Clustering in large networks does not promote upstream reciprocity

Upstream reciprocity (also called generalized reciprocity) is a putative mechanism for cooperation in social dilemma situations with which players help others when they are helped by somebody else. It is a type of indirect reciprocity. Although upstream reciprocity is often observed in experiments, most theories suggest that it is operative only when players form short cycles such as triangles, implying a small population size, or when it is combined with other mechanisms that promote cooperation on their own. An expectation is that real social networks, which are known to be full of triangles and other short cycles, may accommodate upstream reciprocity. In this study, I extend the upstream reciprocity game proposed for a directed cycle by Boyd and Richerson to the case of general networks. The model is not evolutionary and concerns the conditions under which the unanimity of cooperative players is a Nash equilibrium. I show that an abundance of triangles or other short cycles in a network does little to promote upstream reciprocity. Cooperation is less likely for a larger population size even if triangles are abundant in the network. In addition, in contrast to the results for evolutionary social dilemma games on networks, scale-free networks lead to less cooperation than networks with a homogeneous degree distribution.     

Indirect reciprocity under incomplete observation

Indirect reciprocity, in which individuals help others with a good reputation but not those with a bad reputation, is a mechanism for cooperation in social dilemma situations when individuals do not repeatedly interact with the same partners. In a relatively large society where indirect reciprocity is relevant, individuals may not know each other's reputation even indirectly. Previous studies investigated the situations where individuals playing the game have to determine the action possibly without knowing others' reputations. Nevertheless, the possibility that observers of the game, who generate the reputation of the interacting players, assign reputations without complete information about them has been neglected. Because an individual acts as an interacting player and as an observer on different occasions if indirect reciprocity is endogenously sustained in a society, the incompleteness of information may affect either role. We examine the game of indirect reciprocity when the reputations of players are not necessarily known to observers and to interacting players. We find that the trustful discriminator, which cooperates with good and unknown players and defects against bad players, realizes cooperative societies under seven social norms. Among the seven social norms, three of the four suspicious norms under which cooperation (defection) to unknown players leads to a good (bad) reputation enable cooperation down to a relatively small observation probability. In contrast, the three trustful norms under which both cooperation and defection to unknown players lead to a good reputation are relatively efficient.     

Tag-based indirect reciprocity by incomplete social information

Evolution of altruistic behaviour in interacting individuals is accounted for by, for example, kin selection, direct reciprocity, spatially limited interaction and indirect reciprocity. Real social agents, particularly humans, often take actions based on similarity between themselves and others. Although tag-based indirect reciprocity in which altruism occurs exclusively among similar flocks is a natural expectation, its mechanism has not really been established. We propose a model of tag-based indirect reciprocity by assuming that each player may note strategies of others. We show that tag-based altruism can evolve to eradicate other strategies, including unconditional defectors for various initial strategy configurations and parameter sets. A prerequisite for altruism is that the strategy is sometimes, but not always, visible to others. Without visibility of strategies, policing does not take place and defection is optimal. With perfect visibility, what a player does is always witnessed by others and cooperation is optimal. In the intermediate regime, discriminators based on tag proximity, rather than mixture of generous players and defectors, are most likely to evolve. In this situation, altruism is realized based on homophily in which players are exclusively good to similar others.     

Evolution of indirect reciprocity by social information: the role of trust and reputation in evolution of altruism

The complexity of human's cooperative behavior cannot be fully explained by theories of kin selection and group selection. If reciprocal altruism is to provide an explanation for altruistic behavior, it would have to depart from direct reciprocity, which requires dyads of individuals to interact repeatedly. For indirect reciprocity to rationalize cooperation among genetically unrelated or even culturally dissimilar individuals, information about the reputation of individuals must be assessed and propagated in a population. Here, we propose a new framework for the evolution of indirect reciprocity by social information: information selectively retrieved from and propagated through dynamically 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 time, must exceed the cost-to-benefit ratio of the altruistic act. In other words, the benefit received by the trustworthy must out-weigh the cost of helping the untrustworthy.     

Reciprocity on the Hardwood: Passing Patterns among Professional Basketball Players

Past theory and research view reciprocal resource sharing as a fundamental building block of human societies. Most studies of reciprocity dynamics have focused on trading among individuals in laboratory settings. But if motivations to engage in these patterns of resource sharing are powerful, then we should observe forms of reciprocity even in highly structured group environments in which reciprocity does not clearly serve individual or group interests. To this end, we investigated whether patterns of reciprocity might emerge among teammates in professional basketball games. Using data from logs of National Basketball Association (NBA) games of the 2008–9 season, we estimated a series of conditional logistic regression models to test the impact of different factors on the probability that a given player would assist another player in scoring a basket. Our analysis found evidence for a direct reciprocity effect in which players who had “received” assists in the past tended to subsequently reciprocate their benefactors. Further, this tendency was time-dependent, with the probability of repayment highest soon after receiving an assist and declining as game time passed. We found no evidence for generalized reciprocity – a tendency to “pay forward” assists – and only very limited evidence for indirect reciprocity – a tendency to reward players who had sent others many assists. These findings highlight the power of reciprocity to shape human behavior, even in a setting characterized by extensive planning, division of labor, quick decision-making, and a focus on inter-group competition.     

Evolution of Cooperation in Spatial Traveler's Dilemma Game

Traveler''s dilemma (TD) is one of social dilemmas which has been well studied in the economics community, but it is attracted little attention in the physics community. The TD game is a two-person game. Each player can select an integer value between and () as a pure strategy. If both of them select the same value, the payoff to them will be that value. If the players select different values, say and (), then the payoff to the player who chooses the small value will be and the payoff to the other player will be . We term the player who selects a large value as the cooperator, and the one who chooses a small value as the defector. The reason is that if both of them select large values, it will result in a large total payoff. The Nash equilibrium of the TD game is to choose the smallest value . However, in previous behavioral studies, players in TD game typically select values that are much larger than , and the average selected value exhibits an inverse relationship with . To explain such anomalous behavior, in this paper, we study the evolution of cooperation in spatial traveler''s dilemma game where the players are located on a square lattice and each player plays TD games with his neighbors. Players in our model can adopt their neighbors'' strategies following two standard models of spatial game dynamics. Monte-Carlo simulation is applied to our model, and the results show that the cooperation level of the system, which is proportional to the average value of the strategies, decreases with increasing until is greater than the critical value where cooperation vanishes. Our findings indicate that spatial reciprocity promotes the evolution of cooperation in TD game and the spatial TD game model can interpret the anomalous behavior observed in previous behavioral experiments.     

Effect of the presence of empty sites on the evolution of cooperation by costly punishment in spatial games

Cooperation and spiteful behavior are still evolutionary puzzles. Costly punishment, for which the game payoff is the same as that of spiteful behavior, is one mechanism for promoting the evolution of cooperation. A spatially structured population facilitates the evolution of either cooperation or spite/punishment if cooperation is linked explicitly or implicitly with spite/punishment; a cooperator cooperates with another cooperator and punishes/spites the other type of player. Different updating rules in the evolutionary game produce different evolutionary outcomes: with one updating rule—the score-dependent viability model, in which a player dies with a probability inversely proportional to the game score and the resulting unoccupied site is colonized by one player chosen randomly—the evolution of spite/punishment is promoted more than with the other updating rule—the score-dependent fertility model, in which, after a player dies randomly, the site is colonized by a player with a higher game score. If the population has empty sites, spiteful players or punishers should have less chance to interact with others and then spite/punish others. Thus the presence of empty sites would affect the evolutionary dynamics of spite/punishment. Here, we investigated whether the presence of empty sites discourages the evolution of spite/punishment in both a lattice-structured population and a completely mixing population where players interact with others randomly, especially when the score-dependent viability model is adopted. In the lattice-structured population adopting this viability model, the presence of empty sites promoted the evolution of cooperation and did not reduce the effect of spite/punishment. In the completely mixing population, the presence of empty sites did not promote evolution of cooperation by punishment. The evolutionary dynamics of the score-dependent viability model with empty sites were close to those of the score-dependent fertility model.     

How inconsistency between attitude and behavior persists through cultural transmission

Individuals tend to conform their behavior to that of the majority. Consequently, an individual's behavior is not always consistent with his or her attitude, and such inconsistency sometimes causes mental distress. Understanding the mechanism of sustaining inconsistency between attitude and behavior is a challenging problem from the viewpoint of evolutionary theory. We constructed an evolutionary game theory model in which each player has an attitude and behavior toward a single social norm, and the players' attitudes and behaviors are affected by three types of cultural transmission: vertical, oblique, and horizontal. We assumed that strategy is a combination of attitude and behavior and that the process of learning or transmitting the social norm depends on the life stage of each player. Adults play a coordination game in which players whose behaviors match those of the majority obtain a high payoff, which is diminished by any inconsistency between attitude and behavior. The adults' strategies are passed to newborns via vertical transmission, and the frequency of a newborn's replication of strategy is proportional to the corresponding adult's payoff. Newborns imitate behaviors of unrelated adults via oblique transmission. Juveniles change their attitudes or behaviors by observing other juveniles' behaviors or inferring other juveniles' attitudes (horizontal transmission). We conclude that the key factor for sustaining inconsistency between attitude and behavior is the ability of players to infer and imitate others' attitudes, and that oblique transmission promotes inconsistency.     

Is it better to give information, receive it, or be ignorant in a two-player game?

The standard approach in a biological two-player game is toassume both players choose their actions independently of oneanother, having no information about their opponent's action(simultaneous game). However, this approach is not realisticin some circumstances. In many cases, one player chooses hisaction first and then the second player chooses her action withinformation about his action (Stackelberg game). We comparethese two games, which can be mathematically analyzed into twotypes, depending on the direction of the best response function(BRF) at the evolutionarily stable strategy in the simultaneousgame (ESS_sim). We subcategorize each type of game into two cases,depending on the change in payoff to one player, when both playersare at the ESS_sim, and the other player increases his action.Our results show that in cases where the BRF is decreasing atthe ESS_sim, the first player in the Stackelberg game receivesthe highest payoff, followed by both players in the simultaneousgame, followed by the second player in the Stackelberg game.In these cases, it is best to be the first Stackelberg player.In cases where the BRF is increasing at the ESS_sim, both Stackelbergplayers receive a higher payoff than players in a simultaneousgame. In these cases, it is better for both players to playa Stackelberg game rather than a simultaneous game. However,in some cases the first Stackelberg player receives a higherpayoff than the second Stackelberg player, and in some casesthe opposite is true.     

Heterogeneous Coupling between Interdependent Lattices Promotes the Cooperation in the Prisoner’s Dilemma Game

In the research realm of game theory, interdependent networks have extended the content of spatial reciprocity, which needs the suitable coupling between networks. However, thus far, the vast majority of existing works just assume that the coupling strength between networks is symmetric. This hypothesis, to some extent, seems inconsistent with the ubiquitous observation of heterogeneity. Here, we study how the heterogeneous coupling strength, which characterizes the interdependency of utility between corresponding players of both networks, affects the evolution of cooperation in the prisoner’s dilemma game with two types of coupling schemes (symmetric and asymmetric ones). Compared with the traditional case, we show that heterogeneous coupling greatly promotes the collective cooperation. The symmetric scheme seems much better than the asymmetric case. Moreover, the role of varying amplitude of coupling strength is also studied on these two interdependent ways. Current findings are helpful for us to understand the evolution of cooperation within many real-world systems, in particular for the interconnected and interrelated systems.     

The long-term benefits of human generosity in indirect reciprocity

Among the theories that have been proposed to explain the evolution of altruism are direct reciprocity and indirect reciprocity. The idea of the latter is that helping someone or refusing to do so has an impact on one's reputation within a group. This reputation is constantly assessed and reassessed by others and is taken into account by them in future social interactions. Generosity in indirect reciprocity can evolve if and only if it eventually leads to a net benefit in the long term. Here, we show that this key assumption is met. We let 114 students play for money in an indirect and a subsequent direct reciprocity game. We found that although being generous, i.e., giving something of value to others, had the obvious short-term costs, it paid in the long run because it builds up a reputation that is rewarded by third parties (who thereby themselves increase their reputation). A reputation of being generous also provided an advantage in the subsequent direct reciprocity game, probably because it builds up trust that can lead to more stable cooperation.     

Joint Effects of Asymmetric Payoff and Reciprocity Mechanisms on Collective Cooperation in Water Sharing Interactions: A Game Theoretic Perspective

Common-pool resource (CPR) dilemmas distinguish themselves from general public good problems by encompassing both social and physical features. This paper examines how a physical mechanism, namely asymmetric payoff; and a social mechanism, reciprocity; simultaneously affect collective cooperation in theoretical water sharing interactions. We present an iterative N-person game theoretic model to investigate the joint effects of these two mechanisms in a linear fully connected river system under three information assumptions. From a simple evolutionary perspective, this paper quantitatively addresses the conditions for Nash Equilibrium in which collective cooperation might be established. The results suggest that direct reciprocity increases every actor’s motivation to contribute to the collective good of the river system. Meanwhile, various upstream and downstream actors manifest individual disparities as a result of the direct reciprocity and asymmetric payoff mechanisms. More specifically, the downstream actors are less willing to cooperate unless there is a high probability that long-term interactions are ensured; however, a greater level of asymmetries is likely to increase upstream actors’ incentives to cooperate even though the interactions could quickly end. The upstream actors also display weak sensitivity to an increase in the total number of actors, which generally results in a reduction in the other actors’ motivation for cooperation. It is also shown that the indirect reciprocity mechanism relaxes the overall conditions for cooperative Nash Equilibrium.     

Generalized reciprocity in rats

The evolution of cooperation among nonrelatives has been explained by direct, indirect, and strong reciprocity. Animals should base the decision to help others on expected future help, which they may judge from past behavior of their partner. Although many examples of cooperative behavior exist in nature where reciprocity may be involved, experimental evidence for strategies predicted by direct reciprocity models remains controversial; and indirect and strong reciprocity have been found only in humans so far. Here we show experimentally that cooperative behavior of female rats is influenced by prior receipt of help, irrespective of the identity of the partner. Rats that were trained in an instrumental cooperative task (pulling a stick in order to produce food for a partner) pulled more often for an unknown partner after they were helped than if they had not received help before. This alternative mechanism, called generalized reciprocity, requires no specific knowledge about the partner and may promote the evolution of cooperation among unfamiliar nonrelatives.     

Upstream reciprocity and the evolution of gratitude

If someone is nice to you, you feel good and may be inclined to be nice to somebody else. This every day experience is borne out by experimental games: the recipients of an act of kindness are more likely to help in turn, even if the person who benefits from their generosity is somebody else. This behaviour, which has been called ‘upstream reciprocity’, appears to be a misdirected act of gratitude: you help somebody because somebody else has helped you. Does this make any sense from an evolutionary or a game theoretic perspective? In this paper, we show that upstream reciprocity alone does not lead to the evolution of cooperation, but it can evolve and increase the level of cooperation if it is linked to either direct or spatial reciprocity. We calculate the random walks of altruistic acts that are induced by upstream reciprocity. Our analysis shows that gratitude and other positive emotions, which increase the willingness to help others, can evolve in the competitive world of natural selection.     

The influence of cultural framing on play in the trust game: a Maasai example

The effects of cultural framing on behavior in experimental games were explored with a trust game and the Maasai concept of osotua. Maasai use the term osotua to refer to gift-giving relationships based on obligation, need, respect, and restraint. In the trust game, the first player is given money and an opportunity to give any portion of it to the second player. The amount given is then multiplied by the experimenter, and the second player has an opportunity to give any amount back to the first player. Fifty trust games were played by Maasai men at a field site in north central Kenya. Half of the games were played without deliberate framing, and half were framed with the statement, “This is an osotua game.” Compared to games with no deliberate framing, those played within the osotua rhetorical frame were associated with lower transfers by both players and with lower expected returns on the part of the first players. Osotua rhetorical framing is also associated with a negative correlation between amounts given by the first player and amounts returned by the second. These results have implications both for the experimental game method and for our understanding of the relationship between culture and behavior.     

The dependency game: Multiperson reciprocal sharing leads to stable cooperation which can evolve into group formation

In the standard model for reciprocal collaboration, the repeated prisoner's dilemma (PD), it has proved difficult to establish collaboration in larger groups, necessitating the introduction of additional mechanisms such as reputation or assortedness. The problem is corroborated because current multiperson PDs model simultaneous player action, known as a common goods situation, whereas multiperson collaboration could be easier to obtain in a PD with alternate player action, a private goods situation. Here we present such a game, called a dependency game, and show that stable collaboration can be obtained in a 255 player simulation if only players are allowed to remember three previous benefactors, so they can play advanced tit-for-tat. Furthermore, we show that such a freely collaborating population is threatened by assorted strategies, which define groups that parasitize on independent tit-for-tat players. By excluding others, these groups engage in indirect reciprocal behaviour. Our model therefore combines many hitherto separate collaboration-enhancing concepts into one game, and suggests that group formation and collaboration are two separate social phenomena.     

A study of indirect reciprocity involving a reputation system or a simple tag system in a one-shot, multi-player game

The possibility of the evolution of cooperation backed by indirect reciprocity (IDR) in a one-shot, multi-player game is investigated focusing on two mechanisms. First of all, the reputation system with image score (RS with IS), as proposed by Nowak and Sigmund [Nowak, M.A., Sigmund, K., 1998. Evolution of indirect reciprocity by image scoring. Nature 393, 573–577], is investigated in various game structures. A numerical experiment demonstrates that the RS with IS is a robust mechanism for the support of IDR in various dilemma games, but whose effectiveness decreases with an increase in the number of players in a game. It is fair to say that the RS is an information mapping function to relate between player's cooperative fraction on his action and IS. As the second mechanism, a simple tag system which could be applicable to animals having no cognitive complexity is considered to support IDR. Computer simulations of the tag system's strategy for invading a population initially consisting of AllD, AllC, and Random strategies suggest several novel facts. The Tag strategy with plausible crossover and mutation probabilities can only invade to settle down if the game structure is not Trivial and contains a moderate dilemma. The Tag strategy's evolutionary competition takes place mainly in the presence of the AllD strategy. During the competition, the Tag strategy frequently metamorphoses to shake off the AllD strategy, but stops after winning in order to avoid a shrinking payoff due to fragmentation.     

Participation costs can suppress the evolution of upstream reciprocity

Indirect reciprocity, one of the many mechanisms proposed to explain the evolution of cooperation, is the idea that altruistic actions can be rewarded by third parties. Upstream or generalized reciprocity is one type of indirect reciprocity in which individuals help someone if they have been helped by somebody else in the past. Although empirically found to be at work in humans, the evolution of upstream reciprocity is difficult to explain from a theoretical point of view. A recent model of upstream reciprocity, first proposed by Nowak and Roch (2007) and further analyzed by Iwagami and Masuda (2010), shows that while upstream reciprocity alone does not lead to the evolution of cooperation, it can act in tandem with mechanisms such as network reciprocity and increase the total level of cooperativity in the population. We argue, however, that Nowak and Roch's model systematically leads to non-uniform interaction rates, where more cooperative individuals take part in more games than less cooperative ones. As a result, the critical benefit-to-cost ratios derived under this model in previous studies are not invariant with respect to the addition of participation costs. We show that accounting for these costs can hinder and even suppress the evolution of upstream reciprocity, both for populations with non-random encounters and graph-structured populations.     

Participation costs dismiss the advantage of heterogeneous networks in evolution of cooperation

Real social interactions occur on networks in which each individual is connected to some, but not all, of others. In social dilemma games with a fixed population size, heterogeneity in the number of contacts per player is known to promote evolution of cooperation. Under a common assumption of positively biased pay-off structure, well-connected players earn much by playing frequently, and cooperation once adopted by well-connected players is unbeatable and spreads to others. However, maintaining a social contact can be costly, which would prevent local pay-offs from being positively biased. In replicator-type evolutionary dynamics, it is shown that even a relatively small participation cost extinguishes the merit of heterogeneous networks in terms of cooperation. In this situation, more connected players earn less so that they are no longer spreaders of cooperation. Instead, those with fewer contacts win and guide the evolution. The participation cost, or the baseline pay-off, is irrelevant in homogeneous populations, but is essential for evolutionary games on heterogeneous networks.     

Adaptive Dynamics of Extortion and Compliance

Direct reciprocity is a mechanism for the evolution of cooperation. For the iterated prisoner’s dilemma, a new class of strategies has recently been described, the so-called zero-determinant strategies. Using such a strategy, a player can unilaterally enforce a linear relationship between his own payoff and the co-player’s payoff. In particular the player may act in such a way that it becomes optimal for the co-player to cooperate unconditionally. In this way, a player can manipulate and extort his co-player, thereby ensuring that the own payoff never falls below the co-player’s payoff. However, using a compliant strategy instead, a player can also ensure that his own payoff never exceeds the co-player’s payoff. Here, we use adaptive dynamics to study when evolution leads to extortion and when it leads to compliance. We find a remarkable cyclic dynamics: in sufficiently large populations, extortioners play a transient role, helping the population to move from selfish strategies to compliance. Compliant strategies, however, can be subverted by altruists, which in turn give rise to selfish strategies. Whether cooperative strategies are favored in the long run critically depends on the size of the population; we show that cooperation is most abundant in large populations, in which case average payoffs approach the social optimum. Our results are not restricted to the case of the prisoners dilemma, but can be extended to other social dilemmas, such as the snowdrift game. Iterated social dilemmas in large populations do not lead to the evolution of strategies that aim to dominate their co-player. Instead, generosity succeeds.