Repeated Games: A State Space Approach

Abstract

Repeated games, such as the Iterated Prisoner's Dilemma (IPD), are often used as idealised models of social interactions. Here, I develop a state space approach to the study of two-player, repeated games. A strategy is represented by way of a state space, where a player's choice of action depends on the current state, and where the actions performed can cause transitions from one state to another. This kind of representation provides a possible link between a game theoretical analysis and concepts from mechanistically oriented ethology, where an individual's state is viewed as made up of motivational variables. Using the concept of a limit ESS for a game in the extensive form, I derive a number of fundamental results for state space strategies. Conditions ensuring purity of a limit ESS are given, as well as a characterisation of the most important class of pure limit ESSs. The theoretical analysis covers games with and without a role asymmetry and also games where players move alternately. To illustrate the state space approach, I apply the theoretical results to three examples. First, the role symmetric IPD, I find a great number of pure limit ESSs, and relate these to the strategies most frequently studied previously. I also discuss whether there can be randomised limit ESSs, concluding that although this is possible, none have been found so far. Second, as a game possessing a role asymmetry, I study a simplified model of social dominance. I concentrate on the question of whether punishment administered by a dominant can determine the allocation of a resource between the dominant and a subdominant. The game turns out to have limit ESSs with this property, but there are also stable strategies where the dominant lacks control. Third, I analyse an alternating Prisoner's Dilemma, which is a natural model to investigate the evolution of reciprocal altruism. No stable strategy for this game has been described previously. Of the limit ESSs I find, one is of particular interest, in that is closely corresponds to the notion of reciprocal altruism as conceived by Trivers.