more than if he had simply resigned himself to theoriginal time-consistency problem


that the density sums to one and n (2 2 b)(1 2 b) Inthis case it can also be shown that l 5 0 is the onlyequilibrium if M [ 0

c) Finally we provide in the Appendix a simple examplewhere both l 5 0 and l 5 1 coexist as equilibria andwhere either onemdash depending on parameter valuesmdashmaylead to higher ex ante welfare

We have thus far interpreted the ldquoalways face the truthrdquostrategy as an equilibrium sustainable alongside with l Alter-natively it could result from some initial commitment of the typediscussed earlier (chosen before s is observed) which amounts tomaking oneself face steeper costs of self-deception (increasingM(l) for l lN )33 This reinterpretation requires minor modi-cations to (21) but the main conclusions remain unaltered34

The potential multiplicity of equilibria in our model raisesthe issue of coordination among the individualrsquos temporal selvesObserve that Self 1 always values information about the produc-tivity of his own efforts and therefore always ranks equilibria (orcommitment outcomes) in order of decreasing lrsquos When Self 0also prefers the l 5 1 solution it is plausible (we are agnostic onthis point) that the individual will nd ways to coordinate on thisPareto superior outcome When some repression (any l 1) is exante valuable however there is no longer any clearly naturalselection rule In either case our main welfare conclusions re-main unchanged even if one assumes that Self 0 always managesto select his preferred equilibrium First for some range of b or al 5 1 ceases to be an equilibrium even though its still maxi-mizes ex ante welfare Thus once again the individual is trappedin a harmful pattern of systematic denial Conversely for rela-tively high values of a the only equilibrium may be l 5 1 (moregenerally a high l) even though the individual would ex antebe better off it he could manage to repress bad news more easily

Interestingly our multiplicity and welfare results provide arole for parents friends therapists and other benevolent outsideparties to help an individual escape the ldquoself-trapsrdquo [Swann 1996]

33 For instance an individual with b b in Proposition 2 may be worse offwhen memory management is free (M 5 0) than when it is prohibitively costly(M(l) 5 1` for all l THORN 1) For instance specication (a) above shows that suchis always the case when the cost distribution w(c) is uniform

34 Term M(1) in (20)ndash(21) is simply replaced by b2 1 d2 tM (1 2 q) 1 m(1)where M is the up-front cost of the commitment mechanism 2t 0 is the periodwhen the commitment was made and m(1) $ 0 is the cost of perfect recall facedat t 5 0 as a result of this decision (whereas m(l) 5 1` for all l 1)


in which he might be stuck depressive state of low self-esteemchronic blindness to his own failings etc They can make himaware that a better personal equilibrium is feasible and teachhim how to coordinate on it by following certain simple cognitiverules They may also offer a form of informational commitmentserving as the repositories of facts and feelings which the indi-vidual realizes that he has an incentive to forget (ldquoletrsquos talk aboutthat incident with your mother againrdquo) More generally theyallow him to alter the ldquoawarenessrepressionrdquo technology M(l)(and hence the set of feasible equilibria) whether through theirown feedback and questioning or by teaching him certain cue-management techniques Indeed much of modern cognitive ther-apy aims at changing peoplersquos self-image through selective recol-lection and rehearsal of events self-serving attributions aboutsuccess and adversity or conversely helping them ldquosee throughrdquoharmful self-delusions

V VARIANTS AND EXTENSIONS

VA Defensive Pessimism

While people are most often concerned with enhancing andprotecting their self-esteem there are also many instances wherethey seek to minimize their achievements or convince themselvesthat the task at hand will be difcult rather than easy A studentpreparing for exams may thus discount his previous good gradesas attributable to luck or lack of difculty A young researchermay understate the value of his prior achievements comparedwith what will be required to obtain tenure A dieting person wholost a moderate amount of weight may decide that he ldquolooks fatterthan everrdquo no matter what others or the scale may say

Such behavior termed ldquodefensive pessimismrdquo by psycholo-gists can be captured with a very simple variant of our modelThe above are situations where the underlying motive for infor-mation-manipulation is still the same namely to alleviate theshirking incentives of future selves the only difference is thatability is now a substitute rather than a complement to effort ingenerating future payoffs This gives the agent an incentive todiscount ignore and otherwise repress signals of high ability asthese would increase the temptation to ldquocoastrdquo or ldquoslack offrdquo

Substitutability may arise directly in the performance ldquopro-duction functionrdquo which instead of the multiplicative form


p(eu ) 5 ue that we assumed could be of the form p(eu ) withpeu 0 More interestingly it will typically occur when thereward for performance is of a ldquopass-failrdquo nature graduating fromschool making a sale being hired or red (tenure partnership)proposing marriage etc To see this let performance remainmultiplicative in ability and effort p(uee) 5 eue where e is arandom shock with cumulative distribution H(e) The payoff Vhowever is now conditional on performance exceeding a cutofflevel p Self 1rsquos utility function is thus

(22) bdV Preue $ p 2 ce 5 bdV~1 2 H~p ue 2 ce

It is easily veried that if the density h 5 H9 is such thatxh9( x)h( x) 21 on the relevant range of x [ p ue the optimaleffort is decreasing in u Note that these results yield testablepredictions by comparing subjectsrsquo condence-maintenance be-havior across experiments (or careers) where payoffs are comple-ments and substitutes one should be able to distinguish betweenthe motivation-based theory of self-condence and the hedonic orsignaling alternatives

An even simpler form of defensive pessimism arises in situa-tions where the action subject to time inconsistency is such thatthe benets precede the costs One can think of the trade-offbetween the immediate pleasure of smoking drinking spendingfreely etc and the long-term large but uncertain costs of suchbehaviors Suppose for instance that at date 1 the decision is toconsume or not consume The rst option yields utility b but withprobability v entails a cost C at date 2 the second option yieldszero at both dates Clearly if we dene ldquoeffortrdquo e [ 1 2 x asabstinence from consumption c 5 b as its (opportunity) cost andvC as its expected long-term payoff we see that this problem tsexactly with our model Thus to counteract his tendency towardshort-term gratication the agent will try to maintain beliefsthat v and C are high35 Note that these variables are comple-ments to e 5 1 2 x in his utility function If we had framed theproblem in terms of uncertainty over the probability of beingimmune to the health risks (say) from tobacco or alcohol thisprobability 1 2 v would be a substitute with e so the agent wouldlike to understate it to his future selves Whether costs precede or

35 This is the kind of example to which Carrillo and Mariotti [2000] applytheir model of strategic ignorance pointing to studies that suggest that mostpeople actually overestimate the health risks from smoking More generally therole of the timing of costs and benets is emphasized in Brocas and Carrillo [2000]


follow benets thus simply amounts to a relabeling of variablesThe only thing that matters for the direction in which the agentwould like to manipulate his beliefs concerning a variable is itscross-derivative with the decision variable that is being set inef-ciently low due to time inconsistency

VB Self-Esteem as a Consumption Good

We have until now emphasized the value of self-condencefor personal motivation This approach provides an explanation ofboth why and how much people care about their self-image itsvalue arises endogenously from fundamental preferences tech-nological constraints and the structure of incentives As ex-plained earlier the motivation theory also readily extends tosocial interactions

This functional view of self-esteem while pervasive in psy-chology is by no means the only one (eg Baumeister [1998]) Asdiscussed earlier a common and complementary view involvespurely affective concerns people just like to think of themselvesas good able generous attractive and conversely nd it painfulto contemplate their failures and shortcomings Formally self-image is simply posited to be an argument of the utility functionThis potentially allows people to care about a broader set ofself-attributes than a purely motivation-based theory they mayfor instance want to perceive themselves as honest and compas-sionate individuals good citizens faithful spousesmdash or on thecontrary pride themselves on being ruthless businessmen ultra-rational economists irresistible seducers etc There is somewhatof an embarrassment of riches here with few constraints on whatarguments should enter the utility function and with what sign

Let us therefore focus as before on the trait of ldquogeneralabilityrdquo which presumably everyone views as a good This is alsothe type of attribute from which agents are assumed to deriveutility in Weinberg [1999] and Koszegi [1999] as well as in someinterpretations which Akerlof and Dickens [1982] offered for theirmodel of dissonance reduction The trade-off between the costsand benets of information can then be modeled by positingpreferences of the form

(23) Emax $uV 2 c0 1 u~u

where u denotes the individualrsquos self-perceived ability (expected


probability of success) at the time of the effort decision36 The rstterm always generates a demand for accurate information toimprove decision-making Suppose for now that the hedonic valua-tion u(u) is increasing and concave these properties respectivelyimply a positive demand for self-esteem and risk aversion withrespect to self-relevant signals The individual may then onceagain avoid free information or engage in self-handicapping37

Similarly all our results based on memory management on thesupply side carry over to this case Thus ldquopositive thinkingrdquo andsimilar self-deception strategies may be pursued even thoughthey are ultimately detrimental (recall the quotation from Plato)while conversely personal rules not to tamper with the encodingand recall of information such as Darwinrsquos can be valuable Thebasic insight is again one of externalities across informationstates having only good news is not such a great boost to self-esteem once the agent realizes that he would have had reasons tocensor any bad news that might have been received

Unfortunately psychology provides little guidance on whatthe appropriate shape of the hedonic preference function shouldbe (by contrast there is ample evidence of peoplersquos general biastoward short-term rewards tendency to procrastinate etc) It isthus equally likely that u(u) is convex at least over some rangein such cases the individual will be an avid information-seekerchoosing tasks that are excessively hard or risky but very infor-mative as a way of gambling for (self-) resurrection Even mono-tonicity may not be taken for granted since psychologists havedocumented both optimism and defensive pessimism The latterwhether originating from motivation concerns or hedonic ones(lowering onersquos expectations of performance because surprise

36 A more general formulation encompassing both affective and instrumen-tal concerns would be E0[max E1 [uV 2 c]0 1 J(F1)] where F1(u ) denotes theagentrsquos date 0 and date 1 subjective probability distribution over his true abilityThe functional J[ represents either an exogenous hedonic utility or an endoge-nous value function capturing the instrumental value of beliefs for self-motivationor self-presentation (signaling) purposes In our model J(F1) is easily computedand related to b

37 In a different context Rabin [1995] makes beliefs about the negativeexternalities of onersquos actions (on other people animals or the environment) anargument of the utility function and assumes concavity This provides an expla-nation for why people may prefer not to know of the potential harm caused bytheir consumption choices Caplin and Leahy [2001] study a general class ofpreferences where initial perceptions of future lotteries enter into the intertem-poral utility function Depending on whether the dependence is concave or convexa person will choose to avoid information that would make the future lottery morerisky or on the contrary seek out information and situations that increase thestakes


sharpens both the sweetness of success and the bitterness ofdefeat) requires that u(u) be sometimes decreasing

VI CONCLUSION

Building on a number of themes from cognitive and socialpsychology we proposed in this paper a general economic modelof why people value their self-image and of how they seek toenhance or preserve it through a variety of seemingly irrationalbehaviorsmdashfrom handicapping their own performance to practic-ing self-deception through selective memory or awarenessmanagement

This general framework can be enriched in many directionsOn the motivation side we noted earlier that anyone with avested interest in an individualrsquos success (or failure) has incen-tives to manipulate the latterrsquos self-perception Thus in princi-pal-agent relationships or bargaining situations the manage-ment of self-condence will matter even when everyone is fullytime-consistent These issues are explored in Benabou and Tirole[2001] where we examine the provision of incentives by informedprincipals (parents teachers managers) in educational andworkplace environments Because offering rewards for perfor-mance may signal low trust in the abilities of the agent (childstudent worker) or in his suitability to the task such extrinsicmotivators may have only a limited impact on his current perfor-mance and undermine his intrinsic motivation for similar tasksin the futuremdashas stressed by psychologists

Another interesting direction is to further explore the rich setof behavioral implications that arise from the interaction of im-perfect willpower and imperfect memory Thus in Benabou andTirole [2002] we develop a model of self-reputation over onersquoswillpower that can account for the ldquopersonal rulesrdquo (diets exer-cise regimens resolutions moral or religious precepts etc)through which people attempt to achieve self-discipline The sus-tainability of rule-based behavior is shown to depend on theeffectiveness of the individualrsquos self-monitoring (recalling pastlapses and their proper interpretation) which may be subject toopportunistic distortions of memory or inference of the type stud-ied in the present paper The model also helps explain why peoplemay sometimes adopt excessively ldquolegalisticrdquo rules that result incompulsive behavior such as miserliness workaholism oranorexia


APPENDIX

Proof of Proposition 2 For all r and b in [01] let us dene

(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c

which up to a factor of bd measures the incentive to forget badnews UC (uL ur) 2 UT (uL )

LEMMA 1 For all r [ [01] there exists a unique B(r) [ [01]such that x(rB(r)) 5 0 andi) x(rb) 0 for all b B(r) while x(rb) 0 for all b

B(r)ii) B(r) uL u (r) and B(r) is strictly decreasing in r

Proof For any given r it is clear from (A1) that x(rb) 0for b [ [0uL u (r)] while x(r1) 0 Moreover for all b uL u (r) we have

(A2)]x~rb

]b5 d2u ~ruL 2 bu ~rw~bdu ~r

2 d2uLuL 2 buLw~bduL 0

This establishes the existence and uniqueness of the root B(r) [[uL u (r)1] Moreover

(A3)]x~rb

]r5 bd2~uH 2 uLuL 2 bu ~rw~bdu ~r

so ]x(rB(r))]r 0 since B(r)u (r) uL Therefore by theimplicit function theorem B9(r) 0 for all ri

To conclude the proof of Proposition 2 consider the followingcases

a) For b $ B(q) we have for all r [ [q1] b B(r) andtherefore x(rb) 0 Memorizing bad news is thus theoptimal strategy which establishes claim (i) of theProposition

b) For b B(1) we have for all r [ [q1] b B(r) andtherefore x(rb) 0 Forgetting bad news is thus theoptimal strategy which establishes claim (iii)

c) For b [ (B(1)B(q)) there exists by the lemma a uniqueinverse function R(b) [ B2 1(b) such that x(R(b)b) 5 0Moreover R is decreasing and for any r [ (q1) x(rb)


has the sign of R(b) 2 r Therefore the only equilibriumwith r R(b) is r 5 q (or l 5 0) and the only equilibriumwith r R(b) is r 5 1 (or l 5 1) Finally r 5 R(b) is alsoan equilibrium which corresponds to L(b) 5 (1 2qR(b))(1 2 q) Dening b

[ B(1) and b [ B(q) con-

cludes the proof

Proof of Proposition 3 We shall solve for equilibria in termsof the reliability of memory r the recall strategy l is thenobtained by inverting (11) With the assumptions of the proposi-tion the incentive to forget given by (16) equals

(A4) c~rb 5 r~Du S b2d3

c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD

where Du [ uH 2 uL Dening for all b [ [01]

(A5) R~b S 1 2 b

b D 2uL

Du

(A6) V~b ~Du 2S b3d3

2c D S q1 2 qD

It is clear that c(rb) $ 0 if and only if

(A7) P~rb V~b~R~b 2 r~r 2 q $ aq 2 bS r 2 q1 2 rD

Multiplying by (1 2 r) shows that the sign of c(rb) is that of athird-degree polynomial in r with limr N q c(rb) 5 2` since a 0 and limr N 1 c(rb) 5 1` when b 0 Thus for a given b thereare either one or three solutions to c(rb) 5 0 in [q1] ie one orthree equilibria

Let us now specialize (A7) to the case where remembering iscostless but forgetting or repressing is costly b 5 0 Solvingc(rb) 5 0 then reduces to looking for the intersections of thequadratic polynomial P(rb) with the horizontal line aq We shalldenote a [ q2 1 max maxr[[q 1 ] P(rb)0 and aI [ q2 1 maxP(1b)0 a There are several cases to consider

1) For R(b) q or equivalently b R2 1(q) [ b3 it is clearthat P(rb) 0 on [q1] therefore the only equilibriumis r 5 1 Moreover aI 5 a 5 0


2) For q R(b) [ b3 the polynomial P(rb) is positive onr [ [qR(b)] implying a 0 and negative outside

a) If a a then P(rb) 0 on [qR(b)] so the onlyequilibrium is again r 5 1

b) If a a the equation P(rb) 5 aq has two roots r1(a)and r2(a) both in the interval [qR(b)] with r1(a)

r2(a) r1 decreasing and r2 increasing One can associ-ate with these two functions l1(a) and l2(a) by invert-ing (11) Let us now distinguish the following subcases

i) For q R(b) 1 or equivalently b2 [ R2 1(1) b R2 1(q) 5 b3 both r1(a) and r2(a) are in(qR(b)) and represent equilibria On [qr1(a)) and(r2(a)1] we have P(rb) aq hence c(rb) 0This means that the third (and only other) equilib-rium is r 5 1 Furthermore aI 5 0

ii) For 1 R(b) 2 2 q or equivalently b1 [R2 1(2 2 q) b b2 5 R2 1(1) the polynomialP(rb) reaches its maximum at (q 1 R(b)) 2 1Thus P(rb) is positive and hill-shaped on [q1]and aI 5 P(1b) 0 This implies that for aI a a we have q r1(a) r2(a) 1 while for a aI wehave q r1(a) 1 r2(a) In the rst case theequilibria are r [ r1(a)r2(a)1 as in case (i)above In the latter situation the only equilibrium isr 5 r1(a)

iii) For 2 2 q R(b) or equivalently b b1 [R2 1(2 2 q) the polynomial P(rb) is strictly in-creasing on [q1] so the only equilibrium is r 5 r1(a)whenever a aI 5 P(1b) 5 a It is r 5 1 whenevera $ aI

Proof of Proposition 4 Setting l 5 0 r 5 q and M [ 0 in(21) yields

DW~0q 5 ~1 2 q EbduL

bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q

3 ~duL 2 c dF~c 2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q

d~quH 1 ~1 2 quL 2 c dF~c

2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

Dening the function G(Zb) [ 0Z(Z 2 bc) dF(c) we can then

write

(A8) DW~0q 5 b21G~bd~quH 1 ~1 2 quLb

2 qG~bduHb 2 ~1 2 qG~bduLb

Clearly DW(0q) 0 when G is concave in Z and DW(0q) 0when it is convex Indeed bdDW(0q) is (minus) the ex ante valueof information ie of always knowing the true E[uus] rather thanhave only the uninformed prior or posterior u (q) The propositionimmediately follows from the fact that ]2G(Zb)]Z2 5 (2 2b)w(Z) 1 (1 2 b) Zw9(Z)

Welfare Rankings of Multiple Equilibria We construct herea simple example where l 5 0 and l 5 1 coexist as equilibriaand where either one can lead to higher ex ante welfare

First let uL uH and q [ (01) so that uL u (q) 5 quH 1(1 2 q)uL uH For b 1 but not too small we have buL uL bu (q) u (q) buH uH Next let the date 1 cost take twovalues c [ cI c with cI d [ (buL uL ) c d [ (u (q)buH) and p [


Pr[c 5 cI ] [ (01) The l 5 0 strategy is then always anequilibrium since c(qb)bd 5 p(duL 2 cI ) 0 As to l 5 1 itis also an equilibrium whenever c(1b)bd 5 p(duL 2 cI ) 2 (1 2p)(c 2 duL) 0 or

(A9)p

1 2 p

c 2 duL

duL 2 cI r

With this condition both l 5 0 and l 5 1 are equilibria (with amixed-strategy one in between which is always dominated by l5 1 since M [ 0) and l 5 0 yields higher welfare when

DW~0b 5 ~1 2 qp~duL 2 cI 2 q~1 2 p~duH 2 c 0

or

(A10)p

1 2 p S q

1 2 qD S duH 2 cduL 2 cI D r9

Since u (q) dc it is easily veried that r9 r Thus for p(1 2p) [ (r9r) the l 5 0 equilibrium is ex ante superior to the onewith l 5 1 For p(1 2 p) r9 the reverse is true

PRINCETON UNIVERSITY NATIONAL BUREAU OF ECONOMIC RESEARCH AND CENTRE

FOR ECONOMIC POLICY RESEARCH

INSTITUT DrsquoECONOMIE INDUSTRIELLE GREMAQCNRS CERASCNRS ECOLE DES

HAUTES ETUDES EN SCIENCES SOCIALES AND MASSACHUSETTS INSTITUTE OF TECHNOLOGY

REFERENCES

Ainslie G Picoeconomics The Strategic Interaction of Successive MotivationalStates within the Person (Studies in Rationality and Social Change) (Cam-bridge UK Cambridge University Press 1992)

mdashmdash Breakdown of Will (Cambridge UK Cambridge University Press 2001)Akerlof G and W Dickens ldquoThe Economic Consequences of Cognitive Disso-

nancerdquo American Economic Review LXXII (1982) 307ndash319Alloy L T and L Abrahamson ldquoJudgement of Contingency in Depressed and

Nondepressed Students Sadder but Wiserrdquo Journal of Experimental Psy-chology General CVIII (1979) 441ndash485

Arkin R M and A H Baumgardner ldquoSelf-Handicappingrdquo in Attribution BasicIssues and Applications J Harvey and G Weary eds (New York AcademicPress 1985)

Bandura A Self Efcacy The Exercise of Control (New York W H FreemanCompany 1977)

Baumeister R ldquoThe Selfrdquo in The Handbook of Social Psychology D Gilbert SFiske and G Lindzey eds (Boston MA McGraw-Hill 1998)

Benabou R and J Tirole ldquoSelf-Condence Intrapersonal Strategiesrdquo IDEImimeo June 1999

Benabou R and J Tirole ldquoIntrinsic and Extrinsic Motivationrdquo Princeton Uni-versity mimeo December 2001

Benabou R and J Tirole ldquoWillpower and Personal Rulesrdquo CEPR DiscussionPaper No 3143 January 2002

Berglas S and E Jones ldquoDrug Choice as a Self-Handicapping Strategy in


Response to Noncontingent Successrdquo Journal of Personality and Social Psy-chology XXXVI (1978) 405ndash417

Brocas I and J Carrillo ldquoEntry Mistakes Entrepreneurial Boldness and Opti-mismrdquo ULB-ECARE mimeo June 1999

Brocas I and J Carrillo ldquoThe Value of Information When Preferences AreDynamically Inconsistentrdquo European Economic Review XLIV (2000)1104ndash1115

Caplin A and J Leahy ldquoPsychological Expected Utility Theory and AnticipatoryFeelingsrdquo Quarterly Journal of Economics CXVI (2001) 55ndash79

Carrillo J and T Mariotti ldquoStrategic Ignorance as a Self-Disciplining DevicerdquoReview of Economic Studies LXVI (2000) 529ndash544

Crary W G ldquoReactions to Incongruent Self-Experimentsrdquo Journal of ConsultingPsychology XXX (1966) 246ndash252

Darwin F The Life and Letters of Charles Darwin Edited by his son FrancisDarwin (New York D Appleton and Co 1898)

Deci E Intrinsic Motivation (New York Plenum Press 1975)Elster J ldquoMotivated Belief Formationrdquo Columbia University mimeo June 1999Fazio R and M Zanna ldquoDirect Evidence and Attitude-Behavior Consistencyrdquo in

L Berkowitz ed Advances in Experimental Social Psychology Vol 14 (NewYork Academic Press 1981)

Festinger L ldquoA Theory of Social Comparison Processesrdquo Human Relations VII(1954) 117ndash140

Fingarette H ldquoAlcoholism and Self-Deceptionrdquo in Self-Deception and Self-Un-derstanding M Martin ed (Lawrence KS University Press of Kansas 1985)

Freud S A General Introduction to Psychoanalysis (New York Garden CityPublishing Co 1938)

Frey D ldquoThe Effect of Negative Feedback about Oneself and Cost of Informationon Preference for Information about the Source of this Feedbackrdquo Journal ofExperimental Social Psychology XVII (1981) 42ndash50

Gilbert D and J Cooper ldquoSocial Psychological Strategies of Self-Deceptionrdquo inSelf-Deception and Self-Understanding M Martin ed (Lawrence KS Uni-versity Press of Kansas 1985)

Gilovich T How We Know What Isnrsquot So (New York Free Press 1991)Greenier K M Kernis and S Wasschul ldquoNot All High (or Low) Self-Esteem

People Are the Same Theory and Research on the Stability of Self-Esteemrdquoin Efcacy Agency and Self-Esteem M Kernis ed (New York Plenum Press1995)

Greenwald A ldquoThe Totalitarian Ego Fabrication and Revision of Personal His-toryrdquo American Psychology XXXV (1980) 603ndash613

Gur R and H Sackeim ldquoSelf-Deception A Concept in Search of a PhenomenonrdquoJournal of Personality and Social Psychology XXXVII (1979) 147ndash169

Heider F The Psychology of Interpersonal Relations (New York Wiley 1958)James W The Principles of Psychology (Cleveland OH World Publishing 1890)Jones E F Rhodewalt S Berglas and J Skelton ldquoEffects of Strategic Self-

Presentation on Subsequent Self-Esteemrdquo Journal of Personality and SocialPsychology XL (1981) 407ndash421

Kolditz T and R Arkin ldquoAn Impression-Management Interpretation of theSelf-Handicapping Strategyrdquo Journal of Personality and Social PsychologyXLIII (1982) 492ndash502

Korner I Experimental Investigation of Some Aspects of the Problem of Repres-sion Repressive Forgetting Contributions to Education No 970 (New YorkBureau of Publications Teachersrsquo College Columbia University 1950)

Koszegi B ldquoSelf-Image and Economic Behaviorrdquo MIT mimeo October 1999Kunda Z and R Sanitioso ldquoMotivated Changes in the Self-Conceptrdquo Journal of

Personality and Social Psychology LXI (1989) 884ndash 897Laibson D ldquoGolden Eggs and Hyperbolic Discountingrdquo Quarterly Journal of

Economics CXII (1997) 443ndash478mdashmdash ldquoA Cue-Theory of Consumptionrdquo Quarterly Journal of Economics CXVI

(2001) 81ndash119Laughlin H P The Ego and Its Defenses The National Psychiatric Endowment

Fund eds second edition (New York Jason Aaronson Inc 1979)Leary M and D Down ldquoInterpersonal Functions of the Self-Esteem Motive The


Self-Esteem System as Sociometerrdquo in Efcacy Agency and Self-Esteem MKernis ed (New York Plenum Press 1995)

Loewenstein G and D Prelec ldquoAnomalies in Intertemporal Choice Evidenceand Interpretationrdquo Quarterly Journal of Economics CVII (1992) 573ndash597

Mele A ldquoReal Self-Deceptionrdquo Behavioral and Brain Sciences XX (1999) 91ndash136Mischel W E B Ebbesen and A R Zeiss ldquoDeterminants of Selective Memory about

the Selfrdquo Journal of Consulting and Clinical Psychology XLIV (1976) 92ndash103Mullainathan S ldquoA Memory-Based Model of Bounded Rationalityrdquo Quarterly

Journal of Economics CXVII (2002) 735ndash774Murray S L and J G Holmes ldquoSeeing Virtues in Faults Negativity and the

Transformation of Interpersonal Narratives in Close Relationshipsrdquo Journalof Personality and Social Psychology XX (1993) 650ndash663

Nisbett R and T Wilson ldquoTelling More Than We Can Know Verbal Reports onMental Processesrdquo Psychological Review LXXXIV (1977) 231ndash259

OrsquoDonoghue T and M Rabin ldquoDoing it Now or Laterrdquo American EconomicReview LXXXIX (1999) 103ndash124

Phelps E and R Pollack ldquoOn Second-Best National Savings and Game-Equilib-rium Growthrdquo Review of Economic Studies XXXV (1968) 185ndash199

Rabin M ldquoMoral Preferences Moral Rules and Belief Manipulationrdquo Universityof California mimeo April 1995

Rhodewalt F T ldquoSelf-Presentation and the Phenomenal Self On the Stabilityand Malleability of Self-Conceptionsrdquo in Public Self and Private Self RBaumeister ed (New York Springer Verlag 1986)

Salancik G ldquoCommitment and the Control of Organizational Behavior andBeliefrdquo in New Directions in Organizational Behavior B Staw and G Salan-cik eds (Chicago St Clair Press 1977)

Sartre J P The Existential Psychoanalysis (H E Barnes trans) (New YorkPhilosophical Library 1953)

Schacter D Searching for Memory (New York Basic Books 1996)Seligman E Learned Optimism How to Change Your Mind and Your Life (New

York Simon and Schuster 1990)Snyder C ldquoCollaborative Companions The Relationship of Self-Deception and

Excuse Makingrdquo in Self-Deception and Self-Understanding M Martin ed(Lawrence KS University Press of Kansas 1985)

Strotz R ldquoMyopia and Inconsistency in Dynamic Utility Maximizationrdquo Reviewof Economic Studies XXII (1956) 165ndash180

Swann W B Jr Self Traps The Elusive Quest for Higher Self-Esteem (NewYork W H Freeman and Company 1996)

Taylor S E and J D Brown ldquoIllusion and Well-Being A Social PsychologicalPerspective on Mental Healthrdquo Psychological Bulletin CIII (1988) 193ndash210

Weinberg B ldquoA Model of Overcondencerdquo Ohio State University mimeo August1999

Weinstein N ldquoUnrealistic Optimism about Future Life Eventsrdquo Journal ofPersonality and Psychology XXXIX (1980) 806ndash 820

Zuckerman M ldquoAttribution of Success and Failure Revisited or the MotivationalBias Is Alive and Well in Attribution Theoryrdquo Journal of Personality XLVII(1979) 245ndash287


these concerns into the realm of economic analysis and show thatthis has important implications for how agents process informa-tion and make decisions Conversely the tools of economic mod-eling can help shed light on a number of apparently irrationalbehaviors documented by psychologists

Indeed both the demand and the supply sides of self-con-dence appear at odds with economistsrsquo view of human behaviorand cognition Why should people prefer rosy views of themselvesto accurate ones or want to impart such beliefs to their childrenFrom car accidents failed dotcom rms and day trading to thespace shuttle disaster and lost wars the costs of overcondenceare plain for all to see Even granting that some ldquopositive illu-sionsrdquo could be desirable is it even possible for a rational Bayes-ian individual to deceive himself into holding them Finally thewelfare consequences of so-called self-serving beliefs are far fromclear while ldquothinking positiverdquo is often viewed as a good thingself-deception is not even though the former is only a particularform of the latter

To analyze these issues we develop a simple formal frame-work that unies a number of themes from the psychology liter-ature and brings to light some of their economic implications Werst consider the demand side of self-condence and identify inSection I three main reasons why people may prefer optimisticself-views to accurate ones a consumption value a signalingvalue and a motivation value First people may just deriveutility from thinking well of themselves and conversely nd apoor self-image painful Second believingmdashrightly or wronglymdashthat one possesses certain qualities may make it easier to con-vince others of it Finally condence in his abilities and efcacycan help the individual undertake more ambitious goals andpersist in the face of adversity While we shall mostly focus onthis last explanation all three should be seen as complementaryand for many purposes work equally well with the supply side ofour model (self-deception)

The main reason why we emphasize the motivation theory isits substantially broader explanatory power Indeed it yields anendogenous value of self-condence that responds to the situa-tions and incentives which the individual faces in a way that canaccount for both ldquocan-dordquo optimism and ldquodefensiverdquo pessimism Italso readily extends to economic and social interactions (altruisticor not) explaining why people generally prefer self-condent











































(1) u1 1 bdE1u2 5 2c 1 bdu1V













(3) F EcbdV

1


where

(4) F E0

cdV

~c 2 duV dF~u

(5) F EcdV

cbdV

~duV 2 c dF~u






(6) F 5 EcbdV

1

~duV 2 c dF~u 0












(7) E0

cbdV F~u

F~cbdVdu $ S 1 2 b

b D S cdVD









(8) S 1 2 b

b D c 2 dVF E0

cbdV F~u

F~cbdVduG $

h0~F

bdF~cbdV
















































q 1 ~1 2 q~1 2 l


(12) u ~r ruH 1 ~1 2 ruL




(13) bd E0

bdEuus

~dEuus 2 c dF~c






5 bdS EbduL

duL


bdu ~r

~c 2 duL dF~cD



(15) maxl












(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES









































































































(1) u1 1 bdE1u2 5 2c 1 bdu1V













(3) F EcbdV

1


where

(4) F E0

cdV

~c 2 duV dF~u

(5) F EcdV

cbdV

~duV 2 c dF~u






(6) F 5 EcbdV

1

~duV 2 c dF~u 0












(7) E0

cbdV F~u

F~cbdVdu $ S 1 2 b

b D S cdVD









(8) S 1 2 b

b D c 2 dVF E0

cbdV F~u

F~cbdVduG $

h0~F

bdF~cbdV
















































q 1 ~1 2 q~1 2 l


(12) u ~r ruH 1 ~1 2 ruL




(13) bd E0

bdEuus

~dEuus 2 c dF~c






5 bdS EbduL

duL


bdu ~r

~c 2 duL dF~cD



(15) maxl












(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES


































































(3) F EcbdV

1


where

(4) F E0

cdV

~c 2 duV dF~u

(5) F EcdV

cbdV

~duV 2 c dF~u






(6) F 5 EcbdV

1

~duV 2 c dF~u 0












(7) E0

cbdV F~u

F~cbdVdu $ S 1 2 b

b D S cdVD









(8) S 1 2 b

b D c 2 dVF E0

cbdV F~u

F~cbdVduG $

h0~F

bdF~cbdV
















































q 1 ~1 2 q~1 2 l


(12) u ~r ruH 1 ~1 2 ruL




(13) bd E0

bdEuus

~dEuus 2 c dF~c






5 bdS EbduL

duL


bdu ~r

~c 2 duL dF~cD



(15) maxl












(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES


































































(6) F 5 EcbdV

1

~duV 2 c dF~u 0












(7) E0

cbdV F~u

F~cbdVdu $ S 1 2 b

b D S cdVD









(8) S 1 2 b

b D c 2 dVF E0

cbdV F~u

F~cbdVduG $

h0~F

bdF~cbdV
















































q 1 ~1 2 q~1 2 l


(12) u ~r ruH 1 ~1 2 ruL




(13) bd E0

bdEuus

~dEuus 2 c dF~c






5 bdS EbduL

duL


bdu ~r

~c 2 duL dF~cD



(15) maxl












(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES


































































(7) E0

cbdV F~u

F~cbdVdu $ S 1 2 b

b D S cdVD









(8) S 1 2 b

b D c 2 dVF E0

cbdV F~u

F~cbdVduG $

h0~F

bdF~cbdV
















































q 1 ~1 2 q~1 2 l


(12) u ~r ruH 1 ~1 2 ruL




(13) bd E0

bdEuus

~dEuus 2 c dF~c






5 bdS EbduL

duL


bdu ~r

~c 2 duL dF~cD



(15) maxl












(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES

































































(8) S 1 2 b

b D c 2 dVF E0

cbdV F~u

F~cbdVduG $

h0~F

bdF~cbdV
















































q 1 ~1 2 q~1 2 l


(12) u ~r ruH 1 ~1 2 ruL




(13) bd E0

bdEuus

~dEuus 2 c dF~c






5 bdS EbduL

duL


bdu ~r

~c 2 duL dF~cD



(15) maxl












(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES











































































































q 1 ~1 2 q~1 2 l


(12) u ~r ruH 1 ~1 2 ruL




(13) bd E0

bdEuus

~dEuus 2 c dF~c






5 bdS EbduL

duL


bdu ~r

~c 2 duL dF~cD



(15) maxl












(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES


































































(13) bd E0

bdEuus

~dEuus 2 c dF~c






5 bdS EbduL

duL


bdu ~r

~c 2 duL dF~cD



(15) maxl












(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES

































































5 bdS EbduL

duL


bdu ~r

~c 2 duL dF~cD



(15) maxl












(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES



































































(16) c~rb bd EbduL

bd~ruH1~12ruL

~duL 2 c dF~c

1 M9S 1 2 qr1 2 q D 5 0






and b in (01) b


it involves


l ] With

l






to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES



































































to b



FIGURE I







(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES





































































(17) M~l 5 a~1 2 ln l 1 b~1 2 ln~1 2 l


















FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES












































































FIGURE IIcCase b b1























q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES





















































































q 1 p~1 2 q~1 2 l











bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES



































































bdu ~r

~duH 2 c dF~c



(20) W~lr qUT~uHur






or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES
































































or equivalently

(21) DW~lr 5 ~1 2 qS ~1 2 l EbduL

bdu ~r

~duL 2 c dF~c


bduH

~duH 2 c dF~c





2] ln w~c

] ln c

2 2 b

1 2 bfor all c [ 0c







































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES




































































































(23) Emax $uV 2 c0 1 u~u










VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES







































































VI CONCLUSION





APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES
































































APPENDIX


(A1) x~rb EbduL

bdu ~r

~duL 2 c dF~c





(A2)]x~rb




(A3)]x~rb










cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES


































































cludes the proof



c D S ~1 2 buL 2br2

~Du D r

2 arS 1 2 qr 2 qD 1 brS 1 2 q

q~1 2 rD


(A5) R~b S 1 2 b

b D 2uL

Du


2c D S q1 2 qD
















bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES









































































bdu ~q

~duL 2 c dF~c

2 q Ebdu ~q

bduH

~duH 2 c dF~c


5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES
































































5 q E0

bdu ~q

~duH 2 c dF~c 1 ~1 2 q E0

bdu ~q


bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c

5 E0

bdu ~q


2 q E0

bduH

~duH 2 c dF~c

2 ~1 2 q E0

bduL

~duL 2 c dF~c


write








(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES

































































(A9)p

1 2 p

c 2 duL

duL 2 cI r



or

(A10)p

1 2 p S q







REFERENCES
































































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