A zero-sum two-person Perfect Information Semi-Markov game (PISMG) under limiting ratio average payoff has a value and both the maximiser and the minimiser have optimal pure semi-stationary strategies. We arrive at the result by first fixing an arbitrary initial state and forming the matrix of undiscounted payoffs corresponding to each pair of pure stationary strategies of the two players and proving that this matrix has a pure saddle point.

翻译：在极限比率平均收益准则下，零和两人完美信息半马尔可夫博弈具有博弈值，且极大化者与极小化者均存在最优纯半平稳策略。我们通过以下步骤得出该结论：首先固定任意初始状态，构造对应于双方纯平稳策略对的未贴现收益矩阵，并证明该矩阵存在纯鞍点。