Facilitators on networks reveal optimal interplay between information exchange and reciprocity

Reciprocity is firmly established as an important mechanism that promotes cooperation. An efficient information exchange is likewise important, especially on structured populations, where interactions between players are limited. Motivated by these two facts, we explore the role of facilitators in social dilemmas on networks. Facilitators are here mirrors to their neighbors—they cooperate with cooperators and defect with defectors—but they do not participate in the exchange of strategies. As such, in addition to introducing direct reciprocity, they also obstruct information exchange. In well-mixed populations, facilitators favor the replacement and invasion of defection by cooperation as long as their number exceeds a critical value. In structured populations, on the other hand, there exists a delicate balance between the benefits of reciprocity and the deterioration of information exchange. Extensive Monte Carlo simulations of social dilemmas on various interaction networks reveal that there exists an optimal interplay between reciprocity and information exchange, which sets in only when a small number of facilitators occupy the main hubs of the scale-free network. The drawbacks of missing cooperative hubs are more than compensated for by reciprocity and, at the same time, the compromised information exchange is routed via the auxiliary hubs with only marginal losses in effectivity. These results indicate that it is not always optimal for the main hubs to become leaders of the masses, but rather to exploit their highly connected state to promote tit-for-tat-like behavior.

Figure 1

Evolution of cooperation with and without facilitators on the square lattice. Depicted is the rescaled stationary fraction of cooperators ${\rho }_C$ on the whole $T\text{−}S$ parameter plane, as obtained in the absence of facilitators (red dotted lines) and with ${\rho }_F=0.05$ (green solid lines). It can be observed that facilitators do not change the qualitative properties of the solutions, but their presence does shift the survival barrier of cooperators towards harsher conditions, especially in the prisoner's dilemma quadrant (see the text). Here HG denotes the harmony game, SD denotes the snowdrift game, SH denotes the stag-hunt game, and PD denotes the prisoner's dilemma game.

Figure 2

Impact of facilitators on (a) the square lattice and (b) the random regular graph. Depicted is the stationary fraction of cooperators ${\rho }_C$ in dependence on $r$, as obtained for different fractions of facilitators occupying the network (see figure legend for the values of ${\rho }_F$). Due to the qualitatively identical results obtained on the two networks, it can be concluded that the topology of the interaction network does not play a notable role. More precisely, if the network remains degree homogeneous, then the randomness of interactions yields the same results as lattice-type models.

Figure 3

Characteristic distributions of cooperators [dark gray (blue)] and defectors [light gray (red)] on the $L\times L=100\times 100$ square lattice, as obtained for $r=-1$ (left) and $r=1$ (right) if the fraction of facilitators (gray) is sufficiently high for them to split the population in effectively isolated smaller fragments. The left panel depicts the outcome of the most lenient harmony game, yet still some defectors are able to survive. On the contrary, the right panel depicts the outcome of the harshest prisoner's dilemma game, yet cooperators survive. In both panels the fraction of facilitators is ${\rho }_F=0.5$.

Figure 4

Minimal fraction of facilitators that is necessary to avoid the tragedy of the commons (the pure $D$ state) as a function of $r=T-1$ for the prisoner's dilemma: Cooperation becomes viable above this threshold. Comparison of the mean field prediction ${\tilde{\rho }}_F=r/(1+r)$ (dotted line) with the value obtained by numerical simulations on random graphs with regular degree 4 (symbols). When the degree increases, the symbols would move towards the mean field line (not shown here). See also the main text for details.

Figure 5

Impact of facilitators on the scale-free network (a) if their placement is uniformly random regardless of the degree of players or if their placement is limited to players with (b) low or (c) intermediate degree. Depicted is the stationary fraction of cooperators ${\rho }_C$ in dependence on $r$, as obtained for different fractions of facilitators occupying the network (see the legend for the values of ${\rho }_F$). As in Fig. 2, increasing the value of ${\rho }_F$ will significantly extend the region where cooperators and defectors are able to coexist, especially if the facilitators are placed randomly (a). The results are obtained using $N={10}^5$ system size (see the text).

Figure 6

Impact of facilitators on the BA scale-free network if their placement is limited to players with high degree. Depicted is the stationary fraction of cooperators ${\rho }_C$ in dependence on $r$, as obtained for different fractions of facilitators occupying the targeted high-degree nodes (see the legend for the values of ${\rho }_F$ placed at the most connected nodes). These results reveal the existence of the optimal interplay between information exchange and reciprocity (see the main text for details). Compared to the results presented in Fig. 5, only in this particular case is it possible to combine the two effects to arrive at the best conditions for widespread cooperation. The results are obtained using $N={10}^5$ system size.