Final Level of Cooperation

Now we want to compare the simulation results and the two semi-analytical approximations. In Figure 4.8, we consider w= 0, w= 0.05, w= 0.1, and w= 0.5, plotting the fraction of nodes in

state C in the stationary state of the networks against the cost-to-benefit ratio, u. There are three sets of outcomes here: markers are the simulation results, dotted lines are the PA approximation, and the dashed lines are the AME approximation. Although this figure only shows us the final fraction ofC’s, one also knows the final fraction ofD’s simply because the sum ofC and Dratios should always be 1.

In the casew= 0, since there are no strategy updates happening in the networks, nodes couldn’t

change their states during the evolution. Hence, the final fraction ofC andD will remain constant. Since first we set⇢= 0.5, regardless ofu we should get 50% ofC and 50% ofD in the stationary states. In Figure 4.8, simulations, PA, and AME give us the same results, final ratio of C stays constant of 0.5 with regard tou. Therefore, both the PA and AME approximations are satisfying. When w >0, the strategy updating starts playing a role. The greater the cost-to-benefit uis, the stronger the incentive for nodes to defect. Hence, the final fraction of cooperators decreases as u increases. In the casesw= 0.1andw= 0.5, simulation results decrease slowly at first and suddenly

drop to zero around u = 0.2 and u = 0.6, respectively. This phenomenon is captured by AME

approximation not only by the qualitative behavior but also the critical dropping value ofu. On the other hand, although PA gives good qualitative estimation of the behavior, PA lines both drop earlier and fail to provide a good prediction of critical points. Hence, comparing to PA, AME offers better predictions of the simulation. Lastly, in the case w= 0.05, the simulation results gradually

decrease whenuincreases. The gaps between simulation results and PA and AME gradually increase. The PA curve drops to zero aroundu= 0.85, however, the AME curve does not have this qualitative

change. Therefore, throughout all the scenarios listed in this figure, AME always provides fitting qualitative approximations of the simulation results, and it gives accurate predictions at the critical points.

Similarly, we plot the final fraction in the stationary state of cooperatorsCversuswwith different cost-to-benefit ratios,u in Figure 4.9. Here we consider four cases: u= 0,u= 0.01,u= 0.2, and

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Cost−to−benefit ratio, u Fraction of cooperators w = 0 w = 0.05 w = 0.1 w = 0.5

Figure 4.8: Fraction of cooperators versus cost-to-benefit ratiou with different strategy updating rates win stationary states. Markers are the averages of 1,000 simulations results, dotted lines are the semi-analytical results of pair approximation (PA), and the dashed lines are the semi-analytical results of approximate master equations (AMEs). The PA results are different from the ones shown in [46], but we have confirmed the accuracy of our results in personal communication with the authors of that paper.

updating in the dynamics. Hence, all of the nodes only rewire but do not change their states. The final fraction ofC’s remains 50% in the networks. As wincreases, the effect of strategy updating in the dynamics becomes stronger. In the cases u= 0and u= 0.01,u is so small that there is almost no incentive to defect, so in the strategy updating process C’s and D’s are almost indistinguishable. However, because of the effect of rewiring, C would drop its defective neighbor D, reducing the number of partners forD. Comparing withC nodes,Dnodes would not have as many neighbors and hence would have less payoff. This effect makes more D’s change toC’s and leads to the dominance ofC’s whenu is small. In the casesu= 0andu= 0.01, both PA and AME provide accurate results

and behave well qualitatively.

When u= 0.8, asw increases,C nodes become more disadvantaged since the payoff of defecting is high. The final fraction of cooperators decreases to zero aroundw= 0.1, and both PA and AME

capture this critical value ofw well. Although both approximations have humps whenwis small, they still give satisfying predictions and AME does not do worse than PA.

Finally, in the case u = 0.2, the incentive of defecting is not too large. The final fraction of

cooperators would increase at first due to the rewiring effect, that is, C nodes gain more neighbors and have higher payoffs when comparing the payoffs withD nodes. But whenw gets greater, the effect of rewiring diminishes, C nodes would not have more neighbors rewiring to them and start losing payoffto their Dneighbors. This effect makes more C’s becomeD’s and the networks will be dominated by defectors. Both of the semi-analytical approaches capture this qualitative change of the dynamics. The fraction of final cooperators curve drop aroundw= 0.35,w= 0.2, and w= 0.4

for simulation, PA, and AME, respectively. Though not perfect, AME in this case still yields more accurate prediction than PA. Hence, generally speaking, we can see that AME improves the results PA could acquire in this figure.