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 Z, 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, or not,
even when |VR| = 0. In fact, a pseudo-polynomial algorithm for BWR-games would already
imply their polynomial solvability. In this paper, we show that BWR-games with a constant
number of random positions can be solved in pseudo-polynomial time. More precisely, in any
BWR-game with |VR| = O(1), a saddle point in uniformly optimal pure stationary strategies
can be found in time polynomial in |VW| + |VB|, the maximum absolute local reward, and the
common denominator of the transition probabilities
