Zero-sum stochastic games with unbounded costs: Discounted and average cost cases

Zero-sum stochastic games with countable state space and with finitely many moves available to each player in a given state are treated. As a function of the current state and the moves chosen, player I incurs a nonnegative cost and player II receives this as a reward. For both the discounted and average cost cases, assumptions are given for the game to have a finite value and for the existence of an optimal randomized stationary strategy pair. In the average cost case, the assumptions generalize those given in Sennott for the case of a Markov decison chain. Theorems of Hoffman and Karp and Nowak are obtained as corollaries. Sufficient conditions are given for the assumptions to hold. A flow control example illustrates the results.