Grupo Interdisciplinar de Sistemas Complejos-GISC, Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain.
J Theor Biol. 2012 May 7;300:299-308. doi: 10.1016/j.jtbi.2012.02.003.
Recent experimental evidence [Grujić Fosco, Araujo, Cuesta, Sánchez, 2010. Social experiments in the mesoscale: humans playing a spatial Prisoner's dilemma. PLoS ONE 5, e13749] on the spatial Prisoner's Dilemma suggests that players choosing to cooperate or not on the basis of their previous action and the actions of their neighbors coexist with steady defectors and cooperators. We here study the coexistence of these three strategies in the multiplayer iterated Prisoner's Dilemma by means of the replicator dynamics. We consider groups with n=2, 3, 4 and 5 players and compute the payoffs to every type of player as the limit of a Markov chain where the transition probabilities between actions are found from the corresponding strategies. We show that for group sizes up to n=4 there exists an interior point in which the three strategies coexist, the corresponding basin of attraction decreasing with increasing number of players, whereas we have not been able to locate such a point for n=5. We analytically show that in the limit n --> ∞ no interior points can arise. We conclude by discussing the implications of this theoretical approach on the behavior observed in experiments.
最近的实验证据[Grujić Fosco、Araujo、Cuesta、Sánchez，2010. 中尺度的社会实验：人类在空间囚徒困境中进行博弈。PLoS ONE 5，e13749]表明，基于自身和邻居的以往行动选择合作或不合作的玩家与稳定的背叛者和合作者共存。我们通过复制动态研究了多人重复囚徒困境中这三种策略的共存。我们考虑了 n=2、3、4 和 5 名玩家的群体，并计算了每个类型玩家的收益，作为马尔可夫链的极限，其中动作之间的转移概率是从相应的策略中找到的。我们表明，对于 n=4 以内的群体大小，存在一个内部点，三个策略共存，相应的吸引域随着玩家数量的增加而减小，而对于 n=5 我们还没有能够定位到这样的点。我们从理论上证明，在 n→∞的极限下，不可能出现内部点。最后，我们讨论了这种理论方法对实验中观察到的行为的影响。
