Replicator population dynamics of group interactions: Broken symmetry, thresholds for metastability, and macroscopic behavior

The effect of group structure on cooperative behavior is not well understood. In this paper, we study the dynamics of a public goods game involving n-agent interactions. In the proposed setup, the population is organized into groups. We associate the individual fitness to group performance, while the evolutionary dynamics takes place globally. We derive analytical expressions and show that the model exhibits several fixed points, including the symmetric homogeneous states of total cooperation and total defection, which are unstable and stable, respectively. Interestingly, even if both individual and group levels are organized as well-mixed populations, the dynamics displays intermediate values of cooperation under the replicator dynamics. Namely, as soon as one of the groups, at least, is fully cooperative, intermediary fixed points appear for the rest of the groups. In addition to the analytical approach, we have performed numerical simulations that reproduce the internal fixed points obtained theoretically, showing coexisting intermediate levels of cooperation. Potential implications of these results in terms of group selection and the role of social norms are also discussed