Facilitating the evolution of cooperation through altruistic punishment with adaptive feedback

The stability and persistence of cooperation are often threatened by the selfish behavior of self-interested individuals. One possible solution to this problem is to punish self-interested individuals. However, previous research always assumes that the strength of punishment is constant and independent of the game environment, which does not align with many real-world situations. Hence, it remains unclear whether altruistic punishment strategy with adaptive feedback can promote the evolution of cooperation effectively. Here we introduce the punishment strategy with adaptive feedback into the public goods game and develop a coevolutionary game model where the strength of punishment is dynamically adjusted based on the level of cooperation in a population. We adopt a linear feedback form to characterize the impact of strategies on punishment intensity. Through theoretical analysis and numerical calculations, we identify an evolutionary oscillation dynamics, in which the system cycle between high-intensity and low-intensity punishment, as well as cooperation and defection behaviors. Furthermore, we find that a high level of cooperation can be sustained with the use of only a low-intensity punishment.