The shirker's dilemma and the prospect of cooperation in large groups

Cooperation usually becomes harder to sustain as groups become larger because incentives to shirk increase with the number of potential contributors to collective action. But is this always the case? Here we study a binary-action cooperative dilemma where a public good is provided as long as not more than a given number of players shirk from a costly cooperative task. We find that at the stable polymorphic equilibrium, which exists when the cost of cooperation is low enough, the probability of cooperating increases with group size and reaches a limit of one when the group size tends to infinity. Nevertheless, increasing the group size may increase or decrease the probability that the public good is provided at such an equilibrium, depending on the cost value. We also prove that the expected payoff to individuals at the stable polymorphic equilibrium (i.e., their fitness) decreases with group size. For low enough costs of cooperation, both the probability of provision of the public good and the expected payoff converge to positive values in the limit of large group sizes. However, we also find that the basin of attraction of the stable polymorphic equilibrium is a decreasing function of group size and shrinks to zero in the limit of very large groups. Overall, we demonstrate non-trivial comparative statics with respect to group size in an otherwise simple collective action problem.