DEI, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy.
J Theor Biol. 2013 Jun 7;326:70-81. doi: 10.1016/j.jtbi.2012.11.034. Epub 2012 Dec 7.
The fixation of cooperation among unrelated individuals is one of the fundamental problems in biology and social sciences. It is investigated by means of public goods games, the generalization of the prisoner's dilemma to more than two players. In compulsory public goods games, defect is the dominant strategy, while voluntary participation overcomes the social dilemma by allowing a cyclic coexistence of cooperators, defectors, and non-participants. Experimental and theoretical research has shown how the combination of voluntary participation and altruistic punishment-punishing antisocial behaviors at a personal cost-provides a solution to the problem, as long as antisocial punishment-the punishing of cooperators-is not allowed. Altruistic punishment can invade at low participation and pave the way to the fixation of cooperation. Specifically, defectors are overpunished, in the sense that their payoff is reduced by a sanction proportional to the number of punishers in the game. Here we show that qualitatively equivalent results can be achieved with a milder punishing mechanism, where defectors only risk a fixed penalty per round-as in many real situations-and the cost of punishment is shared among the punishers. The payoffs for the four strategies-cooperate, defect, abstain, and cooperate-&-punish-are derived and the corresponding replicator dynamics analyzed in full detail.
非相关个体之间合作的固定是生物学和社会科学的基本问题之一。通过公共物品博弈来研究这个问题，这是对超过两个玩家的囚徒困境的推广。在强制性公共物品博弈中，背叛是占优策略，而自愿参与通过允许合作者、背叛者和非参与者的循环共存来克服社会困境。实验和理论研究表明，自愿参与和利他惩罚——以个人成本惩罚反社会行为——的结合如何提供了解决问题的方法，只要不允许反社会惩罚——对合作者的惩罚。利他惩罚可以在低参与度下入侵，并为合作的固定铺平道路。具体来说，背叛者会受到过度惩罚，也就是说，他们的收益会因与游戏中惩罚者数量成比例的制裁而减少。在这里，我们展示了在更温和的惩罚机制下可以实现定性上等效的结果，在这种机制下，背叛者每轮只承担固定的罚款——就像许多实际情况一样——惩罚的成本由惩罚者共同承担。推导出了四个策略——合作、背叛、弃权和合作-惩罚——的收益，并详细分析了相应的复制者动力学。
