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We develop value iteration-based algorithms to solve in a unified mannerdifferent classes of combinatorial zero-sum games with mean-payoff typerewards. These algorithms rely on an oracle, evaluating the dynamic programmingoperator up to a given precision. We show that the number of calls to theoracle needed to determine exact optimal (positional) strategies is, up to afactor polynomial in the dimension, of order R/sep, where the "separation" sepis defined as the minimal difference between distinct values arising fromstrategies, and R is a metric estimate, involving the norm of approximate suband super-eigenvectors of the dynamic programming operator. We illustrate thismethod by two applications. The first one is a new proof, leading to improvedcomplexity estimates, of a theorem of Boros, Elbassioni, Gurvich and Makino,showing that turn-based mean-payoff games with a fixed number of randompositions can be solved in pseudo-polynomial time. The second one concernsentropy games, a model introduced by Asarin, Cervelle, Degorre, Dima, Horn andKozyakin. The rank of an entropy game is defined as the maximal rank among allthe ambiguity matrices determined by strategies of the two players. We showthat entropy games with a fixed rank, in their original formulation, can besolved in polynomial time, and that an extension of entropy games incorporatingweights can be solved in pseudo-polynomial time under the same fixed rankcondition.

Universal Complexity Bounds Based on Value Iteration for Stochastic Mean Payoff Games and Entropy Games