Dynamic scale return coefficient with environmental feedback promotes cooperation in spatial public goods game

In the traditional spatial public goods game (SPGG), the total contributions of each group are linearly amplified by the same enhancement factor, which does not coincide with the real situation. Although some literature has considered the non-linearity and heterogeneity of scale returns between groups, the differences are completely generated by stochastic mechanisms. In addition, the coefficient will not change once assigned in previous studies. In this paper, we consider an environmental feedback mechanism and introduce a dynamic scale return coefficient into the payoff function. We observe that people who have a good record of cooperation working together can produce greater synergies and can have a larger return coefficient. We use the concept of reputation to describe individuals' historical cooperation records. Therefore, we assume that the scale return coefficient is governed by the difference between the average reputation of the group and the whole population, which will also evolve over time. In addition, we set a reputation amplitude to control the extent to which reputation differences between groups affect payoffs. By simulation experiments, we reveal that dynamic scale return coefficient based on reputation difference between groups helps to promote the evolution of cooperation. Moreover, the magnitude of the reputation amplitude directly affects the level of cooperation and the emergence of cooperative clusters. Our model provides a new way to introduce the environmental feedback mechanism to payoffs function and facilitate the in-depth study of the impact of reputation on cooperation.