Effects of migration in the spatial ultimatum game

We develop a simple yet effective model to study the effects of migration on the evolution of fairness in the spatial ultimatum game in dependence on the population density. The evolution of fairness is first studied with different population densities whenever migration is absent. We find that sparsity may not favor the evolution of fairness in the population except a very small specific region of the population density. Then, we endow the population with the capability of migration where dissatisfied individuals tend to move away and search for new partners to interact. We find that, in sharp contrast with the population without migration, the evolution of fairness is influenced fundamentally. Remarkably, there exists a certain low population density maximally promoting the evolution of fairness as well as a certain high population density maximally inhibiting the evolution of fairness. Surprisingly, the variations of both average offer and acceptance level of the population with population densities are analogous to the sinusoidal functions, which present the novel “sine-like” phenomena. In addition, we investigate the effect of noise on the evolution of fairness as a function of the population density. Simulation results evidence that there also exist similar “sine-like” phenomena for different noise effects. Our work may pave the way for understanding the role of migration on the evolution of fairness in the spatial ultimatum game.