Effect of fairness-based sympathy and retaliation on cooperation in multi-player dilemma games

Fairness preference is common in real society and natural communities. We consider this preference and explore the effect of individual sympathy and retaliation caused by payoff inequality on the evolution of cooperation in multi-player dilemma games with a structured population. Specifically, by setting payoff-dependent heterogeneously sympathy threshold and reputation-based intensity of sympathy, we make high-payoff players more likely to perceive payoff lead, and the payoff redistribution proportion is determined by the reputation of low-payoff players. While all players have the same perception of payoff lag and retaliate against high-payoff players by damaging their payoff with a cost. We find that even if only a few players have this preference, cooperation can be effectively promoted regardless of whether sympathy and retaliation are considered separately or together. Although retaliation is indeed a Pareto loss behavior that damages social welfare, the reduction of the payoff gap can increase the frequency of cooperators. Moreover, a relatively small upper limit of reputation enables high-payoff players to more accurately identify defectors in the process of payoff redistribution. Moreover, high positive correlations between strategy, payoff, and reputation convince us that sympathy groups and retaliation groups are mainly composed of cooperators and defectors, respectively. By analyzing the underlying micro process of the redistribution of players’ payoffs, we observe that the reduction of payoff gaps increases the probability that defectors imitate the cooperative strategy, which further expands the areas of cooperative groups. These findings enrich our knowledge of the positive effect of sympathizing with the poor and retaliating with the rich caused by payoff inequality on cooperation.