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Semi-Markov games are investigated under discounted and limiting average payoff criteria. The issue of the existence of the value and a pair of stationary optimal strategies are settled; the optimality equation is studied and under a natural ergodic condition the existence of a solution to the optimality equation is proved for the limiting average case. Semi-Markov games provide useful flexibility in constructing recursive game models. All the work on Markov/semi-Markov decision processes and Markov (stochastic) games can be viewed as special cases of the developments in this paper.

Abstract. A bidirected tree is a tree in which each edge is replaced by two arcs in either direction. Formulas are obtained for the determinant and the inverse of a bidirected tree, generalizing well-known formulas in the literature.