Heterogeneous investments induced by historical payoffs promote cooperation in spatial public goods games

Public goods game (PGG) is one of the most widely used metaphors for studying the cooperation emerged in the situation of N-person social dilemma. In the traditional version of spatial PGGs, each cooperative individual contributes to the groups he/she participates in with an equal investment. In this work, we introduce the heterogeneous investments induced by historical payoffs into spatial PGGs. We denote the number of considered historical steps by the memory length, M. We use a parameter ω to control different weights of the historical payoffs when affecting the heterogeneous investments. In particular, positive ω indicates that earlier historical payoffs have higher weights to decide the present investments and vice versa. Another tunable parameter, α, is introduced to characterize the heterogeneous strength. The results show that heterogeneous investments induced by historical payoffs can significantly promote cooperation. Moreover, we find that both small M and small ω are favorable to the evolution of cooperation. We also find that stronger heterogenous strength of investments can further improve cooperation. Besides, we check the robustness of the results on different networks and find that these conclusions are robust to the network topology.