REPEATED GAMES - A STATE-SPACE APPROACH

Repeated games, such as the Iterated Prisoner's Dilemma (IPD), are oft en used as idealised models of social interactions. Here, I develop a state space approach to the study of two-player, repeated games. A str ategy is represented by way of a state space, where a player's choice of action depends on the current state, and where the actions performe d can cause transitions from one state to another. This kind of repres entation provides a possible link between a game theoretical analysis and concepts from mechanistically oriented ethology, where an individu al's state is viewed as made up of motivational variables. Using the c oncept of a limit ESS for a game in the extensive form, I derive a num ber of fundamental results for state space strategies. Conditions ensu ring 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 p layers move alternately. To illustrate the state space approach, I app ly the theoretical results to three examples. First, for the role symm etric IPD, I find a great number of pure limit ESSs, and relate these to the strategies most frequently studied previously. I also discuss w hether there can be randomised limit ESSs, concluding that although th is is possible, none have been found so far. Second, as a game possess ing 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 domina nt and a subdominant. The game turns out to have limit ESSs with this property, but there are also stable strategies where the dominant lack s control. Third, I analyse an alternating Prisoner's Dilemma, which i s a natural model to investigate the evolution of reciprocal altruism. No stable strategy for this game has been described previously. Of th e limit ESSs I find, one is of particular interest, in that it closely corresponds to the notion of reciprocal altruism as conceived by Triv ers. (C) 1997 Academic Press Limited.