We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph , with local rewards , and three types of positions: black , white , and random forming a partition of V. It is a long-standing open question whether a polynomial time algorithm for BWR-games exists, even when . 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.