Abstract
We study subgame phi -maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, phi denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame phi -maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by phi . First, we provide necessary and sufficient conditions for a strategy to be a subgame phi -maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame phi -maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function phi ^* with the following property: if a player has a subgame phi ^*-maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame phi -maxmin strategy for every positive tolerance function phi is equivalent to the existence of a subgame maxmin strategy.
Highlights
We consider two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function
The equalizing condition requires that for every strategy of the other player, a subgame φ-maxmin strategy almost surely results in a play with an eventually good enough payoff, where eventually good enough means being at least the lower value in very deep subgames up to the allowed tolerance level
We prove that for a positive tolerance function φ, a subgame φ-maxmin strategy exists if every play is either a point of upper semicontinuity of the payoff function or if the sequence of tolerance levels which occur along the play has a positive lower bound
Summary
We consider two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. The central topic of this paper is the concept of a subgame φ-maxmin strategy This is a strategy of the maximizing player that guarantees the lower value at every subgame within the allowed tolerance level. The equalizing condition requires that for every strategy of the other player, a subgame φ-maxmin strategy almost surely results in a play with an eventually good enough payoff, where eventually good enough means being at least the lower value in very deep subgames up to the allowed tolerance level. The existence of subgame maxmin strategies follows by a result of Laraki, Maitra, and Sudderth ([10]) Another is the case when along each play tolerance levels remain bounded away from zero.
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