A Potential Reduction Algorithm for Two-Person Zero-Sum Mean Payoff Stochastic Games

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real ε, let us call a stochastic game ε-ergodic, if its values from any two initial positions differ by at most ε. The proposed new algorithm outputs for every ε> 0 in finite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an ε-range, or identifies two initial positions u and v and corresponding stationary strategies for the players proving that the game values starting from u and v are at least ε/ 24 apart. In particular, the above result shows that if a stochastic game is ε-ergodic, then there are stationary strategies for the players proving 24 ε-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (Stochastic games with finite state and action spaces. PhD thesis, Centrum voor Wiskunde en Informatica, Amsterdam, 1980) claiming that if a stochastic game is 0-ergodic, then there are ε-optimal stationary strategies for every ε> 0. The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game.