Núñez Rodríguez Irene, Neves Armando G M
Departamento de Matemática, UFMG, Belo Horizonte, Brazil.
J Math Biol. 2016 Dec;73(6-7):1665-1690. doi: 10.1007/s00285-016-1009-1. Epub 2016 Apr 19.
We study a population of individuals playing the infinitely repeated prisoner's dilemma under replicator dynamics. The population consists of three kinds of individuals adopting the following reactive strategies: ALLD (individuals which always defect), ATFT (almost tit-for-tat: individuals which almost always repeat the opponent's last move) and G (generous individuals, which always cooperate when the opponent cooperated in the last move and have a positive probability q of cooperating when their opponent has defected). Our aim is studying in a mathematically rigorous fashion the dynamics of a simplified version for the computer experiment in Nowak and Sigmund (Nature 355:250-253, 1992) involving 100 reactive strategies. We see that as the generosity degree of the G individuals varies, equilibria (rest points) of the dynamics appear or disappear, and the dynamics changes accordingly. Not only we prove that the results of the experiment are true in our simplified version, but we also have complete control on the existence or non-existence of the equilbria for the dynamics for all possible values of the parameters, given that ATFT individuals are close enough to TFT. For most values of the parameters the dynamics can be completely determined.
我们研究了在复制者动态下玩无限重复囚徒困境的个体群体。该群体由采用以下反应策略的三种个体组成：ALLD（总是背叛的个体）、ATFT（近似针锋相对：几乎总是重复对手上一步行动的个体）和G（慷慨的个体，当对手在上一步合作时总是合作，当对手背叛时以正概率q合作）。我们的目标是以数学上严格的方式研究Nowak和Sigmund（《自然》355:250 - 253，1992）中涉及100种反应策略的计算机实验简化版本的动态。我们发现，随着G个体的慷慨程度变化，动态的均衡点（静止点）会出现或消失，并且动态也会相应改变。我们不仅证明了实验结果在我们的简化版本中是正确的，而且鉴于ATFT个体足够接近TFT，对于参数的所有可能值，我们还能完全控制动态均衡的存在与否情况。对于大多数参数值，动态可以被完全确定。
