Two‐Person Zero‐Sum Stochastic Games with Semicontinuous Payoff

Consider a two‐person, zero‐sum stochastic game with Borel state space S, finite action sets A,B, and Borel measurable law of motion q. Suppose the payoff is a bounded function f of the infinite history of states and actions that is measurable for the product of the Borel σ‐field for S and the σ‐fields of all subsets for A and B, and is lower semicontinuous for the product of the discrete topologies on the coordinate spaces. Then the game has a value and player II has a subgame perfect optimal strategy.