Completely Mixed Strategies for Two Structured Classes of Semi-Markov Games, Principal Pivot Transform and Its Generalizations

Abstract

In this article, we revisit the applications of principal pivot transform and its generalization for a rectangular matrix (in the context of vertical linear complementarity problem) to solve some structured classes of zero-sum two-person discounted semi-Markov games with finitely many states and actions. The single controller semi-Markov games have been formulated as a linear complementarity problem and solved using a stepwise principal pivoting algorithm. We provide a sufficient condition for such games to be completely mixed. The concept of switching controller semi-Markov games is introduced and we prove the ordered field property and the existence of stationary optimal strategies for such games. Moreover, such games are formulated as a vertical linear complementarity problem and have been solved using a stepwise generalized principal pivoting algorithm. Sufficient conditions are also given for such games to be completely mixed. For both these classes of games, some properties analogous to completely mixed matrix games, are established.