On the computational complexity of solving stochastic mean-payoff games
  We consider some well-known families of two-player, zero-sum, perfect information games that can be viewed as special cases of Shapley's stochastic games. We show that the following tasks are polynomial time equivalent:   - Solving simple stochastic games.   - Solving stochastic mean-payoff games with rewards and probabilities given in unary. - Solving stochastic mean-payoff games with rewards and probabilities given in binary. 
Vladimir Gurvich and Peter Bro Miltersen