Effects of exposure-based reward and punishment on the evolution of cooperation in prisoner's dilemma game

Reward and punishment are important factors in maintaining cooperation among selfish individuals. This paper explores the effects of exposure-based reward and punishment on the evolution of cooperation. In this model, players can choose to expose their opponent's cooperation or defection status, leading to the identification of four strategies: exposure cooperation, exposure defection, ordinary cooperation, and ordinary defection. Exposing one's behavior incurs a cost, and exposed cooperators are rewarded while exposed defectors are punished. The exposure is, in turn, rewarded for exposing the defection. The evolutionary results show that three states emerge after the population stabilizes: only ordinary defection remains, exposure cooperation coexists with exposure defection, and all four strategies coexist. Cooperative behavior can occur when the exposure cost is less than the punishment for defection. Simply rewarding exposed cooperators is insufficient to promote cooperation. Simultaneous reward and punishment can significantly promote cooperation. There exists an optimal minimum reward threshold for cooperation, which varies nonlinear with increasing punishment. Larger temptation to defect and smaller cooperation reward favor the survival of exposure-oriented players and can effectively discourage second-order free-riding behavior. Our study provides insights into effective solutions to social dilemmas.