  We develop value iteration-based algorithms to solve in a unified manner
different classes of combinatorial zero-sum games with mean-payoff type
rewards. These algorithms rely on an oracle, evaluating the dynamic programming
operator up to a given precision. We show that the number of calls to the
oracle needed to determine exact optimal (positional) strategies is, up to a
factor polynomial in the dimension, of order R/sep, where the "separation" sep
is defined as the minimal difference between distinct values arising from
strategies, and R is a metric estimate, involving the norm of approximate sub
and super-eigenvectors of the dynamic programming operator. We illustrate this
method by two applications. The first one is a new proof, leading to improved
complexity estimates, of a theorem of Boros, Elbassioni, Gurvich and Makino,
showing that turn-based mean-payoff games with a fixed number of random
positions can be solved in pseudo-polynomial time. The second one concerns
entropy games, a model introduced by Asarin, Cervelle, Degorre, Dima, Horn and
Kozyakin. The rank of an entropy game is defined as the maximal rank among all
the ambiguity matrices determined by strategies of the two players. We show
that entropy games with a fixed rank, in their original formulation, can be
solved in polynomial time, and that an extension of entropy games incorporating
weights can be solved in pseudo-polynomial time under the same fixed rank
condition.
