  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
$r: E \to \ZZ$, 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 $|V_R|=0$.
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.
