Effects of group size and noise on cooperation in population evolution of dynamic groups

We consider a large population of \(\textit{M}\) agents, who are randomly selected to form size-distributed groups from time to time, and the grouped agents engage in the Public Goods Game (PGG). The size \(\textit{g}\) of the group is within \(\textit{g}_\textit{l}\) and \(\textit{g}_\textit{h}\), where \(\textit{g}_\textit{l}\) and \(\textit{g}_\textit{h}\) are the lower and upper limits of the group size, respectively. Players have two strategies to choose, i.e., to cooperate (C), or to defect (D). Based on the dynamic grouping, we investigate the cooperative behavior of the system, and the results show that the frequency of cooperation is greatly affected by the noise intensity and group size distribution. In the evolutionary process, the payoffs of the cooperators (defectors) mainly depend on the strategy selection implemented by the death-birth process. For \(\textit{g}\in \text {[1,3]}\), different noise intensities \(\textit{K}\) induce different dynamic behaviors together with the multiplication factor \(\textit{r}\). For \(\textit{g}\in \text {[1,5]}\), the system may evolve to one of the bistable states (one is the totally cooperative state and the other is a mixed state with cooperators and defectors) starting from different initial concentrations of cooperation. The results of numerical computation seems to fit well with the simulation data. Furthermore, for \(K=1.0\) and \(g \in [1,5]\), we still observe the phenomenon of hysteresis effect where the system just reaches to the totally cooperative state slowly after a period of delay with increasing multiplication factor r. In addition, when one D-player tries to invade the C-population, there exists a critical game parameter \(\textit{r}_\textit{invade}\), below which the C-population will be invaded. We also study how the critical game parameter relies on the noise intensity and the group size distribution.