on the evolution of overconfidence and entrepreneurs

On the Evolution of Overcon denceand Entrepreneurs

Antonio E BernardoUCLA Anderson Graduate School of Management

Los Angeles CA 90095antoniobernardoandersonuclaedu

Ivo WelchYale School of Management and NBER

New Haven CT 06520-8200ivowelchyaleedu

This paper explains why seemingly irrational overcondent behavior canpersist Information aggregation is poor in groups in which most individ-uals herd By ignoring the herd the actions of overcondent individuals(ldquoentrepreneursrdquo) convey their private information However entrepreneursmake mistakes and thus die more frequently The socially optimal proportionof entrepreneurs trades off the positive information externality against highattrition rates of entrepreneurs and depends on the size of the group on thedegree of overcondence and on the accuracy of individualsrsquo private infor-mation The stationary distribution trades off the tness of the group againstthe tness of overcondent individuals

Starting any company is really hard to do so you canrsquot beso smart that it occurs to you that it canrsquot be done

mdashKathryn Gould Foundation Capital Menlo Parkin GSB Chicago Magazine 21-3 Summer 1999

1 Introduction

According to DeBondt and Thaler (1995) ldquoPerhaps the most robustnding in the psychology of judgment is that people are overcon-dentrdquo Such overcondence induces individuals to undertake ventures

The second author may maintain a Web page at httpwelchsomyaleedu for addi-tional information (if available) We are grateful for comments from Jennifer AakerRobert Boyd Kent Daniel David Hirshleifer Jack Hirshleifer Josh Lerner SteveLippman John Mamer Bill McKelvey Terry Odean Richard Roll Larry SamuelsonLars Stole Avanidhar Subrahmanyam Karen Stephenson Richard Thaler Karen VanNuys and workshop participants at the University of Minnesota and Harvard BusinessSchool

copy 2001 Massachusetts Institute of TechnologyJournal of Economics amp Management Strategy Volume 10 Number 3 Fall 2001 301ndash330

302 Journal of Economics amp Management Strategy

that more rational individuals might not undertake For exampleovercondence among economic entrepreneurs has been documentedby Cooper et al (1988) In their sample of 2994 entrepreneurs 81believe their chances of success are at least 70 and 33 believe theirchances are a certain 100 In reality about 75 of new businessesno longer exist after ve years Busenitz and Barney (1997) comparedentrepreneursrsquo and managersrsquo assessments on a set of real-world ques-tions (eg whether cancer or heart disease is the leading cause ofdeath in the United States) Entrepreneurs and managers were aboutequal in their accuracy but the level of condence of entrepreneurs intheir own answers was dramatically higher The question our papertries to address is if economic principles can offer an explanation forsuch relatively common overcondent behavior which has clearly andreproducibly been documented in laboratory settings to be irrationalIn addition while overcondence and entrepreneurship are impor-tant phenomena in themselves there is another motivation for study-ing overcondence Some recent work in economics and nance (egDelong et al 1991 Daniel et al 1998 Odean 1998) relies on over-condence as an underlying primitive assumption often with littletheoretical justication as to why such irrational behavior can persist

Our paper offers a simple explanation for the presence ofovercondence and entrepreneurs Our main argument is that over-condent entrepreneurs (independent spirits innovators leaders changeagents or even dissidents) are less likely to imitate their peers and morelikely to explore their environment Entrepreneurial activity can thusprovide valuable additional information to their social group

Our point holds when individual actions can convey valuableprivate information and when information aggregation within theoverall group is otherwise poor Our specic modeling frameworkis built on the concept of informational cascades introduced in Welch(1992) Banerjee (1992) and Bikhchandani et al (1992) In this contextindividuals can observe one another and typically end up followingthe same action yet information aggregation is poor because rationalnonentrepreneurial individuals who follow ldquothe herdrdquo reveal nothingabout their private information From a social perspective cascadeslead to a suboptimal level of information disclosure experimentationand exploration of the environment

When overcondent entrepreneurial individuals instead followtheir own information downweighting the information in the herdtheir actions in effect broadcast their private information to the restof their group Unknowingly overcondent entrepreneurs behavealtruistically making irrational choices that are to their own detri-ment but help their groups Because the herd carries relatively lit-tle information this is only mildly individually suboptimal Still we

The Evolution of Overcondence 303

would not expect entrepreneurs to reect very deeply on their actionsInstead we would expect them to be either socially or biologicallyldquoprogrammedrdquo to overestimate the quality of their own informationIndeed the presence of such overcondent individuals who act ontheir own information and who irrationally ignore the actions of otherindividuals in the group has already been demonstrated in laboratorysettings by Anderson and Holt (1996) Our model can easily be cali-brated to generate benets to their group that are larger by a factorof 100 than the cost to the individual

In Section 2 we identify conditions under which the benetsof entrepreneurship to the group are high and the costs to individ-ual entrepreneurs are low We then show that when groups com-pete and inferior groups disappear groups with some entrepreneurialactivity may gain enough of an evolutionary advantage to permitentrepreneurs to survive in equilibrium Our paper therefore arguesthat groups with some overcondent individuals have an evolution-ary advantage over groups without such individuals In Section 3we derive a stationary distribution in which overcondence persistsacross generations This distribution trades off the relative tness ofthe group against the relative tness of the (altruistic) individuals whoare overcondent

Our paper identies some of the forces important to the relativebenets and costs of being an entrepreneur and to being a group cul-ture society or rm that fosters or handicaps entrepreneurship Forexample we nd that overcondenceentrepreneurship can be usefulif groups are large enough to benet from the positive informationexternality if individuals have low-precision information and if over-condence is moderate rather than extreme There are of course otherimportant aspects to entrepreneurship that are not modeled by ourpaper and not every entrepreneur behaves irrationally ex ante (egManove 1988)

There are surprisingly few papers that explicitly adopt evolu-tionary selection (eg Becker 1976 Ainslie 1975 Hirshleifer 1977Hirshleifer 1987 Hirshleifer and Martinez-Coll 1988 Waldman 1994Rogers 1994 Hirshleifer and Luo 2001 Wang 2001) and fewer yetthat invoke group selection As far as we know models of group selec-tion have appeared only outside economics Section 3 discusses thehistory of arguments pro and con group selection We believe a devi-ation from the individual optimization paradigm in our context to benecessary

1 There are many well-documented psychological inference biasesthat are intrinsically difcult to defend as being in the interest


of the individual Although some biases can be explained withindividual-centric explanations (eg Hirshleifer 1987) explanationsfor inference biases should recognize that they are inference distortionsand individual behavior should follow directly from the inferenceprocess This is perhaps best to explain in the context of over-condence When cornered most economists tend to argue thatwell-documented overcondence (or other biases) could possiblybe directly linked to behavior that could enhance individual sur-vival for example an increase in aggressive behavior (see Sec 4)To defend such an argument one would have to show (i) an empir-ical linkage between aggression and overcondence (ii) why ag-gressive behavior is optimal in an environment and (iii) why it isthe distorted inference process that creates aggression In contrastour paperrsquos explains ldquofollowing onersquos own informationrdquo directly

2 Homo sapiens is unusual We are constantly judging how altruis-tic our peers are and we are constantly judged by our peers Theability to exclude individuals from membership in a society espe-cially when coupled with our long-term memory can be a power-ful force towards social behavior It should not be surprising thatbehavior that enhances group survival can play a role in certainsocial situations yet the fact that humans can behave in a sociallyvaluable manner is often neglected in economics When altruisticbehavior is entertained the economic literature often simply entersit directly into the utility function We believe group selection canoffer a model-disciplining mechanism about which irrational andnear-rational behavior may reasonably enter a utility function andwhich may not

3 Groups can evolve mechanisms that are surprisingly effective atovercoming a public goods problem For example the costs tobeing overcondent can be trivial and individual costs can beorders of magnitudes less than the benets to the group In manysituations it is difcult to imagine that alternative mechanisms(eg cultural mechanisms such as large-scale coordination by cred-ible communication) have lower social or individual costsmdashasidefrom the fact that they were not feasible when evolution shapedour psyche

4 Because economics has been faced with such puzzling psychologi-cal biases it has developed a chasm between a ldquobehavioral litera-turerdquo which takes documented psychological biases as primitivesbut rarely offers an explanation for why these biases are so per-vasive and a ldquorational literaturerdquo which de facto argues that onlyindividually rational behavior can survive in an evolutionary ormarket setting and which consequently often tends to discounteven near-rational behavior The use of optimal group-selection


mechanisms offends both camps equally but (or perhaps because)it holds the promise of reconciliation between them

In the end one goal of our paper is to develop a disciplinedapproach to the investigation of seemingly irrational inferences andbehavior based on group selection principles In particular the pointof our paper is not to show that informational cascades can (and havebeen documented to) be broken by overcondent behavior but thatovercondent behavior creates a large positive externality on pub-lic information aggregation and small costs on its perpetrators In agroup selection framework this allows overcondence to survive Assuch our theory has the potential both to explain why we are overcon-dent and to offer new insights into the question under what circum-stances overcondence is most likely to be useful and thus appearOur paper does not propose to deemphasize self-interested behav-ior and individual selectionmdashindeed that is the stronger force whenpayoffs are equal But group selection in which the cost to the irra-tional individual is very low and the benet to the group is very highcan help us understand documented individually irrational biases that areotherwise difcult to explain

Our paper now proceeds as follows The formal model in Section 2derives the socially optimal proportion of entrepreneurs It is pur-posely kept as simple and focused as possible It ends with a briefsummary of factors inuencing the trade-off between the informa-tional externality and entrepreneurial attrition Section 3 derives thestationary distribution ie the trade-off between intergroup and intra-group selection It also discusses arguments pro and con groupselectionmdashfamiliar to readers of the biology literaturemdashas they per-tain to our model Section 4 discusses alternative explanations forthe presence of overcondence and entrepreneurs Briey overcon-dence could also be explained as a signal that helps individuals con-vince others of high ability entrepreneurship could also be explainedas a high-risk but value-maximizing alternative These explanationsare not only different from those advanced in our model but also(and more importantly) are complementary to our own explanationSection 5 concludes

2 The Model

We now develop a simple model to illustrate that overcondence canimpose only small costs on entrepreneurs (individuals that put toomuch weight on their own information) but provide large benets inrevealing their private information to their groups Although our spe-cic model is based on the cascades framework it could have equally


TABLE IInformation Structure

Value State

Signal Value h 1 h 1

H p qL q p

well been based on a different framework (eg a two-armed-banditsearch model) The basic intuition and comparative statics would besimilar Our goal is to show that overcondent behavior can create anexternality that improves public information aggregation

21 Available Information

Assume that members of a group of N risk-neutral individuals choosein sequence whether or not to take an action with uncertain value h The action is costless and the true value of h is either 1 or 1 eachwith prior probability 12 No individual can observe the true valueof h but each individual can privately observe an iid signal thatis correlated with h For simplicity we assume that if h 1 theneach individual observes a private signal H (high) with probability1 gt p gt 1 2 and a signal L (low) with probability q 1 p Thus ifh 1 individuals are more likely to observe H Likewise if h 1then individuals observe the signal L with probability p and the signalH with probability q In this setting a higher value of p implies thatthe signal is more informative Table I summarizes the informationstructure

The group of individuals is sequenced randomly and exoge-nously Each individual chooses a publicly visible action either toadopt (A) to reject (R) or to abstain from decision Adopting (reject-ing) has higher expected payoffs if it is more likely that the state ish 1 (h 1) Without loss of generality we assume that the indi-vidual abstains if and only if she is indifferent between adopting andrejecting There are two types of individuals in this model who differonly in one respect

Normal individuals are fully rational in that they base their decisionsoptimally on both publicly available information and their individ-ual private informationEntrepreneurs base their decisions on both publicly available and theirown information but they do not put enough weight (in Bayesianterms) on the public information This denition of overcondence


conforms to the denition employed in recent nance models (Danielet al 1998) and to the ndings in Anderson and Holt (1996) Indi-viduals are either relatively more skeptical about external informationor relatively more overcondent about internalized information1

For computational simplicity we assume that the type of eachindividual is public knowledge2 We dene cedil to be the proportion ofentrepreneurs in the group Note that cedil is not necessarily the inci-dence of overcondent actionsmdasheven an overcondent entrepreneurcan nd himself in a situation in which the public information is sooverwhelming that even he follows it

22 Normal Individualsrsquo Decision Rule

The decision rule for normal individuals optimally uses their privateinformation signal and information contained in the decisions of indi-viduals that arrived earlier The payoff and information structure issuch that normal individuals adopt if h 1 is more likely thanh 1 In our setup this occurs when individuals can infer thatmore H -signals have been observed than L-signals

Let Sn be the number of H -signals less the number of L-signalsthat can be inferred by all individuals from the actions of the rst narrivals Thus Sn Sn 1 1 if all individuals can infer that the nthindividualrsquos signal was H Sn Sn 1 1 if all individuals can inferthat the nth individualrsquos signal was L and Sn Sn 1 if an individualcannot infer anything about the nth individualrsquos signal It is straight-forward to show that within our information structure the state ofinformation at any stage n is completely summarized by Sn

The normal individualsrsquo optimal decision rule is as follows Thenth individual adopts if (i) Sn 1 0 and she observes H or (ii) Sn 1 2

1 Ross and Sicoly (1979) outline why information availability and attribution canhave an egocentric bias and much of their argument would naturally apply to an ego-centric bias on judgment about the relative precision of own vs othersrsquo informationIn updating a mean estimate the lower relative subjective variance about the internal-ized estimate than that of the external estimate would then lead subjects to put toomuch weight on their internal information and too little weight on external informa-tion While such behavior (by security analysts) has been documented in Abarbanelland Bernard (1992) and Batchelor and Dua (1992) experimental studies of overcon-dence that we are aware of show only that subjectsrsquo internal information generatesassessments with too narrow a range We are unaware of experimental studies that testthe relative overcondence about internalized vs new external information

2 When the types of individuals are unknown each conforming individual couldbe an entrepreneur or a normal individual This causes the public state Sn (denedbelow) to drift (slowly) as more individuals conform The algebra gets more complex(see Anderson and Holt 1996) but the intuition of our paper remains informationaggregation is poor and overcondent entrepreneurs can provide useful informationto their group


and she observes L Stated differently the nth individual adopts ifSn 1

To understand the information content of past decisions con-sider the cascade scenario in Bikhchandani et al (1992) in whichthere are no entrepreneurs (cedil 0) If the rst arrival observes thesignal H then the conditional expected value of adopting is p q gt 0and therefore the rst individual adopts Even though the rst indi-vidualrsquos information is private all individuals can infer from the rstindividualrsquos action that she observed the signal H and thus S1 1Now suppose the second individual observes the signal L Condi-tional on the sequence of signals HL the expected value of adoptingis 0 and the individual is indifferent between adopting and reject-ing and by assumption the individual abstains Consequently if therst individual adopts and the second abstains then all individualsknow that HL has occurred thus S2 0 However if the second indi-vidual also adopts all individuals know that HH has occurred thusS2 2 and the third individual adopts regardless of her private infor-mation Because this action is uninformative the fourth individualalso adopts regardless of her private information and this continuesfor all future individuals Similarly when Sn 1 2 all subsequentarrivals reject Sn 2 and Sn 2 are absorbing states and allfuture arrivals will conform adopting or rejecting respectively Thegroup gets entrenched in good or bad cascades In a good cascadeeveryone gets locked into adopting (rejecting) if h 1 (h 1) In abad cascade everyone gets locked into rejecting (adopting) if h 1( h 1) The probability of such a bad cascade can be quite high forexample if p 051 it approaches 48 even in large groups

23 Entrepreneursrsquo Decision Rule

Entrepreneurs also use both publicly available information and theirown private information but place too much weight on the latter Weassume that entrepreneurs believe their signal has precision p gt pFor a given p this leads them to follow their own signal if Sn 1 lt kand to behave like normal individuals and follow the crowd if Sn 1

k where the critical state k increases monotonically with p [The con-dition that entrepreneurs always follow their own information if andonly if Sn 1 lt k is equivalent to pk 1 (pk 1 qk 1) lt p lt pk (pk qk )]Once the public information becomes sufciently overwhelming evenif every individual is overcondent irrationally overcondent actionscease in our model and entrepreneurs suppress their eagerness toignore the public information in favor of their more limited privateinformation If p 1 the critical state k is innity and public infor-mation never tempers overcondent actions If p p then k 2 and


entrepreneurs are like normal individuals regardless of the currentstate

The decision rule for entrepreneurs is similar to those of normalindividuals The entrepreneur ignores public information and followsthe private information (adopt if H reject if L) if the state of informa-tion prior to their arrival satises Sn 1 lt k However if Sn 1 kentrepreneurs follow their predecessor The information states k and

k are absorbing states because once they are reached no one caninfer the information signals of subsequent individuals In effect hav-ing entrepreneurs expands the ldquononcascade action intervalrdquo betweenthe absorbing cascades states from 2 and 2 to k and k Once

k or k is reached the informational cascade becomes unbreakableeven in the presence of entrepreneurs

24 Payoffs

We now dene the payoffs and ex ante welfare for both types of indi-vidual and for the group overall We begin with the normal typesLet EumlVR n( ) denote the random payoff to the nth arrival if she is anormal type and let E[EumlVR n(cedil)] denote its unconditional expectationBecause the model is symmetric we only have to consider the caseswhen she adopts First suppose the true value state is h 1 in whichcase adopters get a payoff of 1 Rational types adopt if Sn 1 whichoccurs with probability Pr(Sn 1 h 1) Yet if the true value state ish 1 adopters receive a payoff of 1 which occurs with probabilityPr(Sn 1 h 1) Thus

E[EumlVR n ( )] Pr[Adopt h 1]Pr[ h 1]

Pr[Adopt h 1]Pr[ h 1]

12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ]

12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ] (1)

because Pr(Sn 1 h 1) Pr(Sn 1 h 1) Let EumlVOC n( ) denote the random payoff to the nth arrival if she

is an entrepreneur An entrepreneur adopts either if she receives aprivate high signal and the state is not above k or below k or ifthe critical state k that produces an adopt cascade has already beenreached

Pr(Adopt h 1) p Pr( Sn 1 lt k h 1)

Pr(Sn 1 k h 1) (2)


and

Pr(Adopt h 1) q Pr( Sn 1 lt k h 1)

Pr(Sn 1 k h 1) (3)

Thus an entrepreneur expects to receive

E[EumlVOC n(cedil)]12

[pPr( Sn 1 lt k h 1)

qPr( Sn 1 lt k h 1)

Pr(Sn 1 k h 1)

Pr(Sn 1 k h 1)] (4)

No closed-form solution exists for these probabilities for p( 1

2 1) but we can derive a recursion formula to compute them numer-ically (in the Appendix)

The overall group payoff is the expected payoff to individualsin the group

E[EumlV ( )] cedil1N

N

Sn 1

E[EumlVOC n( )] (1 )1N

N

Sn 1

E[EumlVR n ( )]

cedilE[EumlVOC( )] (1 )E[EumlVR( ) ] (5)

From a group perspective the presence of entrepreneurs in the grouphelps in that it releases more information which makes it more likelythat most individuals choose correctly it hurts in that entrepreneursmake more frequent mistakes which hurts themselves and thus low-ers the average group payoff

25 The Solution The Socially OptimalProportion of Entrepreneurs (cedil )

We begin by determining the optimal proportion of entrepreneursfrom a social welfare perspective cedil The social welfare function isdened as E[EumlV ( )] in (5) which is maximized by some cedil The fol-lowing proposition describes the effect of entrepreneursmdasha positiveexternalitymdashon normal individuals

Proposition 1

1 The (ex ante) probability that a normal individual makes an incor-rect decision decreases with increasing proportion of entrepreneurs in thegroup ( ) with increasing degree of overcondence among entrepreneurs(k(p p)) and with increasing size of the group (N ) 2 The limiting probability (n ) of being in an incorrect cascadeequals q2 (p2 q2) for cedil 0 and qk (pk qk ) for all cedil gt 0


05 06 07 08 09 1Private Signal Precision (p)

0

025

05

075

1

Pro

babi

lity

Correct Choice No EntrepreneursIncorrect Choice No EntrepreneursCorrect Choice 5 Entrepreneurs Modest OCIncorrect Choice 5 Entrepreneurs Modest OCCorrect Choice 5 Entrepreneurs Extreme OCIncorrect Choice 5 Entrepreneurs Extreme OC

FIGURE 1 THE PROBABILITIES THAT NORMAL INDIVIDUALSLATE IN THE QUEUE END UP IN A CORRECT AND IN ANINCORRECT INFORMATIONAL CASCADE AS A FUNCTION OFTHE PRIVATE SIGNAL PRECISION (p) IN A GROUP OF N 250INDIVIDUALS

Proof See Appendix A1 u

Figure 1 shows the effect of the proportion of entrepreneurs( ) on the decisions of the normal individuals Having entrepreneursshifts probability mass from being in an incorrect cascade to being ina correct cascade

The probability of not being in a cascade approaches 0 veryrapidly For example the 10th individual is in a cascade with prob-ability 975 if p 06 99 if p 075 and 9998 if p 09 The50th individual is in a cascade with probability in excess of 9999999for these three prsquos The innermost two lines are the probabilities ofending up in a right or a wrong cascade in the straight informational-cascade-without-entrepreneurs scenario The outer lines are the sameprobabilities in the presence of 5 entrepreneurs with modest over-condence [k(p p) 12] and extreme overcondence [k(p p) ]


0 10 20 30 40 50Degree of Overconfidence (k)

0

025

05

075

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs

Signal Precision p=051 Group Size N=50Signal Precision p=051 Group Size N=500Signal Precision p=060 Group Size N=50Signal Precision p=060 Group Size N=500

FIGURE 2 THE OPTIMAL PROPORTION OF ENTREPRENEURS (cedil )AS A FUNCTION OF THE DEGREE OF OVERCONFIDENCE (k)FOR TWO VALUES OF INDIVIDUALSrsquo SIGNAL PRECISION p ANDGROUP SIZE N

26 Comparative Statics

There are three parameters in our model that inuence the optimal pro-portion of entrepreneurs (cedil) the group size N the information pre-cision p and the degree of overcondence [k(p p)] Unfortunatelythe effects of the three parameters on cedil cannot generally be foundanalytically However numerical simulations show that certain direc-tional inuences of the parameters p p and N on the solution cedil

are presentmdashand to the extent that we cover the relevant parameterspace we can conjecture that they are pervasive

261 Degree of Overcon dence (p ) Figure 2 shows thatthe optimal proportion of entrepreneurs in the group decreases as thedegree of overcondence increases When overcondence is extremeentrepreneurs make more mistakes but provide more information thanmoderately overcondent entrepreneurs However the marginal valueof the extra information is small because it arrives when the pub-lic state of information is already very informative Thus it is ben-ecial to the group to have fewer entrepreneurs as overcondence


becomes more extreme The gure also shows that if overcondenceis modest it can be socially optimal for the group to consist entirelyof entrepreneurs When overcondence is modest entrepreneurs pro-vide extra information when it is most valuable and make few extramistakes relative to normal types

We also compared social welfare for all degrees of overcon-dence (p ) The social welfare function is typically highest for inter-mediate levels of overcondence For xed p greater overcondencep is better for larger N For xed N less overcondence p is betterfor larger p Interestingly Hirshleifer and Luo (2001) and Wang (2001)demonstrate a similar result in different settings (Obviously hold-ing everything else constant increasing N and p improves the socialwelfare function)

262 Signal Precision Figure 3 shows that the optimal pro-portion of entrepreneurs decreases in the private signal precision (p)mdashexcept in rare border cases (not plotted) Holding group size constant


0

01

02

03

04

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs

Population N=50 Extreme OCPopulation N=100 Extreme OCPopulation N=250 Extreme OCPopulation N=500 Extreme OCPopulation N=250 Medium OC (k=12)Population N=50 Modest OC (k=7)

Jump to 1

FIGURE 3 THE OPTIMAL PROPORTION OF ENTREPRENEURS(cedil ) AS A FUNCTION OF EACH INDIVIDUALrsquoS PRIVATE SIGNALPRECISION (p)


the optimal proportion of entrepreneurs decreases with increasing pri-vate signal precision over a certain range the optimal proportion ofentrepreneurs can be either one or zero Typically when p is low thereare more bad cascades in the absence of information aggregation andthe cost of being an entrepreneur is lower This favors the presence ofentrepreneurs The gure also shows that large groups require moreentrepreneurs than small groups when the private signal precision iseither very low or very high Thus the effect of group growth onentrepreneurship depends on signal precision for very low and veryhigh private information precision group growth translates into anincreased optimal proportion of entrepreneurs For intermediate pri-vate information precision group growth translates into a smalleroptimal proportion of entrepreneurs

263 Group Size (N) Figure 4 shows that the effect of thegroup size (N ) on the optimal cedil is ambiguous and depends on thelevel of overcondence A higher p would shift all the functions towards

1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs

Modest Overconfidence (k=4)Medium Overconfidence (k=10)High Overconfidence (k=25)Extreme Overconfidence (k=Infinite)

FIGURE 4 OPTIMAL PROPORTION OF ENTREPRENEURS (cedil ) ASA FUNCTION OF GROUP SIZE N FOR VARIOUS DEGREES OFOVERCONFIDENCE (k) AND HOLDING PRIVATE SIGNAL PRECI-SION CONSTANT AT p 06


the southeast When overcondence is innite the optimal propor-tion of entrepreneurs is zero for small N then reaches a maximumand nally asymptotes towards zero When the overcondence k isnite the optimal proportion closely tracks the innite-overcondenceproportion below a critical population size and quickly converges to100 just above that size

First consider the case where entrepreneurs exhibit extreme over-condence Such entrepreneurs provide a positive externality onlyto the normal types in the group because all other entrepreneurscompletely ignore the public information When the group is smallentrepreneurs provide few benets because there are few individualsfollowing them who can take advantage of the positive informationexternality Consequently a small proportion of entrepreneurs is opti-mal Similarly if the group is very large even a small proportion ofentrepreneurs can represent a large absolute number of entrepreneurs(additional pieces of information) almost assuring a correct cascadeAgain a small proportion of entrepreneurs is optimal The optimalproportion of entrepreneurs is highest for intermediate group sizesNow consider the case where entrepreneurs are not perfectly but onlymoderately overcondent (k ) The above arguments continue tobe truemdashexcept that entrepreneurs act like normal individuals oncea sufcient number of other entrepreneurs have appeared At thatpoint entrepreneurs make no additional mistakes (relative to nor-mal individuals) and thus impose no extra costs on the populationBecause the payoffs to both types are increasing in the probability ofbeing in the correct cascade which increases with the proportion ofentrepreneurs cedil 1 for sufciently large N

It is noteworthy that the positive information externality workswell only when groups are sufciently large There is a minimumgroup size necessary for groups to benet from overcondent behav-ior This suggests economies of scale small groups are intrinsicallypoorly suited towards taking advantage of the information external-ity and may be poorly equipped eg to deal with new situationsin which information aggregation is especially important They can-not afford to risk the loss of entrepreneurs in ordinary situations Thesteep slope at the minimum N also suggests that tests of our theorywould do well to focus on situations in which group size increasedfrom a very small to a slightly larger number of entrepreneurs (hold-ing relatedness constant) In such cases we would expect to see anexplosion of nonconforming irrational behavior

27 Omitted In uences

Although the formal model was based on the specic concept of infor-mational cascades the point of our paper is to argue that overcondence


could have evolved as a device that helps groups to overcome poorinformation aggregation and to explore their environment betterOffering only a model this paper has to ignore a number of otherinuences that can be important To the extent that other factors canreduce information aggregation or increase the usefulness of the infor-mation the marginal value of having more overcondence wouldincrease and we would expect to see more entrepreneurs

Experimentation Information aggregation is particularly poor whena situation is unique and choices are discrete so that individualscannot repeat and experiment with different choices As reasonedabove because information aggregation is poorer when experimen-tation is not feasible we would expect to see more entrepreneursand overcondenceCommunication Information aggregation is particularly poor whendirect talk (conversation) fails when it is too cumbersome and costlyor when it is not credible It is better when there is much trust andcoordination Finally there should be more overcondence in socialspeciessocieties then in solitary speciessocietiesOrdering Information aggregation is particularly poor if the mostinformed (possibly mostprestigious) individual actsrst and therebyinduces all subsequent individuals to conform (see Zhang 1997)Information Costs Information aggregation is particularly poor ifindividuals have to purchase information instead of being freelyendowed with it (To remedy this we would have to dene anentrepreneur as someone who altruistically chooses to purchaseinformation and act on it)Memory Information aggregation is particularly poor if individualscan only observe the most recent individuals rather than everyoneThen the information in the dissent of a single entrepreneur may bequickly forgotten it would take a set of consecutive entrepreneursto signal to the group that a different action would be betterChanging environment Information aggregation is more valuable ifthe environment is stable enough for the actions of previous indi-viduals to be informative

In our informational cascade model agents observe the actionsof the other agents but not their information But noncascade settingscan offer similar ndings as long as the informational group bene-ts are large relative to the costs to the entrepreneur For exampleBolton and Harris (1999) examine a two-armed bandit problem withN agents in which each agent observes the actions and informationgenerated by the experimentation of others They demonstrate thatfree-rider problems yield suboptimal experimentation (relative to thesocial optimum) The presence of overcondent individuals would


help to mitigate these free-rider problems Cao and Hirshleifer (2000)extend the simple cascades model to allow early adopters to com-municate the payoffs (but not the signals) they received They showthat informational cascades will still occur with positive probabilityMoreover they show that it is possible that observing payoffs andactions of predecessors can reduce average welfare compared to thesimple case where only actions can be observed Again the presenceof overcondent individuals would help to mitigate these free-riderproblems

3 The Public-Goods Problem andIntragroup Survival

31 A Stationary Distribution

In the previous model selection occurred only at the group level ineffect assuming that all individuals within a group were clones In thissection we sketch an environment in which the public-goods problem(suboptimal information acquisition from a social perspective becauseit is in everyonersquos interest to have someone else acquire the infor-mation) exposes individuals to both group and individual selectionThis combined with the fact that the benets to the group can beseveral orders of magnitude greater than the costs to the individualallows entrepreneurs to survive robustly in equilibrium For exampleif the signal precision is p 051 and the group contains 500 individ-uals the expected group benet to having a rst entrepreneur withk(p p) 4 is approximately 114 times larger than the expected costto this individual

An entrepreneur can benet from such large incremental grouppayoffssurvival to the extent that her genes are ldquoin the same boatrdquo(likely to cosurvive) with those of her other group members in at leastthree ways

311 Indirect Genetic Bene ts(ie KinshipRelatedness) Hamiltonrsquos rule (eg Smith 1989pp 169f Boyd and Silk 1997 pp 260ff) is commonly used in evo-lutionary genetics to show that a small set of related altruists caninitially increase in numbers when they appear within a large popu-lation of normal individuals We now show that a variant of this rulecan apply to our entrepreneurs

Consider a scenario in which the signal precision is p 051and the population consists of a large number of groups each withN 500 normal individuals and no entrepreneurs Now suppose thatone group appears that contains a (family with a) mutation that gives


rise to 20 moderately overcondent individuals with k(p p) 43 Ina population of many groups composed of normal individuals theaverage tness of normal individuals in the population is not muchinuenced by the presence of 20 entrepreneurs but the average t-ness of the entrepreneurs in the populationmdash100 of whom enjoy thebenets of the presence of the other entrepreneursmdashdoes increase dra-matically It is this disproportionate benet that allows entrepreneursto increase in frequency in the overall population of both groups eventhough their frequency in their own group may decrease4

In this scenario the expected marginal cost of being the twenti-eth entrepreneur is 00035 The expected marginal benet to the other499 individuals from having this twentieth entrepreneur totals 011Yet only 19 499 38 of the group are other entrepreneurs (the coef-cient of relatedness)5 Furthermore each of the 19 entrepreneurs gar-ners slightly less benet from the presence of this entrepreneur than anormal individual because they tend to act more based on their owninformation than on public information Adding up the expected ben-et to entrepreneurs gives a total gain to the other 19 entrepreneursof 000755 from the presence of the marginal 20th entrepreneur Forthe group of twenty individuals the total cost of overcondence toall entrepreneurs is 0129 the total benet is 0156 Consequently inthe next generation entrepreneurial types displace some normal types

3 In standard biology models simultaneous within-group appearance is assumed inthat members of sibgroups interact only with other members of their sibgroup (althoughthese members need not be of the same genotype) Similarly in Eshel et al (1998)altruists survive only if they are clustered together with other altruists but the methodto produce spacially correlated distributions of types is different Altruists can appearby imitating other altruists around them which allows one altruist to be more likely tobenet other altruists

4 Both the consideration of small changes in characteristics (such as our introduc-tion of just modest overcondence) and the consideration of a simultaneous appearanceof just a few altruistic individuals within the same group are standard practice inevolutionary biology If group benets were not captured disproportionately by otherentrepreneurs altruism (overcondence) would quickly disappear Put differently ifthe main beneciaries were egoists the altruistic type would likely disappear beforeit could help enough another altruistic types to garner a survival benet Such aproximity-of-types condition is necessary in calculations that employ Hamiltonrsquos rule toshow that biological altruism can increase and is usually accomplished by computingpayoffs over paired sibgroups

5 Relatedness on the order of 5ndash10 is not implausible Because marriage occursprimarily in proximity (geographical cultural) there is more genetic similarity amonggroups than suggested by gene dispersion by random mating This is readily visi-ble in some persistent local regional national ethnic and racial physical traits Forexample the gene for dark skin is ubiquitous in sub-saharan Africa and nonexistentin Europe For a more random set of polymorphic genestraits subject more to geneticdrift than selection Lewontin (1974) estimates that 854 of the genetic variance amonghumans is between individuals 83 between populations and 63 is between racesCavalli-Sforza (1969) nds that individuals in alpine villages in the Parma valley ofItaly display a genetic similarity of about 3


within the overall population (It is not important to the argument thatnormal types from the group containing the entrepreneurs will alsodisplace normal types in other groups)

312 Direct Transfers As in all public-goods problems itis in the grouprsquos interest to nd a mechanism to enhance the inter-nalization of the positive spillover provided by entrepreneurs In thenational economic sphere internalizing mechanisms could be patentand copyright protection or even public recognition and social stand-ing In a rm or institution a governing body might be able to directlysubsidize or discourage entrepreneurial activity In small social groupsindividuals displaying no intention to explore the environment couldbe ostracized (Hirshleifer and Rasmusen 1989)

It is reasonable to presume that the within-group sharing arrange-ment is itself subject to evolutionary pressures and therefore likely toevolve towards solutions favoring the outcome that enhances groupsurvival Consequently we would expect to see group institutionsevolve that facilitate long-run solutions closer to the group-optimal cedilYet if optimal institutions can evolve they could potentially reducethe need for biological irrational overcondence and augmententrepreneurship with incentives (transfer subsidies) that are optimalfrom a group perspective6

313 Direct Payoff Participation Groups can share in theirsuccess through economies of scale and equitable distribution eg intheir joint hunting of large prey or conquest of new territory Whilethis cannot in itself stop the shrinking proportion of entrepreneurs itcan increase the absolute number of entrepreneurs To the extent thatgroups with entrepreneurs can maintain relative faster growth thangroups without them an equilibrium can emerge This is exploredbelow

314 Summary In sum the large discrepancy between groupbenets and individual costs makes it easy to construct models inwhich overcondent entrepreneurs can survive The next subsectionsketches one such possible model

6 The issue of socialcultural mechanisms as replacements for biological mecha-nisms is itself rather interesting and the subject of much debate between sociologistsand biologists (eg Rogers 1988) For our paper we note that the evolution of socialinstitutions is fairly recent in the history of Homo sapiens while imitation has beendocumented in many vertebrate species (see eg Gibson and Hoglund 1992) Finallysocial mechanisms could also exert pressure towards other non-overcondence-basedmechanisms that help to resolve the information aggregation problem differently egwith culture and conversation


32 A Displacement Model

321 Distribution of Types within Groups We nowsketch a model to compute an equilibrium distribution of entrepre-neurs In each generation t we pit two groups (g [A B]) against oneanother and we pit individuals within groups against one anotherWe assume that each group consists of N individuals drawn froman underlying population of groups Let f t( ) denote the probabilitydensity of -groups in the population at generation t and dene cedilA t

and B t to be the realized proportion of entrepreneurs in groups Aand B Groups and individuals compete a large number of times ineach generation so that their payoffs are the expected payoffs com-puted in Section 2 overcondent individuals in group g receive pay-offs E[VOC( g t)] normal individuals receive E[VR( g t)] and the aver-age group payoff is g tE[VOC( g t )] (1 g t)E[VR( g t) ]

322 The Contest The winning group displaces the losinggroup in the next generation Within the winning group individu-als survive in proportion to their relative payoffs Consequently theproportion of entrepreneurs in the next generation t 1 is

t 1(cedilA t B t)g tE[VOC( g t)]

w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)

where g denotes the winning group t is a subscript for the genera-tion and w modulates the relative efciency by which overcondentindividuals are replaced by normal individuals from one generation tothe next For example assume groups A and B are of size N 100 thesignal precision is p 06 and overcondence is a modest k(p p) 4Also suppose groups A and B with A 005 and B 01 respec-tively compete Entrepreneurs in group A expect to receive a payoffof 01644 and normal individuals to receive 02417 for a group aver-age of 02378 Entrepreneurs in group B expect to receive 02126 andnormal types to receive 02656 for a group average of 02603 Con-sequently group B survives but entrepreneurs within this group arein a less favorable position This latter effect is a standard within-group replicator dynamic The group competition however counter-balances the individual selection pressure With a coefcient w of 1entrepreneurs constitute 01 02126 02603 82 of the populationin the next generation which is larger than the 75 it was in theprevious generation

One can compute a matrix of the resulting proportion of entrepre-neurs t 1 as a function of the proportion of entrepreneurs in groupsA and B This matrix has certain general properties The proportionof entrepreneurs in the next generation declines in cells close to the


diagonal of the matrix because when the two groups have a similarproportion of overcondent types only individual selection pressureremains Group selection plays no role when both groups are of equalquality However the generational decline in entrepreneurs is zero ifthere are either no or only entrepreneurs (and it is small nearby) Insum if the frequency of entrepreneurs across groups has no variancethen group effects cannot persist and only pure groups have a chanceof survival

On moving away from the diagonal cellsmdashie increasing the dif-ference in the proportion of entrepreneurs in the two groupsmdashas onegroup gets closer to the optimal proportion of entrepreneurs than itscompetitor it tends to win When the group with more entrepreneursis tter than its competition group selection favors more entrepre-neurs to counterbalance individual selection in favor of more normaltypes Thus in such matrix cells the next generation may or may notcontain a greater proportion of entrepreneurs than the average pro-portion from both groups in the previous generation Finally in off-diagonal matrix cells in which the group with fewer entrepreneurs istter both individual and group selection effects reduce the propor-tion of entrepreneurs in the next generation

323 Equilibrium Distributions A probability distributionover -groups f t(cedil) is a stationary distribution if 7

f t 1( ) f t ( ) (7)

The degenerate distribution cedil 1 with probability 1 (only entrepre-neurs) is a stationary distribution However as explained above noother degenerate distribution is necessarily an equilibrium becausetwo equal competing groups face no group pressure and the propor-tion of types would change in favor of normal individuals in the nextgeneration

324 Computing a Stationary Distribution To illustratean equilibrium consider the previously used example with groups ofsize N 100 entrepreneurs of type k(p p) 4 and information pre-cision p 06 There are N 1 101 possible group arrangementsDene frac14i t to be the probability that cedil i N for i 0 1 2 N To compute eg the number of groups with a proportion t 1 8 100of entrepreneurs in the next generation we need to consider all pos-sible competitive scenarios in this generation that can produce 8 100

7 This is a different denition from the standard evolutionarily stable strategy (ESS)denition because we are also concerned with across-group dynamics and not onlywith within-group dynamics


entrepreneurs For example we showed above that if cedilA t 005 andB t 01 then the next generation will have a proportion t 1 0082

of entrepreneurs To maintain discreteness we assume that the nextgeneration contains a proportion 8 100 of entrepreneurs with proba-bility 80 and 9 100 with probability 20 The probability that cedilA t

005 and B t 01 groups meet is frac145 tfrac1410 t consequently this scenariocontributes probability mass frac145 t frac1410 t 08 to the probability of hav-ing frac148 t 1 in the next generation and frac145 t frac1410 t 02 to the probabilityof having frac149 t 1 in the next generation In general to nd frac14x t 1 onemust sum probabilities over all possible competitive scenarios Thus

frac14x t 1

100

Si 0

100

Sj 0

frac14i tfrac14j th(x i j ) (8)

where h( ) apportions probability mass to adjacent -fractions to main-tain the discreteness of the distribution

More specically dene yi j int[ t 1( i t j t) N ] the integerportion of the number of entrepreneurs and mi j t 1( i t j t) yi j the remainder Then

h(x i j )

8gtlt

gt

1 mi j if x yi j mi j if x yi j 10 otherwise

(9)

The stationary distribution requires that frac14x t 1 frac14x t for all x Thesolution to this system of N 1 nonlinear equations and unknownsis not necessarily unique Evolution need not necessarily lead to a uniqueoutcome Figure 5 graphs a set of viable stationary distributions in theexample case In this case the group-optimal cedil is 1 (allentrepreneurs)mdashand the distribution cedil 1 with probability 1 is alsothe Pareto-optimal stationary distribution

This is a general result

Proposition 2 If the group-optimal proportion of entrepeneurs cedil is1 a (Pareto-dominating) equilibrium exists in which no groups with anynormal types can survive

Groups with only entrepreneurs face no individual selection pres-sure and eventually wipe out all other groups consequently normalindividuals have no chance to replicate (This is not an artifact of oursevere penalty for the losing group even if the losing group shrankonly slowly the cedil 1 group which does not experience internal


0 10 20 30 40 50 60 70 80 90 100Groups With Number of Entrepreneurs

0

01

02

03

04

05

06

07

08

Pro

babi

lity

FIGURE 5 SOME STATIONARY STRATEGY DISTRIBUTIONS (frac14 )WHEN THE POPULATION SIZE N IS 100 SIGNAL PRECISION p IS051 AND OVERCONFIDENCE k(p p) IS A MODEST 4

selection pressure against entrepreneurs would still end up eventu-ally displacing all other groups)

However when cedil 1 selection pressure against entrepreneursis strong For example Figure 6 graphs a set of equilibria when p051 k 12 and N 100 (These parameters imply that entrepreneurstend to act with close to extreme overcondence) The socially optimalproportion of entrepreneurs is 425 Yet the Pareto-best distributionof groups contains on average only about 1 entrepreneur per groupThe gure shows that entrepreneurs survive in the Pareto-preferredequilibrium but the average frequency of entrepreneurs in this sta-tionary distribution is ldquoonlyrdquo about 1ndash2mdashan order of magnitudelower than the group optimum of cedil 0425

These examples show that overcondence and entrepreneurshipcan survive in an evolutionary setting There is a large parameterspace in which either all individuals can end up being entrepreneursor only a certain proportion within the population can end up beingentrepreneurs Unfortunately a thorough comparative statics analysisis not possible because a unique stationary distribution does not exist



0

01

02

03

04

05

06P

roba

bilit

y

FIGURE 6 SOME STATIONARY STRATEGY DISTRIBUTIONSWHEN THE POPULATION SIZE N IS 100 THE SIGNAL PRECISIONp IS 051 AND THE OVERCONFIDENCE k IS A LARGE 12

33 Group Selection in the Social Sciences

It is generally recognized that although genes are the unit of biolog-ical selection they require vehicles of selection These vehicles are thedegree to which genes nd themselves ldquoin the same boatrdquo with othergenes as far as survival is concerned The vehicles can be cells (egcancer cells) individuals kin villages tribes nationalities ethnicitiesraces or even species The appropriate question is not whether groupvehicles are logically possible but whether selection at a higher orga-nizational level can be sufciently important to overwhelm selectionat a lower organizational level The empirical evidence indicates thatgroup selection can be an important force especially in human socialstructures Yet Wilson and Sober (1994) lament that ldquothe most recentdevelopments in biology have not yet reached the human behavioralsciences which still know group selection primarily as the bogey manof the 60rsquos and 70rsquosrdquo They also argue that ldquosocial structures havethe effect of reducing tness differences within groups concentratingnatural selection (and functional organization) at the group levelrdquo


4 Alternative Theories ExplainingOvercon dence and Entrepreneurs

Our theory has argued that overcondent entrepreneurs are usefulbecause they broadcast information and thus break the poor informa-tion aggregation intrinsic to conformity

We are aware of only one alternative explanation for seeminglyirrational overcondence proposed in Trivers (1985) and Hirshleifer(1997) when trying to deceive others that they are of higher abilityindividualsrsquo credibility is enhanced if they are themselves convincedof this higher ability One concern with this argument is that the ben-et to overcondence rests on onersquos own inability or on other indi-vidualsrsquo willingness to let themselves be deceived Yet if discoverycosts are not too high those nonentrepreneurial individuals who seethrough the deception will be more likely to survive than individualswho buy into the ldquoovercondence equals abilityrdquo argument Fortu-nately our own argument for the presence of overcondence is syn-ergistic groups that allow themselves to be ldquodeceivedrdquomdashpermittingentrepreneurs to procreate as frequentlymdashreceive extra informationwhich in turn enhances the grouprsquos chances for survival The will-ingness to be deceived may in effect be a transfer of resources fromnormal types to entrepreneurs

The alternative prevailing view of entrepreneurship is thatentrepreneurs are tempted by high payoffs associated with noncon-forming This view considers the innovation to be an activity in theentrepreneur rsquos self-interest In such a setting if there are diminishingreturns to entrepreneurial activity an optimal interior proportion ofentrepreneurs may arise8 This hypothesis is testably different fromour hypothesis in which entrepreneurs are overcondent make mis-takes on average and suffer in terms of expected payoffs More real-istically entrepreneurship is likely to be the outcome of both factors(1) lower risk aversion and the lure of payoffs substantially higherthan those available to the majority and a higher tendency for somesuch risk-loving individuals to survive within their group and (2) agenetic overcondence of entrepreneurs and a higher tendency forgroups with some such individuals to survive

In addition to the informational externality and the above-mentioned risky benets there are certainly other real-world facets of

8 Models in which individuals can either follow or learn (but not both) and inwhich the marginal costsbenets to the two activities are equal in equilibrium can befound in Boyd and Richerson (1988) and Rogers (1988) This typically results in learningthat is suboptimal from a group perspective Biological models in which some but notall individuals pursue an activity in their own self-interest and in which an optimuminterior proportion develops can be found in Smith (1974) Cornell and Roll (1981) andWeibull (1995)


entrepreneurship that we have omitted Still the informational bene-ts of entrepreneurship discussed in our paper are likely to persist ina much richer model than is considered in our paper

5 Conclusion

Our paper argues that overcondence (and with it certain forms ofentrepreneurship) can persist because overcondent behavior broad-casts valuable private information to the groupmdashinformation thatwould be lost if rational individuals instead just ldquofollowed the herdrdquoWe explored the costs and benets to individuals and groups in asimple setting A group with too few entrepreneurs falls too easilyinto an incorrect choice (in which the entire crowd follows the wrongpath) because of poor aggregation of information across individualsA group with too many entrepreneurs has too many individuals rely-ing only on their own information and making mistakes too oftenand thus suffers from high attrition The social optimum trades offthe information externality against this attrition We also identieda set of inuences (eg group size information precision degree ofovercondence type of decision etc) that inuences the optimal pro-portion of entrepreneurs (degree of overcondence) in groups

Unfortunately individual selection tends to discriminate againstentrepreneurs From an economic perspective this is a particularlysevere form of a public-goods problem We showed that the bene-ts to the group can easily be two orders of magnitude larger thanthe costs to the entrepreneur We briey discussed multiple mecha-nisms by which entrepreneurs may participate in such large payoffsand offered one such model that trades off the tendency of overcon-dent individuals to disappear against the tendency of groups withoutovercondent individuals to disappear We showed that there are sit-uations in which groups exclusively consisting of entrepreneurs candrive out mixed groups and other situations in which entrepreneurscan survive in mixed groups

We have mixed feelings about group selection which is rarelyfound in economic theory today Clearly the rst best set of explana-tions for behavior patterns should be based on individual rationalityBut when there is no rational explanation for a widely observed devia-tion from rationality the next best set of explanations should be basedon arguments in which this behavior is helpful to the individualrsquosgroup (and preferably of limited harm to the individual altruist him-self) Having invoked such a group selection argument allowed us toexplain and justify the presence of an otherwise seemingly irrationalunjustiable and ubiquitous behavior patternmdashovercondencemdashwhile


still imposing a discipline on the types of ldquoreasonablerdquo behavioralanomalies we would permit in a model

In conclusion there are many facets to the presence and bene-ts of entrepreneurs the entrepreneurial spirit and overcondencenot all of which are captured by our model But we believe thatour theory has captured one important aspect of overcondence andentrepreneurial culturemdasha possibly rare but persistent presence ofindividuals who provide information to their groupmdashin a simple rea-sonable and intuitive model

Appendix Recursion Formula and Proofs

A1 Recursion Formula for State-Time Probabilities

Let frac14ns denote the probability of being in state s after n signals that can

be inferred if the true state is h 1 These probabilities are required tocompute E[EumlVR n ( )] and E[EumlVOC n ( )] in equations (1) and (4) respec-tively The recursion formulae for deriving these probabilities are asfollows

frac14n0 pfrac14n 1

1 qfrac14n 11

frac14n1 cedilpfrac14n 1

2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10


3 qfrac14n 11 (1 )frac14n 1

2

frac14n2 qfrac14n 1

3 pfrac14n 11 (1 )frac14n 1

2

frac14ns qfrac14n 1

s 1 cedilpfrac14n 1s 1 (1 )frac14n 1

s for 2 lt s lt k 1 (A1)

frac14nk 1 cedilqfrac14n 1

k 2 (1 )frac14n 1k 1

frac14nk 1 cedilpfrac14n 1

k 2 (1 )frac14n 1k 1

frac14nk cedilqfrac14n 1

k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k

with the starting value given by frac1400 1 if k 4

A2 Proof of Proposition 1

Proof of Part 1 Fix n and let X i 1 (X i 1) if the ith signal is high(low) and it can be publicly inferred Fix k(p p) k let Y1(n) denotethe random number of publicly inferred signals by the nth arrival ifcedil 1 and let Y2(n) denote the random number of publicly inferred


signals by the nth arrival if cedil 2 Without loss of generality let1 gt 2

The state of publicly observed information at the nth arrival isS1

n S Y1 (n)i 1 X i if cedil 1 and S2

n S Y2 (n)i 1 X i if cedil 2

The ex ante probability that a rational individual makes an incor-rect decision if she is the nth arrival is equal to the probability that sheaccepts the project when the project is an incorrect one ( h 1) plusthe probability that she does not accept the project when the projectis a good one (h 1) Thus the probability of making an incorrectdecision when cedil j is

Pr[Sjn 1 h 1] Pr[ h 1] Pr[Sj

n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)

If 1 gt 2 then Y1(n) is stochastically larger than Y2(n) iePr[Y1(n) gt a] Pr[Y2(n) gt a] a and n Consequently

12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)

where the inequality follows from the fact that f ( ) is a decreasingfunction because the Xi are iid with E[Xi ] gt 0 and Y1(n) is stochas-tically larger than Y2(n) See Ross (1983 Proposition 812)

For xed simply let Y1(n) denote the random number of pub-licly inferred signals by the nth arrival if k k1 let Y2(n) denotethe random number of publicly inferred signals by the nth arrival ifk k2 and wlog let k1 gt k2 Now the remainder of the proof isidentical to the proof above

Finally since Xi are iid with E[X i] gt 0 the probability of mak-ing an incorrect decision decreases with each successive arrival whencedil and k are xed Thus the greater is N the smaller is the ex anteprobability that a rational type will make an incorrect decision u

Proof of Part 2 Follows directly from the solution to the gamblerrsquos ruinproblem (see Ross 1983 p 115f) u


References

Abarbanell JS and VL Bernard 1992 ldquoTests of Analystsrsquo OverreactionUnderreactionto Earnings Information as an Explanation for Anomalous Stock Price Behaviorrdquo TheJournal of Finance 47(3) 1181ndash1208

Ainslie G 1975 ldquoSpecious Reward A Behavioral Theory of Impulsivenessrdquo Psycho-logical Bulletin 82(4) 463ndash496

Anderson LR and CA Holt 1996 ldquoInformation Cascades in the Laboratoryrdquo AmericanEconomic Review 87(5) 847ndash862

Banerjee AV 1992 ldquoA Simple Model of Herd Behaviorrdquo Quarterly Journal of Economics107(3) 797ndash818

Batchelor R and P Dua 1992 ldquoConservatism and Consensus-Seeking among EconomicForecastersrdquo Journal of Forecasting 11(2) 169ndash181

Becker GS 1976 ldquoAltruism Egoism and Genetic Fitness Economics and SociobiologyrdquoJournal of Economic Literature 14(3) 817ndash826

Bikhchandani S D Hirshleifer and I Welch 1992 ldquoA Theory of Fads FashionCustom and Cultural Change as Informational Cascadesrdquo Journal of Political Econ-omy 100(5) 992ndash1026

Bolton P and C Harris 1999 ldquoStrategic Experimentationrdquo Econometrica 67(2)349ndash374

Boyd R and PJ Richerson 1988 ldquoAn Evolutionary Model of Social Learnings TheEffects of Spatial and Temporal Variationrdquo in TR Zentall J Bennett and G Galefeds Social Learning Hillsdale NJ Lawrence Erlbaum Associates

and JB Silk 1997 How Humans Evolved New York WW NortonBusenitz LW and JB Barney 1997 ldquoDifferences between Entrepreneurs and Managers

in Large Organizations Biases and Heuristics in Strategic Decision-Makingrdquo Journalof Business Venturing 12(1) 9ndash30

Cao H and D Hirshleifer 2000 ldquoConversation Observational Learning and Informa-tional Cascadesrdquo Working Paper Ohio State University

Cavalli-Sforza LL 1969 ldquoGenetic Drift in an Italian Populationrdquo Scientic American221 30

Cooper AC CA Woo and W Dunkelberg 1988 ldquoEntrepreneurs Perceived Chancesfor Successrdquo Journal of Business Venturing 3 97ndash108

Cornell B and R Roll 1981 ldquoStrategies for Pairwise Competitions in Markets andOrganizationsrdquo Bell Journal of Economics 12(1) 201ndash216

Daniel K D Hirshleifer and A Subrahmanyam 1998 ldquoA Theory of OvercondenceSelf-Attribution and Security Market Under- and Over-reactionrdquo The Journal ofFinance 53(5) 1839ndash1886

DeBondt WFM and RH Thaler 1995 ldquoFinancial Decision-Making in Markets andFirms A Behavioral Perspectiverdquo in R Jarrow V Maksimovic and W Ziemba edsFinance Elsevier North Holland Amsterdam Handbook in Operations Research andManagement Science Vol 9 Chap 13 385ndash410

Delong BJ A Shleifer L Summers and RJ Waldmann 1991 ldquoThe Survival of NoiseTraders in Financial Marketsrdquo Journal of Business 64(1) 1ndash20

Eshel I L Samuelson and A Shaked 1998 ldquoAltruists Egoists and Hooligans in aLocal Interaction Modelrdquo American Economic Review 88(1) 157ndash179

Gibson RM and J Hoglund 1992 ldquoCopying and Sexual Selectionrdquo TREE 7(7)229ndash232

Hirshleifer D 1997 ldquoAn Adaptive Theory of Self Deceptionrdquo Working Paper Universityof Michigan Graduate School of Business

and G Luo 2001 ldquoOn the Survival of Overcondent Traders in a CompetitiveSecurities Marketrdquo Journal of Financial Markets 4 73ndash84


and E Rasmusen 1989 ldquoCooperation in a Repeated Prisonersrsquo Dilemma withOstracismrdquo Journal of Economic Behavior and Organization 12(1) 87ndash106

Hirshleifer J 1977 ldquoEconomics from a Biological Viewpointrdquo Journal of Law and Eco-nomics 20(1) 1ndash52

1987 ldquoOn the Emotions as Guarantors of Threats and Promisesrdquo in J Dupre edThe Latest on the Best Essays on Evolution and Optimality Cambridge MA The MITPress 307ndash326

and JC Martinez-Coll 1988 ldquoWhat Strategies Can Support the EvolutionaryEmergence of Cooperationrdquo Journal of Conict Resolution 32(2) 367ndash398

Lewontin RC 1974 The Genetic Basis of Evolutionary Change New York ColumbiaUniversity Press

Manove M 1998 ldquoEntrepeneurs Optimism and the Competitive Edgerdquo WorkingPaper Boston University

Odean T 1998 ldquoVolume Volatility Price and Prot When All Traders Are AboveAveragerdquo The Journal of Finance 53(6) 1887ndash1934

Rogers AR 1988 ldquoDoes Biology Constrain Culturerdquo American Anthropologist 819ndash831 1994 ldquoEvolution of Time Preference by Natural Selectionrdquo American Economic

Review 84(3) 461ndash481Ross M and F Sicoly 1979 ldquoEgocentric Biases in Availability and Attributionrdquo The

Journal of Personality and Social Psychology 37 322ndash336Ross SM 1983 Stochastic Processes New York WileySmith JM 1974 ldquoThe Theory of Games and the Evolution of Animal Conictsrdquo Journal

of Theoretical Biology 209ndash219 1989 Evolutionary Genetics Oxford Oxford University Press

Trivers R 1985 Social Evolution Menlo Park CA BenjaminCummingsWaldman M 1994 ldquoSystematic Errors and the Theory of Natural Selectionrdquo American

Economic Review 84(3) 482ndash497Wang A 2001 ldquoOvercondence Investor Sentiment and Evolutionrdquo Working Paper

Rice University Journal of Financial Intermediation forthcomingWeibull JW 1995 Evolutionary Game Theory Cambridge MA The MIT PressWelch I 1992 ldquoSequential Sales Learning and Cascadesrdquo Journal of Finance 47(2)

695ndash732Wilson DS and E Sober 1994 ldquoRe-introducing Group Selection to the Human Behav-

ioral Sciencesrdquo Behavioral and Brain Sciences httpwwwcogscisotonacukbbsScientic American July 1996

Zhang J 1997 ldquoStrategic Delay and the Onset of Investment Cascadesrdquo Rand Journalof Economics 28(1) 188ndash205


that more rational individuals might not undertake For exampleovercondence among economic entrepreneurs has been documentedby Cooper et al (1988) In their sample of 2994 entrepreneurs 81believe their chances of success are at least 70 and 33 believe theirchances are a certain 100 In reality about 75 of new businessesno longer exist after ve years Busenitz and Barney (1997) comparedentrepreneursrsquo and managersrsquo assessments on a set of real-world ques-tions (eg whether cancer or heart disease is the leading cause ofdeath in the United States) Entrepreneurs and managers were aboutequal in their accuracy but the level of condence of entrepreneurs intheir own answers was dramatically higher The question our papertries to address is if economic principles can offer an explanation forsuch relatively common overcondent behavior which has clearly andreproducibly been documented in laboratory settings to be irrationalIn addition while overcondence and entrepreneurship are impor-tant phenomena in themselves there is another motivation for study-ing overcondence Some recent work in economics and nance (egDelong et al 1991 Daniel et al 1998 Odean 1998) relies on over-condence as an underlying primitive assumption often with littletheoretical justication as to why such irrational behavior can persist

Our paper offers a simple explanation for the presence ofovercondence and entrepreneurs Our main argument is that over-condent entrepreneurs (independent spirits innovators leaders changeagents or even dissidents) are less likely to imitate their peers and morelikely to explore their environment Entrepreneurial activity can thusprovide valuable additional information to their social group

Our point holds when individual actions can convey valuableprivate information and when information aggregation within theoverall group is otherwise poor Our specic modeling frameworkis built on the concept of informational cascades introduced in Welch(1992) Banerjee (1992) and Bikhchandani et al (1992) In this contextindividuals can observe one another and typically end up followingthe same action yet information aggregation is poor because rationalnonentrepreneurial individuals who follow ldquothe herdrdquo reveal nothingabout their private information From a social perspective cascadeslead to a suboptimal level of information disclosure experimentationand exploration of the environment

When overcondent entrepreneurial individuals instead followtheir own information downweighting the information in the herdtheir actions in effect broadcast their private information to the restof their group Unknowingly overcondent entrepreneurs behavealtruistically making irrational choices that are to their own detri-ment but help their groups Because the herd carries relatively lit-tle information this is only mildly individually suboptimal Still we
















2 The Model




Value State


H p qL q p


























24 Payoffs




12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ]

12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ] (1)




Pr(Sn 1 k h 1) (2)


and


Pr(Sn 1 k h 1) (3)




qPr( Sn 1 lt k h 1)

Pr(Sn 1 k h 1)

Pr(Sn 1 k h 1)] (4)





N

Sn 1


N

Sn 1

E[EumlVR n ( )]





Proposition 1




0

025

05

075

1

Pro

babi

lity








0

025

05

075

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs












0

01

02

03

04

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs


Jump to 1





1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs








































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References
























































2 The Model




Value State


H p qL q p


























24 Payoffs




12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ]

12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ] (1)




Pr(Sn 1 k h 1) (2)


and


Pr(Sn 1 k h 1) (3)




qPr( Sn 1 lt k h 1)

Pr(Sn 1 k h 1)

Pr(Sn 1 k h 1)] (4)





N

Sn 1


N

Sn 1

E[EumlVR n ( )]





Proposition 1




0

025

05

075

1

Pro

babi

lity








0

025

05

075

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs












0

01

02

03

04

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs


Jump to 1





1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs








































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References











































Value State


H p qL q p


























24 Payoffs




12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ]

12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ] (1)




Pr(Sn 1 k h 1) (2)


and


Pr(Sn 1 k h 1) (3)




qPr( Sn 1 lt k h 1)

Pr(Sn 1 k h 1)

Pr(Sn 1 k h 1)] (4)





N

Sn 1


N

Sn 1

E[EumlVR n ( )]





Proposition 1




0

025

05

075

1

Pro

babi

lity








0

025

05

075

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs












0

01

02

03

04

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs


Jump to 1





1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs








































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References





























































24 Payoffs




12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ]

12

[Pr(Sn 1 h 1) Pr(Sn 1 h 1) ] (1)




Pr(Sn 1 k h 1) (2)


and


Pr(Sn 1 k h 1) (3)




qPr( Sn 1 lt k h 1)

Pr(Sn 1 k h 1)

Pr(Sn 1 k h 1)] (4)





N

Sn 1


N

Sn 1

E[EumlVR n ( )]





Proposition 1




0

025

05

075

1

Pro

babi

lity








0

025

05

075

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs












0

01

02

03

04

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs


Jump to 1





1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs








































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References










































and


Pr(Sn 1 k h 1) (3)




qPr( Sn 1 lt k h 1)

Pr(Sn 1 k h 1)

Pr(Sn 1 k h 1)] (4)





N

Sn 1


N

Sn 1

E[EumlVR n ( )]





Proposition 1




0

025

05

075

1

Pro

babi

lity








0

025

05

075

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs












0

01

02

03

04

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs


Jump to 1





1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs








































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References











































0

025

05

075

1

Pro

babi

lity








0

025

05

075

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs












0

01

02

03

04

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs


Jump to 1





1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs








































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References











































0

025

05

075

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs












0

01

02

03

04

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs


Jump to 1





1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs








































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References














































0

01

02

03

04

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs


Jump to 1





1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs








































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References












































1 10 100 1000 10000 100000Group Size (N)

0

02

04

06

08

1

Opt

imal

Pro

port

ion

of E

ntre

pren

eurs








































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References














































































w

g tE[VOC( g t )]w

(1 g t )E[VR( g t) ] w (6)







f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References













































f t 1( ) f t ( ) (7)








frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References













































frac14x t 1

100

Si 0

100

Sj 0




h(x i j )

8gtlt

gt


(9)







0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References











































0

01

02

03

04

05

06

07

08

Pro

babi

lity







0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References











































0

01

02

03

04

05

06P

roba

bilit

y













5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References


















































5 Conclusion











frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References
















































frac14n0 pfrac14n 1

1 qfrac14n 11


2 qfrac14n 10

frac14n1 qfrac14n 1

2 pfrac14n 10



2

frac14n2 qfrac14n 1


2

frac14ns qfrac14n 1




k 2 (1 )frac14n 1k 1


k 2 (1 )frac14n 1k 1


k 1 frac14n 1k

frac14nk pfrac14n 1

k 1 frac14n 1k











n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References
















































n 0 h 1] Pr[ h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1]12 Pr[Sj

n 1 h 1] 12 Pr[Sj

n 0 h 1](A2)


12 Pr[S1

n 1 h 1] 12 Pr[S1

n 0 h 1]12 E[1(S1

n 1 h 1) 1(S1n 0 h 1) ]

12 E[E[1(S1

n 1 h 1) 1(S1n 0 h 1) Y1(n)]]

12 E[f (Y1(n)) ]

12 E[f (Y2(n)) ]

12 Pr[S2

n 1 h 1] 12 Pr[S2

n 0 h 1] (A3)






References










































References









































on the evolution of overconfidence and entrepreneurs

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