Evolution of Indirect Reciprocity by Social Information: The Role of

Transcription

1 1 Title: Evolution of indirect reciprocity by social inforation: the role of Trust and reputation in evolution of altruis Author Affiliation: Mojdeh Mohtashei* and Lik Mui* *Laboratory for Coputer Science, MIT, 200 Technology Square, Cabridge, MA Corresponding Author: Mojdeh Mohtashei Laboratory for Coputer Science, MIT 200 Technology Square, Roo 419 Cabridge, MA Tel: (617) Fax: (617) Eail: Running Headline: Evolution of indirect reciprocity by social inforation Evolution of Indirect Reciprocity by Social Inforation: The Role of

2 2 Trust and Reputation in Evolution of Altruis The coplexity of huans cooperative behavior cannot be fully explained by theories of kin selection and group selection. If reciprocal altruis is to provide an explanation for altruistic behavior, it would have to depart fro direct reciprocity, which requires dyads of individuals to interact repeatedly. For indirect reciprocity to rationalize cooperation aong genetically unrelated or even culturally dissiilar individuals, inforation about the reputation of individuals ust be assessed and propagated in a population. Here we propose a new fraework for the evolution of indirect reciprocity by social inforation: inforation selectively retrieved fro and propagated through dynaically evolving networks of friends and acquaintances. We show that for indirect reciprocity to be evolutionarily stable, the differential probability of trusting and helping a reputable individual over a disreputable individual, at a point in tie, ust exceed the cost-to-benefit ratio of the altruistic act. In other words, the benefit received by the trustworthy ust out-weigh the cost of helping the untrustworthy. Keywords: Reciprocity, Social inforation, Trust, Reputation, Collective eory, Growth, Order. 1. Introduction Throughout social history, huans have exhibited large scale cooperative behavior toward individuals or groups, with who they are neither acquainted nor share genetic, phenotypic, or cultural traits. Such behavior is difficult to account for by theories of kin selection (Hailton, 1963, Hailton, 1964), group selection (Willias, 1971, Wilson & Sober, 1994), or direct reciprocation (Axelrod & Hailton, 1981, Axelrod, 1984, Axelrod & Dion, 1988). Alexander (1987) coined the ter indirect reciprocity to refer to the coonly practiced act of cooperation in huan societies, where the donor of a good deed does not necessarily expect to be rewarded by the recipient but perhaps by another individual who ay be the recipient of other good deeds by other donors 1. Several authors have since attepted to foralize the echaniss by which indirect reciprocity can evolve. Boyd and Richerson (1989) developed a atheatical odel of circular reciprocity where the donor of a good deed is to be rewarded by the last individual in a ring of n reciprocating individuals. Their results suggested that indirect reciprocity is unlikely to be iportant unless interacting groups are relatively sall. Alexander, however, has further hypothesized that indirect reciprocity is a

3 3 consequence of direct reciprocity occurring in the presence of others (Alexander, 1987). Those who observe direct reciprocation between individuals will then be in the position of assessing the reputation of ebers of a population. Hence, reputation is a key concept in Alexander s preise and he conjectures that indirect reciprocity requires that the reputation and status of ebers of a group be continually assessed and reassessed. Pollock and Dugatkin (1992) investigated the significance of observation in guiding behavioral choice by studying a variant of tit-for-tat (TFT) where players behave like TFT in the absence of inforation about a new co-player but defect if the co-player has been observed defecting in his last interaction. They showed that when TFT fails to be evolutionarily stable, its variant is indeed evolutionarily stable. To our knowledge, this is the first work to have introduced and utilized the notion of reputation, although binary and inial, in the context of a theoretical fraework for reciprocity and evolution of cooperation. Following Alexander s conjecture and studies by Pollock and Dugatkin, Nowak and Sigund (1998a, 1998b) developed a odel of indirect reciprocity by iage scoring to study the role of observers in assessing the reputation of donating players. Under this odel, every player has an iage score. When a player is selected as donor, his 2 iage score is odified locally by the recipient of the action and a few randoly selected observers. Thus, different individuals ay have different perceptions about a sae player. Cooperators are rewarded for their altruistic acts through increases of their iage scores for their recipients and observers. Every player also has a nueric strategy. A potential donor cooperates only if the recipient s iage is at least as high as his own strategy. The underlying preise in this fraework is that if the inforation about the iage of ebers of a population can be obtained, then an infored donor only helps those who are likely to help others. This in turn will iprove the reputation of the donor thereby introducing feedback into the syste: Although an altruistic act entails a cost, if a potential donor helps today, his reputation will iprove, which then increases the likelihood that others will help hi in the future (Ostro, 1998) 3. Indeed as baseline experientation, Nowak and Sigund (1998a) siulated a population wherein everyone s iage is broadcasted

4 4 globally to all ebers of the population. Under this odel, cooperation evolves irrespective of the population size. This result well deonstrates the role of reputation-based inforation in the evolution of indirect reciprocity. When inforation is locally available to observers, however, in order for this inforation to be of utility, potential donors would have to interact with observed agents since there is no other echanis for the transfer of inforation in the population. Nowak and Sigund s results suggest that when inforation is localized, cooperation can be established. However, a greater level of interactions per generation, or a larger nuber of observers, is needed for cooperation to be sustainable in a larger population. 2. Social Inforation: Trust, Reputation, and Collective Meory Although aking observations is one of the echaniss for acquiring inforation, it is not the principal anner by which huans process inforation about others actions. Furtherore, inforation is seldo randoly propagated in populations. Ebedded in every social network is a web of trust with nodes representing ebers of the web and edges representing the aount of trust between pairs of friends or acquaintances. When faced with social dileas, such as to cooperate or not, individuals ake use of social inforation ebedded in their social networks to reduce uncertainty. A key echanis by which individuals acquire inforation about others is by seeking the opinions of trusted friends and reputable acquaintances. Parents are the first ebers of such trust-based networks who provide their children with instruction and advice. Because it is not possible for one to observe or reeber all possible events even in sall populations, the inforation content of such a web of trust serves as one s collective eory. Such a trust-based body of inforation sees to be a fundaental eleent of social inforation and an inherent aspect of the processes by which huans ake decisions. Aside fro this trust-based coponent of social inforation which assues direct links between ebers of a social network, there is another vital, while not as apparent, eleent of social

5 5 inforation consequent of indirect links between ebers of different social networks, where one attepts to seek the opinion of a reputable k-degree acquaintance who is not a direct acquaintance but is connected through a chain of k other individuals. In other words, trust can be inferred in a transitive anner using the notion of reputation: If A trusts B, who trusts C, even if A has not et C before, it is quite likely that A will cooperate with C if C s good reputation is counicated to A by B. Such a process of decision aking not only requires the ability to induce friendship and ake acquaintances, it also entails the cognitive ability to learn and counicate. What sees coon in ost odels of evolution of altruis in huan populations is the players passive treatent of inforation and their liited ability to reeber only the near past. Even if we assue that huan eory is not functional on a longer tieline, it is iportant to note the existence of a collective eory upon which huans routinely rely to gain ore inforation and ake decisions ore effectively. Therefore, as long as we can counicate with our acquaintances, even if everyone in our social web can only reeber one past action, at any point in tie we have access to a body of inforation uch richer than the inforation content of our own personal experiences and eory. The resulting network then consists of any sall but densely connected clusters of close ties consisting of friends and faily, connected to each other by occasional links between few ebers of different clusters (Granovetter, 1973, Granovetter, 1985) 4. Therefore the underlying network connectivity, which in turn defines the routes of inforation transfer, contains order. This results in selective retrieval and disseination of inforation in populations, a property far fro rando acquisition of inforation routinely adopted in the literature. Another property of social networks which is often ignored in odels of evolution of cooperation is dynaics and growth. Even if we assue closed populations of constant size, due to interaction between ebers of the population, new links are created and new

6 6 acquaintances are ade at all ties 5. This eans that: (1) The topology of a social network changes over tie aking the network a dynaic entity; and (2) As new links are added to the network the average path between two individuals becoes shorter over tie, increasing the probability that any two randoly selected individuals know each other thereby causing growth in the network. In the next section we develop a odel of indirect reciprocity by social inforation in which we ake the following assuptions: (1) Players can counicate and inquire inforation about the reputation of their co-players. (2) Inforation about the reputation of coplayers is not obtained randoly, but rather players selectively acquire inforation fro their acquaintance networks by taking advantage of the collective eory of the social networks to which they belong. (3) Inforation is not propagated randoly in a population; new inforation resulting fro new interactions odifies the content of the collective eory of a recipient and is therefore selectively propagated through the recipient acquaintance network. (4) A social network is an evolving dynaic entity; with new interactions, new links are created, which in turn increases the likelihood of any two randoly selected players to know each other. 3. Model: Siulation Environent To ake clear the concepts introduced thus far, in this section we develop a siulation environent for the evolution of indirect reciprocity by social inforation. Consider a population of n individuals divided into non-overlapping groups of acquaintances of the sae size. Here, we odel the underlying graph structure of a group as a clique. Every generation consists of a fixed nuber of rounds,. In every round two players are selected at rando, one as donor and the other as recipient. The donor has the option of helping or defecting upon the recipient. If the donor cooperates it will cost hi a value of c and the recipient receives a benefit value of b (b>c). At the beginning of each generation every player is born into a unique clique of acquaintances. At the end of each generation every one dies and produces offspring in proportion to the total payoff they receive throughout the generation. We assue that every

7 7 agent j possesses a nueric strategy, k j, and has an iage score, s ij, for agent i (see Nowak & Sigund, 1998a). At the start of a generation all players have iage score of zero. A donor perfors one of the two actions, cooperate or defect. If a donor cooperates, his iage score is increased by one unit; otherwise it is decreased by one unit. A potential donor cooperates if the iage score of the recipient is at least as high as his own strategy. Acquaintanceship is defined as a one-way (unidirectional) or a two-way (bidirectional) relation. An acquaintance set is then the union of one-way and two-way acquaintances. An initial clique is a bidirectional network. Once the donor perfors an action, he becoes a oneway acquaintance of the recipient and her acquaintances if he is not already. By definition of collective eory all the donor s acquaintances will also becoe one-way acquaintances of the recipient and her acquaintances. However, this is a one-way relationship because the donor gains no new inforation about the recipient by donating to her. But the recipient who observes the donor in action can use this inforation to her advantage in future encounters. This eans that in future rounds if the recipient is chosen as donor, in addition to her current group of acquaintances she will also inquire the opinion of her newly acquired acquaintance, the donor. Note that the newly acquired acquaintance ay be untrustworthy if he defected upon the recipient, in which case the recipient ay consider perforing the opposite of what the donor recoends. Here, however, we do not ipose a distribution upon trust and assue that one equally trusts all one s acquaintances (see Section 4 for further elaboration). Upon perforing his action, the donor s iage score is updated for his acquaintances and the recipient. However, once the recipient has a new iage score for the donor, this knowledge will becoe part of the collective eory ebedded in the recipient s acquaintance set. Therefore, a new link is created between two players in every round of the gae if the donor (or any of his acquaintances) is not in the recipient s acquaintance set already. However, this is a one-way relationship since the donor is being added to the recipient s acquaintance set but not vice versa. Such a one-way link ay becoe bidirectional in a future round under three

8 8 conditions: (1) If the sae two players are selected again in opposite roles; (2) If the donor is selected to play as recipient against a donating eber of the recipient s acquaintance set; (3) If a eber of the donor s acquaintance set is selected to play as recipient against a donating eber of the recipient s acquaintance set (including the recipient herself). To avoid the effects of direct reciprocity on the evolution of cooperation here we assue that the probability of the sae pair being selected again, i.e. condition (1), is negligible. Conditions (2) and (3) are siply iplications of the notion of the collective eory. Therefore when a one-way link is created between a recipient and a donor, by definition all ebers of the recipient s acquaintance set are also linked to the donor and his acquaintance set in a one-way anner. If a potential donor does not know the iage score of the recipient, he will ake use of the social inforation available to hi by asking all his acquaintances if they have ever played in the recipient role against the current recipient, i.e. if the current recipient is a one-way acquaintance of the donor s acquaintance set. If no inforation is learned, the donor will assue an iage score of zero; otherwise, he assesses the reputation of the recipient by adding up her iage scores, provided by ebers of his acquaintance set, and dividing by the total nuber of encounters. He then copares the final score to his own strategy. He cooperates if the final score is at least as large as his own strategy and defects otherwise. Therefore, the outcoe will depend on the probability of knowing the recipient s iage, which is derived fro the collective eory ebedded in the donor s acquaintance set. As the gae continues and players eet over tie, the underlying topology of the social network also evolves, which causes the probability of knowing a randoly selected recipient to increase over the lifetie of a generation. Figure 1 illustrates the steady rise of this probability in a siulation environent of a hypothetical population of 100 individuals with an initial clique size of four, i.e. 3 friends per agent, over a generation consisting of encounters. In Section 5 we derive this dynaic quantity analytically (see Equations (1)-(3) in Methods). 4. Siulation Results

9 9 Figure 2 shows the results of coputer siulations for varying population sizes with the initial acquaintance clique size of four (three initial friends per agent). Every generation consists of a fixed nuber of rounds,. Children inherit neither the iage score of their parents nor their parental acquaintance structure. They only inherit the strategy of their parents unless they are subject to utation. At the beginning of each generation all players are randoly assigned to unique cliques of the sae size. The gae is played for any generations to subject the population to selective pressure. We say that cooperation is established if the average winning strategy (k) for all individuals at the end of the gae is less than or equal zero. We find that under our fraework even with as few as three initial acquaintances 6 cooperation is evolved and sustained after the gae is played for any generations (see Figure 2). It is interesting to note the effect of the size of the initial social clique on the evolution of indirect reciprocity. Figure 3 shows the results of coputer siulations for a population of 100 individuals under varying initial clique sizes. With ore initial acquaintances cooperation is clearly evolved and sustained in a ore secure anner. Under siilar conditions, in a population of all loners, i.e. when the initial clique size is one and no one is initially born into a social group, cooperation fails to evolve. Due to inherent growth factor in the network connectivity, however, even a slight increase in the nuber of rounds per generation can help a ixture of cooperating strategies to be established in a population of all-loners. Therefore either a generation ust live long enough so that in the absence of initial social structures, over tie, interactions between individuals would create interconnected social networks for inforation to transfer, or if the generation life is short then the existence of initial trust relations gives the inforation transfer a head start. When inforation is scarce, in the absence of a dynaically evolving network of acquaintances, cooperators (players with k 0 ) are less fit and can be easily defeated by defectors. As soon as we allow for channels of inforation to evolve by letting individuals ake new connections as they interact, the likelihood of disseination of inforation about

10 10 players reputation is increased, thereby increasing the likelihood of discriinators to rightfully discriinate against exploiters. Cooperators benefit fro this situation: Discriinators donate to exploiters less and less, causing exploiters average fitness and progeny nuber to decrease; at the sae tie, discriinators donate to cooperators ore and ore, causing cooperators average fitness and progeny nuber to increase. This is why cooperation can evolve under this scenario despite the fact that discriinators are the only ones who can ake use of inforation. Evolution of cooperation is then a consequence of infored discriination. Phrasing after Alexander: indirect reciprocity is a consequence of direct reciprocity occurring in the presence of others, we add that indirect reciprocity is also a consequence of selective inquiry about direct reciprocity. Variation on the Thee In the fraework presented here we assue that a potential donor takes the advice of all his acquaintances at their face values and siply adds up the iage scores reported by the. In particular, even when a donor defects upon his recipient, he still becoes a one-way acquaintance of the recipient. In future rounds, the recipient ay inquire the opinion of such a istrusted acquaintance. Although in social networks clusters of acquaintanceship appear to capture the underlying graph structure of a clique (Granovetter, 1973, Granovetter, 1985), i.e. it ay quite be likely that any two acquaintances of an individual are also acquaintances, it is not realistic to assue that one equally trusts all one s friends and acquaintances. In a odel of social networks capturing the possibility of isinforation or istrust, links in the graph ust have weights associated with the representing the aount of failiarity or trust between pairs of acquaintances. Furtherore, a pair of acquaintances ay have different values of trust for each other. These properties, naely varying degrees of trust for one s acquaintances and asyetry in pair-wise trust relations should be captured both in the initial cliques and new links resulting fro acquiring new acquaintances. But this is a subject for future works. 5. Methods: Analytic Results

11 11 Let A be the average nuber of acquaintances per player at round i,0 i, where is the i nuber of rounds per generation. This variable represents the average network connectivity per player at round i (including self). Let A 0 be the initial clique size at the beginning of each generation, and define q i as the probability that in round i the donor knows the recipient. Thus in effect, q i represents the average aount of inforation available to discriinators at round i since defectors and unconditional cooperators do not ake use of inforation to ake decisions. If we assue that only one donor-recipient pair interacts in each round, as in Sections 3 and 4, then for 1 i we have: q i = A n 1 i 1 1 (1). Equation (1) asserts that for a discriinator the likelihood of knowing the opponent s reputation at round i is one less than the average nuber of acquaintances per player fro the previous round (assuing that a player cannot play as both recipient and donor) over n 1 possible ways a recipient can be selected fro a population of size n (excluding the donor). The average network connectivity per player at round i can then be expressed as the following recurrence: A A q A n i 1 i = i 1 + (1 i ) (2). This eans that a new link is created between two players in every round of the gae if the donor or his acquaintances do not know the recipient, i.e. if the donor is not in the recipient s acquaintance set already, in which case the donor is added to the recipient s acquaintance set, resulting in Ai 1 new links in the recipient s acquaintance set or A / i 1 n new links per player. If we rewrite equation (2) using equation (1) we get: A i 2 i 1 n A = Ai 1 n 1 n( n 1) (3).

12 12 2 Define X = A / n. Then equation (3) can be rewritten in canonical for i i n X i = X i 1 (1 X i 1 ), which is the failiar logistic equation. As n, the growth n 1 n * rate, 1. Therefore, for n 2 the syste is stable with the nontrivial fixed point A = n n 1. This is the axiu network connectivity per player, i.e. the axiu nuber of acquaintances an individual can have in a population of size n (including self). In the reainder of this section, we will show how qi can be used to provide a qualitatively sensible interpretation of a tie-varying upper bound on the cost-to-benefit ratio of the altruistic act in order for cooperation to evolve. A Siple Three-Strategy Gae: Here we odify the basic odel of indirect reciprocity with incoplete inforation, proposed by Nowak and Sigund (1998b), according to the assuptions underlying the notion of collective eory to derive analytic conditions under which indirect reciprocity can evolve. Unlike their fraework, wherein all players are paired up to play as either recipients or donors in each round, here we assue that only one donor-recipient pair will be selected to play in a round. Later in this section we ake explicit coparison between our results and those of Nowak and Sigund s (1998a, b) for a twostrategy gae. Consider a population of size n consisting of three types of players: defectors who never help, cooperators who always help, and discriinators who only help players with good iage. Let x denote the frequency of discriinators, y be the frequency of unconditional defectors, and z be that of unconditional cooperators. As in Nowak and Sigund s (1998a, b) for a discriinating donor, a recipient has a good iage if she is known to have cooperated last tie; otherwise she has a bad iage. We odify the odel of indirect reciprocity by Nowak and Sigund by iposing a social structure on acquaintance relations so that players can acquire inforation through their acquaintance network. If discriinating donors do not learn new inforation about their recipients by asking their acquaintances, they always cooperate.

13 13 This eans that in the absence of inforation defectors can be istaken for good scorers by discriinators. Therefore at the beginning of each generation, when no inforation about players behaviors is available, all players are assued to be good scorers. Let g i be the proportion of good scorers at round i. Then g = x ( i) + y ( i) + z ( i), where x ( i), y ( i), z ( i) i g g g g g g represent the respective frequency of good scoring discriinators, defectors, and cooperators at round i. For cooperators zg ( i) = z for 1 i since they always help. For defectors and discriinators xg (1) good iage. = x and y (1) g = y, since at the beginning of the gae everyone has a For defectors, 1 yg ( i) = ( 1 ) yg ( i 1), 2 i. This is because the proportion of n good scoring defectors in round i changes only if a good scoring defector is selected to play as donor with probability yg ( i 1), in which case the nuber of good scoring defectors is reduced by one (or by 1/ n in proportion). Hence for 1 i we have: i 1 1 yg ( i) = 1 y n (4). The proportion of good scoring discriinators that changes in every round due to selection of a discriinating donor is x. There are two conditions under which the proportion n of good scoring discriinators in round i can change: (1) If a bad scoring discriinator is selected to play as donor and he cooperates, which happens with probability ( x xg i ) qi 1 qi 1gi 1 ( 1) (1 + ), in which case the nuber of good scoring discriinators will increase by one (or by 1/ n in proportion); (2) If a good scoring discriinator is selected to play as donor and defects, which happens with probability x ( i 1) q (1 g ), in which case the nuber of good scoring discriinators will g i 1 i 1 decrease by one (or by 1/ n in proportion). Therefore we have: 2 x x xg ( i) = 1 xg ( i 1) + (1 qi 1 + qi 1 gi 1) n n (5).

14 14 Let Fx ( i), Fy ( i), Fz ( i) denote the respective expected payoff for discriinators, unconditional defectors and unconditional cooperators at round i. Then we have: 1 xg ( i) Fx ( i) = c( qig i + 1 q i ) + b ( x + z ) bqix 1 n x b yg ( i) Fy ( i) = z + x(1 qi ) + xqi n y (7) (6) 1 Fz ( i) = { c + b( x + z)} (8). n Because only one donor-recipient pair interacts in each round and therefore the payoff ust be averaged over the population of interacting strategies, the cost c and benefit b are divided by the respective nuber of discriinators, defectors and cooperators, xn, yn, zn. At the sae tie, each equation is ultiplied by the respective probability of selecting a discriinator, defector and cooperator, x, y, z. Hence the ratio 1/ n appears in all three equations. Next, we noralize the payoffs by subtracting equation (7) fro all three equations to get: 1 ( ) ( ) ˆ xg i yg i Fx ( i ) = c ( qig i + 1 q i ) + bq ix n x y 1 ( ) ˆ yg i Fz ( i ) = c + bqi x 1 (10). n y (9) Therefore the total expected payoffs for discriinators and cooperators over one generation consisting of rounds are: 1 xg( i) yg( i) Px = c + c qi (1 gi) + bx qi n i= 1 i= 1 x y (11) 1 yg ( i) Pz = c + bx qi 1 n i= 1 y (12). Note that P z > 0 iff

15 15 c b < yg ( i) x qi 1 i= 1 y (13) and P x > 0 iff c b < x i= 1 xg( i) yg( i) qi x y q (1 g ) i= 1 i i (14). Inequality (13) asserts that for cooperators to attain a positive fitness over a generation, the average per-round probability of a discriinator defecting on a defector ust exceed the cost-to-benefit ratio of the altruistic act. This is a sensible condition since in the absence of discriinators, unconditional cooperators will be less fit and easily defeated by defectors. In other words, for cooperators to prosper, they need discriinators who can use inforation to rightfully deny defectors help. However, a reasonable precondition to discriinators defecting on defectors is for the to survive, i.e. to attain positive fitness. Therefore, if inequality (14) is satisfied, so is inequality (13) in which case a ixture of cooperating strategies will be established (unconditional cooperators and discriinators). To ake intuitive sense of the upper bound cross ultiply inequality (14) and divide both sides by to get: 1 The ratio 1 bx xg( i) yg( i) c 1 qi(1 gi) qi < i= 1 i= 1 x y i = 1 (15). q (1 g ) is the average per round probability of a discriinator not i i cooperating. This is because 1 is the proportion of bad scorers at round i, which is the gi probability that a randoly selected recipient is a bad scorer, and q i is the probability that a 1 discriinator knows the recipient. The ter 1 qi(1 gi) is then the average per round i = 1 probability of a discriinator cooperating. The ratio x xg ( i) yg ( i) qi( ) is the average per x y round differential probability of a discriinator being helped over a defector. Inequality (15) i= 1

16 16 then asserts that for cooperators to outperfor the defectors, the average per round differential benefit received by a discriinator over a defector ust exceed the average per round cost to a discriinator for cooperating. In other words, trust pays off only if it is placed upon the trustworthy. Two-Strategy Gae: To copare the outcoe of the approach presented here with that of the two-strategy odel of incoplete inforation proposed by Nowak and Sigund (1998b), we first derive a condition analogous to inequality (15) for a gae with two strategies consisting of discriinators and defectors, where g = x ( i) + y ( i). For a gae with two strategies inequality (15) reduces to: i g g 1 b y g( i) c 1 qi(1 gi) qi gi < i= 1 i= 1 y (16). Inequality (16) has a siilar interpretation to inequality (15): For discriinators to outperfor the defectors, the average per round differential benefit received by a discriinator over a defector ust exceed the average per round cost to a discriinator for cooperating. If we rewrite inequality (16) in ters of the cost-to-benefit ratio we derive that for a gae with two strategies we ust have: c b < 1 yg ( i) qi gi i 1 y = 1 1 qi(1 gi ) i= 1 (17). For a two-strategy gae, under siilar assuptions about the population coposition, Nowak and Sigund derived the condition that for discriinators to be evolutionarily stable we ust have: q > c / b, where q is a constant representing the probability of knowing the coplayer s iage score. Nowak and Sigund aptly pointed out the siilarity between their result and Hailton s rule for altruis through kin selection, where the paraeter for genetic relatedness is replaced by q, i.e. the likelihood of knowing the opponent s reputation. However, when we ipose a constantly evolving social structure under selectivity and order in

17 17 inforation transfer, the paraeter of failiarity, whether genetic, cultural, or siply due to observation or interaction, is replaced by a tie-varying ratio that aounts to the average differential likelihood of knowing and helping a trustworthy individual (discriinator) over an untrustworthy individual (defector) out of the proportion of those that a discriinator helps at a point in tie. As we have shown, this dynaic quantity can be derived fro the collective knowledge ebedded in the social network of acquaintances once we reove randoness in observation and introduce a trust-based bias in the selectivity and propagation of inforation. 6. Discussion We proposed a new fraework for the evolution of indirect reciprocity by social inforation and its vital constituents, trust and reputation, under two organizing principles of social networks: order and growth (Barabasi, 2002). We defined collective eory as the inforation content of networks of friends and acquaintances, and argued that at any tie the aount of inforation available to an individual is uch larger than that of one s own eory. Such body of inforation can be retrieved by counicating to friends and acquaintances. This would then iply that in social networks: (1) Inforation is selectively retrieved fro and propagated through networks of acquaintances due to the existing clustering effect of social ties. (2) Dynaics and growth constantly odify the topology of the underlying network thus odifying the content of the collective eory. These observations are a significant departure fro the prevailing view of social networks as rando and static entities, which can be utilized to identify new sources of inforation thereby increasing the average aount of inforation available at any point in tie and the likelihood for infored cooperation. Under these assuptions, we analytically derived the condition under which indirect reciprocity can evolve. Our results suggest that for cooperators to be evolutionarily stable, on average the benefit received by a trusting and reputably cooperative individual ust out-weigh the cost of trusting and helping a disreputable individual. In other words, trust pays off only if it is placed upon the trustworthy. But to ake

18 18 the right decision as to who to trust, inforation on trustworthiness ust be obtained by counicating to others. Hence, iplicit in our fraework is the preise that the evolution of huan language has played a critical role in the evolution and sustenance of indirect reciprocity in huan populations. Are soe populations ore cooperative than others due to different processes of inforation transfer? Different political and econoical structures, aong any other factors, appear to result in different underlying networks of inforation transfer in different societies. Under ore rigid econoic and political conditions, where the processes of inforation transfer are rather stagnant and the cost of trust and cooperation is too high (soeties one s life), people tend to distrust strangers. Lack of trust and cooperation then results in disreputability, which introduces negative feedback into the network driving the syste into alldefection. By analogy, less rigidity in the political and econoic infrastructure should result in people exhibiting ore cooperative behavior due to the relative openness of channels of inforation and lower costs of trusting and cooperating with strangers. However, this question should be put to test ore rigorously through further experiental and theoretical studies (See Henrich et al., 2001, Henrich et al., 2002) 7. Have huans becoe increasingly ore cooperative throughout their social history? If the availability of inforation about others was the single driving force behind evolution of cooperation, and if, on average, inforation has becoe ore available due to higher degree of obility and technological advanceent throughout huan history then one ay also expect to observe higher levels of cooperation over tie. Our goal here, however, was to study the evolution of cooperation through indirect reciprocity by social inforation, a task uch ore odest than providing a sensible answer to the posed question. Nevertheless we find the question worthy of note and leave it open for future studies. REFERENCES

19 19 Alexander, R.D. (1987) The Biology of Moral Systes. New York: Aldine de Gruyter. Axelrod, R. (1984) The Evolution of Cooperation. New York: Basic Books. Axelrod, R. & Hailton, W.D. (1981) The evolution of cooperation. Science 211, Axelrod, R. & Dion, D. (1988) The further evolution of cooperation. Science 242, Barabasi, A.-L. (2002) Linked: The New Science of Networks. Cabridge, MA: Perseus Publishing. Boyd, R. & Richerson, P.J. (1989) Evolution of indirect reciprocity. Social Networks 11, Granovetter, M.S. (1973) The strength of weak ties. A. J. Soc. 78, Granovetter, M.S. (1985) Econoic action and social structure: the proble of ebeddedness. A. J. Soc. 91, Hailton, W.D. (1963) The evolution of altruistic behavior. A. Nat. 97, Hailton, W.D. (1964) The genetical evolution of social behavior. J. Theor. Biol. 7, Henrich, J., Boyd, R., Bowles, S., Caerer, C., Fehr, E., Gintis, H., McElreath, R., Alvard, M., Barr, A., Ensinger, J., Hill, K., Gil-White, F., Gurven, M., Marlowe, F., Patton, J.Q., Sith, N. & Tracer, D. (2002) 'Econoic an' in cross-cultural perspective: behavioral experients in 15 sall-scale societies. Unpublished Manuscript. Henrich, J., Boyd, R., Bowles, S., Gintis, H. & Fehr, E. (2001) In search of hoo econoicus: experients in 15 sall-scale societies. A. Econ. Rev. 91, Nowak, M.A. & Sigund, K. (1998a) Evolution of indirect reciprocity by iage scoring. Nature 393, Nowak, M.A. & Sigund, K. (1998b) The dynaics of indirect reciprocity. J. Theor. Biol. 194, , doi: /jtbi Ostro, E. (1998) A behavioral approach to the rational-choice theory of collective action. A. Pol. Sci. Rev. 92, Pollock, G. & Dugatkin, L.A. (1992) Reciprocity and the eergence of reputation. J. Theor. Biol. 159, Trivers, R. (1971) The evolution of reciprocal altruis. Q. Rev. Biol. 46, Willias, G.C. (1971) Group Selection (Aldine-Atherton, Chicago). Wilson, D.S. & Sober, E. (1994) Reintroducing group selection to the huan behavioral sciences. Behav.Brain Sci. 17,

20 20 Footnotes 1. Trivers, who introduced the notion of reciprocal altruis in his seinal paper in 1971, referred to this type of reciprocation as generalized altruis in order to differentiate it fro direct reciprocity. 2. Throughout this article, we use the ale pronoun for donor and feale pronoun for recipient. 3. Elinor Ostro in her 1998 paper deonstrates the reinforcing relationship between trust, reputation, and reciprocity, and how reciprocity affects the level of cooperation which affects the overall net benefit in a society. 4. Granovetter, in his influential paper The strength of weak ties, for the first tie noted the existence of order in social networks, in which sall clusters of close ties, organized into cliques, are connected by links of acquaintanceship. 5. In gae theoretic odels with non-overlapping generations no one dies before the end of the generation. Hence we assue that links are only added and not reoved. 6. Sociologists estiate that everyone knows about 150 people on a regular basis and 15 people on an intiate basis. 7. The preliinary results of a series of experiental field studies by the authors suggest that the ore arket integrated a society the ore cooperative its people are. It is iportant to note, however, that in these studies cooperation is defined in econoic ters. Figure 1 Dynaics of acquaintanceship. The probability of knowing a randoly selected co-player s iage score increases with nuber of rounds. Initially a

21 21 population of n=100 individuals is divided into non-overlapping cliques of size 4, i.e. 3 friends per agent. As the gae is continued, agents eet and ake new connections. The average nuber of acquaintances per player increases with every round of the gae thereby increasing the probability of knowing the opponent s iage score. Here a generation consists of =10,000 nuber of rounds. Towards the end of the generation life the probability of knowing a randoly selected eber of the population approaches 1, an indication that if a generation lives long enough one will eventually have et everyone in the population. The rise in the probability of failiarity is consistent with the analytic result in Equation (1) in the Methods section. Figure 2 Evolution of indirect reciprocity by trust and reputation under varying population sizes. A population of n individuals is organized into non-overlapping cliques of size 4, representing 3 initial acquaintances per player at the beginning of each generation. Modeling after Nowak and Sigund s siulation fraework (12), the strategy k ranges fro -5 to 6 where k=-5 represents unconditional cooperators, k=6 represents defectors, and k=0 represents the ost discriinating. The iage scores range fro -5 to 5. A potential donor cooperates only if the iage score of the recipient is at least as large as his own strategy. The children inherit the strategies of their parents unless they are subject to utation at a rate of We sapled the frequency distribution of strategies over 10 6 generations for population sizes n=52, n=100, and n=200. Each generation consists of =10n nuber of rounds. Cooperation is evolved and sustained in populations after the gae is played for any generations. Figure 3 Evolution of indirect reciprocity by trust and reputation under varying initial clique sizes. We conducted the sae siulation experient as in Figure 2 except we experiented with different initial clique sizes of 1, 3, and 5, for a population of n=100

22 22 individuals. As the initial clique size increases, cooperation is evolved and sustained ore securely. In a population of all loners, i.e. when the initial clique size is one and no one is born into social groups, defectors win.

23 23 Pr(knowing co-player iage score) Rounds Figure 1

24 n= Frequency n=100 Frequency Frequency n= Strategy, k Figure 2

25 initial clique size=1 0.2 Frequency initial clique size=3 0.2 Frequency initial clique size=5 Frequency Figure Strategy, k