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A usual criticism of the theory of infinitely repeated games is that it does not provide sharp predictions since there may be a multiplicity of equilibria. To address this issue, we present experimental evidence on the evolution of cooperation in infinitely repeated prisoner's dilemma games as subjects gain experience. We show that cooperation may prevail in infinitely repeated games, but the conditions under which this occurs are more stringent than the subgame perfect conditions usually considered or even a condition based on risk dominance. (JEL C71, C73)

We study the experimental play of the repeated prisoner's dilemma when intended actions are implemented with noise. In treatments where cooperation is an equilibrium, subjects cooperate substantially more than in treatments without cooperative equilibria. In all settings there was considerable strategic diversity, indicating that subjects had not fully learned the distribution of play. Furthermore, cooperative strategies yielded higher payoffs than uncooperative strategies in the treatments with cooperative equilibria. In these treatments successful strategies were \"lenient\" in not retaliating for the first defection, and many were \"forgiving\" in trying to return to cooperation after inflicting a punishment.

It is generally believed that in a situation where individual and collective interests are in conflict, the availability of optional participation is a key mechanism to maintain cooperation. Surprisingly, this effect is sensitive to the use of microscopic dynamics and can easily be broken when agents make a fully rational decision during their strategy updates. In the framework of the celebrated prisoner's dilemma game, we show that this discrepancy can be fixed automatically if we leave the strict and frequently artifact condition of a fully occupied interaction graph, and allow agents to change not just their strategies but also their positions according to their success. In this way, a diluted graph where agents may move offers a natural and alternative way to handle artifacts arising from the application of specific and sometimes awkward microscopic rules.

Reciprocity and repeated games have been at the center of attention when studying the evolution of human cooperation. Direct reciprocity is considered to be a powerful mechanism for the evolution of cooperation, and it is generally assumed that it can lead to high levels of cooperation. Here we explore an openended, infinite strategy space, where every strategy that can be encoded by a finite state automaton is a possible mutant. Surprisingly, we find that direct reciprocity alone does not lead to high levels of cooperation. Instead we observe perpetual oscillations between cooperation and defection, with defection being substantially more frequent than cooperation. The reason for this is that \"indirect invasions\" remove equilibrium strategies: every strategy has neutral mutants, which in turn can be invaded by other strategies. However, reciprocity is not the only way to promote cooperation. Another mechanism for the evolution of cooperation, which has received as much attention, is assortment because of population structure. Here we develop a theory that allows us to study the synergistic interaction between direct reciprocity and assortment. This framework is particularly well suited for understanding human interactions, which are typically repeated and occur in relatively fluid but not unstructured populations. We show that if repeated games are combined with only a small amount of assortment, then natural selection favors the behavior typically observed among humans: high levels of cooperation implemented using conditional strategies.

We study prisoners' dilemmas played in continuous time with flow payoffs accumulated over 60 seconds. In most cases, the median rate of mutual cooperation is about 90 percent. Control sessions with repeated matchings over eight subperiods achieve less than half as much cooperation, and cooperation rates approach zero in one-shot sessions. In follow-up sessions with a variable number of subperiods, cooperation rates increase nearly linearly as the grid size decreases, and, with one-second subperiods, they approach continuous levels. Our data support a strand of theory that explains how capacity to respond rapidly stabilizes cooperation and destabilizes defection in the prisoner's dilemma.

In the repeated prisoner's dilemma, predictions are notoriously difficult. Recently, however, Blonski, Ockenfels, and Spagnolo (2011)—henceforth, BOS—showed that experimental subjects predictably cooperate when the discount factor exceeds a particular threshold. I analyze individual strategies in four recent experiments to examine whether strategies are predictable, too. Behavior is well summarized by \"Semi-Grim\" strategies: cooperate after mutual cooperation, defect after mutual defection, randomize otherwise. This holds both in aggregate and individually, and it explains the BOS-threshold: Semi-Grim equilibria appear as the discount factor crosses this threshold, and then, subjects start cooperating in round 1 and switch to Semi-Grim in continuation play.

Social interaction is a fundamental part of our daily lives; however, exactly how our brains use social cues to determine whether to cooperate without being exploited remains unclear. In this study, we used an electroencephalography (EEG) hyperscanning approach to investigate the effect of face-to-face contact on the brain mechanisms underlying the decision to cooperate or defect in an iterated version of the Prisoner's Dilemma Game. Participants played the game either in face-to-face or face-blocked conditions. The face-to-face interaction led players to cooperate more often, providing behavioral evidence for the use of these nonverbal cues in their social decision-making. In addition, the EEG hyperscanning identified temporal dynamics and inter-brain synchronization across the cortex, providing evidence for involvement of these regions in the processing of face-to-face cues to read each other's intent to cooperate. Most notably, the power of the alpha frequency band (8–13Hz) in the right temporoparietal region immediately after seeing a round outcome significantly differed between face-to-face and face-blocked conditions and predicted whether an individual would adopt a ‘cooperation’ or ‘defection’ strategy. Moreover, inter-brain synchronies within this time and frequency range reflected the use of these strategies. This study provides evidence for how the cortex uses nonverbal social cues to determine other's intentions, and highlights the significance of power in the alpha band and inter-brain phase synchronizations in high-level socio-cognitive processing.

The production of public goods by the contribution of individual volunteers is a social dilemma because an individual that does not volunteer can benefit from the public good produced by the contributions of others. Therefore it is generally believed that public goods can be produced only in the presence of repeated interactions (which allow reciprocation, reputation effects and punishment) or relatedness (kin selection). Cooperation, however, often occurs in the absence of iterations and relatedness. We show that when the production of a public good is a Volunteer's Dilemma, in which a fixed number of cooperators is necessary to produce the public good, cooperators and defectors persist in a mixed equilibrium, without iterations and without relatedness. This mixed equilibrium is absent in the N-person Prisoner's Dilemma, in which the public good is a linear function of the individual contributions. We also show that the Prisoner's Dilemma and the Volunteer's Dilemma are the two opposite extremes of a general public goods game, and that all intermediate cases can have a mixed equilibrium like the Volunteer's Dilemma. The coexistence of cooperators and defectors, therefore, is a typical outcome of most social dilemmas, which requires neither relatedness nor iterations.

We use a novel experimental design to reliably elicit subjects’ strategies in an infinitely repeated prisoner’s dilemma experiment with perfect monitoring. We find that three simple strategies represent the majority of the chosen strategies: Always Defect, Tit-for-Tat, and Grim. In addition, we identify how the strategies systematically vary with the parameters of the game. Finally, we use the elicited strategies to test the ability to recover strategies using statistical methods based on observed round-by-round cooperation choices and find that this can be done fairly well, but only under certain conditions.

Dilemmas in cooperation are one of the major concerns in game theory. In a public goods game, each individual cooperates by paying a cost or defecting without paying it, and receives a reward from the group out of the collected cost. Thus, defecting is beneficial for each individual, while cooperation is beneficial for the group. Now, groups (say, countries) consisting of individuals also play games. To study such a multi-level game, we introduce a hierarchical game in which multiple groups compete for limited resources by utilizing the collected cost in each group, where the power to appropriate resources increases with the population of the group. Analyzing this hierarchical game, we found a hierarchical prisoner's dilemma, in which groups choose the defecting policy (say, armament) as a Nash strategy to optimize each group's benefit, while cooperation optimizes the total benefit. On the other hand, for each individual, refusing to pay the cost (say, tax) is a Nash strategy, which turns out to be a cooperation policy for the group, thus leading to a hierarchical dilemma. Here the group reward increases with the group size. However, we find that there exists an optimal group size that maximizes the individual payoff. Furthermore, when the population asymmetry between two groups is large, the smaller group will choose a cooperation policy (say, disarmament) to avoid excessive response from the larger group, and the prisoner's dilemma between the groups is resolved. Accordingly, the relevance of this hierarchical game on policy selection in society and the optimal size of human or animal groups are discussed.