Institut de Mathématiques Appliquées (IMA), Université de Lausanne, CH-1015, Lausanne, Switzerland E-mail:
Evolution. 2012 Mar;66(3):623-636. doi: 10.1111/j.1558-5646.2011.01504.x. Epub 2011 Dec 21.
Public goods games are models of social dilemmas where cooperators pay a cost for the production of a public good while defectors free ride on the contributions of cooperators. In the traditional framework of evolutionary game theory, the payoffs of cooperators and defectors result from interactions in groups formed by binomial sampling from an infinite population. Despite empirical evidence showing that group-size distributions in nature are highly heterogeneous, most models of social evolution assume that the group size is constant. In this article, I remove this assumption and explore the effects of having random group sizes on the evolutionary dynamics of public goods games. By a straightforward application of Jensen's inequality, I show that the outcome of general nonlinear public goods games depends not only on the average group size but also on the variance of the group-size distribution. This general result is illustrated with two nonlinear public goods games (the public goods game with discounting or synergy and the N-person volunteer's dilemma) and three different group-size distributions (Poisson, geometric, and Waring). The results suggest that failing to acknowledge the natural variation of group sizes can lead to an underestimation of the actual level of cooperation exhibited in evolving populations.
公共物品博弈是社会困境的模型，其中合作者为公共物品的生产付出代价，而叛徒则搭合作者的便车。在传统的进化博弈论框架中，合作者和叛徒的收益来自于从无限人口中进行二项式抽样形成的群体中的相互作用。尽管有经验证据表明，自然界中的群体大小分布高度异质，但大多数社会进化模型假设群体大小是恒定的。在本文中，我取消了这个假设，并探讨了群体大小随机对公共物品博弈进化动态的影响。通过对 Jensen 不等式的直接应用，我表明一般非线性公共物品博弈的结果不仅取决于平均群体大小，还取决于群体大小分布的方差。这个一般结果通过两个非线性公共物品博弈（带有折扣或协同作用的公共物品博弈和 N 人志愿者困境）和三个不同的群体大小分布（泊松分布、几何分布和瓦伦分布）来说明。结果表明，不承认群体大小的自然变化可能导致对进化群体中实际合作水平的低估。
