The optimality equation and \(\varepsilon\)-optimal strategies in Markov games with average reward criterion (Q1812295)

This paper deals with two-person zero-sum Markov games on a Borel state space with the expected average reward criterion. The assumptions imposed on the model are of Lyapunov-type. The value of the game and \(\varepsilon\)-optimal strategies for the players are obtained. The approach relies on the existence of solutions to a parametrized family of functional equations. Then the continuity and monotonicity properties of these solutions yield the optimalize equation. Other methods are presented in [\textit{A. Jaskiewicz} and \textit{A. S. Nowak}, Math. Methods Oper. Res. 54, No. 2, 291--301 (2001; Zbl 1102.91305), \textit{O. Vega-Amaya}, SIAM J. Control Optimization 42, No. 5, 1876--1894 (2003; Zbl 1125.91308)] (see also the references therein), where the geometric ergodicity properties of the induced Markov chain are used. The value of the game and optimal strategies for the players are proved by a vanishing discount approach [A. Jakiewicz et al. (loc. cit.)] and the fixed-point argument [O. Vega-Amaya (loc. cit.)].