Zero-determinant strategies are memory-one strategies in repeated games which unilaterally enforce linear relations between expected payoffs of players. Recently, the concept of zero-determinant strategies was extended to the class of memory-$n$ strategies with $n\geq 1$, which enables more complicated control of payoffs by one player. However, what we can do by memory-$n$ zero-determinant strategies is still not clear. Here, we show that memory-$n$ zero-determinant strategies in repeated games can be used to control conditional expectations of payoffs. Equivalently, they can be used to control expected payoffs in biased ensembles, where a history of action profiles with large value of bias function is more weighted. Controlling conditional expectations of payoffs is useful for strengthening zero-determinant strategies, because players can choose conditions in such a way that only unfavorable action profiles to one player are contained in the conditions. We provide several examples of memory-$n$ zero-determinant strategies in the repeated prisoner's dilemma game. We also explain that a deformed version of zero-determinant strategies is easily extended to the memory-$n$ case.
翻译:零确定性战略是多次游戏中的记忆-一战略,这些策略单方面强制实施球员预期报酬之间的线性关系。最近,零确定性战略的概念扩大到了记忆-一美元战略的类别,使对一个球员的支付控制更为复杂。然而,我们通过记忆-一美元零确定性战略可以做的仍然是不明确的。在这里,我们表明,重复游戏中的记忆-一美元零确定性战略可以用来控制有条件的付款期望。同样,它们也可以用来控制偏向组合中的预期付款,因为在这个组合中,具有重大偏向功能价值的行动史更加权重。控制有条件的付款预期对于加强零确定性战略是有用的,因为玩家可以选择这样的条件,只有对一个球员的不易变行动简介才包含在条件中。我们提供了在重复的囚犯两难游戏中的一些记忆-一美元零确定性战略的例子。我们还解释,在重复的两难性游戏中,对零确定性战略的变换式是容易的零决定性战略。我们还解释,对零决定性战略的错误式的错误式是案例。