Canonical forms of two-person zero-sum limit average payoff stochastic games

We consider two-person zero-sum stochastic games with perfect information and, for each k ∈ Z+, introduce a new payoff function, called the k-total reward. For k = 0 and 1 they are the so called mean and total rewards, respectively. For all k, we prove solvability of the considered games in pure stationary strategies, and show that the uniformly optimal strategies for the discounted mean payoff (discounted 0-reward) function are also uniformly optimal for k-total rewards if the discount factor is close enough (depending on k) to 1. We also demonstrate that the k-total reward games form a proper subset of the (k + 1)-total reward games for each k. In particular, all these classes contain mean-payoff games. This observation implies that, in the non-zero-sum case, Nash-solvability fails for all k.