Controlling conditional expectations by zero-determinant strategies

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.