Probabilistic reward or punishment promotes cooperation in evolutionary games

Reward and punishment are crucial for the emergence and sustainability of cooperation in evolutionary games. In this work, we introduce a third-party agent, who plays the role of a judge to reward cooperators and punish defectors, to study the impact of reward and punishment on the evolution of cooperation. The introduced righteous agent is different from the cooperators and defectors in the traditional games and it exists as a judge independent of the processes of the games. In each round of the evolutionary game, each player has a half chance to confront the righteous agent. If the player is a cooperator, it's possible for it to obtain an extra profit. On the contrary, when a defector meets the righteous agent, its earnings may be reduced. The simulation results show that the introduction of the righteous agent in the evolutionary game favors the evolution of cooperation. The robustness of the promoting effect is tested for different complex topologies for the prisoner's dilemma game. The enhancement effects are confirmed in the snowdrift game as well, which may imply that the facilitation effects show a high degree of universality independent of the structure of the applied spatial networks and the potential evolutionary game models. Our conclusion may be conducive to interpret the emergence and sustainability of cooperation within the structured populations.