In this paper we extend the classical concept of positional strategies for a mean payoff game to a general mixed stationary strategy approach, and prove the existence of mixed stationary Nash equilibria for an arbitrary m-player mean payoff game on graphs. Traditionally, a positional strategy represents a pure stationary strategy in a classical mean payoff game, where a Nash equilibrium in pure stationary strategies in general may not exist. Based on a constructive proof of the existence of specific equilibria for an m-player mean payoff game we propose a new approach for determining the optimal mixed stationary strategies. Additionally we characterize and extend the general problem of the existence of pure stationary Nash equilibria for some special classes of mean payoff games.