Unbending strategies shepherd cooperation and suppress extortion in spatial populations

Evolutionary game dynamics on networks typically consider the competition among simple strategies such as cooperation and defection in the Prisoner's Dilemma and summarize the effect of population structure as network reciprocity. However, it remains largely unknown regarding the evolutionary dynamics involving multiple powerful strategies typically considered in repeated games, such as the zero-determinant (ZD) strategies that are able to enforce a linear payoff relationship between them and their co-players. Here, we consider the evolutionary dynamics of always cooperate (AllC), extortionate ZD (extortioners), and unbending players in lattice populations based on the commonly used death-birth updating. Out of the class of unbending strategies, we consider a particular candidate, PSO Gambler, a machine-learning-optimized memory-one strategy, which can foster reciprocal cooperation and fairness among extortionate players. We derive analytical results under weak selection and rare mutations, including pairwise fixation probabilities and long-term frequencies of strategies. In the absence of the third unbending type, extortioners can achieve a half-half split in equilibrium with unconditional cooperators for sufficiently large extortion factors. However, the presence of unbending players fundamentally changes the dynamics and tilts the system to favor unbending cooperation. Most surprisingly, extortioners cannot dominate at all regardless of how large their extortion factor is, and the long-term frequency of unbending players is maintained almost as a constant. Our analytical method is applicable to studying the evolutionary dynamics of multiple strategies in structured populations. Our work provides insights into the interplay between network reciprocity and direct reciprocity, revealing the role of unbending strategies in enforcing fairness and suppressing extortion.