Is cooperation sustained under increased mixing in evolutionary public goods games on networks?

A qualitative difference between evolutionary public goods games in a single well-mixing population on the one hand and in neighborhoods of interaction networks on the other hand is the possibility of sustained cooperation within subpopulations. Compared with well-mixed populations, networks model local rather than global interactions by restricting them to social neighborhoods. In this work, we propose an evolutionary game model that is able to capture the effect of long-range links mixing local neighborhood and global group interactions in a finite networked population. We derived dynamical equations for the evolution of cooperation under weak selection by employing the mean-field and pair approximation approach. Using properties of Markov processes, we can approach a theoretical analysis of the effect of the density of mixing link. We find a rule governing the emergence and stabilization of cooperation, which shows that the positive or negative effect of mixing-link density for fixed group size depends on the global benefit in the public goods game. With mutations, we study the average abundance of cooperators and find that increasing mixing links promotes cooperation in strong dilemmas and hinders cooperation in weak dilemmas. These results are independent of whether strategy transfer is allowed via mixing links or not.