Evolutionary Justifications for Overconfidence

Transcription

1 Evolutionary Justifications for Overconfidence Kim Gannon Hanzhe Zhang August 29, 2017 Abstract This aer rovides evolutionary justifications for overconfidence. In each eriod, layers are airwise matched to fight for a resource and there is uncertainty about who wins the resource if they engage in the fight. Players have different confidence levels about their chance of winning although they actually have the same chance of winning in reality. Each layer may know or may not know her oonent s confidence level. We characterize the evolutionarily stable equilibrium, reresented by layers strategies and distribution of confidence levels. Furthermore, we characterize the evolutionary dynamics and the rate of convergence to the equilibrium. Under different informational environments, majority of layers are overconfident, i.e. overestimate their chance of winning. Gannon: Massachusetts Institute of Technology, gannonki@mit.edu. Zhang: Deartment of Economics, Michigan State University; hanzhe@msu.edu. We thank Provost s Undergraduate Research Initiative rogram in the College of Social Science and Dean s Assistantshi at Michigan State University for its funding of this roject. We thank comments and suggestions from Jon Eguia and Ryan Yuhao Fang.

2 1 Introduction A erson is overconfident when he or she overestimates his or her own ability. There is overwhelming evidence of oulation-wide overconfidence: Oskam (1965), Kidd (1970), Svenson (1981), Cooer et al. (1988), Bondt et al. (1989), Russo and Schoemaker (1992), Babcock and Loewenstein (1997), Guthrie et al. (2001). The overwhelming evidence suggests that erhas overconfidence is an evolutionarily desirable trait. This aer rovides evolutionary game-theoretic justifications for overconfidence. We show that in a resource-fighting game, with layers of heterogeneous confidence levels about their own chance of winning the game, layers with overconfident beliefs survive. The key imrovement is that we do not need to assume any bias or constraint in Nature, in contrast to the majority of revious work. Rather, strategic interactions themselves result in the survival of biased beliefs and, in articular, that of overconfident beliefs. The intuition for evolutionarily otimal overconfident beliefs is that rational layers do not necessarily get the most ayoff in strategic settings. In a game where layers fight for resources, a rational erfect Bayesian layer should not fight an overconfident layer, but an overconfident layer will fight a rational erfect Bayesian layer who will back down as a result of the overconfident layer s aggression. As a result, a rational layer chooses not to fight with the overconfident layer and the overconfident layer wins when the two encounter. Over the long run, there will be a distribution of layers with heterogeneous beliefs about the state of the world. We show that, generically, over the long run, the majority of layers will be overconfident. We show the robustness of the result both when a layer knows his oonent s tye and when a layer does not know his oonent s tye. After a brief literature review, the aer is organized as follows. Section 2 rovides a motivating examle. Section 3 describes the general model. Section 4 solves the general model when confidence tyes are observed. Section 5 solves the general model when confidence tyes are not observed. Section 6 concludes. 1.1 Literature Review Biased Beliefs of a Single Decision Maker When a layer dearts from being a risk-neutral utility maximizer, having a biased belief may correct other inherent biases. Zhang (2013) shows that when a decision maker is riskaverse, the evolutionarily otimal belief is ositively biased. Herold and Netzer (2015) show that non-bayesian robability weighting is a way to correct for S-shaed value function that arises in rosect theory, and make the general oint that non-bayesian robability weighting is the correction to biases in non-linear utilities. 1 The common thread in all these works is that there is some constraint in Nature, and evolution works to find the second best 1 The natural question why Bayesian and linear utilities do not evolve to survive and dominate other sub-otimal second-best robability weighting and strategies is addressed in Ely (2011). He gives analogies in Microsoft Office udates: the system is not comletely overhauled to reach the first-best solution; rather, atches and what he calls kludges are added over time. Similarly, evolution works by adding atches rather than comletely overhauling the system. 1

3 solution. One a riori bias results in an ex ost bias in another dimension (e.g. risk aversion results in overconfident, S-shaed utility function results in non-linear robability weighting). Benabou and Tirole (2002) and Comte and Postlewaite (2004) also oint out the desirable effects of self-ercetion for a single decision maker who exhibits other constraints. Zabojnik (2004), Benoit and Dubra (2011), and Harris and Hahn (2011) try to rationalize observed overconfidence, but the emirical evidence for overconfidence is too overwhelming to be disregarded Biased Beliefs in Strategic Interactions Güth and Yaari (1992) and Güth (1995) develoed the so-called indirect evolutionary aroach which we use in this aer: layers choose actions to maximize their erceived utilities but their actual ayoffs and fitness differ from their erceived utilities. Heifetz et al. (2007a) and Heifetz et al. (2007b) show in general that the rational layers are not the ones who survive in an evolutionary rocess. Because the games are assumed to be dominance solvable and have a unique Nash equilibrium, only one tye of layer survives in the limit. Instead, in this aer, we deal with games with multile equilibria and a distribution of layers with different belief tyes will survive. Dekel et al. (2007) deal with two by two games otentially with multile equilibria and characterize stable distributions of references. The game we consider in this aer does not have a stable distribution according to their definition of stability (Proosition 3 in Dekel et al. (2007)). For our urose of showing the limit distribution of overconfidence, we do not need to restrict our attention to any articular equilibrium. The aer most closely related to the current aer is Johnson and Fowler (2011). However, their aer treats biased estimation of winning robability as a mistake" and also does not discuss the cases where the layers tyes are not observed at all. Other aers also mention the advantage of being an overconfident tye in Tullock contests: see Ludwig et al. (2011) Market Selection Hyothesis The market selection hyothesis literature investigates the survival of layers with different beliefs in a cometitive market, instead of airwise strategies interactions in the indirect evolutionary games. Blume and Easley (1992) show that when layers maximize their discounted log utilities of consumtion, the layers who have correct beliefs about the state of the world will dominate the asset market in the long run. Sandroni (2000) endogenizes the savings decision in Blume and Easley (1992) and shows that regardless of the layers utility functions, the layers with correct beliefs will dominate. A crucial assumtion is that markets are comlete. Subsequent aers deal with incomlete markets, and find that layers with non-bayesian beliefs may survive: Mailath and Sandroni (2003), Blume and Easley (2009b), Blume and Easley (2009a), Blume and Easley (2010), Beker and Chattoadhyay (2010), Coury and Sciubba (2012), Condie and Phillis (2016). 2

4 2 Motivating Examle Let s use a motivating examle with secific ayoffs to illustrate the key insight of the aer. The rest of the aer generalizes the key insight in the same game with general ayoffs. Two layers (e.g. two countries, two lions, or two firms) both want an indivisible resource (e.g. a land, a lioness, or a market). If they do not Fight for it, neither would get the resource and both get a ayoff of r = 3. If one of them Fights, the one who Fights gets R = 6 and the one who does Not Fight gets r = 3. If both Fight, they engage in a costly encounter that costs c = 2 to each and the winner gets R = 6 and the loser gets r = 3. (The ayoffs R = 6, r = 3, and c = 2 used in the motivating examle are the smallest integers that satisfy the general game described in the next section.) In an evolutionary context, they are naturally occurring games in the nature when two arties of the same secifies or two different secifies fight for resources such as food, land, and money. consider these ayoffs as their fitness levels, the (exected) number of offsring. The game is summarized as follows, where 1 i, 1 j denote that i or j wins the resource. i \ j Fight Not Fight Fight 1 i j 3 2, 1 j i 3 2 6, 3 Not Fight 3, 6 3, 3 Everyone in the oulation actually has the same chance of winning, that is, Pr(i wins) = Pr(i loses) = 0.5. A layer with the correct belief that he wins with robability 0.5 erceives his exected utility to be the same as the fitness ayoff, i \ j Fight Not Fight Fight (0.5)(6) + (0.5)(3) 2 = Not Fight 3 3 A layer may not have the correct belief that he wins with robability of 0.5. A layer with belief that he wins with robability θ i [0, 1] erceives his exected utility to be i \ j Fight Not Fight Fight (θ i )(6) + (1 θ i )(3) 2 = 3θ i Not Fight 3 3 Note that when the oonent Fights, a tye θ i > 2/3 layer s erceived exected utility when he Fights is greater than his utility when he does Not Fight. In articular, Fight is a strictly dominant strategy for a confidence tye θ i > 2/3. In other words, a sufficiently overconfident layer always Fights regardless of the situation. 2.1 Confidence Tyes are Observed First, suose that each layer knows her oonent s confidence tye. She chooses her action based on her own confidence tye and her oonent s observed confidence tye. Suose the layers are either rational with confidence levels of 0.5 or overconfident with confidence levels of

5 2.1.1 Static Pairwise Game (a) Rational Player versus Rational Player When two layers who have correct beliefs about their chance of winning lay the game, they are laying the actual game i \ j Fight Not Fight Fight 2.5, 2.5 6, 3 Not Fight 3, 6 3, 3 There are three equilibria: (F, NF), (NF, F), and ( 6 F+ 1 NF, 6 F+ 1 NF). In the two ure-strategy equilibria, the one who fights gets 6 and the one who does not fight gets 3; in the mixed-strategy equilibrium, both layers get an exected ayoff of 3. (b) Overconfident Player versus Rational Player Now suose an overconfident layer i with belief θ i = 0.8 and a rational layer j with belief θ j = 0.5 lay against each other and their beliefs are observable and common knowledge. The game they believe they are laying is then θ i = 0.8 \ θ j = 0.5 Fight Not Fight Fight 3.4, Not Fight 3 3 The overconfident layer i has a dominant strategy of Fight. The rational layer j on the other hand has an exected utility of 2.5 from fighting when i fights. When i fights, j s best resonse is Not Fight. As a result, the equilibrium is that i lays Fight and j lays Not Fight, and i the overconfident layer gets 6 and j the Bayesian layer gets 3. (c) Overconfident Player versus Overconfident Player If two overconfident tye θ i = θ j = 0.8 layers encounter each other, their erceived exected utilities are whereas their true exected ayoffs are i \ j Fight Not Fight Fight 3.4, 3.4 6, 3 Not Fight 3, 6 3, 3 i \ j Fight Not Fight Fight 2.5, 2.5 6, 3 Not Fight 3, 6 3, 3 They both lay Fight and get an actual ayoff of

6 2.1.2 Evolutionarily Stable Distribution of Confidence Consider a distribution of layers with different beliefs. For examle, start with half a oulation of rational tye 0.5 layers and half a oulation of overconfident tye 0.8 layers. They randomly match to lay the game just described knowing the belief tye of the oonent. Within a generation, layers lay the game reeatedly. Their average ayoffs determine the growth rate of the oulation. We focus on an evolutionarily stable equilibrium such that (1) two layers meet and lay one of the Nash equilibria of the erceived game, and (2) the average fitness of the two tyes of layers is the same. Formally, an evolutionarily stable equilibrium secifies that (1) each tye θ i layer s robability σ θi (θ j ) of Fight when he encounters a tye θ j layer, and (2) roortion of layers are overconfident and roortion (1 ) of layers are rational, such that the exected ayoffs are the same for the two tyes of layers. When two rational layers match, each layer s ayoff is x between 3 and 6 deending on the equilibrium they lay, but the sum of the ayoffs is less than 9 for the two layers, so each rational layer has an equilibrium ayoff smaller than 4.5 on average. When an overconfident layer lays with a rational layer, the overconfident layer lays Fight and gets 6 and the rational layer lays Not Fight and gets 3. When two overconfident layers meet, they both lay Fight. Ex ost one gets 4 and one gets 1; ex ante two layers average ayoff is 2.5. Suose in the evolutionarily stable distribution, a roortion of layers are overconfident and roortion 1 are rational. Overconfident layers have an average ayoff of Rational layer have an average ayoff of ()(2.5) + (1 )(6). ()(3) + (1 )x where x is between 3 and 4.5. In the stable equilibrium, there is a roortion of overconfident layers. Overconfident and rational layers have the same exected ayoff: ( )(2.5) + (1 )(6) = ( )(3) + (1 )x = 6 x 6.5 x = x. Since 3 x 4.5, = = 6 7. That is, in an evolutionary stable equilibrium, between 75% and 85.7% of the layers are overconfident (recall 75% of Harvard undergraduates believe that they are better their median Harvard eer Möbius et al. (2012) and 88% of drivers believe they have above -median driving skills in Svenson (1981)), and only between 14.3% and 25% of the layers are rational and have correct beliefs Evolutionary Dynamics of Confidence With our simle model, we can also investigate numerically how long it takes for the oulation to reach the stable distribution of overconfidence levels. Figure illustrates the 5

7 evolutionary dynamics when initially there is an equal roortion of rational and overconfidence layers is 0.5. Then, for each succeeding generation, the roortion of overconfident layers is calculated as the roortion of exected number of offsring to overconfident layers over the entire oulation. The exected number of offsring is calculated based on the robability of encountering a layer of each tye and the ayoff under each match: t ( t 1 ) = t 1 [6(1 t 1 ) t 1 ] t 1 [6(1 t 1 ) t 1 ] + (1 t 1 )[4.5(1 t 1 ) + 3 t 1 ]. Here, the exected ayoff to a rational layer who lays a rational layer is taken to be 4.5 based on the robability that the layer lays one of the ure strategy equilibria or the mixed strategy equilibrium. Generally, the roortions tend to be equal within three decimal laces of each other beginning around the fortieth generation. We take the roortion 50 =, the limit of this sequence. Evolution of equilibrium roortion of overconfident layers generation Figure 1: Evolutionary dynamics when 0 = 0.5, R = 6, r = 3, and c = 2. The figure illustrates 50 generations of the evolutionary dynamics of confidence when confidence tyes are always observed, the initial roortion of overconfident layers is 0 = 0.5, the winner s ayoff is R = 6, the loser s ayoff is r = 3, and cost of fighting is c = 2. The roortion of overconfident layers in the oulation aroaches 0.75, the evolutionarily stable roortion, in aroximately 30 generations. Next, we calculate the rate of convergence of the sequence {x t } by fixed oint iteration. This method is based on the linear order convergence of { t } and assumes that the deviation e t = t λρ, [ 0 ρ, 0 + ρ] for 0 { t }. Here, λ t is the convergence rate. We then calculate λ by curve-fitting and transforming the coefficient on t. log(e t ) = t log(λ) + log(ρ) 6

8 From this analysis, we estimate that the roortion of overconfident layers in a oulation converges to = , our true equilibrium distribution as calculated in the revious subsection. Our curve-fitting rocedure gives a coefficient of , yielding a growth rate of λ k = (e ) k = k. The deviation between t and the limit decreases by aroximately 13.4% each generation. 2.2 Confidence Tyes are Not Observed Second, suose that a layer cannot observe his oonent s confidence tye but knows the distribution of confidence tyes in a eriod. Each layer lays against a oulation of layers with heterogeneous confidence levels. Each layer chooses an action based on his own confidence level and the aggregate distribution of confidence levels and strategies. Suose that there are just two tyes of agents as in the revious case: the overconfident tye, whose believed robability of victory is 0.8, and the rational tye, whose is Static Pairwise Game The overconfident tye, regardless of the situation, Fights, as Fighting is an overconfident layer s dominant strategy. A rational layer weighs the benefits and costs of Fighting. Given that roortion t of layers is overconfident, a rational layer s exected utility of Fighting is t [0.5(6) + 0.5(3) 2] + (1 t )[σ(0.5, t, Fight) (2.5) + (1 σ(0.5, t, Fight))(6)] }{{} σ t = [1 (1 t )(1 σ t )](2.5) + (1 t )(1 σ t )(6) vs 3 if he chooses Not Fight. If (1 t )(1 σ t ) = 1/7, then the rational layer is indifferent between Fighting and Not Fighting. Therefore, the robability of a rational layer Fighting when the roortion of overconfident layers is t is σ t σ(0.5, t, Fight) = 1 1 7(1 t ). This distribution is indeendent across time eriods. Therefore, when each layer cannot observe the other layers individual tyes but an aggregate distribution of other layers confidence tyes, there exist only trivial evolutionary dynamics. Nonetheless, extremely under-confident layers will be driven out of the market Evolutionarily Stable Distribution of Confidence As above, we focus on the evolutionary stable equilibrium. In equilibrium, a tye θ layer lays Fight with robability σθ. Proortion of the oulation is overconfident, such that, in equilibrium, (1) σθ is best resonding to the situation in which a layer is laying with robability an overconfident layer who adots strategy σ0.8 and with robability 1 a rational layer who adots strategy σ0.5, and (2) the exected ayoffs of the two tyes of layers are the same. 2 2 in σ θ indicates that the layer receives no information about his oonent s tye. 7

9 When an overconfident layer lays Fight, his exected utility is u 0.8 = [(σ 0.8 )(3.4) + (1 σ 0.8 )(6)] + (1 )[(σ 0.5 )(3.4) + (1 σ 0.5 )(6)] > 3 where 3 is the exected utility of Not Fight. Therefore, an overconfident layer always lays Fight, and thus σ 0.8 = 1. A rational layer laying Fight gets an exected utility of u 0.5 = (2.5) + (1 )[(σ 0.5 )(2.5) + (1 σ 0.5 )(6)]. In order for a rational layer to be indifferent between Fight and Not Fight, [1 (1 )(1 σ 0.5)](2.5) + (1 )(1 σ 0.5)(6) = 3 (1 )(1 σ 0.5) = 1/7. 1 1/7 and 1 σ 0.5 1/7, so 6/7 and σ 0.5 6/7. There could be roortion [0, 6/7] of overconfident layers. Unless = 0, the average confidence level, ( )(0.8) + (1 )(0.5) strictly exceeds the rational level of 0.5, and the majority of layers are weakly overconfident. We will show that the claim that majority of layers are weakly overconfident holds in general, even when there are under-confident layers Evolutionary Dynamics of Confidence Now let s consider the evolution of distribution of overconfident layers, t ( t 1 ). Consider random airwise matching between all the layers. The exected ayoff of an overconfident layer is t (2.5) + (1 t )[(σ t )(2.5) + (1 σ t )(6)] [( ) ] 1 1 = t (2.5) + (1 t ) 1 (2.5) + 7(1 t ) 7(1 t ) (6) [ ] 3.5 = t (2.5) + (1 t ) = 3. 7(1 t ) The exected ayoff of a rational layer is t [(σ t )(2.5) + (1 σ t )(3)] + (1 t )[(σ 2 t )(2.5) +(σ t )(1 σ t )(6) + (1 σ t )(σ t )(2) + (1 σ t ) 2 (3)] = 3. In this examle, the roortion of overconfident layers in a oulation t is constant at 3 does not deend on that from a revious eriod t 1. Therefore, if we assume that the ayoff to each layer is the number of offsring in the next generation, the distribution of overconfident layers in a oulation does not change over time: t = t 1 (3) t 1 (3) + (1 t 1 )(3) = t 1(3) 3 = t 1. Therefore, under the arameters of this game, a oulation s distribution of overconfident layers will not change over time if oonents confidence tyes are never known. 8

10 3 General Model There is a continuum of layers with heterogeneous beliefs θ [0, 1] about their chance of winning. Let µ denote the measure of layers with different beliefs. The fitness ayoff is i \ j Fight Not Fight Fight 1 i R + 1 j r c, 1 j R + 1 i r c R, r Not Fight r, R r, r where r < R and 1(R r) < c < R r. 1(R r) < c imlies that a rational layer would 2 2 not fight if the oonent fights. c < R r imlies that a layer who believes he always wins would refer to fight even if the oonent fights, i.e. a layer of tye θ = 1 has a strictly dominant strategy of Fight. With robability q, a layer observes his oonent s lay. 4 Confidence Tyes are Observed Suose for now that oonents confidence tyes are always observable. In this section, we solve for the airwise static games, layers ayoffs, and the equilibrium distribution, and the evolutionary dynamics. 4.1 Pairwise Static Games Two layers of confidence confidence tyes θ i [0, 1] and θ j [0, 1] lay the following erceived game, i \ j Fight Not Fight Fight θ i R + (1 θ i )r c, θ j R + (1 θ j )r c R, r Not Fight r, R r, r Any tye θ layer who decides to Fight receives a minimum exected ayoff of θr+(1 θ)r c (in the case that the oosing layer fights for sure). The alternative is Not Fight, and the tye θ layer gets r. Consequently, a layer will definitely Fight if θr + (1 θ)r c > r, which rearranges to θ > c R r = θ where we call θ the critical tye. Since c > (R r)/2 by assumtion, the critical tye θ > 1/2. Any layer with tye θ < θ is called under-critical and any layer with tye θ θ is called over-critical. Since θ > 1/2, an over-critical layer is necessarily overconfident, a rational layer is necessarily under-critical, but some under-critical layers are also overconfident. For any over-critical θ θ layer, Fight is a dominant strategy. For any under-critical θ < θ layer, Fight is not a dominant strategy. In articular, for a rational θ = 1/2 < θ layer, Fight is never a dominant strategy. 9

11 (a) Under-critical Tye θ i < θ Player versus Under-critical Tye θ j < θ Player. Suose that two under-critical layers encounter each other. In articular, two rational θ = 1/2 layers encountering each other is a secial case. There are three equilibria: two ure-strategy equilibria (Fight, Not Fight) and (Not Fight, Fight) and one mixed-strategy equilibrium (σ θ i (θ j ), σ θ j (θ i )). In the two ure-strategy Nash equilibria, one layer gets R and the other gets r, and the total ayoff is R + r. On average, the two layers on average get a ayoff (R + r)/2. In the unique mixed-strategy Nash equilibrium, each erson gets r in exectation. Therefore, each θ i, θ j < θ layer gets a ayoff x [r, (R + r)/2]. (b) Overconfident θ i θ Player versus Tye θ j < θ Player. It is a dominant strategy to lay Fight. Given that a tye θ i θ layer lays Fight, the θ j < θ layer lays Not Fight. The unique Nash equilibrium is (Fight, Not Fight), and the tye θ i > θ layer gets R and the tye θ j < θ layer gets r. (c) Overconfident θ i θ Player versus Overconfident θ j θ Player. Two overconfident layers both choose Fight and they end u with an exected ayoff of (R+r)/2 c. 4.2 Evolutionarily Stable Distribution of Confidence Definition 1. An equilibrium consists of a CDF F with associated PDF f on [0, 1] reresenting the distribution of tyes, and a strategy σθ : [0, 1] [0, 1] such that 1. For each θ i, θ j [0, 1], σθ i (θ j ) arg max σ[σθ σ [0,1] j (θ i )(θ i R + (1 θ i )r c) + (1 σθ j (θ i ))R] + (1 σ)r, or more loosely, to incororate asymmetric strategies, (σ θ i (θ j ), σ θ j (θ i )) N (θ i, θ j ), where N (θ i, θ j ) denotes the set of Nash equilibria in the erceived game layed between tyes θ i and θ j layers. 2. π θ equalizes across all θ, where π θi = ˆ 1 0 [σ θ i (θ j )σ θ j (θ i )( 1 2 R r c) + σ θ i (θ j )(1 σ θ j (θ i ))R + (1 σ θ i (θ j ))r]f (θ j )dθ j. We can show that the stable roortion of overconfident layers is always above 1/2. Suose is the roortion of layers with tyes above θ. Then, an overconfident θ θ layer gets a ayoff of π θ>θ = [(R + r)/2 c] + (1 )R and a tye θ < θ layer gets a ayoff of π θ θ = r + (1 )x where x [r, (R + r)/2]. In equilibrium, the two equate, [(R + r)/2 c] + (1 )R = r + (1 )x 10

12 which simlifies to Since c < R r, = R x R+r 2 + c x > 1 R x + R r x = R x R + 1(R r) x = 1 (R r) 2 R + 1 (R r) x R+r Since x (R + r)/2, 1 1 (R r) 2 > 1 R + 1(R r) x 1 (R r) 2 R + 1(R r) 1(R + r) = That is, in equilibrium, it always holds that more than half of the agents have a confidence level above θ > (R r)/c > 0.5. The exact distribution of confidence levels deends on the initial distribution one starts with, but we have shown that regardless of the initial distribution, at least half of the oulation have strictly ositively biased beliefs. We can also look at the change in the evolutionarily stable roortion of overconfident layers. The stable roortion of overconfident layers increases when the winning ayoff R increases, when the losing ayoff r decreases, and when the cost of fighting c decreases. Proortion of Overconfident Players vs R Proortion of Overconfident Players vs r Proortion of Overconfident Players vs c R r c Figure 2: Evolutionarily stable roortion of overconfident layers, when R, r, and c vary. Our numerical analysis also allows variation on various arameters within the model. Figure 2 notes the differences in the evolutionarily stable roortion of overconfident layers for different values of R holding constant the values of r and c. Indeed, we note that with a value R less than 5, the long run roortion of overconfident layers lies beneath the initial roortion 0.5. At R = 5, the value of remains constant at 0.5, and any value of R greater than 5 yields an increase in. We note that around roughly R = 7.5, the roortion of overconfident layers begins to aroach 1 in the long run. Figure 3 illustrates the change in the stable roortion of overconfident layers for different values of winning ayoff R, losing ayoff r, cost c, initial roortion of overconfident layers. 11

13 Evolutionary dynamics varying with R Evolutionary dynamics varying with L R r generation Evolutionary dynamics varying with c generation Evolutionary dynamics varying with initial roortion c generation generation Figure 3: Evolutionary dynamics, when R, r, c and 0 vary. 12

14 4.3 Evolutionary Dynamics Figure 4 lots the number of time eriods necessary for the oulation to reach within 0.01 of the evolutionarily stable distribution. Somewhat surrisingly, the number of generations to reach within 0.01 of the equilibrium roortion is not monotonic in R, r, c, 0. The variations of convergence rates deserve attention from future research. Time of convergence vs R Time of convergence vs r Time of convergence vs c R r Time of convergence vs initial roortion c initial roortion Figure 4: Number of generations to reach equilibrium, when R, r, c and 0 vary. 13

15 5 Confidence Tyes are Not Observed Suose that layers cannot observe their oonent s confidence tye. We are interested in characterizing the equilibrium distribution of confidence levels and also the evolutionary dynamics. Definition 2. An equilibrium consists of an equilibrium distribution of tyes reresented by a CDF F and PDF f on [0, 1] and an equilibrium strategy σθ [0, 1] such that 1. For each θ i [0, 1], σ θ i arg max σ [0,1] ˆ 1 2. π θ equalizes across all θ [0, 1], where π θi = ˆ { } σ[σθ j (θ i R + (1 θ i )r c) + (1 σθ j )R] + (1 σ)r f (θ j )dθ j. [σ θ i σ θ j ( 1 2 R r c) + σ θ i (1 σ θ j )R + (1 σ θ i (θ j ))r]f (θ j )dθ j. By the following argument, we will see that overconfident layers will always make u more than half of the oulation in the stable equilibrium. There is some cutoff tye who is indifferent between Fight and Not Fight. Let s call the tye θ. Any tye θ > θ layer lays Fight and any tye θ < θ layer always lays Not Fight. Let denote the roortion of tye θ > θ layers, roortion of tye θ layers, and 1 roortion of tye θ < θ layers. When a θ layer chooses to fight, his exected utility is It must hold for θ, u θ = r. u θ = ( + σ )[θr + (1 θ)r c] + (1 σ )R ( + σ )[θ R + (1 θ )r c] + (1 σ )R = r Second, the fitness level of all surviving confidence tyes must be the same, we must have Therefore, θ = 1/2. The equation simlifies to ( + σ )( 1 2 R r c) + (1 σ )R = r. + σ = 1 2 R r (R r) + c. Since c > R r, + σ > 2/3. If there is no atom at θ = 1/2, then at least 2/3 of the oulation have overconfident beliefs. There is no non-trivial evolutionary dynamics. Rational layers all best resond by mixing between Fighting and Not Fighting. The rational layers and overconfident layers get the same exected ayoffs because the rational layers mix. The extremely underconfident layers who do not Fight at all are driven out of the market over time, but anyone who mixes between Fighting and Not Fighting survives the market. 14

16 6 Conclusion This aer rovides evolutionary justifications for overconfident beliefs. We show that in a class of resource-fighting games, under different settings of observability of layers beliefs, layers with overconfident beliefs are more likely to survive and thrive. Two comments are in order to defend our simle assumtions. First, we only consider layers of the same ability in the model. One justification for this is that layers of lower abilities will be driven out of the market and cometition rior to any belief heterogeneity. Belief heterogeneity only lays a significant role when layers of the same ability comete. Second, we also try to be agnostic about equilibrium selection and restrictions on evolutionary stability. Although we have multile equilibria and unidentified distributions of confidence in the general model, the main oint that majority of layers are overconfident in equilibrium holds with very little assumtion. References Babcock, Linda and George Loewenstein, Exlaining Bargaining Imasse: The Role of Self-Serving Biases, Journal of Economic Persectives, Winter 1997, 11, Beker, Pablo and Subir Chattoadhyay, Consumtion Dynamics in General Equilibrium: A Characterisation when Markets are Incomlete, Journal of Economic Theory, 2010, 145 (6), Benabou, Roland and Jean Tirole, Self-Confidence and Personal Motivation, Quarterly Journal of Economics, 2002, 117 (3), Benoit, Jean-Pierre and Juan Dubra, Aarent Overconfidence, Econometrica, Setember 2011, 79 (5), Blume, Lawrence and David Easley, Evolution and Market Behavior, Journal of Economic Theory, 1992, 58 (1), and, The Market Organism: Long-Run Survival in Markets with Heterogeneous Traders, Journal of Economic Dynamics and Control, 2009, 33 (5), and, Market Selection and Asset Pricing, in Handbook of Financial Markets: Dynamics and Evolution, Elsevier, 2009, and, Heterogeneity, Selection, and Wealth Dynamics, Annual Review of Economics, 2010, 2 (1), Bondt, Kenneth A, Jeffrey A Frankel, Maria Sohia Aguirre, Reza Saidi, Paul R Krugman, and Maurice Obstfeld, Forward Discount Bias: Is it an Exchange Risk Premium?, Quarterly Journal of Economics, 1989, 104 (9), Comte, Olivier and Andrew Postlewaite, Confidence-Enhanced Performance, American Economic Review, 2004, 94 (5),

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