Disbandment rules that most facilitate the evolution of cooperation

Cooperation is considered a mysterious phenomenon from the perspective of adaptive evolution. However, if an individual can separate from an unsatisfactory group and join another, then this can facilitate positive assortment between cooperative types and promote the evolution of cooperation. What kind of disbandment rule most facilitates the evolution of cooperation? A previous study investigated exogenous disbandment rules and showed that, when games are played between two players, a rule where heterogeneous groups disband facilitates the evolution of cooperation. However, in groups of more than two individuals, a rule strictly requiring homogeneity applied if and only if the expected number of rounds played in a group was greater than some critical value. How large is the critical value? In this study, we make a mathematical analysis using evolutionary game theory. Our results show that the critical number of rounds increases greatly as the group size increases. Consequently, for species with large group sizes, e.g., Homo sapiens, under plausible parameter values, the strict homogeneity rule is unlikely to facilitate the evolution of cooperation. We find instead that a disbandment rule that requires a threshold level of homogeneity outperformed the strict homogeneity rule. Furthermore, we calculate the position of internal equilibria at which cooperators and defectors coexist and show that the initial evolution of cooperation is most encouraged when cooperators are tolerant (intolerant) of defectors if the benefit-to-cost ratio is large (small).