A pseudo-polynomial algorithm for mean payoff stochastic games with perfect information and a 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 → ℝ, and three types of vertices: black VB , white V W , and random V R forming a partition of V. It is a long-standing open question whether a polynomial time algorithm for BWR-games exists, or not. In fact, a pseudo-polynomial algorithm for these games would already imply their polynomial solvability. In this paper, we show that BWR-games with a constant number of random nodes can be solved in pseudo-polynomial time. That is, for any such game with a few random nodes |VR| = O(1), a saddle point in pure stationary strategies can be found in time polynomial in |VW| + |VB|, the maximum absolute local reward R, and the common denominator of the transition probabilities. © 2013 Springer-Verlag.