The semi-algebraic theory of stochastic games

The asymptotic behavior of the min–max value of a finite-state zero-sum discounted stochastic game, as the discounted rate approaches 0, has been studied in the past using the history of real-closed fields. We use the theory of semi-algebraic sets and mappings to prove some asymptotic properties of the min–max value, which hold uniformly for all stochastic games in which the number of states and players' actions are predetermined to some fixed values. As a corollary, we prove a uniform polynomial convergence rate of the value of the N-stage game to the value of the nondiscount game, over a bounded set of payoffs.