Cooperative Behavior and the Frequency of Social Interaction John Duffy and Jack Ochs Department of Economics University of Pittsburgh Pittsburgh, PA 15260 USA This Draft: April 27, 2005 Abstract: We report results from an experiment that examines play in an indefinitely repeated, 2-player Prisoner’s Dilemma game. Each experimental session involves N subjects and a sequence of indefinitely repeated games. The main treatment consists of whether agents are matched in fixed pairings or matched randomly in each indefinitely repeated game. Within the random matching treatment, we vary the information that players have about their opponents. Contrary to a theoretical possibility suggested by Kandori (1992), a cooperative norm does not emerge in the treatments where players are matched randomly. On the other hand, in the fixed pairings treatment, the evidence suggests that a cooperative norm does emerge as players gain more experience. JEL Codes: C72, C73, C78, C92, D83. Keywords: Cooperation, Repeated Prisoner’s Dilemma, Folk Theorem, Information, Matching.1“Sometimes cooperation emerges where it is least expected.” -Robert Axelrod, The Evolution of Cooperation (1984, p. 73). 1. Introduction When individuals have an incentive to deviate from cooperation, a social norm of cooperation can only be sustained by the threat that a deviation will trigger a sufficiently unfavorable future response. In games where agents are not anonymous and their actions are common knowledge, it is possible to directly punish an agent who is observed to deviate from the social norm. However, when each agent is anonymous and interacts only at randomly determined times with any other particular agent in the population, a deviating agent cannot be directly punished. Nevertheless, a deviator could be indirectly punished if the deviation were to trigger a contagious reaction that destroyed the social norm of cooperation. If the threat of such a reaction were credible, and if the consequences of the eventual destruction of the norm were sufficiently severe, then the threat might sustain a social norm of cooperation. Kandori (1992) shows that there exist conditions that make such a threat credible. He analyzes a situation in which a fixed, finite population of anonymous agents are repeatedly randomly paired to play a prisoners’ dilemma game. He shows that under certain conditions a social norm of cooperation can be supported by a sub-game perfect equilibrium strategy in which anyone who has never experienced a defection, chooses `Cooperate’ while anyone who has either experienced a defection or has defected in the past chooses `Defect’. To sustain cooperation via the threat of triggering a contagious reaction, it is necessary that once the process is triggered, the infection must ‘infect’ a sufficiently large percentage of the population fast enough that no one who has been ‘infected’ would gain an advantage by deviating from the strategy in order to slow the process down, while no one who has not yet been ‘infected’ would get a large enough one time gain to deviate and trigger the contagion process. Consequently, as Kandori shows, in informationally sparse environments the existence of a cooperative norm supported entirely by the threat of contagion will only exist for small populations. We may not see such size populations in natural settings. But in laboratory experiments there are typically small numbers of subjects participating in any one experimental session. Experimenters know that as subjects acquire experience with a game of strategy their choice of2action often changes. Sometimes it is this learning behavior itself that is of direct interest. In other experiments, the experimenter is most interested in how subjects who are experienced with a given game play that game. In either case, it is necessary to give subjects repeated opportunities to play the same game. To avoid repeated game effects it is common practice to randomly match the players in an anonymous setting so that pairs of players do not meet repeatedly. Therefore, as Ellison (1994) notes, Kandori’s result has potentially important implications for the interpretation of behavior exhibited in laboratory experiments.1 Can we be confident that the typical anonymous, random matching protocol used by experimenters actually controls for the possibility that subjects treat the entire session as one large supergame? We cannot be confident that such schemes do what they are supposed to do without direct empirical evidence comparing the repeated play of a game under different matching protocols. It is in response to this challenge that the experiment discussed below is designed. 2. Related Work 2.1 Indefinitely Repeated Stage Games Kandori’s (1992) theorem applies to indefinitely repeated games of local interaction with minimal observation of the past actions of individuals with whom one is currently playing a stage game. Our experiment is designed, therefore, to study the behavior of individuals drawn from a fixed population who play an indefinite sequence of a two person –Prisoner’s Dilemma under different matching protocols and different amounts of information transmission. The objective of varying the matching protocol (fixed pairings versus random matching) is to determine empirically how much difference in the level of cooperative play is associated with these different matching protocols. The objective of varying the information transmitted to players (under the random matching protocol only) is to determine whether information on the 1 As Ellison (1994, pp. 568-69) observes, “…in experimental economics it is a well-recognized concern that subjects who are asked to play a game several times may treat the situation as a repeated game. To avoid repeated game effects it is common practice to randomly match the players in an anonymous setting so that pairs of players do not meet repeatedly. The results here suggest, however, that given moderate population sizes random matching may not solve the problem.3payoff or action history of an opponent prior to play of the stage game has any affect on the level of cooperative play. By definition, indefinitely repeated stage games have no (predictable) last stage. Therefore, if cooperative play is reciprocated in the early stages, a belief in future reciprocity will be reinforced and cooperative behavior may be