Inequal dependence on members stabilizes cooperation in spatial public goods game

For many years, social diversity has been recognized as one of the important factors in the emergence of cooperation. In PGG that characterizes collective social dilemmas, the difference in individual contributions to collective output usually arises from diversity of endowments and productivity, or from heterogeneous investments, but this difference may also be caused by the unequal dependence of collective output on members. In real collective actions, individuals may not contribute homogeneous currencies or goods, but instead divide labor to complete a series of tasks. Some individuals who master critical resources and skills may have a greater impact on collective output. In this work, a heterogeneous population containing elites and ordinary people plays spatial public goods game. The peculiarity is that ordinary people can make full contributions only when there are cooperative elites within the team. Otherwise, a part of the work will not be able to be carried out, and the cooperation contributions of ordinary people will be greatly reduced. Simulations under multiple social structures consistently show that the presence of a small number of elites can greatly stabilize cooperation. On the contrary, an excess of elites undermines the stable cooperative structure between elites and ordinary people.