# 5G Price Competition with Social Equilibrium Optimality for Social Networks
## 6 Numerical Results  
We need to expand the settings to get numerical results. In Example 1, the fraction of time restricted to the 5G wireless supplier's frequency band is the resource that is being sold, i.e., \(R_{m}=1\) for \(m\in\mathcal{M}\). Let \(W_{m}=700\) MHz, \(m\in\mathcal{M}\). \(W_{m}\) means the 5G wireless suppliers' bandwidth. \(a_{n}\text{log}(\sum_{m=1}^{M}r_{nm}c_{nm}+1)\) is user \(n\)'s utility function, in which the spectral efficiency \(c_{nm}\) from the Shannon formula \(\frac{1}{2}Wlog(1+\frac{E_{b}/N_{0}}{W}|h_{nm}|^{2})\), \(a_{n}\) is the \"willingness to pay\" factor, which we assume is the same among users, \(r_{nm}\) is allocated time fraction, and \(E_{b}/N_{0}\) is the transmit power divided by thermal noise. Suppose that the coding choices and modulation are perfect, with a continuum of values supplying a steady communication rate. Users are placed, then, uniformly in field that is \(500\times 500\) square meters. We only provide these parameters to elucidate our point; we can change the numbers and the theory still applies to the parameters. Let us think of an example in the 5G background with 5 suppliers and 20 users. In Figures 2 and 3, the equilibrium prices are represented as dashed lines. The competition among users can influence the equilibrium. Notice that supplier \(d\) offers a higher price than supplier \(a\). Therefore, although supplier \(d\) can provide better resource quality, the user's choice will also be affected by the equilibrium prices. Meanwhile, supplier \(b\) has the most buyers, its price is the highest, as Figures 2 and 3 shows. As for the convergence time in the discrete-time version: Suppose there are only five suppliers, then the number of users increases from 20 to 100. At each condition, we repeat the experiments 2000 times with random locations of suppliers and users. That way, we can obtain the average convergence speed and plot it. We define convergence as the number of iterations after which \(\varepsilon R_{m}\) is larger than the gap between demand and supply. For different \(\varepsilon\), Table 1 shows the average convergence time. Generally, if \(\varepsilon=10^{-3}\), the time to convergence is 2000-4000. If \(\varepsilon=10^{-4}\), the time to convergence is 3000-6000. Figure 3: Evolution of prices. The chart shows that when the equilibrium state is reached, the price of supplier \(d\) is approximately 1.5 times higher than that of supplier \(a\). Figure 2: Evolution of difference between demand and supply. In Table 2, we change the number of suppliers and see how the average convergence time changes. Now, let \(\varepsilon=10^{-4}\), and let it be constant. The update rates decide the convergence time: with low rates, the variables are likely to stabilize, and they will not take long to converge. In contrast, with high rates, the variables may converge rapidly. Based on Section 4's theoretical analysis, to obtain the algorithm's global convergence, let us distribute update variables randomly. Generally, the algorithm will iterate many times to converge if the ratio of users per supplier is too low or too large. At last, Table 3 presents the average convergence time when there are five suppliers with the standard variation. If the number of users is not 20, it does not affect the convergence time variance. If the number of users per supplier is under 4, update rates significantly impact the algorithm. In Table 2, we find that demands and prices vibrate and converge slowly in such instances.