A Nested Family of $k$-total Effective Rewards for Positional Games

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