Completely mixed strategies for single controller unichain semi-Markov games with undiscounted payoffs

Abstract

Zero-sum two-person finite undiscounted (limiting ratio average) semi-Markov games are considered where the transition probabilities and the transition times are controlled by a fixed player in all states. We prove that if such a game is unichain, the value and optimal stationary strategies can be obtained from an optimal solution of a linear programming algorithm for the associated undiscounted unichain single controller stochastic game (obtained by a data transformation method). The single controller undiscounted unichain semi-Markov games have been formulated as a linear complementarity problem and solved using a stepwise principal pivoting algorithm. We provide necessary and sufficient conditions for such games to be completely mixed (i.e., every optimal stationary strategy for each player assigns a positive probability to every action in every state). Some properties analogous to completely mixed matrix games are also established in this paper.