Evolution of cooperation in well-mixed N-person snowdrift games

We study the time evolution of cooperation in a recently proposed NN-person evolutionary snowdrift game, by focusing on the details of the evolutionary dynamics. It is found that the analytic solution for the equilibrium fraction of cooperators as given previously by the replicator dynamics stems from a balance between the terms: the cost to contribute to a common task and the risk in refusing to participate in a common task. Analytic expressions for these two terms are given, and their magnitudes are studied over the whole range of parameter space. Away from equilibrium, it is the imbalance between these terms that drives the system to equilibrium. A continuous time first-order differential equation for the degree of cooperation is derived, for arbitrary interacting group size NN and cost-to-benefit ratio. Analytic solutions to the time evolution of cooperation for the cases of N=2N=2 and N=3N=3 are obtained, with results in good agreement with those obtained by numerical simulations. For arbitrary NN, numerical solutions to the equation give the time evolution of cooperation, with the long time limit giving the equilibrium fraction of cooperators.