Cooperation dilemmas in multi-stage public goods games with finite populations

Cooperation in the real world is often based on continuous inputs in multiple stages, and the realization of cooperative outcomes depends on individual strategic choices in multiple successive stages. However, this mechanism of stage benefit accumulation and delayed distribution may compress the viability of cooperation and exacerbate the evolutionary pressure faced by cooperators. In this paper, we construct a two-stage public goods game model based on finite populations, and examine the effects of different selection intensities and stage synergy factors on the evolution of the system. It is found that under the setting of cumulative gains and delayed allocation, it is difficult for cooperators to resist the joint exploitation by defectors at different stages, and the system is more inclined to evolution to a mixed-strategy pattern dominated by first-stage cooperation only. It is difficult to change the cooperation disadvantage by enhancing the first-stage synergies alone, and it is also difficult to realize the full cooperation steady state by enhancing the two-stage synergy factors at the same time. In contrast, weak selection or smaller group sizes can help alleviate the pressure of strategic exploitation and provide a narrow space for cooperation.