Overpunishing is not necessary to fix cooperation in voluntary public goods games

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.

► Altruistic punishment and voluntary participation allow the fixation of cooperation in public goods games.
► This was shown in the absence of antisocial punishment and with sanctions proportional to the number of punishers.
► The original model is modified by introducing a milder punishing mechanism based on a fix sanction.
► The two models are dynamically equivalent, so overpunishment is not necessary to solve the social dilemma.
► The emergence of altruistic punishment and the ultimate fixation of cooperation is also clarified.