Memory-two zero-determinant strategies in repeated games

Repeated games have provided an explanation how mutual cooperation can be achieved even if defection is more favorable in a one-shot game in prisoner's dilemma situation. Recently found zero-determinant strategies have substantially been investigated in evolutionary game theory. The original memory-one zero-determinant strategies unilaterally enforce linear relations between average payoffs of players. Here, we extend the concept of zero-determinant strategies to memory-two strategies in repeated games. Memory-two zero-determinant strategies unilaterally enforce linear relations between correlation functions of payoffs and payoffs at the previous round. Examples of memory-two zero-determinant strategy in the repeated prisoner's dilemma game are provided, some of which generalize the Tit-for-Tat strategy to memory-two case. Extension of zero-determinant strategies to memory-$n$ case with $n\geq 2$ is also straightforward.重复博弈（Repeated Games）理论已为囚徒困境（prisoner's dilemma）情境下，即便单次博弈（one-shot game）中背叛更具优势时仍能达成互惠合作的现象提供了合理解释。近年来，零行列式策略（zero-determinant strategies）已在演化博弈论领域得到广泛研究。初始的记忆一型零行列式策略（memory-one zero-determinant strategies）可单方面强制约束博弈参与者平均收益间的线性关系。本文将零行列式策略的概念拓展至重复博弈中的记忆二型策略范畴。记忆二型零行列式策略（memory-two zero-determinant strategies）可单方面强制约束收益相关函数与上一轮收益间的线性关系。本文还提供了重复囚徒困境博弈中记忆二型零行列式策略的实例，其中部分策略将一报还一报（Tit-for-Tat）策略拓展至记忆二型场景。将零行列式策略拓展至n≥2的记忆n型场景同样较为直观。