Evolutionary behavior of generalized zero-determinant strategies in iterated prisoner’s dilemma

• Studied how generalized ZDS competes in well-mixed and structured populations.
• Presented expected payoffs for ZDS against AllC, AllD, TFT, and WSLS strategies.
• For finite and well-mixed populations, explained phase diagrams analytically.
• For square lattice, explained results by payoff band theory qualitatively.

We study the competition and strategy selections between a class of generalized zero-determinant (ZD) strategies and the classic strategies of always cooperate (AllC), always defect (AllD), tit-for-tat (TFT), and win-stay-lose-shift (WSLS) strategies in an iterated prisoner’s dilemma comprehensively. Using the generalized ZD strategy, a player could get a payoff that is χχ (χ>1χ>1) times that of his opponent’s, when the payoff is measured with respect to a referencing baseline parameterized by 0≤σ≤10≤σ≤1. Varying σσ gives ZD strategies of tunable generosity from the extortionate-like ZD strategy for σ≪1σ≪1 to the compliance-like strategy at σ≈1σ≈1. Expected payoffs when ZD strategy competes with each one of the classic strategies are presented. Strategy evolution based on adopting the strategy of a better performing neighbor is studied in a well-mixed population of finite size and a population on a square lattice. Depending on the parameters, extortion-like strategies may not be evolutionarily stable despite a positive surplus over cooperative strategies, while extortion-like strategies may dominate or coexist with other strategies that tend to defect despite a negative surplus. The dependence of the equilibrium fraction of ZD strategy players on the model parameters in a well-mixed population can be understood analytically by comparing the average payoffs to the competing strategies. On a square lattice, the success of the ZD strategy can be qualitatively understood by focusing on the relative alignments of the finite number of payoff values that the two competing strategies could attain when the spatial structure is imposed. ZD strategies with properly chosen generosity could be more successful in evolutionary competing systems.