Nested Family of Cyclic Games with $k$-total Effective Rewards

We consider Gillette’s two-person zero-sum stochastic games with perfect information. For each k ∈ Z+ we introduce a payoff function, called the k-total reward. For k = 0 and 1 these payoffs are known as mean payoff and total reward, respectively. We restrict our attention to the deterministic case, the so called cyclic games. For all k, we prove the existence of a saddle point which can be realized by pure stationary strategies. We also demonstrate that k-total reward games can be embedded into (k+ 1)-total reward games. In particular, all of these classes contain mean payoff cyclic games.