A Potential Reduction Algorithm for Two-person Zero-sum Mean Payoff Stochastic Games

Abstract

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 $24ε$-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (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.