The Semi-Algebraic Theory of Stochastic Games

Abstract

The asymptotic behavior of the min-max value of a finite-state zero-sum discounted stochastic game, as the discount rate approaches 0, has been studied in the past using the theory 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.