Reputation update of responders efficiently promotes the evolution of fairness in the ultimatum game

Recently, exploration on how to sustain cooperation based on indirect reciprocity has obtained considerable concern. However, thus far few studies have investigated the effects of indirect reciprocity with shared social norms on fairness in the ultimatum game (UG), which has been a prominent paradigm for studying the evolution of fairness. In this paper, we introduce indirect reciprocity with social norms into the UG. We consider three scenarios, (1) reputation of proposers updates alone, (2) reputation of responders updates alone, and (3) reputation of both proposers and responders updates. By a two-timescale theoretical analysis, we find that reputation update of proposers alone leads to a higher level of fairness than the one without reputation under lots of social norms, and that reputation update of responders alone generates a very high level of fairness under image scoring. For the majority of the third-order social norms, the effect of proposers’ reputation update on fairness can be enhanced when it is combined with reputation update of responders. The excellent positive effect of responders’ reputation update on fairness cannot be enhanced when it is combined with reputation update of proposers under all social norms. The key factor for the high level of fairness is up to the dominance of the strategies with rejecting unfair offers in the pairwise competition, suggesting the behavior of altruistic punishment can emerge in the population.