Zero-Sum Average Cost Semi-Markov Games with Weakly Continuous Transition Probabilities and a Minimax Semi-Markov Inventory Problem

Under the framework given by a growth condition, a Lyapunov property and some continuity assumptions, the present work shows the existence of lower semicontinuous solutions to the Shapley equation for zero-sum semi-Markov games with Borel spaces, weakly continuous transition probabilities and possible unbounded payoff. It is also shown the existence of stationary optimal strategies for the minimizing player and stationary ε-optimal strategies for the maximizing player. These results are proved using a fixed-point approach. Moreover, it is shown the existence of a deterministic stationary minimax strategy for a minimax semi-Markov inventory problem under mild assumptions on the demand distribution.