On Markov Games with Average Reward Criterion and Weakly Continuous Transition Probabilities

Abstract

In this paper we consider two‐person zero‐sum stochastic games with the average reward criterion and Borel state and action spaces. A geometric drift condition is assumed. We show that the optimality (Shapley) equation has a unique solution if the transition probability function is weakly continuous, the stage reward is lower semicontinuous, and the set‐valued mappings of admissible actions satisfy some semicontinuity assumptions. Furthermore, the minimizing player has an optimal stationary strategy and the maximizing player has an ε‐optimal stationary strategy for every $\varepsilon > 0 $.