Memory-two zero-determinant strategies in repeated games

TL;DR

This paper extends zero-determinant strategies from memory-one to memory-two in repeated games, enabling unilateral control over payoff correlations and generalizing strategies like Tit-for-Tat.

Contribution

It introduces the concept of memory-two zero-determinant strategies, expanding the framework to control payoff relations in repeated games beyond memory-one.

Findings

Memory-two zero-determinant strategies enforce linear relations between payoff correlations.

Examples include generalizations of Tit-for-Tat to memory-two.

Extension to memory-n strategies is straightforward.

Abstract

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.