Enhanced cooperation in multiplayer snowdrift games with random and dynamic groupings

An analytically tractable generalization of the $N$-person snowdrift (NSG) game that illustrates how cooperation can be enhanced is proposed and studied. The number of players competing within a NSG varies from one time step to another. Exact equations governing the frequency of cooperation $f_c\left(r\right)$ as a function of the cost-to-benefit ratio $r$ within an imitation strategy updating scheme are presented. For group sizes $g$ uniformly distributed within the range $g\in [1,g_m]$, an analytic formula for the critical value $r_c\left(g_m\right)$, below which the system evolves into a totally cooperative (AllC) state, is derived. In contrast, a fixed group size NSG does not support an AllC state. The result $r_c\left(g_m\right)$ requires the presence of sole-player groups and involves the inverse of the harmonic numbers and, more generally, the inverse first moment of the group size distribution. For $r>r_c\left(g_m\right)$, the equation that determines the dynamical mixed states $f_c\left(r\right)$ is given, with exact solutions existing for $g_m\le 5$. The exact treatment allows the study of the phase boundary between the AllC state and the mixed states. The analytic results are checked against simulation results and exact agreements are demonstrated. The analytic form of the critical $r_c\left(g_m\right)$ illustrates the necessity of having groups of a sole player in the evolutionary process. This result is supported by simulations with group sizes excluding the sole groups for which no AllC state emerges. A physically transparent picture of the importance of the sole players in inducing an AllC state is further presented based on the last surviving pattern before the AllC state is attained. The exact expression $r_c\left(g_m\right)$ turns out to remain valid for nonuniform group-size distributions. Our analytical tractable generalization, therefore, sheds light on how a competing environment with variable group sizes could enhance cooperation and induce an AllC state.

Figure 1

The dependence of $f_c$ on $r$ for four different values of $g_m$. The competing group sizes are randomly selected within the range of $g=1$ to $g=g_m$ during evolutionary process. Results obtained by the theory (curves) and results obtained by simulations (symbols) are shown for comparison. There is an AllC phase with $f_c=1$ for $r<r_c$, in contrast to the NSG.

Figure 2

Phase diagram on the $g_m-r$ plane. The phase boundary as obtained by theory (curve) and by simulations (symbols) are shown. The phase boundary separates a homogeneous phase with a totally cooperative population (AllC phase) from a mixed dynamical phase characterized by a coexistent population with cooperators and defectors ($C+D$ phase).

Figure 3

The dependence of $f_c$ on $r$ for several distributions with identical mean group size of 4. Results obtained by the theory (curves) and results obtained by simulations (symbols) are shown for comparison. The case of $g=4$ refers to the case of NSG with a fixed group size of 4. The presence of $g=1$ groups induces the extinction of $D$-strategy for $r<r_c$ and leads to an AllC state.