A Convex Programming-based Algorithm for Mean Payoff Stochastic Games with Perfect Information

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 → ZZ, 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, even when |VR|=0. In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this short note, we show that BWR-games can be solved via convex programming in pseudo-polynomial time if the number of random positions is a constant.