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

To verify the above inference, we remove a number of edges in the square lattice and randomly add the same number of edges. During this randomizing process, the network connectivity is always guaranteed and the self-connections and multi-connections are not allowed. The number of removed edges, A, can be used to quantify the strength of delocalization. As shown in Fig. 4 , with the increasing of A, the self-protection strategy gets promoted and the clusters of uninfected laissez-faire individuals are fragmented into small pieces. When A gets larger and larger, the strategy distribution pattern becomes closer and closer to that of ER, BA and well-mixed networks. The gradually changing process in Fig. 4 clearly demonstrates that the main reason resulting in the quantitative differences is the structural localization effects. In a word, the ER, BA and well-mixed display essentially the same results since they do not have many localized clusters. Figures 5 and 6 report the degree and width of the Braess's paradox for well-mixed networks 25 (Figures S6, S7 and S8 present the degree and width of the Braess's paradox for the other three kinds of networks under investigation). The degree of the Braess's paradox is defined as D R 5 R max 2 R initial , where R max is the maximal value of epidemic size and R initial is the value of epidemic size when the Braess's paradox starts to happen. Fig. 5(a) presents an illustration about the definition of D R , and Fig. 5(b) plots the value of D R for different parameters. Each subpanel in Fig. 5(b) is associated with a given (l, m) pair with b and c being two variables. Analogously, the width of the Braess's paradox is defined as D d 5 d end 2 d start , where d start is the starting point corresponding to R initial and d end is the right point when the Braess's paradox disappears. Fig. 6(a) present an illustration about the definition of D d . Figure 6 (b) plots the value of D d in the similar way to Fig. 5(b) . The simulation results indicate that when l is very small (e.g., l 5 0.2 in the top panels of Fig. 5(b) and Fig. 6(b) ), the Braess's paradox disappears, and with the increasing of the value of l, the Braess's paradox becomes more obvious. The parameters b and c also affect the existence of Braess's paradox, for instance, when b and c are all close to 1, the Braess's paradox Lastly, we present an approximation analysis based on the meanfield theory for well-mixed networks (see analysis in the Methods section), which could reproduce the counter-intuitive phenomenon. Figure 7 compares the analytical prediction with simulation, indicating a good accordance.