Approximation Schemes for Stochastic Mean Payoff Games with Perfect Information and a Few Random Positions

We consider two-player zero-sum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWR-games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games, stochastic parity games, and Markov decision processes. They can also be used to model parlor games such as Chess or Backgammon. It is a long-standing open question if a polynomial-time algorithm exists that solves BWR-games. In fact, it has been observed that a pseudo-polynomial algorithm for these games with an arbitrary number of random nodes would already imply their polynomial solvability. Currently, only one class is known to have such a pseudopolynomial algorithm, namely BWR-games with constant number of random nodes. We show that the existence of a pseudo-polynomial algorithm for BWR-games with a constant number of random vertices implies the existence of absolute and relative polynomial-time approximation schemes. In particular, we obtain absolute and relative approximation schemes for BWR-games with a constant number of random vertices, assuming the the smallest positive probability at a random node is bounded from below by a polynomial.