A Pseudo-Polynomial Algorithm for Mean Payoff Stochastic Games with Perfect Information and a Few Random Positions

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 longstanding 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.