Cooperation dynamics in nonlinear spatial public goods games with endogenous synergy and discounting feedback

The synergy and discounting effects were initially proposed to characterize how the benefit nonlinearly accumulates in social dilemmas, however, these effects do not always act statically and may be affected by individuals' previous actions. To capture this feedback mechanism, we introduce an endogenous control variable based on historical records to the spatial public goods games (SPGG), which allows the payoff to be synergistically enhanced or discounted depending on the current and historical level of cooperation. We explore cooperation dynamics in a lattice network structured population playing this nonlinear game. The simulation results reveal that the synergy and discounting feedback based on historical information which allows the group to obtain a synergistic payoff when there are more cooperators than the historical average level is conducive to the emergence and prosperity of cooperation. Moreover, the length of the endogenous feedback accelerates the transition rate of strategy during the evolutionary process, and the length of feedback demonstrates a strong role in ending the coexistence state of cooperation and defection, while the amplitude efficiently regulates the intensity of feedback. Additionally, we find that endogenous synergy and discounting feedback are more effective in promoting the spread of cooperative strategy compared with the exogenous model. This work may contribute to expanding our understanding of the feedback-based endogenous synergy and discounting effect in promoting cooperation.