Evolutionary stability of first-order-information indirect reciprocity in sizable groups

Theoretical Population Biology 73 (2008) 426–436 www.elsevier.com/locate/tpb

Evolutionary stability of first-order-information indirect reciprocity in sizable groups Shinsuke Suzuki ∗ , Eizo Akiyama Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8573, Japan Received 19 July 2007 Available online 23 December 2007

Abstract Indirect reciprocity is considered as a key mechanism for explaining the evolution of cooperation in situations where the same individuals interact only a few times. Under indirect reciprocity, an individual who helps others gets returns indirectly from others who know her good reputation. Recently, many studies have discussed the effect of reputation criteria based only on the former actions of the others (first-order information) and of those based also on the former reputation of opponents of the others (second-order information) on the evolution of indirect reciprocity. In this study, we investigate the evolutionary stability of the indirectly reciprocal strategy (discriminating strategy: DIS), which cooperates only with opponents who have good reputations, in n (>2)-person games where more than two individuals take part in a single group (interaction). We show that in n-person games, DIS is an evolutionarily stable strategy (ESS) even under the image-scoring reputation criterion, which is based only on first-order information and where cooperations (defections) are judged to be good (bad). This result is in contrast to that of 2-person games where DIS is not an ESS under reputation criteria based only on first-order information. c 2007 Elsevier Inc. All rights reserved.

Keywords: ESS; Indirect reciprocity; Image scoring; Reputation; n-person prisoner’s dilemma

1. Introduction Indirect reciprocity is considered to be a key mechanism for explaining the evolution of cooperation in situations where the same individuals interact only a few times and where the reputations of individuals affect the decision-making process (Alexander, 1987; Nowak and Sigmund, 2005). Under indirect reciprocity, an individual who helps others gets returns indirectly from others who know her good reputation in the community. Nowak and Sigmund (1998a,b) have formalized a mathematical model of indirect reciprocity as an evolutionary 2person giving game involving image scoring that evaluates the reputations of individuals based on their past behaviors. In their model, pairs of individuals interact only a few times and all individuals are informed about their partners’ reputations, which reflect the partners’ past behaviors. They have shown that, in this model, an indirectly reciprocal strategy called dis∗ Corresponding author.

E-mail address: [email protected] (S. Suzuki). c 2007 Elsevier Inc. All rights reserved. 0040-5809/$ - see front matter doi:10.1016/j.tpb.2007.12.005

criminating strategy (DIS), which cooperates only with opponents having good reputations, can evolve against unconditionally defective strategy. Since the seminal works by Nowak & Sigmund, many theoretical and experimental studies on indirect reciprocity have been conducted (e.g. Leimar and Hammerstein (2001), Wedekind and Milinski (2000), Lotem et al. (1999), Milinski et al. (2001, 2002), Panchanathan and Boyd (2003, 2004), Fishman (2003), Mohtashemi and Mui (2003), Ohtsuki (2004), Ohtsuki and Iwasa (2004), Brandt and Sigmund (2004, 2005, 2006), Suzuki and Toquenaga (2005), Takahashi and Mashima (2006), Chalub et al. (2006) and Pacheco et al. (2006)). One of the most important issues regarding the studies of indirect reciprocity has been how people judge (define) others’ goodness (e.g. Leimar and Hammerstein (2001), Milinski et al. (2001), Panchanathan and Boyd (2003), Ohtsuki and Iwasa (2004, 2006, 2007), Brandt and Sigmund (2004), Bolton et al. (2005) and Takahashi and Mashima (2006)). To put it more precisely, a controversial issue in indirect reciprocity is whether people use only first-order information, which are the former actions of the others, or use in addition second-order information, which are the former reputation of opponents of

S. Suzuki, E. Akiyama / Theoretical Population Biology 73 (2008) 426–436

the others. Using second-order information, each individual can distinguish between defections to bad individuals and those to good individuals. In other words, second-order information enables us to distinguish between justified and unjustified defections. (Note that image scoring given in Nowak and Sigmund (1998a,b) is a reputation criterion based only on firstorder information, i.e., others’ past actions.) Many theoretical studies (e.g. Leimar and Hammerstein (2001), Panchanathan and Boyd (2003), Ohtsuki and Iwasa (2004, 2006) and Brandt and Sigmund (2004)), using 2-person games, have pointed out that, under image scoring that is based only on first-order information, the indirectly reciprocal strategy, DIS, is not an evolutionarily stable strategy (ESS). The reason that DIS is not an ESS is as follows: DIS strategists hurt each other in response to error defections because a DIS strategist does not cooperate with individuals who defected out of error, causing her own reputation to become bad. As a result, DIS strategists defect against each other. On the contrary, individuals who adopt an unconditionally cooperative strategy called ALLC always intend to cooperate. Therefore ALLC strategists do not hurt others except erroneous defections and so are not hurt by DIS strategists. Consequently, in the population mostly consisting of DIS, the fitness of DIS is less than that of ALLC, so the DIS population can be invaded by ALLC. On the other hand, several studies (e.g. Leimar and Hammerstein (2001), Panchanathan and Boyd (2003), Brandt and Sigmund (2004) and Takahashi and Mashima (2006)) have stated that DIS can be an ESS under a reputation criterion that reflects not only first-order information but also second-order information. Especially, Panchanathan and Boyd (2003) have shown that, in the presence of implementation error causing unintentional defection, DIS can be an ESS under standing reputation criterion under which an individual’s defections to bad individuals are not unjustified (Sugden, 1986). Moreover, Ohtsuki and Iwasa (2004, 2006) have shown that, in the presence of implementation error and objective perception error, under the eight reputation criteria called leading eight where an individual’s defections to bad individuals are not unjustified, a strategy forming a cooperative society can be an ESS. Furthermore, Takahashi and Mashima (2006) have shown that, in the presence of subjective perception error, DIS can be an ESS under the reputation criteria where cooperations with bad individuals are unjustified.1 Here, note that most of the above studies on indirect reciprocity have presumed dyadic interactions (i.e., 2-person games). However, in the real world, more than two individuals often take part in a single group or interaction (e.g. cooperative management of common-pool resources (Hardin, 1968), environmental problems such as global warming or air pollution (Milinski et al., 2006), and predator inspection in a school of fish (Dugatkin, 1990)). Therefore, we believe that 1 However, experimental studies have not been able to conclude whether people actually use only first-order information or also use second-order information. For instance, Milinski et al. (2001) have found that people do not actually use second-order information, while Bolton et al. (2005) have found that they do.

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not only 2-person games but also n (>2)-person games should be considered as models of interactions in human societies or ecosystems. The evolutionary stability of DIS, which cooperates only when all the opponents in the group have good reputations, in n-person games has been investigated in Suzuki and Akiyama (2007a,b).2 They have shown that in a population consisting only of DIS, ALLC, and ALLD (unconditionally defective strategy), DIS can be an ESS under image scoring, which reflects only first-order information. Although Suzuki and Akiyama (2007a,b) have analyzed the population consisting only of the three strategies (DIS, ALLC and ALLD),3 of course, possible strategies we can adopt in the real world are not limited to the three strategies. In order to discuss the evolutionary stability of DIS more properly, we need to consider broader strategy space than only the three strategies. In this study, we analyze the evolutionary stability of DIS under image scoring in the situation where more than two individuals take part in a single group (i.e., n-person games) and where the population consists of all the possible strategies that decide their own action based on the number of opponents in the group whose reputation is good. Clearly, in the absence of error, DIS is not an ESS because ALLC is an alternative best reply to DIS in the DIS population. Therefore, in this study, we focus on the case with implementation error causing unintentional defection. 2. Model Consider a population composed of an infinite number of individuals. Each individual in the population has her own reputation either G (good) or B (bad). Each generation comprises a number of rounds. After the first round, each of the subsequent rounds occurs with probability w (0 < w < 1), i.e., the expected value of the number of rounds in a generation is 1/(1−w). At the beginning of each generation, the reputation of each individual is G. In each round, all individuals are divided randomly into groups, each of which comprises n (≥3) individuals, and play an n-person prisoner’s dilemma game in each group. In this game, each of the individuals chooses either to “cooperate (C)” or “defect (D)”. The payoffs for a cooperator, V (C | k), and that for a defector, V (D | k), where k is the number of opponents cooperating in the group, satisfy the following conditions (e.g. Boyd and Richerson (1988), Molander (1992) and Eriksson and Lindgren (2005)): • Each individual is better off choosing D regardless of the choices of the other members in the group (dominance of D): V (D | k) > V (C | k).

(1)

2 Furthermore Suzuki and Akiyama (2005) have used agent-based computer simulation to investigate the evolution of indirect reciprocity in groups of various sizes. However they did not mention the evolutionary stability. 3 Suzuki and Akiyama (2007a,b) have investigated not only evolutionary stability but also evolutionary dynamics in the population of DIS, ALLC and ALLD.

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Table 1 Payoff of 2-person prisoner’s dilemma game (b > c > 0)

Player 1

Cooperation Defection

Player 2 Cooperation

Defection

(b − c, b − c) (b, −c)

(−c, b) (0, 0)

• The payoff for each individual increases as the number of cooperators increases (monotonicity): V (D | k) > V (D | k − 1),

V (C | k) > V (C | k − 1). (2)

• The average payoff of individuals in the group increases with the number of cooperators (efficiency of cooperation): (k + 1)V (C | k) + (n − k − 1)V (D | k + 1) > kV (C | k − 1) + (n − k)V (D | k).

(3)

In this paper, we assume that the payoff functions for cooperators and defectors are, as in Joshi (1987), Eriksson and Lindgren (2005) and Lindgren and Johansson (2001), etc., calculated as a linear combination of the payoffs against the n − 1 opponents in 2-person prisoner’s dilemma games whose payoff is given in Table 1: bk −c n−1 bk V (D | k) = , n−1 V (C | k) =

(4) (5)

where b > c > 0 and we divided the linear combination of the payoffs by n − 1 in order to compare results from different group sizes. Of course, these payoff functions satisfy the conditions (1)–(3). In this game, b represents the benefit of cooperation to the recipients and c denotes the cost to the donor. Note that this game is a straightforward expansion of the 2-person prisoner’s dilemma (giving) game used in many studies regarding indirect reciprocity (e.g. Nowak and Sigmund (1998a,b) and Panchanathan and Boyd (2003)). Moreover, implementation error is introduced by the parameter  (0 <   1). With a small probability , an individual who intends to cooperate fails to cooperate due to a lack of resources or a mistake.4 In other words, an individual who intends to cooperate succeeds in cooperation with a probability ˆ = 1 −  (0  ˆ < 1). In this study, we mainly use the probability of the success, ˆ . In the numerical calculations in this study, we fix the small probability  at 0.01 and so ˆ = 0.99.

each individual? In this study, we adopt “image scoring” as a reputation criterion, which prescribes how to judge the reputation of others on the basis of the others’ past action. Under image scoring, which was first used in Nowak and Sigmund (1998a,b), the reputation of an individual who has defected becomes B and that of an individual who has cooperated becomes G. This is a simple reputation criterion that requires only the past action of opponents (first-order information) (cf., standing reputation criterion given in Leimar and Hammerstein (2001) and Panchanathan and Boyd (2003) uses second-order information). 2.2. Strategies In this study, individuals are supposed to use a strategy that prescribes how to decide their own action based on their opponents’ reputations. Such a strategy is represented by a vector p = ( p0 , . . . , pn−1 ), where pi ∈ {0, 1}, in which pi represents an action of the individual when the number of opponents who have reputation G is i, and 0 and 1 mean defection and cooperation, respectively. For example, in the four-person case (i.e., n = 4), a p = (0, 1, 0, 0) strategist cooperates only when one opponent has reputation G, and a p = (0, 0, 1, 1) strategist cooperates only when more than one opponents have reputation G. In total, there are 2n different strategies. 3. Evolutionary stability of indirect reciprocity Here we discuss the evolutionary stability of the indirectly reciprocal strategy pˆ = (0, . . . , 0, 1), which is called discriminating strategy (DIS). pˆ strategists cooperate only when all the n − 1 opponents have reputation G. For this purpose, we consider the situation where a small number of p strategists enter a population dominated by pˆ strategists and where the frequency of the invaders is sufficiently small for us to presume that each individual always interacts only with incumbent pˆ strategists. In the rest of this paper, we always consider this situation (detailed explanations are described in Appendix A). ˆ be the fitness for p strategists in this situation. Let W (p|p) In this section, we demonstrate the inequality ˆ p) ˆ > W (p|p) ˆ W (p|

ˆ (for all p 6= p).

(6)

If the inequality holds, pˆ is an ESS. 3.1. Frequency of individuals with reputation G

2.1. Reputation criterion In this model, the reputation of opponents affects the decision-making process. How is the reputation assigned to 4 We, as in Panchanathan and Boyd (2003), Fishman (2003) and Brandt and Sigmund (2004), do not consider errors that cause unintentional cooperation, i.e., an individual who intends to defect never fails to defect. Furthermore, perception error (Ohtsuki and Iwasa, 2004; Takahashi and Mashima, 2006) is not included.

With gˆ t , we represent the frequency of individuals with reputation G among incumbent pˆ strategists, and with gt , that among invader p strategists. Under image scoring, only the individuals who have cooperated at round t − 1 have reputation G at round t. In other words, to derive the frequency of individuals with reputation G, it is necessary to derive the probability that each individual cooperates. The possible situations that each individual faces are the following n cases: there is no opponent with G reputation; there

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is one; · · ·; and there are n − 1 opponents with G reputation. Since we are considering the case where each individual always interacts with only incumbent pˆ strategists, the probability that an individual belongs to a group with i opponents who have reputation G is represented as n−1 Ci gˆ ti (1− gˆ t )n−1−i . Therefore, the probability that each individual faces each of the above n situations at round t is represented as an n-dimensional vector, s(t) = (s0 (t), . . . , sn−1 (t)) 
n−1 = 1 − gˆ t ,..., i n−1 Ci gˆ t

1 − gˆ t


n−1−i

 , . . . , gˆ tn−1 .

(7)

Using s(t), we represent the probability that an incumbent pˆ strategist cooperates at round t as ˆ pˆ · s(t), and the probability that an invader p strategist cooperates at round t as ˆ p · s(t). Consequently, the frequency of individuals with reputation G among incumbent pˆ strategists at round t is ( 1 if t = 1 gˆ t = (8) ˆ pˆ · s(t − 1) if t ≥ 2. n−1 Since pˆ = (0, . . . , 0, 1), gˆ t = ˆ gˆ t−1 for t ≥ 2. Solving (n−1)t−1 −1

this recursive equation, we get gˆ t = ˆ n−2 . Considering 0  ˆ < 1, clearly limt→∞ gˆ t = gˆ ∞ = 0. Likewise, the frequency of individuals with reputation G among invader p strategists at round t is ( 1 if t = 1 gt = (9) ˆ p · s(t − 1) if t ≥ 2.

Next, we derive the expected payoff at round t for invader p strategists. For this purpose, we need to get the probabilities that an invader p strategist cooperates at round t and that an opponent of the focal invader p strategist cooperates at round t. The probability that an invader p strategist cooperates at round t is ˆ p · s(t). Moreover, we consider the probability that an opponent of the focal invader p strategist cooperates at round t. The probability that an opponent of the focal p strategist faces a situation where no opponent has reputation G is (1 − gt )(1 − gˆ t )n−2 and the probability that she faces a situation where all the n − 1 opponents have reputation G is gt gˆ tn−2 . Furthermore, the probability that she faces a situation where i (0 < i < n − 1) opponents have reputation G is
n−2−i (1 − gt )n−2 Ci gˆ ti 1 − gˆ t
n−1−i + gt ·n−2 Ci−1 gˆ ti−1 1 − gˆ t . (11) Note that in this study, we consider the case for n ≥ 3. The first-term indicates the case in which the focal p strategist in the group has reputation B and the second-term indicates the case in which the focal p strategist in the group has reputation G. Therefore the probability that, at round t, an opponent of the focal invader p strategist faces each of the n situations – there is no opponent with G reputation; there is one; · · ·; and there are n − 1 opponents with G reputation – is represented as an 0 n-dimensional vector s0 (t) = (s00 (t), . . . , sn−1 (t)), 
n−2 s0 (t) = (1 − gt ) 1 − gˆ t ,
n−2
n−3 , + gt 1 − gˆ t (1 − gt ) (n − 2)gˆ t 1 − gˆ t ...,

Considering gˆ ∞ = 0 and Eq. (9), gt in the limit of t → ∞, is ˆ p0 where p0 is the first component of p.

(1 − gt )n−2 Ci gˆ ti 1 − gˆ t

3.2. Expected payoff at round t

...,

Here we derive the expected payoff at round t for incumbent pˆ strategists and for invader p strategists. Recall that the payoff for an individual is determined by the probability that the focal individual cooperates and by the probability that an opponent of the focal individual cooperates (see Eqs. (4) and (5)). Therefore, in order to derive the expected payoff at round t for incumbent pˆ strategists, we need to get the probability that an incumbent pˆ strategist cooperates and the probability that an opponent of the focal pˆ strategist cooperates at round t. (Note that the opponent always has pˆ strategy since we consider the situation where each individual always interacts with only incumbent pˆ strategists.) As we mentioned above, the probability that the focal pˆ strategist cooperates is ˆ pˆ · s(t). Furthermore, an opponent of the focal pˆ strategist cooperates only when the focal pˆ strategist and the other n − 2 opponents have G reputation, the probability of which is gˆ t · gˆ tn−2 = gˆ tn−1 . So, the probability that an opponent of the focal pˆ strategist cooperates is ˆ gˆ tn−1 = ˆ pˆ · s(t). Consequently, the expected payoff at round t for incumbent pˆ strategists is ˆ p) ˆ = ˆ bpˆ · s(t) − ˆ cpˆ · s(t) = ˆ pˆ · s(t)(b − c). Wt (p|

(10)


n−2−i

+ gt ·n−2 Ci−1 gˆ ti−1 1 − gˆ t


n−1−i

,


(1 − gt ) gˆ tn−2 + gt (n − 2)gˆ tn−3 1 − gˆ t ,  gt gˆ tn−2 .

(12)

Therefore, the probability that an opponent of the focal invader p strategist cooperates at round t is ˆ pˆ · s0 (t). Note that an opponent of the focal invader p strategist always has pˆ strategy. Consequently, the expected payoff at round t for invader p strategists is ˆ = ˆ bpˆ · s0 (t) − ˆ cp · s(t) Wt (p|p)
= ˆ bpˆ · s0 (t) − cp · s(t) .

(13)

In the next section, using Eqs. (10) and (13), we will derive the fitness for two types of individuals. 3.3. Fitness for incumbent pˆ strategists and for invader p strategists ˆ p), ˆ We define the fitness for incumbent pˆ strategists, W (p| ˆ as the expected value and that for invader p strategists, W (p|p), of the total payoff received by the two types of individuals,

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Fig. 1. Regions in (c/b, w) space where pˆ is an ESS (ˆ = 0.99). Region I: pˆ cannot resist invasion by cooperative strategies. Region II: pˆ is an ESS. Region III: pˆ cannot resist invasion by defective strategies. (a) group size, n = 3, (b) n = 5, (c) n = 7, and (d) n = 10.

ˆ p) ˆ and respectively, during a generation. In this case, W (p| ˆ are respectively calculated by W (p|p) ˆ p) ˆ = W (p|

∞ X

ˆ p), ˆ w t−1 Wt (p|

(14)

ˆ w t−1 Wt (p|p).

(15)

t=1

ˆ = W (p|p)

∞ X t=1

3.4. Evolutionary stability of pˆ (numerical investigation) Here we investigate the evolutionary stability of pˆ using numerical calculation.5 First, we derive the parameter space for c/b and w where ˆ p) ˆ > W (p|p) ˆ holds for all p 6= p, ˆ i.e., pˆ is an ESS. W (p| Note that c/b represents the cost-to-benefit ratio of cooperation and w indicates the probability that each of subsequent rounds occurs. Fig. 1 shows the parameter space for various group sizes, n. As the figure shows, between small and large costto-benefit ratio of cooperation (region II in the figure), there exist a range of the medium ratio where the indirectly reciprocal strategy pˆ is an ESS. Specifically, if the cost-to-benefit ratio is large (region III in the figure), pˆ cannot resist invasion by ˆ and, if the ratio is extremely more defective strategies than p, 5 We cannot get the exact calculation of W (p| ˆ p) ˆ and W (p|p). ˆ Therefore, PT t−1 W (p| ˆ p) ˆ = ˆ we obtain those values approximately by W (p| t ˆ p) t=1 w PT t−1 ˆ = ˆ and W (p|p) w W (p| p) with finite T . This approximation does t t=1 not change the results essentially as far as T is sufficiently large, because ˆ p) ˆ and Wt (p|p) ˆ are bounded. Throughout this study, 0 < w < 1, and Wt (p| we set T = 10 000.

small (region I in the figure), pˆ cannot resist invasion by more ˆ For example, in the case of n = 3, cooperative strategies than p. strategies (0, 0, 0) and (0, 1, 0) can invade the population of pˆ strategy when c/b = 10/11, w = 0.90 and ˆ = 0.99, which corresponds to region III in the figure. Furthermore, strategies (0, 1, 1) and (1, 1, 1) can invade the population of pˆ when c/b = 1/20, w = 0.90 and ˆ = 0.99, which corresponds to region I in the figure. Under the medium cost-to-benefit ratio of cooperation (region II in the figure), pˆ is an ESS. Moreover the range becomes larger as the probability that each of the subsequent rounds occurs, w, increases. That is, under image scoring, the strategy pˆ is an ESS for some parameter ranges. Furthermore, comparing Fig. 1(a), (b), (c) and (d), we find that, as group size increases, the size of the region I where the pˆ population can be invaded by cooperative strategies decreases and the size of the region III where the population can be invaded by defective strategies increases. Note that, here, we describe the effect of group size, n, on the evolutionary stability of pˆ for n > 2. In the case of n = 2, it has been shown that pˆ is never an ESS (Panchanathan and Boyd, 2003; Ohtsuki and Iwasa, 2004, ). Comparison between the case of n > 2 and that of n = 2 will be discussed in Section 4. Regarding the evolutionary stability against defective strategies, incumbent pˆ strategists are exploited by defective invader strategists at the first round where all individuals have reputation G. This is a disadvantage of incumbent pˆ strategists. On the other hand, pˆ strategists are not exploited by defective invader strategists in early rounds except the first round because the defective invaders have B reputation after the first round. Furthermore, in the early rounds, pˆ strategists can cooperate with each other when the group consists only of pˆ

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strategists with reputation G.6 This an advantage of incumbent pˆ strategists over defective invader strategists. However, as group size increases, it becomes more likely that a group comprises some pˆ strategists with reputation B who defected by error. In other words, the probability that a group comprises only pˆ strategists with reputation G, which is gˆ tn−1 , becomes small with the increase in group size, n. That is, the advantage of incumbent pˆ strategists decreases as group size increases. Therefore, the size of the region III where the population of pˆ can be invaded by defective strategies increases as group size increases. Regarding the stability against cooperative strategies, in early rounds except the first round, cooperative invader strategists are more likely to be cooperated by incumbent pˆ strategists than incumbent pˆ strategists are. (This is an advantage of cooperative invader strategists over incumbent pˆ strategists.) This is because the frequency of individuals with reputation G among cooperative invader strategists is greater than that among incumbent pˆ strategists. For example, for n = 3, at round 3, the frequency of individuals with reputation G among invader unconditional cooperators (p = (1, . . . , 1)) and that among incumbent pˆ strategists are ˆ and ˆ 3 , respectively (see Eqs. (9) and (8)). However, in late rounds where no incumbent pˆ strategist cooperates (see footnote 6), cooperative invader strategists are no longer cooperated by incumbent pˆ strategists and so get lower fitness than that for pˆ strategists because of their excessive cooperativeness. This is a disadvantage of cooperative invader strategists. The larger the group size is, the more rapidly the rounds, where no pˆ strategist has reputation G and cooperates with others, come (see Eq. (8)). This is because the frequency of individuals with reputation G among pˆ strategists, gˆ t , rapidly approaches zer o with time especially for large group size, n (see Eq. (8)). In other words, the number of the early rounds where cooperative invaders earn higher fitness than that for pˆ strategists becomes small as group size increases. That is, the opportunity for invader cooperative strategists to get higher payoff than that for incumbent pˆ strategists decreases as group size increases. Hence, the size of the region I where the pˆ population can be invaded by cooperative strategies decreases with the increase in group size. In Fig. 2, the size of the region II where pˆ is an ESS is plotted as a function of group size. As shown in the figure, there exists a group size (n = 8) where the parameter space for w and for c/b where pˆ is an ESS is the largest. This is because, when group size is too large, the size of the region where the population of pˆ can be invaded by defective strategies is large, and when group size is too small, the size of the region where the population can be invaded by cooperative strategies is large. 6 In late rounds, no individual is cooperated regardless of her strategy. This is because the frequency of individuals with reputation G among incumbent pˆ strategists at round t converges to zer o, i.e., limt→∞ gˆ t = gˆ ∞ = 0 (see Eq. (8)). Note that, for n > 2, each individual belongs to a group with two or more incumbent pˆ strategists whose reputation is B in the late rounds. (This is because each individual interacts only with incumbent pˆ strategists.) In this group, the incumbent pˆ strategists never cooperate in response to another pˆ strategist’s B reputation.

Fig. 2. Size of the region in (c/b, w) space where pˆ is an ESS (ˆ = 0.99). The size of the region is plotted as a function of group size, n.

With the intermediate group size, the region where pˆ is an ESS is the largest. The large parameter space implies that pˆ is likely to be observed in the real world. In other words, it is expected that pˆ is hardly observed in the real world when the parameter space is extremely small. That is, pˆ is likely to be observed in the community where the number of individuals engaged in a single interaction (game) is neither large nor small. So far, we have investigated the evolutionary stability of pˆ using numerical calculation in the case where group size, n, is at most 10. One may ask what if group size further increased. However, it is difficult to investigate the case for larger group size by numerical calculation, because, for large group size, we have to check the stability of pˆ against a huge number of strategies in order to examine the evolutionary stability of pˆ and it requires huge computational time. For instance, if group size, n = 20, we have to check the stability against about million strategies. For investigation of the case for larger group size, in the next section, we will analytically examine the evolutionary stability of pˆ with an approximation. 3.5. Evolutionary stability of pˆ (analytical investigation) Here we analytically investigate the evolutionary stability of ˆ p) ˆ and Wt (p|p) ˆ pˆ with an approximation. Assuming that Wt (p| (t ≥ 3) are approximately the same as those in the limit of ˆ p) ˆ and W (p|p) ˆ are round t → ∞, W (p| ˆ p) ˆ = W1 (p| ˆ p) ˆ + w · W2 (p| ˆ p) ˆ + W (p|

w2 ˆ p), ˆ W∞ (p| 1−w

(16)

ˆ = W1 (p|p) ˆ + w · W2 (p|p) ˆ + W (p|p)

w2 ˆ W∞ (p|p). 1−w

(17)

This approximation is valid especially for large group size, because the frequency of individuals with reputation G among incumbent pˆ strategists, gˆ t , rapidly reaches gˆ ∞ (=0) for large group size (see Eq. (8)). The validity of the approximation for large group size is shown in Appendix B and can be also confirmed in Fig. 4. Using the approximation, we analytically show that ˆ p) ˆ > W (p|p) ˆ W (p|

ˆ (for all p 6= p),

(18)

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Fig. 3. Regions in (c/b, w) space where pˆ is an ESS for large group size given analytically with the approximation (ˆ = 0.99). Region II: pˆ is an ESS. Region III: pˆ cannot resist invasion by defective strategies. (a) group size, n = 20, and (b) n = 50.

Fig. 4. Size of the region in (c/b, w) space where pˆ is an ESS (ˆ = 0.99). The size of the region is plotted as a function of group size, n. Unfilled circles represent the values given by numerical calculation, and filled circles indicate the values given analytically with the approximation.

if and only if c/b < wˆ n−1 /(1 + w ˆ n−1 ). (Details of the analysis are given in Appendix C.) In other words, for large group size, the strategy pˆ is an ESS under image scoring if the cost-to-benefit ratio of cooperation is sufficiently small. The condition of the cost-to-benefit ratio is illustrated in Fig. 3. As shown in the figure, the region where pˆ is an ESS becomes larger as the cost-to-benefit ratio decreases or as the probability that each of the subsequent rounds occurs, w, increases. Furthermore, integrating the results with those for smaller group size given by numerical calculation, we demonstrate the effect of group size on the size of the region where pˆ is an ESS in Fig. 4. The figure shows that, for large group size, the region becomes smaller as group size increases but not so drastically. Moreover, the figure also shows the validity of the approximation given in Eqs. (16) and (17) for large group size (see the range for 8 ≤ n ≤ 9). 3.6. Frequency of cooperation in the pˆ population ˆ In So far, we have analyzed the evolutionary stability of p. this section, we examine how much cooperation is formed in the pˆ population. In Fig. 5, we show, using numerical calculation, the frequency of cooperation over a generation as a function of the probability that each of the subsequent rounds occurs, w. Note that the frequency is independent of the cost or benefit

Fig. 5. Frequency of cooperation over a generation for various group size in the pˆ population (ˆ = 0.99). The frequency of cooperation is plotted as a function of the probability that each of the subsequent rounds occurs, w.

of cooperation. As the figure shows, cooperation is formed to some degree when both w and group size, n, are small. In the pˆ population, all pˆ strategists come to defect in the long run because the frequency of individuals having reputation G decreases to zer o over time in response to error defections (see Eq. (8)). When w is small, it is likely that a generation terminates before the retaliative defections spread over the population, and so the frequency of cooperation is retained high.7 For example, in the case of n = 3, the frequency of cooperation is about 0.6 when w = 0.8 (i.e., the expected number of rounds in a generation is 1/(1 − w) = 5), and the frequency is about 0.4 when w = 0.9 (i.e., the expected number of rounds is 1/(1 − w) = 10). Moreover, the smaller the group size, n, is, the higher the frequency of cooperation formed in the pˆ population becomes. This is because, as n decreases, the decrease in the frequency of individuals having G reputation becomes smaller (see Eq. (8)). In summary, in the pˆ population, cooperation is formed under image scoring to some degree when both w and n are small. 4. Discussion In this paper, we have analyzed the evolutionary stability of indirectly reciprocal strategy, DIS (pˆ strategy) which 7 Note that we consider the probability of error defection, , is very small (ˆ is very close to 1).

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cooperates only when all the opponents in the group have a good reputation, under image scoring. Under image scoring which reflects only first-order information, cooperations (defections) are judged to be good (bad) unconditionally. In particular, we have focused on n-person games (>2) where n individuals take part in a single group, and in the case with implementation error. The analysis has shown that, in n-person games, DIS can be an ESS even under image scoring, which reflects only first-order information, whereas DIS can never be an ESS in 2-person games (e.g. Panchanathan and Boyd (2003), Ohtsuki and Iwasa (2004), Brandt and Sigmund (2004) and Takahashi and Mashima (2006)). In other words, in nperson games, first-order information is sufficient for indirect reciprocity to be maintained stably. Specifically, in n-person games, DIS is an ESS among all the possible strategies that decide their own action based on the number of opponents having good reputation, within a range of the cost-to-benefit ratio of cooperation, even under image scoring. If the cost-to-benefit ratio is large, DIS cannot resist invasion by defective strategies and, if the ratio is extremely small, DIS cannot resist invasion by cooperative strategies. Under the medium cost-to-benefit ratio of cooperation, DIS is an ESS. Moreover, in the DIS population, cooperation is formed and maintained to some degree when both the probability that each of the subsequent rounds occurs, w, and group size, n, are small. Here, we describe why DIS can be an ESS under image scoring in n-person games whereas DIS is never in 2-person games. In the population of DIS, as shown in Eq. (8), the frequency of good individuals decreases rapidly over time and eventually becomes zer o in response to error defection (this also holds in 2-person games). This indicates that incumbent DIS strategists defect in late rounds. Only in a first few rounds, they reciprocate with each other. Therefore, in order for other strategists to invade the DIS population, they have to adopt either of the following two ways: (i) by defecting, exploiting incumbent DIS strategists at the first round where all individuals have good reputation; or (ii) by cooperating, being cooperated by incumbent DIS in late rounds while incumbent DIS defect against each other. Regarding the way (i), there is no difference between 2- and n-person games, i.e., a few defective strategists can invade the population of DIS if the cost-tobenefit ratio of cooperation, c/b, is sufficiently large. On the other hand, there is a difference in the way (ii). In 2-person games, in late rounds where incumbent DIS defect against each other, a few cooperative invader strategists having good reputation can receive cooperation by DIS strategists because of their good reputation. Panchanathan and Boyd (2003) have shown that, for this reason, a few cooperative invader strategists get higher fitness than incumbent DIS strategists and so DIS is not an ESS in 2-person games. However, in n-person games, a few cooperative invader strategists cannot receive cooperation by DIS strategists in late rounds even though they have good reputation. This is because, in n-person games, a few cooperative invader strategists mostly belongs to a group with two or more incumbent DIS strategists (note that the share of the invader is assumed to be very small). In this group,

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the incumbent DIS strategists defect in response to another DIS strategist’s bad reputation. Therefore, a few cooperative strategists cannot invade the population of DIS in n-person games. In other words, DIS in n-person games is more robust against invasion by cooperative strategists than that in 2-person games. Hence, DIS can be an ESS in n-person games while this cannot be in 2-person games. Acknowledgments We thank Karthik Panchanathan and anonymous two reviewers for their useful comments. This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology and a Grant-in-Aid for Scientific Research (S), 2005–2009, 17103002, and for JSPS Fellows, 2006–2007, 183845. Appendix A In this study, to derive the condition for pˆ strategy to be an ESS, we assume that the frequency of invaders is sufficiently small for us to presume that each individual always interacts only with incumbent pˆ strategists. Here we explain the assumption in detail. Let δ be the share of invader p strategists. Then, expected payoffs for incumbent pˆ and invader p strategists at round t in the population are respectively ˆ ˆ 0) + · · · +n−1 C j δ n−1− j (1 − δ) j Wt (p|δ) = δ n−1 Wt (p|δ, ˆ ˆ n − 1) × Wt (p|δ, j) + · · · + (1 − δ)n−1 Wt (p|δ, n−1 X n−1− j ˆ = (1 − δ) j Wt (p|δ, j), (A.1) n−1 C j δ j=0

Wt (p|δ) = δ n−1 Wt (p|δ, 0) + · · · +n−1 C j δ n−1− j (1 − δ) j × Wt (p|δ, j) + · · · + (1 − δ)n−1 Wt (p|δ, n − 1) n−1 X n−1− j = (1 − δ) j Wt (p|δ, j), (A.2) n−1 C j δ j=0

where Wt (p|δ, j) indicates the expected payoff for a p strategist at round t when the share of invader in the population is δ and when the number of pˆ strategists in her group is j. (Note that n−1− j (1 − δ) j denotes the probability that an individual n−1 C j δ belongs to a group with j incumbent pˆ strategists and n − 1 − j invader p strategists.) In this case, the fitness for pˆ and p strategists, defined as expected total payoff during a generation, are ˆ W (p|δ) =

∞ X

ˆ w t−1 Wt (p|δ),

(A.3)

w t−1 Wt (p|δ).

(A.4)

t=1

W (p|δ) =

∞ X t=1

ˆ there exists a Here, pˆ is an ESS if and only if, for all p 6= p, ¯ ˆ δ¯ > 0 such that W (p|δ) > W (p|δ) for δ ∈ (0, δ). In this paper, we consider the case of δ → 0, and then the fitness for incumbent pˆ strategists and that for

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P∞ t−1 ˆ n − 1) invader converge to Wt (p|0, t=1 w P∞p strategists and t=1 w t−1 Wt (p|0, n − 1) respectively (see Eqs. (A.1)– (A.4)). This means that we focus only on the situation where the frequency of the invader is sufficiently small for us to presume that each individual always interacts only with incumbent ˆ strategists. Under this case, P p we derive the condition that P ∞ ∞ t−1 W (p|0, t−1 W (p|0, n − 1) for ˆ w n − 1) > t t t=1 t=1 w ˆ If this condition holds, there exists a δ¯ > 0 such all p 6= p. ¯ that is, pˆ ˆ that W (p|δ) > W (p|δ) for all p 6= pˆ for δ ∈ (0, δ), is an ESS. This is because, if we consider sufficiently small δ, all the terms except for the last term of the expected payoffs at round t given in Eqs. (A.1) and (A.2) are sufficiently small to be negligible. Appendix B In this section, we show the validity of the approximation ˆ p) ˆ and given in Eqs. (16) and (17), i.e., assumption that Wt (p| ˆ (t ≥ 3) are approximately the same as those in the Wt (p|p) limit of round t → ∞, for large group size, n. In Fig. 6, we compare the evolutionary stability of pˆ given analytically under the approximation with the stability given numerically without the approximation, for group size, n = 15 and 20. The figure shows that the approximation given in Eqs. (16) and (17) does not change essentially the evolutionary stability of pˆ for large group size. In other words, the approximation is valid for large group size. Specifically, dividing both expected payoff at round t for incumbent pˆ strategists and that for invader p strategists by b, we can rewrite the expected payoffs given in Eqs. (10) and (13) as ˆ p) ˆ = ˆ pˆ · s(t)(1 − c/b) Wt (p| = ˆ gˆ tn−1 (1 − c/b), (B.1)   c ˆ = ˆ pˆ · s0 (t) − p · s(t) Wt (p|p) b ! n−1 cX n−2 i n−1−i , pi Ci gˆ t (1 − gˆ t ) = ˆ gt gˆ t − b i=0 n−1 (B.2) which does not affect the evolutionary phenomena of the game (Houfbauer and Sigmund, 1996, ). In this case, the difference between the expected payoff at round t for incumbent pˆ strategists and that for invader p strategists is ˆ p) ˆ − Wt (p|p) ˆ = ˆ gˆ tn−1 (1 − c/b) Wt (p| − ˆ

gt gˆ tn−2

" = ˆ

gˆ tn−2



n−1 cX − pi Ci gˆ ti (1 − gˆ t )n−1−i b i=0 n−1

!

gˆ t , rapidly approaches zer o with time, especially for large group size, n (see Eq. (8)). This indicates that, for large n, ˆ p) ˆ and Wt (p|p) ˆ (t ≥ 3) is almost the difference between Wt (p| the same as that in the limit of round t → ∞. Therefore, the approximation given in Eqs. (16) and (17) is valid for large group size. Appendix C Here we analytically investigate the evolutionary stability of pˆ with the approximation given in Eqs. (16) and (17).

 c gˆ t − gˆ t − gt b

# n−1 cX i n−1−i + pi Ci gˆ t (1 − gˆ t ) . b i=0 n−1

Fig. 6. Regions in (c/b,w) space where pˆ is an ESS. The separatrix indicates the boundary between region II and III given analytically with the approximation. The left-hand side of the separatrix is region II where pˆ is an ESS, and the right-hand side is region III where pˆ cannot resist invasion by defective strategies. On the other hand, filled and unfilled circles indicate the evolutionary stability of pˆ given by numerical calculation without the approximation. The filled circles represent that pˆ is an ESS under the parameter, c/b and w, and the unfilled circles denote that pˆ cannot resist invasion by defective strategies, i.e., pˆ is not an ESS. (a) group size, n = 15, and (b) n = 20.

(B.3)

Here, it is important to note that the frequency of individuals with reputation G among incumbent pˆ strategists at round t,

C.1. Evolutionary stability of pˆ against invasion by p with pn−1 = 1 ˆ p) ˆ > W (p|p) ˆ For this purpose, we first show that W (p| ˆ with pn−1 = 1, hereafter called pC = for all p (6= p) C , 1). ( p0C , . . . , pn−2

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Substituting Eq. (10) into Eq. (16), the fitness for incumbent pˆ strategists is ˆ p) ˆ = ˆ (b − c) + w ˆ pˆ · s(2)(b − c) W (p| w2 ˆ pˆ · s(∞)(b − c) + 1−w = ˆ (b − c) + w ˆ pˆ · s(2)(b − c),

(C.1)

where s(∞) represents s(t) in the limit of t → ∞. Note that since gˆ ∞ = 0, pˆ · s(∞) = 0. On the other hand, substituting Eq. (13) into Eq. (17), the fitness for invader pC strategists is ˆ = ˆ (b − c) + w ˆ [bpˆ · s0 (2) − cpC · s(2)] W (pC |p) w2 + ˆ [bpˆ · s0 (∞) − cpC · s(∞)] 1−w = ˆ (b − c) + w ˆ [b(p) · s0 (2) − cpC · s(2)] −

w2 ˆ cp0C , 1−w

(C.2)

w2 ˆ p) ˆ − W (p |p) ˆ = w ˆ cs(2) · (p − p) ˆ + W (p| ˆ cp0C . 1−w (C.3) C

ˆ The difference (Eq. (C.3)) is 0 if and only if pC = p, i.e., pC = (0, . . . , 0, 1); otherwise the difference is positive.8 Consequently, ˆ p) ˆ > W (pC |p) ˆ W (p|

ˆ (for all pC 6= p),

ˆ = ˆ b − w ˆ cp D · s(2) − W (p D |p) = ˆ b − w ˆ cp D · s(2) −

w2 ˆ cp D · s(∞) 1−w w2 ˆ cp0D . 1−w

(C.4)

(C.6)

Note that s(∞) = (1, 0, . . . , 0). The difference between the fitness for invader p0 and p D strategists is ˆ − W (p D |p) ˆ = w ˆ cp D · s(2) + W (p0 |p)

where s(∞) and s0 (∞) indicate s(t) and s0 (t) respectively in the limit of t → ∞. Note that since we are considering the case 0 for n ≥ 3, sn−1 (∞) = 0 and so pˆ · s0 (∞) = 0. Furthermore, s(∞) = (1, 0, . . . , 0) because gˆ ∞ = 0. Therefore, pC · s(∞) = p0C . Since both types of individuals intend to cooperate at the first round, gˆ 2 = g2 and s(2) = s0 (2). Considering s(2) = s0 (2), the difference between the fitness for pˆ and pC strategists is C

Note that since p0 strategists never cooperate and so gt = 0 for t ≥ 2, incumbent pˆ strategists never cooperate with them. D = 0 and Eq. (9), the frequency On the other hand, by pn−1 of individuals with reputation G among p D strategists is as follows: g2 = 0 and g∞ = ˆ p0D . Therefore, substituting Eq. (13) into Eq. (17), the fitness for invader p D strategists is

w2 ˆ cp0D . 1−w

(C.7)

Since si (2) > 0 for all i (0 ≤ i ≤ n − 1), the difference (Eq. (C.7)) is 0 if and only if p D = p0 , i.e., p D = (0, . . . , 0); ˆ > otherwise the difference is positive.9 Therefore, W (p0 |p) ˆ for all p D 6= p0 . W (p D |p) ˆ p) ˆ > W (p D |p) ˆ for all p D if and only if Hence, W (p| 0 ˆ p) ˆ > W (p |p). ˆ The difference between W (p| ˆ p) ˆ given in W (p| ˆ given in Eq. (C.5) is Eq. (C.1) and W (p0 |p) ˆ p) ˆ − W (p0 |p) ˆ = ˆ (b − c) + w ˆ pˆ · s(2)(b − c) − ˆ b W (p| = ˆ (b − c) + w ˆ n (b − c) − ˆ b. (C.8) Note that pˆ · s(2) = ˆ n−1 because of gˆ 2 = ˆ and of the definition of s(2) given in Eq. (7). Therefore, the inequality ˆ p) ˆ > W (p0 |p) ˆ holds if and only if W (p| c/b <

w ˆ n−1 . 1 + w ˆ n−1

(C.9)

ˆ p) ˆ > W (p D |p) ˆ for all p D if and only if Consequently, W (p| n−1 n−1 c/b < wˆ /(1 + w ˆ ).

where pC is a strategy p with pn−1 = 1.

C.3. Evolutionary stability of pˆ against invasion by all the p strategists

C.2. Evolutionary stability of pˆ against invasion by p with pn−1 = 0

ˆ p) ˆ > W (p|p) ˆ for all p (6= p) ˆ with We have shown that W (p| ˆ p) ˆ > W (p|p) ˆ for all p with pn−1 = 0 pn−1 = 1 and that W (p| if and only if c/b < wˆ n−1 /(1 + w ˆ n−1 ). Consequently,

ˆ p) ˆ > Next, we show the condition that the inequality W (p| ˆ holds for all p with pn−1 = 0, hereafter called p D = W (p|p) D , 0). To show this, we first show that the fitness ( p0D , . . . , pn−2 0 for the p = (0, . . . , 0) strategy, which is one of the p D strategies, is the largest among the fitness for the p D strategies, ˆ > W (p D |p) ˆ for all p D 6= p0 . i.e., W (p0 |p) Substituting Eq. (13) into Eq. (17), the fitness for invader p0 strategists is

ˆ p) ˆ > W (p|p) ˆ W (p|

ˆ (for all p 6= p),

(C.10)

if and only if c/b < wˆ n−1 /(1 + w ˆ n−1 ). References

(C.5)

Alexander, R.D., 1987. The Biology of Moral Systems. Aldine de Gruyter, New York. Bolton, G.E., Katok, E., Ockenfels, A., 2005. Cooperation among strangers with limited information about reputation. J. Public. Econ. 89, 1457–1468.

8 Note that since all the components of s(2) are positive and those of pC − pˆ are non-negative, these two vectors are not orthogonal.

9 Note that since all the components of s(2) are positive and those of p D are non-negative, these two vectors are not orthogonal.

ˆ = ˆ b. W (p0 |p)

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