Stochastic Games with General Payoff Functions

We consider multiplayer stochastic games with finitely many players and actions, and countably many states, in which the payoff of each player is a bounded and Borel-measurable function of the infinite play. By using a generalization of the technique of Martin [Martin DA (1998) The determinacy of Blackwell games. J. Symb. Log. 63(4):1565–1581] and Maitra and Sudderth [Maitra A, Sudderth W (1998) Finitely additive stochastic games with Borel measurable payoffs. Internat. J. Game Theory 27:257–267], we show four different existence results. In each stochastic game, it holds for every ε > 0 that (i) each player has a strategy that guarantees in each subgame that this player’s payoff is at least his or her maxmin value up to ε, (ii) there exists a strategy profile under which in each subgame each player’s payoff is at least his or her minmax value up to ε, (iii) the game admits an extensive-form correlated ε-equilibrium, and (iv) there exists a subgame that admits an ε-equilibrium.