Evolutionary dynamics of the traveler's dilemma and minimum-effort coordination games on complex networks

The traveler's dilemma game and the minimum-effort coordination game are social dilemmas that have received significant attention resulting from the fact that the predictions of classical game theory are inconsistent with the results found when the games are studied experimentally. Moreover, both the traveler's dilemma and the minimum-effort coordination games have potentially important applications in evolutionary biology. Interestingly, standard deterministic evolutionary game theory, as represented by the replicator dynamics in a well-mixed population, is also inadequate to account for the behavior observed in these games. Here we study the evolutionary dynamics of both these games in populations with interaction patterns described by a variety of complex network topologies. We investigate the evolutionary dynamics of these games through agent-based simulations on both model and empirical networks. In particular, we study the effects of network clustering and assortativity on the evolutionary dynamics of both games. In general, we show that the evolutionary behavior of the traveler's dilemma and minimum-effort coordination games on complex networks is in good agreement with that observed experimentally. Thus, formulating the traveler's dilemma and the minimum-effort coordination games on complex networks neatly resolves the paradoxical aspects of these games.

Figure 1

Long-term average claims ${\overline{x}}_{\infty }\left(R\right)$ in the TD game (calculated over the last 20% of 20 000 generations) versus the reward or punishment parameter $R$, on (a) a complete network, (b) a random regular network with degree $k=4$, (c) degree $k=8$, (d) an exponential network with mean degree $k=4$, (e) mean degree $k=8$, (f) a scale-free network with mean degree $k=4$, (g) mean degree $k=8$, (h) a two-dimensional lattice network with $k=4$ neighbors, and (i) $k=8$ neighbors. Parameter values: $n=10000$ and $\mu =0.05$. Each data point shown in the plots was obtained by averaging over ten separate runs of the model.

Figure 2

Values of the cooperation index $\Lambda$ in the TD game on scale-free networks with (a) clustering coefficient $C=0.0,0.1,0.2,0.3,0.4,0.5$ and on scale-free networks with (b) assortativity coefficient $r=-0.1,0.0,0.1,0.2$. Parameter values: $n=10000$ and $\mu =0.05$. Each data point in the figure represents the mean value of $\Lambda$ obtained from 20 different runs of the model, and the error bars (which are approximately the same size as the point markers) show the variance in $\Lambda$.

Figure 3

Long-term average claims ${\overline{x}}_{\infty }\left(R\right)$ in the TD game (calculated over the last 20% of 20 000 generations) versus the reward or punishment parameter $R$ on (a) the coauthorship network and on (b) a random network with the same degree sequence as the coauthorship network but with no clustering or assortativity. (c) Number of individuals making a certain claim during the last generation in the TD game on the coauthorship network. Parameter values: $n=5835$ and $\mu =0.05$. Each data point shown in the plots was obtained by averaging over ten separate runs of the model.

Figure 4

Long-term average effort levels ${\overline{x}}_{\infty }\left(\kappa \right)$ in the MEC game (calculated over the last 20% of 20 000 generations) versus the effort cost parameter $\kappa$ on (a) a complete network, (b) a random regular network with degree $k=4$, (c) degree $k=8$, (d) an exponential network with mean degree $k=4$, (e) mean degree $k=8$, (f) a scale-free network with mean degree $k=4$, (g) mean degree $k=8$, (h) a two-dimensional lattice network with $k=4$ neighbors, and (i) $k=8$ neighbors. Parameter values: $n=10000$ and $\mu =0.05$. Each data point shown in the plots was obtained by averaging over ten separate runs of the model.

Figure 5

Variation in the cooperation index $\Lambda$ in the MEC game on scale-free networks with (a) clustering coefficient $C=0.0,0.1,0.2,0.3,0.4,0.5$ and (b) scale-free networks with assortativity coefficient $r=-0.1,0.0,0.1,0.2$. Parameter values: $n=10000$ and $\mu =0.05$. Each data point in the figure represents the mean value of $\Lambda$ obtained from 20 different runs of the model, and the error bars (which are approximately the same size as the point markers) show the variance in $\Lambda$.

Figure 6

Long-term average effort levels ${\overline{x}}_{\infty }\left(\kappa \right)$ in the MEC game (calculated over the last 20% of 20 000 generations) versus the effort cost parameter $\kappa$ on (a) the coauthorship network and (b) a random network with the same degree sequence as the coauthorship network but without clustering or assortativity. (c) Number of individuals making a certain effort during the last generation in the MEC game on the coauthorship network. Parameter values: $n=5835$ and $\mu =0.05$. Each data point shown in the plots was obtained by averaging over ten separate runs of the model.