Evolution of cooperation with nonlinear environment feedback in repeated public goods game

The mutual feedback between the environment and the strategy is a significant mechanism in the evolution of cooperation. A feedback-evolving repeated public goods game model with nonlinear effects is explored. The players' behaviors in the current round may influence the environment, leading the environment feedback on the behavior by the modulation of payoffs in the form of positive or negative. Under the former case, the groups realize the synergy effect of payoffs in the next round. Vice versa, the discount effect will be instead. The introduction of nonlinear effects enables heterogeneous feedback of the environment on different groups without introducing too many environmental states. The results show the enhancing role of the nonlinear effects on the evolution of cooperation compared to linear switching of enforcement factors. Compared to static game environments, the dynamic feedback scenario can realize better performance in the evolution of cooperation. The effect is more pronounced as the synergy factor increases or the discount factor decreases. The results are verified when the game environment changes between two or three states and under different environment transition structures. In addition, under the scenarios of probabilistic transition, the results are explored by taking into account the cases of state-independent and state-dependent, where both the homogeneity and heterogeneity transition probability are considered in the state-independent case, respectively. The probabilistic transition results further emphasized the transition probability's significant role in the evolution of cooperation when cooperation behaviors are more likely incentivized to interact in the valuable environment.