Abstract
Repeated games have provided an explanation of how mutual cooperation can be achieved even if defection is more favourable in a one-shot game in the Prisoner’s Dilemma situation. Recently found zero-determinant (ZD) strategies have substantially been investigated in evolutionary game theory. The original memory-one ZD strategies unilaterally enforce linear relationships between average pay-offs of players. Here, we extend the concept of ZD strategies to memory-two strategies in repeated games. Memory-two ZD strategies unilaterally enforce linear relationships between correlation functions of pay-offs and pay-offs of the previous round. Examples of memory-two ZD strategy in the repeated Prisoner’s Dilemma game are provided, some of which generalize the tit-for-tat strategy to a memory-two case. Extension of ZD strategies to memory-n case with n ≥ ~2 is also straightforward.
Highlights
Repeated games offer a framework explaining forward-looking behaviours and reciprocity of rational agents [1,2]
Since it was pointed out that game theory of rational agents can be applied to evolutionary behaviour of populations of biological systems [3], evolutionary game theory has investigated the condition where mutualism is maintained in conflicts [4,5,6,7,8,9,10,11]
We extend the concept of ZD strategies to memory-two strategies in repeated games
Summary
Repeated games offer a framework explaining forward-looking behaviours and reciprocity of rational agents [1,2]. Linera algebraic properties of ZD strategies in general multi-player multi-action games with many ZD players were investigated in [30], which found that possible ZD strategies are constrained by the consistency of the linear pay-off relationships. Another extension is ZD strategies in repeated games with imperfect monitoring [30,31,32], where possible ZD strategies are more restricted than ones in perfect monitoring cases. ZD strategies are not necessarily a rational strategy, they contain the TFT strategy in the Prisoner’s Dilemma game [18], which returns the opponent’s previous action, and ZD strategies form a significant class of memory-one strategies.
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