Enhanced cooperation and harmonious population in an evolutionary N-person snowdrift game

The N-person evolutionary snowdrift game (NESG) is generalized to study the effects of the additional benefit to all agents in a competing group of size N resulting from an earlier completion of a task when more agents are willing to share the work. Following replicator dynamics, an equation that can be used to solve for the steady state frequency of cooperation x∗ in a well-mixed population as a function of the parameters representing the cost-to-benefit ratio c/b, additional reward w/b, and N is derived. Cooperation is enhanced in general for w≠0 and a stable state with all cooperative agents (AllC state) emerges for small groups N and small c/b. In contrast, such a harmonious AllC state does not exist in the original NESG for c≠0. The condition for the existence of an AllC state is estimated to be (N−1)c<w and good agreement is found when compared with numerical solutions. The observed disappearance of an AllC state and the drop in x∗ with N are discussed in terms of the change in the stability of the AllC state as N increases. Previous results in NESG are recovered as the w=0 case of our model. Thus, the pleasure of enjoying the completed task earlier promotes cooperation and the effects are more pronounced when small-group collective interactions are considered.