Bach L A, Helvik T, Christiansen F B
Department of Theoretical Ecology, Ecology Building, Lund University, SE-223 62 Lund, Sweden.
J Theor Biol. 2006 Jan 21;238(2):426-34. doi: 10.1016/j.jtbi.2005.06.007. Epub 2005 Jul 20.
An evolutionary game of individuals cooperating to obtain a collective benefit is here modelled as an n-player Prisoner's Dilemma game. With reference to biological situations, such as group foraging, we introduce a threshold condition in the number of cooperators required to obtain the collective benefit. In the simplest version, a three-player game, complex behaviour appears as the replicator dynamics exhibits a catastrophic event separating a parameter region allowing for coexistence of cooperators and defectors and a region of pure defection. Cooperation emerges through an ESS bifurcation, and cooperators only thrive beyond a critical point in cost-benefit space. Moreover, a repelling fixed point of the dynamics acts as a barrier to the introduction of cooperation in defecting populations. The results illustrate the qualitative difference between two-player games and multiple player games and thus the limitations to the generality of conclusions from two-player games. We present a procedure to find the evolutionarily stable strategies in any n-player game with cost and benefit depending on the number of cooperators. This was previously done by Motro [1991. Co-operation and defection: playing the field and the ESS. J. Theor. Biol. 151, 145-154] in the special cases of convex and concave benefit functions and constant cost.
个体通过合作获取集体利益的进化博弈在此被建模为一个n人囚徒困境博弈。参照诸如群体觅食等生物学情形，我们引入了获得集体利益所需合作者数量的阈值条件。在最简单的版本，即一个三人博弈中，复杂行为表现为复制者动态呈现出一个灾难性事件，该事件将允许合作者和背叛者共存的参数区域与纯背叛区域分隔开来。合作通过一个进化稳定策略（ESS）分岔出现，并且合作者仅在成本效益空间中的一个临界点之外才能繁荣发展。此外，动态的一个排斥性不动点充当了在背叛群体中引入合作的障碍。这些结果说明了两人博弈和多人博弈之间的质的差异，从而也说明了两人博弈结论普遍性的局限性。我们提出了一种在任何n人博弈中找到进化稳定策略的程序，其中成本和效益取决于合作者的数量。此前Motro [1991年。合作与背叛：参与竞争与进化稳定策略。《理论生物学杂志》151卷，第145 - 154页] 在凸效益函数和凹效益函数以及恒定成本的特殊情况下做过此项工作。
