Rare Third-Party Punishment Promotes Cooperation in Risk-Averse Social Learning Dynamics

Abstract

Third-party punishment is a common mechanism to promote cooperation in humans. Theoretical models of evolution of cooperation predict that punishment maintains cooperation if it is sufficiently frequent. On the other hand, empirical studies have found that participants frequently punishing others do not success in comparison with those not eager to punish others, suggesting that punishment is suboptimal and thus should not be frequent. That being the case, our question is what mechanism, if any, can sustain cooperation even if punishment is rare. The present study proposes that one possible mechanism is risk-averse social learning. Using the method of evolutionary game dynamics, we investigate the effect of risk attitude of individuals on the question. In our framework, individuals select a strategy based on its risk, i.e., the variance of the payoff, as well as its expected payoff; risk-averse individuals prefer to select a strategy with low variable payoff. Using the framework, we examine the evolution of cooperation in two-player social dilemma games with punishment. We study two models: cooperators and defectors compete, while defectors may be punished by an exogenous authority; and cooperators, defectors, and cooperative punishers compete, while defectors may be punished by the cooperative punishers. We find that in both models, risk-averse individuals achieve stable cooperation with significantly low frequency of punishment. We also examine three punishment variants: in each game, all defectors are punished; only one of defectors is punished; and only a defector who exploits a cooperator or a cooperative punisher is punished. We find that the first and second variants effectively promote cooperation. Comparing the first and second variants, each can be more effective than the other depending on punishment frequency.