Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels

We study subgame φ-maxmin strategies in two-player zero-sum stochastic games with finite action spaces and a countable state space. Here φ denotes the tolerance function, a function which assigns a non-negative tolerated error level to every subgame. Subgame φ-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 φ. First, we provide necessary and sufficient conditions for a strategy to be a subgame φ-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 φ-maxmin strategy. Finally, we show the possibly surprising result that the existence of subgame φ-maxmin strategies for every positive tolerance function φ is equivalent to the existence of a subgame maxmin strategy