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.

随着团体规模的扩大，合作通常会变得更加难以维持，因为随着集体行动的潜在贡献者数量的增加，逃避责任的动机也会增加。但情况总是如此吗？在这里，我们研究了一种二元行动合作困境，其中只要不超过给定数量的参与者逃避代价高昂的合作任务，就可以提供公共物品。我们发现，在合作成本足够低时存在的稳定多态平衡下，合作的概率随着群体规模的增大而增加，并且当群体规模趋于无穷大时达到极限1。然而，增加群体规模可能会增加或减少在这种均衡下提供公共物品的可能性，具体取决于成本价值。我们还证明，在稳定的多态平衡（即他们的适应度）下，个体的预期收益随着群体规模的大小而减少。对于足够低的合作成本，在大群体规模的限制下，提供公共物品的概率和预期收益都收敛于正值。然而，我们还发现，稳定多态平衡的吸引力盆地是群体大小的递减函数，并且在非常大的群体的极限下缩小到零。总的来说，我们在一个简单的集体行动问题中展示了关于群体规模的重要比较静态。