Departamento de Matemática Aplicada, Universidade Federal Fluminense, R. Mário Santos Braga, s/n, Niterói RJ, 24020-140, Brasil.
J Theor Biol. 2009 Oct 21;260(4):581-8. doi: 10.1016/j.jtbi.2009.07.010. Epub 2009 Jul 16.
In the animal world, performing a given task which is beneficial to an entire group requires the cooperation of several individuals of that group who often share the workload required to perform the task. The mathematical framework to study the dynamics of collective action is game theory. Here we study the evolutionary dynamics of cooperators and defectors in a population in which groups of individuals engage in N-person, non-excludable public goods games. We explore an N-person generalization of the well-known two-person snowdrift game. We discuss both the case of infinite and finite populations, taking explicitly into consideration the possible existence of a threshold above which collective action is materialized. Whereas in infinite populations, an N-person snowdrift game (NSG) leads to a stable coexistence between cooperators and defectors, the introduction of a threshold leads to the appearance of a new interior fixed point associated with a coordination threshold. The fingerprints of the stable and unstable interior fixed points still affect the evolutionary dynamics in finite populations, despite evolution leading the population inexorably to a monomorphic end-state. However, when the group size and population size become comparable, we find that spite sets in, rendering cooperation unfeasible.
在动物世界中，执行对整个群体有益的特定任务需要该群体中的几个个体合作，这些个体通常共同承担完成任务所需的工作量。研究集体行动动态的数学框架是博弈论。在这里，我们研究了在一个群体中，合作者和破坏者的进化动态，该群体中的个体参与了 N 人非排他性公共物品博弈。我们探讨了广为人知的两人雪堆博弈的 N 人推广。我们讨论了无限和有限种群的情况，明确考虑了可能存在一个阈值，超过该阈值就会实现集体行动。虽然在无限种群中，N 人雪堆博弈（NSG）导致合作者和破坏者之间稳定共存，但引入阈值会导致与协调阈值相关的新内部平衡点的出现。尽管进化使种群不可避免地走向单态终结状态，但稳定和不稳定内部平衡点的指纹仍然会影响有限种群中的进化动态。然而，当群体规模和种群规模变得相当时，我们发现尽管存在合作，但合作变得不可行。
