Cooperation dynamics under fine redistribution and mercenary punishment in public goods games

Punishment is widely regarded as an effective way to curb defection in public goods games. However, its cooperation-promoting effect largely depends on whether punishers can be sustainably supported. We consider a public goods game comprising cooperators, defectors, and punishers. In this game, the fine imposed on defectors is no longer considered a net loss to the system but is regarded as a shared and redistributable resource within the population. We focus on three key incentive parameters: the fine redistribution coefficient, the baseline reward provided to punishers, and the fine intensity. By combining replicator dynamics, local stability analysis, and center manifold theory, we derive the stability conditions for the defector-dominated equilibrium and the coexistence equilibrium on the cooperator–punisher boundary. The research results indicate that the redistribution coefficient primarily determines when the defector-dominated equilibrium loses stability, while the baseline reward exhibits a non-monotonic effect: moderate support helps maintain the punishment mechanism and promote cooperation, while excessive support may increase the burden on the cooperators and weaken the cooperative outcome. Additionally, the fine intensity is a prerequisite for the effective functioning of the redistribution mechanism, as a weak fine cannot form a sufficient resource pool to simultaneously support the survival of the punishers and compensate the cooperators. Numerical simulations further reveal the bistability and initial-state dependence. These findings suggest that effective cooperation is not only influenced by the punishment itself, but also by how the resources generated by the punishment are reinvested into the cooperative system.