Severe Punishment for Those Who Were Caught

Cooperation among unrelated individuals is pervasive in human societies, while natural selection favors the survival of the fittest. The theoretical explanation for this puzzle is a challenge across disciplines. Among other mechanisms that can promote cooperation is costly punishment, in which altruistic individuals privately bear the cost to punish defection. But it decreases the welfare of both the punisher and the punished, and defectors are hard and costly to be caught in many cases. Moreover, punishment may trigger a chain of retaliation between the punisher and the punished and induce second-order free riding - those who never punish will be better off than punishers. In this article we aim to get rid of these drawbacks by refining the mechanism of punishment. First we define an extended Public Goods game in which the defector is caught and punished with a certain probability and the cost of punishment is equally shared between the remainder of the group. Then we derive the fixation probability for one mutant cooperator to invade and take over the whole population. Our analysis shows that, when the probability for punishment is above a threshold, natural selection favors cooperating replacing defecting. Both analytical results and computer simulations show that severe punishment for a small fraction of defectors is enough for one mutant cooperator to become fixed with an advantageous probability in the population of defectors. In addition, this way of punishment can considerably decrease the total cost of inhibiting defection, especially for large populations.