Group-level imitation may stabilize cooperation

Stabilizing cooperation among self-interested individuals presents a fundamental challenge in evolutionary theory and social science. While classical models predict the dominance of defection in social dilemmas, empirical and theoretical studies have identified various mechanisms that promote cooperation, including kin selection, reciprocity, and spatial structure. In this work, we investigate the role of selective imitation in the evolutionary dynamics of cooperation within an Optional Public Goods Game. We introduce a model where individuals belong to distinct groups and adapt their strategies based solely on comparisons within their own group. We identify different dynamical regimes, including stable fixed points, limit cycles, and Rock-Scissors-paper-type oscillations. Our analysis goes beyond numerical simulations and derives explicit constants of motion, providing conditions for the emergence of these regimes. Grounded in a replicator-type framework, it further reveals that such group-level imitation can stabilize cooperative behavior for certain initial group compositions. In other words, restricting imitation to group-level interactions mitigates the destabilizing effects of global competition and offers an explanation for the resilience of cooperation when social imitation is limited to a group, without invoking explicit spatial interaction structure or group-level reproduction.