Coexistence of strong and weak social ties leads to fair allocations of resources in finite populations

The ultimatum game, characterizing the allocation behaviors in a two-side context, has been a natural and influential paradigm for understanding the evolution of fairness. It allows two players to split a sum of endowment, where the proposer and the responder gain benefit if they reach an agreement. Motivated by the diversity of genetic and social relationships in real society, we study the effects of strong and weak social ties on the evolution of fairness on the circle and the square lattice, where each player owns a single strong social tie and the remaining weak ones in his own eyes. By using the Monte Carlo simulation, we demonstrate that the coexistence of strong and weak social ties is sufficient to generate fair allocations of resources in finite populations under certain conditions.