Equilibrium Payoffs for Pure Strategies in Repeated Games

Equilibrium payoffs corresponding to subgame perfect equilibria in pure strategies are characterized for infinitely repeated games with discounted payoffs. The equilibrium payoff set of a game is a fixed-point of set-valued operators introduced in the paper. The new operator formalism is utilized in showing the folk theorem for repeated games with unequal but constant discount rates. When the players become more patient, the equilibrium payoff set converges to the fixed-point of an asymptotic operator. This limit set corresponds to the subgame perfect equilibria of a continuous-time repeated game. It is shown that the limit set is convex, which implies that pure strategies are sufficient in obtaining all payoffs in the limit. However, this set differs from the set of all feasible and individually rational payoffs, when the discount rates are not equal. The limit sets for constant discount rates can be used in analyzing the outer limit of equilibrium payoffs when the discount factors increase but discount rates are not fixed.