Rules for the evolution of multiplayer cooperation under networks of communities.

We assume a large number M of communities and that they are composed of at least two individuals (Q ≥ 2). These conditions guarantee that ρC > ρneutral > ρD. We denote the harmonic series as . Under Q = 1, the derived conditions are the following: cooperation never evolves under the CPD, TVD, SH, FSH, and TS (assuming that L ≥ 2), cooperation evolves for V/K > 1 under the PD, PDV, VD and S, and both strategies are neutral under the HD. These results are valid under arbitrary values of w and M, and they are the same under all six dynamics.