On the existence of good stationary strategies for nonleavable stochastic games

This paper discusses the problem regarding the existence of optimal or nearly optimal stationary strategies for a player engaged in a nonleavable stochastic game. It is known that, for these games, player I need not have an -optimal stationary strategy even when the state space of the game is finite. On the contrary, we show that uniformly -optimal stationary strategies are available to player II for nonleavable stochastic games with finite state space. Our methods will also yield sufficient conditions for the existence of optimal and -optimal stationary strategies for player II for games with countably infinite state space. With the purpose of introducing and explaining the main results of the paper, special consideration is given to a particular class of nonleavable games whose utility is equal to the indicator of a subset of the state space of the game.