Braess's Paradox in Epidemic Game: Better Condition Results in Less Payoff

Zhang, Hai-Feng; Yang, Zimo; Wu, Zhi-Xi; Wang, Bing-Hong; Zhou, Tao

We first study the model in square lattices with von Neumann neighborhood and periodic boundary conditions. Figure 1 (a) presents the effect of the successful rate of self-protection, d (see Methods section for the description of the model and parameters), on the decision makings of individuals and the epidemic size. Clearly, as the increasing of d, the condition gets better and better (the efficiency of selfprotection gets improved as the increase of d). A counter-intuitive phenomenon is observed when d lies in the middle range (,[0.3, 0.4]), during which a better condition leads to a larger epidemic size. One may think that though the epidemic size becomes larger, the system payoff (the sum payoff of all individuals) could still get higher since individuals pay less in choosing self-protection than vaccination. However, as shown in Fig. 1 (b) and Fig. 1 (c), the system payoff is strongly negatively correlated with the epidemic size. That is to say, a better condition (i.e., a larger d) could result in worse performance in view of both the larger epidemic size and the less system payoff (In Fig. S1 , we report the epidemic size as a function of d in square lattices with different sizes, the simulation results indicate that our main results are insensitive to the network sizes). This is very similar to the so-called Braess's Paradox, which states that adding extra capacity to a network when the moving entities selfishly choose their route, can in some cases reduce overall performance [20] [21] [22] . Figure 2 shows the strategy distribution patterns of four representative cases. When d is small, it is unwise to take self-protection because of its low efficiency, and people prefer to take vaccination or laissez faire. As shown in Fig. 2(b) , there are only two strategies, vaccination and laissez faire, and thus d has no effect on the epidemic size. Meanwhile, one can find that the infected laissez-faire individuals (dark red) and uninfected laissez-faire individuals (light red) are isolated by the vaccinated individuals and form respective percolating clusters. That is to say, the vaccinated individuals play the role of firewall in preventing the contagion of disease to the whole system. Of course, this kind of partial separation can only be possible when the number of vaccinated individuals is considerable, otherwise, the number of vaccinated individuals is insufficient to cut off the spreading paths of the disease. When d gets larger (,[0.3, 0.4]), the advantage of self-protection starts to appear, so more individuals will take self-protection and fewer individuals take vaccination or laissez faire. However, low efficiency of self-protection (i.e., small d) cannot offset the losses coming from the reduced vaccinated individuals, which leads to the increase of the epidemic size (see the dark red points in Fig. 2 (c)) as well as the decrease of the system payoff. Also as shown in Fig. 2(c) , the light-red percolating cluster (i.e., uninfected laissez-faire individuals) is fragmented into pieces due to the decrease of irrelevant individuals (the irrelevant individuals include both vaccinated and successful self-protective individuals, see Methods for the definition), which is also a reason of the decrease of the fraction of laissez-faire individuals: being laissez-faire becomes more risky now. When d is large, the superiority of self-protection becomes more striking and no one takes vaccination, then the epidemic size decreases as d incre