A New Condition and Approach for Zero-Sum Stochastic Games with Average Payoffs

Abstract

This article deals with discrete-time two-person zero-sum stochastic games with Borel state and action spaces. The optimality criterion to be studied is the long-run expected average payoff criterion, and the (immediate) payoff function may have neither upper nor lower bounds. We first replace the optimality equation widely used in the previous literature with two so-called optimality inequalities, and give a new set of conditions for the existence of solutions to the optimality inequalities. Then, from the optimality inequalities we ensure the existence of a pair of average optimal stationary strategies. Our new condition is slightly weaker than those in the previous literature, and as a byproduct some interesting results such as the convergence of a value iteration scheme to the value of the discounted payoff game is obtained. Finally, we first apply the main results in this article to generalized inventory systems, and then further provide an example of controlled population processes for which all of our conditions are satisfied, while some of conditions in some of previous literature fail to hold.