Cooperation dynamics driven by inequality-averse reward in structured populations

Inequality is ubiquitous in social and economic interactions and can fundamentally alter the evolutionary prospects of cooperation. Here, we study a spatial public goods game in which fairness concerns are implemented through an inequality-averse rewarding rule. Players with inequality aversion redistribute payoffs locally by rewarding neighbors whose public-goods payoffs are lower than their own, while paying a corresponding cost whenever they are better off. This mechanism is embedded into the evolutionary dynamics by extending the strategy space to four types-standard cooperators and defectors, and inequality-averse cooperators and defectors-and by adopting a three-stage updating process consisting of public-goods interaction, payoff redistribution via reward, and Fermi imitation. Extensive Monte Carlo simulations reveal that increasing reward intensity induces a cascade of nonequilibrium phase transitions in the joint parameter space of reward strength and synergy factor, progressively enlarging the region where inequality-averse cooperation dominates. We further show that the coupling between reward strength and synergy factor governs both the stability and composition of mixed phases: weak rewards allow persistent free-riding, intermediate rewards create transient niches for inequality-averse defectors, whereas strong rewards stabilize cooperative clustering and suppress defection. These results demonstrate how redistribution motivated by inequality aversion reshapes local payoff landscapes and promotes cooperative order in structured populations, providing a mechanism-based perspective on designing fairness-driven incentives in collective-action systems.