Rolling horizon procedures in Semi-Markov Games: The Discounted Case
We study the properties of the rolling horizon and the approximate rolling horizon procedures for the case of two-person zero-sum discounted semi-Markov games with infi nite horizon, under several assumptions on the reward function, when the state space is a borelian set and the action spaces are considered compact. Under suitable conditions, we prove that the equilibrium is the unique solution of its dynamic programming equation, and we prove bounds which imply the convergence of the procedures when the horizon length tends to in finity. The approach is based on the formalism for Semi-Markov games developed by Luque-Vásquez, together with extensions of the results of Hernández-Lerma and Lasserre for Markov Decision Processes and Chang and Marcus for Markov Games, both in discrete time. In this way we generalize the results on the rolling horizon and approximate rolling horizon procedures previously obtained for discrete-time problems.