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 $r:E\to \text{ZZ}$, and three types of positions: black $V_B$, 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, even when $\mathrm{\mid }{V}_R\mathrm{\mid }=0$. 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.