Evolutionary dynamics of fairness on graphs with migration.

Individual migration plays a crucial role in evolutionary dynamics of population on networks. In this paper, we generalize the networked ultimatum game by diluting population structures as well as endowing individuals with migration ability, and investigate evolutionary dynamics of fairness on graphs with migration in the ultimatum game. We first revisit the impact of node degree on the evolution of fairness. Interestingly, numerical simulations reveal that there exists an optimal value of node degree resulting in the maximal offer level of populations. Then we explore the effects of dilution and migration on the evolution of fairness, and find that both the dilution of population structures and the endowment of migration ability to individuals would lead to the drop of offer level, while the rise of acceptance level of populations. Notably, natural selection even favors the evolution of self-incompatible strategies, when either vacancy rate or migration rate exceeds a critical threshold. To confirm our simulation results, we also propose an analytical method to study the evolutionary dynamics of fairness on graphs with migration. This method can be applied to explore any games governed by pairwise interactions in finite populations.