Zero-sum semi-Markov games with state-action-dependent discount factors

Abstract

Semi-Markov model is one of the most general models for stochastic dynamic systems. This paper deals with a two-person zero-sum game for semi-Markov processes. We focus on the expected discounted criterion with state-action-dependent discount factors. The state and action spaces are both Polish spaces, and the reward rate function is ω-bounded. We first construct a fairly general model of semi-Markov games under a given semi-Markov kernel and a pair of strategies. Next, based on the standard regularity condition and the continuity-compactness condition for semi-Markov games, we derive a “drift condition” on the semi-Markov kernel and suppose that the discount factors have a positive lower bound, under which the existence of the value function and an optimal pair of stationary strategies of our semi-Markov game are proved by using a general Shapley equation. Moreover, in the scenario of finite state and action spaces, a value iteration-type algorithm for approximating the value function and an optimal pair of stationary strategies is developed. The convergence and the error bound of the algorithm are also proved. Finally, we conduct numerical examples to demonstrate the main results.