Cooperation and age structure in spatial games

We study the evolution of cooperation in evolutionary spatial games when the payoff correlates with the increasing age of players (the level of correlation is set through a single parameter, $\alpha$). The demographic heterogeneous age distribution, directly affecting the outcome of the game, is thus shown to be responsible for enhancing the cooperative behavior in the population. In particular, moderate values of $\alpha$ allow cooperators not only to survive but to outcompete defectors, even when the temptation to defect is large and the ageless, standard $\alpha =0$ model does not sustain cooperation. The interplay between age structure and noise is also considered, and we obtain the conditions for optimal levels of cooperation.

Figure 1

Asymptotic fraction of cooperators ${\rho }_{\mathrm{C}}$ as a function of the temptation $T$ for different values of $\alpha$ and $S$, with $K=0.1$ and $A_{\max }=100$. The topmost row, with $S=0$, is the weak PD game while the second and third row have $S>0$, that is, the SD game. Since for higher values of $S$ it is better, against a defector, to cooperate, the overall increase in cooperation is not a surprise. Notice that for the original model, $\alpha =0$, cooperation vanishes for not too large temptations and cooperators never fully occupy the system, ${\rho }_{\mathrm{C}}<1$. For finite $\alpha$ and small values of $T$, cooperators fully invade the system and there is an optimum $\alpha$ for which cooperation is enhanced. Apart the crossing between some of the curves for $S>0$, the overall behavior is the same in all cases. The last row presents results for the weak PD on a random regular graph (the previous figures were for the square lattice). Cooperation is enhanced when compared with the original $\alpha =0$ case as well, showing that persistent neighbors (in contrast to well-mixed ones) is essential, while the small loops present in the square lattice, but absent in the random graph, do not qualitatively change the picture.

Figure 2

Phase diagram showing the critical threshold (red box symbols) above which cooperators go extinct, along with the pure C and coexisting phases. As $\alpha$ increases, there is at first a fast increase both in the cooperators dominated phase and in the coexistence one. After the optimal cooperation happening at $\alpha \simeq 2.5$, further increasing $\alpha$ is detrimental for cooperation, although the decrease is slow.

Figure 3

Time evolution of cooperation for different values of $\alpha$, $T=1.125$, $S=0$, and $K=0.1$. Since the initial state is random and cooperators are not yet organized in compact groups, many of them are easy prey for defectors and there always is an initial decrease on their number. For $\alpha =0$ and 20, cooperators do not recover from the initial downfall, the curves are monotonously decreasing and they get extinct. Inset: the fraction of agents that did not change strategy up to the time $t$ [the persistence $P\left(t\right)$]. Notice that for $\alpha =0$ the decay is exponential while it does become stretched for $\alpha \ne 0$. Moreover, for the cases presenting cooperation at long times ($\alpha =1$ and 2), the persistence decays more slowly, showing that the initially formed clusters are more resistent (the long lasting agents are cooperators).

Figure 4

Lattice snapshots for different values of $\alpha$ on a ${400}^2$ square lattice, $S=0$, $T=1.2$, and $K=0.1$, showing the clusterized cooperators after ${10}^5$ MCS.

Figure 5

Stationary average size of cooperators clusters for a ${400}^2$ lattice with $T=1.2$, $S=0$, and $K=0.1$. Inset: corresponding number of cooperators clusters, $N_{\mathrm{C}}$.

Figure 6

Full $K-T$ phase diagram for $\alpha =0$, 2, and 20 and $S=0$. Green circles and red boxes mark the border between pure C and D phases and the mixed C+D phase, respectively. For $\alpha =0$ (top panel) [56], the presence of some uncertainty in the strategy dynamics ($K>0$) leads to an optimal level of cooperation (the upper curve has a maximum at $K\simeq 0.3$). For small $\alpha$ (middle panel), the optimal noise threshold shifts toward zero but, when compared with the $\alpha =0$ case, is larger for all values of $K$. Upon further increasing $\alpha$, as shown in the bottom panel, the system becomes once again more noise tolerant. However, there is a huge difference between all these cases: the pure C phase is greatly enhanced at the expense of the coexistence region whenever $\alpha \ne 0$. Indeed, only for $\alpha =0$ the C+D regions is prominent. Moreover, the large $\alpha$ case also presents an optimal value of $T$ for the transition between the pure and mixed phases ($K\simeq 0.25$). Thus, by adjusting the level of heterogeneity one can either increase the amount of cooperation (small $\alpha$) or the robustness to noise (large $\alpha$).