Reacting Differently to Adverse Ties Promotes Cooperation in Social Networks

We investigate how diversity in individual responses to unwanted interactions affects the evolution of cooperation modeled as a 2-person prisoner’s dilemma. We combine adaptive networks and evolutionary game theory, showing analytically how the coevolution of social dynamics, network dynamics, and behavioral differences benefit the entire community even though myopic individuals still act in their own interest. As defectors are wiped out, surviving cooperators maintain the full diversity of behavioral types, establishing cooperation as a robust evolutionary strategy. The present framework can be used in other problems where the feedback between topology and dynamics affects the overall behavior of the system.

Figure 1

Transition probabilities and stationary distributions for the PD with $M=2$ types of individuals. Solid black (dashed grey) arrows indicate those transitions, computed analytically (numerically), for which the fixation probability is greater than neutral fixation, ${\rho }_N=1/N$ (see main text for details). Large (small) circles indicate the analytical (numerical) values for the stationary distribution. FC’s remain in the population while $D$’s dominate due to the direct FD-SD transition and the transient nature of SC ($N=100$, $T=2.1$, $R=2$, $S=0.9$, $P=1$, $\alpha =0.4$, $\beta =0.01$, ${\gamma }_S=0.25$, and ${\gamma }_F=0.75$).

Figure 2

Impact of behavioral differences. (a) Population spends more time in a cooperative state (of any type) when the number of types increases ($M$), irrespective of the temptation to defect $T$ in the PD ($N=100$, $R=2$, $P=1$, $S=3-T$, $\alpha =0.4$, $\beta =0.01$, $\delta =0.25$). (b) Most types of $C$’s end up equally represented, whereas only the slower $D$’s survive. In contrast to Fig. 1, the surviving $D$’s exhibit behavioral differences, which acts to inhibit the dominance of their slowest type, providing an escape hatch for cooperation ($M=50$, $T=2.1$, $R=2$, $P=1$, $S=0.9$).

Figure 3

Transition probabilities and stationary distributions for the PD game with $M=3$ (a) and $M=10$ (b) types of individuals. $C$’s and $D$’s are numbered according to increasing $\gamma$, $C1$ ($D1$) being the slowest $C$ ($D$). We use the same notation of Fig. 1. Increasing the number of possible types splits the outflow of fast $D$’s toward other types among a wide range of different possibilities. Cooperation increases, since only few types of $D$’s are evolutionarily stable, whereas the majority of $C$ types work as flow sinks. ($N=100$, $T=2.1$, $R=2$, $S=0.9$, $P=1$, $\alpha =0.4$, $\beta =0.01$, $\delta =0.25$).