Fostering cooperative evolution through probabilistic punishment and environmental feedback in public goods game

Punishing selfish individuals is regarded as an effective method to maintain social cooperation. In reality, the corresponding punishment probability should vary with different game environments. However, most current research treats this probability as a constant or exogenously given. In this paper, based on the public goods game, we design an environmental feedback mechanism and establish a feedback evolutionary game model. The model assumes that the probability of punishing defectors will change with the proportion of cooperators, ultimately influencing individual decision-making. Through theoretical analysis and numerical simulations, we obtain three stable states of the system under different parameter conditions: a state of complete defection with low punishment probability, a state of complete cooperation with high punishment probability, and a bistable state. Our research results indicate that the environmental feedback mechanism plays a crucial role in promoting long-term social stability and sustainable development.