Evolution of cooperation in an n-player game with opting out.

How cooperation could have evolved has been one of the central topics in evolutionary biology. When cooperators are likely to interact with other cooperators, and defectors are likely to interact with other defectors, positive assortment is created, facilitating the evolution of cooperation. Cooperation is observed not only in dyadic interactions but also, sometimes, in sizable groups. Previous studies have found that the opting out rule in which the group is disbanded if and only if the group is heterogeneous, especially facilitates the evolution of cooperation compared to the other opting out rules in multi-player games when the number of rounds is sufficiently large. However, the dynamics between the cooperators and defectors under such an opting out rule have been investigated only in the case where group size is relatively small (e.g., four). In addition, the effect of group size on the evolution of cooperation has not been explored, and considering that humans interact within larger groups, investigation in such large groups is essential. Here, through further algebraic analyses, it is newly revealed that there can be four internal equilibria when the group size is larger than four. In addition, while the effect of group size on cooperation is negative in the case of common goods, it is not straightforward (i.e., can be positive) in the case of public goods.