A pseudo-polynomial algorithm for mean payoff stochastic games with perfect information and few random positions

Abstract

We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph G = (V;E), with local rewards r : E Z, and three types of positions: black VB, white VW, and random VR forming a partition of V . It is a long- standing open question whether a polynomial time algorithm for BWR-games exists, or not, even when |VR| = 0. In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this paper, we show that BWR-games with a constant number of random positions can be solved in pseudo-polynomial time. More precisely, in any BWR-game with |VR| = O(1), a saddle point in uniformly optimal pure stationary strategies can be found in time polynomial in |VW| + |VB|, the maximum absolute local reward, and the common denominator of the transition probabilities.