服从半群上对策的最优策略
Optimal strategies for a game on amenable semigroups
摘要：
半群博弈是一个定义在半群S上的两人零和博弈：参与人1和2分别选择S中的元素x和y，参与人1得到一个由从S到[-1，1]的函数f定义的收益f（xy）。如果半群在Day和von Neumann的意义下是顺从的，则可以扩展经典策略集，即S上的可数可加概率测度，以自然的方式包括一些可数可加测度。这个扩展博弈有一个值，参与者有最优策略。这个定理推广了紧致群上或具有特定支付的正整数上的乘法博弈的先前结果。我们还证明了扩展允许的策略集的过程保持经典解：如果一个半群博弈有一个经典解，这个解也解决了扩展的博弈。
英文摘要：
The semigroup game is a two-person zero-sum game defined on a semigroup S as follows: Players 1 and 2 choose elements x and y in S, respectively, and player 1 receives a payoff f(xy) defined by a function f from S to [-1,1]. If the semigroup is amenable in the sense of Day and von Neumann, one can extend the set of classical strategies, namely countably additive probability measures on S, to include some finitely additive measures in a natural way. This extended game has a value and the players have optimal strategies. This theorem extends previous results for the multiplication game on a compact group or on the positive integers with a specific payoff. We also prove that the procedure of extending the set of allowed strategies preserves classical solutions: if a semigroup game has a classical solution, this solution solves also the extended game.
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