The combination of social reward and punishment is conducive to the cooperation and heterogeneity of social relations

Individual behaviors and social relations influence each other. However, understanding the underlying mechanism remains challenging. From social norms controlling human behavior to individual management of interpersonal relationships, rewards and punishments are some of the most commonly used measures. Through simulating the weak prisoner's dilemma in finite populations, we find that neither a simple reward measure nor a pure punishment mechanism can extensively promote cooperation. Instead, a combination of appropriate punishment and reward mechanisms can promote cooperation's prosperity regardless of how large or small the temptation to defect is. In addition, the combination spontaneously produces inhomogeneities in social relations and individual influence, which support the continued existence of cooperative behavior. Finally, we further explain how cooperators establish a sustainable existence under the combination by investigating the social relations at different moments in a small system. These results demonstrate that dispensing rewards and punishments impartially in society is essential to social harmony.