Social imitation versus strategic choice, or consensus versus cooperation, in the networked Prisoner's Dilemma

The interplay of social and strategic motivations in human interactions is a largely unexplored topic in collective social phenomena. Whether individuals' decisions are taken in a purely strategic basis or due to social pressure without a rational background crucially influences the model outcome. Here we study a networked Prisoner's Dilemma in which decisions are made either based on the replication of the most successful neighbor's strategy (unconditional imitation) or by pure social imitation following an update rule inspired by the voter model. The main effects of the voter dynamics are an enhancement of the final consensus, i.e., asymptotic states are generally uniform, and a promotion of cooperation in certain regions of the parameter space as compared to the outcome of purely strategic updates. Thus, voter dynamics acts as an interface noise and has a similar effect as a pure random noise; furthermore, its influence is mostly independent of the network heterogeneity. When strategic decisions are made following other update rules such as the replicator or Moran processes, the dynamic mixed state found under unconditional imitation for some parameters disappears, but an increase of cooperation in certain parameter regions is still observed. Comparing our results with recent experiments on the Prisoner's Dilemma, we conclude that such a mixed dynamics may explain moody conditional cooperation among the agents.

Figure 1

Final active bonds density (top) and cooperator density (bottom) as a function of the dynamics mixing parameter $q$ and the punishment $\varepsilon$ for a system on an ER random network of size $N=3000$ and average degree $\langle k\rangle =8.48$. Caveat: the interval of parameter space investigated is $(\varepsilon ,q)\in [0,0.5]\times (0,1)$; notice that $q=0$ and $q=1$ are excluded.

Figure 2

(a),(b) Final active bonds density and (c),(d) cooperator density as a function of the mixing parameter $q$ and $\varepsilon$ for a system on networks of size $N=3000$. The network is (a),(c) an ER and (b),(d) scale free, with average degree $\langle k\rangle =5.14$. Caveat: the interval of parameter space investigated is $(\varepsilon ,q)\in [0,0.5]\times (0,1)$; notice that $q=0$ and $q=1$ are excluded.

Figure 3

Time evolution of the average active links for a system on networks of size $N=3000$; (a),(c) a random network and (b),(d) a scale free network, both with $\langle k\rangle =5.14$. (a),(b) Low value of $\varepsilon$; (c), (d) higher punishment parameter. In each subfigure, the curve with the smallest value of $q$ decays as a power law; the other curves (a) stay on a plateau and (c),(d) decay exponentially; (b) the curve for $q=0.7$ remains on a plateau, while the one for $q=0.95$ finally decays exponentially.

Figure 4

Dependence of the time needed to reach the final state $\tau$ as a function of the system size $N$. Operatively, $\tau$ was measured as the time when the active link density dropped below $0.07$. The networks have a (a) random or (b) scale free topology with average degree $\langle k\rangle =5.14$. The punishment has been set at $\varepsilon =0.05$ and $q=0.5$ and 0.7, respectively. In the exponential fits $\tau \sim exp\left(\gamma N\right)$, we have estimated $\gamma \simeq 0.0032\pm 0.0008$ for the random networks and $\gamma \simeq 0.0038\pm 0.0001$ for the scale free networks.

Figure 5

Final cooperator density for a system on a (a) random or (b) scale free network of size $N=3000$ and average degree $\langle k\rangle =5.14$. The punishment parameter is $\varepsilon =0.05$. $q=0$ means a purely strategy update of the PD; note that there is a slight increase of $n_c^{\infty }$ as soon as $q$ becomes larger than zero.

Figure 6

Time evolution of average active links for a system on a random network of size $N=3000$ and average degree $\langle k\rangle =8.48$ with punishment $\varepsilon =0.1$ (top figures), and $\varepsilon =0.4$ (bottom figures). Column (a): REP update rule. Column (b): MOR update rule.

Figure 7

Final cooperator density as a function of $q$ and $\varepsilon$ for a system on (a),(c) a random or (b),(d) scale free network of size $N=3000$ and average degree $\langle k\rangle =8.48$ and $\langle k\rangle =5.14$, respectively. (a),(b): REP update rule; (c),(d): MOR update rule. Caveat: the interval of parameter space investigated is $(\varepsilon ,q)\in [0,0.5]\times (0,1)$; notice that $q=0$ and $q=1$ are excluded.

Figure 8

Final cooperator density for a system on a random network of size $N=3000$ and average degree $\langle k\rangle =8.48$. The punishment parameter is $\varepsilon =0.10$. (a) REP update rule, (b) MOR rule. Note that since the final configuration is static, $n_c^{\infty }$ corresponds to the fraction of realizations ending in full cooperation.

Figure 9

Evidence for moody conditional cooperation on systems given by (a),(c) a random network and (b),(d) a scale free network, of size $N=3000$ and average degree $\langle k\rangle =5.14$. (a),(b) Frequency of cooperation after defection as a function of the fraction of cooperating neighbors (see text) in the previous round. (c),(d) Frequency of cooperation after cooperation as a function of the fraction of cooperating neighbors in the previous round. Note that the histograms are similar regardless of the network topology. The payoff matrix is the same as in [24] for comparison, with $\varepsilon =0$, $T=1.4$, and the payoff to a cooperator facing a defector vanishing as well; UI update rule.

Figure 10

Time evolution of the fractions of cooperators and active links on a scale free network of size $N=3000$ and average degree $\langle k\rangle =5.14$. The payoff matrix is the same as in [24] for comparison, with $\varepsilon =0$, $T=1.4$, and the payoff to a cooperator facing a defector vanishing as well; UI update rule.