## 5 Numerical Experiments
### Performance of the Algorithm  
We now show the performance of the algorithm described in Section 3 to solve a linear-quadratic network Nash equilibrium problem (Example 1) and a linear-quadratic network GNEP (Example 2). We implemented three different versions of the algorithm according to the Lipschitz estimate used. We considered the estimate \(\Lambda_{0}\) provided in [12], and the new estimates \(\Lambda_{1}\) and \(\Lambda_{2}\) introduced in Theorems 2 and 3, respectively. **Example 1**. : _We consider a linear-quadratic network Nash equilibrium problem as in the framework of Section 4.2, with a network of 8 nodes (players) as shown in Figure 2.__The spectral radius of the adjacency matrix \(G\) is \(\rho(G)\simeq 3.1019\). We set the function \(k(t)=t+5\), with \(t\in[0,15]\). We set two different values for parameter \(\phi\): \(\phi=0.1/\rho(G)\) or \(\phi=0.5/\rho(G)\), so that there exists a unique Nash equilibrium in both cases. We set the upper bounds \(U_{i}(t)=\min\{2i+(t+5)/2,18\}\) for any \(i=1,\ldots,8\). We recall that the map \(F\) in (28) satisfies the assumptions of Theorems 2 and 3 with constants \(\alpha=1-\phi\rho(G)\), \(L_{x}=\|I-\phi G\|_{2}\), \(L_{t}=\sqrt{8}\), \(L=\max\{L_{x},L_{t}\}\), and the feasible region_  
\[A(t)=\prod_{i=1}^{8}[0,U_{i}(t)]\]  
_satisfies Assumption (3) with constant \(M=\|(L_{1},\ldots,L_{8})\|_{2}\), where \(L_{i}\) is the Lipschitz constant of the function \(U_{i}\) for any \(i=1,\ldots,8\). We run the algorithm to approximate the fifth component of the solution vector \(x(t)\), due to the special position of the player 5 in the network. To evaluate the function \(x_{5}(t)\) during the execution of the algorithm, i.e., to find Nash equilibria of the problem, we exploited the ad hoc algorithm proposed in [39]._  
_Table 2 shows the number of iterations performed (or Nash equilibria computed) by the three versions of the algorithm (based on the estimates \(\Lambda_{0}\), \(\Lambda_{1}\) and \(\Lambda_{2}\)) for different values of \(\phi\) and the error tolerance \(\epsilon_{tot}\). We set the maximum number of iterations equal to \(10^{6}\). Results show that the algorithm based on the estimate \(\Lambda_{2}\) outperforms the other two versions based on the estimates \(\Lambda_{0}\) and \(\Lambda_{1}\). Table 3 shows the convergence of the approximate mean value of \(x_{5}(t)\) (computed by the algorithm based on \(\Lambda_{2}\)) for different values of \(\phi\)._ \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{3}{c}{\(\phi=0.1\rho(G)\)} & \multicolumn{3}{c}{\(\phi=0.5/\rho(G)\)} \\ \cline{2-7} \(\epsilon_{tot}\) & **Alg with \(\Lambda_{0}\)** & **Alg with \(\Lambda_{1}\)** & **Alg with \(\Lambda_{2}\)** & **Alg with \(\Lambda_{0}\)** & **Alg with \(\Lambda_{1}\)** & **Alg with \(\Lambda_{2}\)** \\ \hline
1 & 5107 & 1799 & 640 & 15,383 & 5246 & 1714 \\
0.5 & 10,213 & 3597 & 1277 & 30,765 & 10,490 & 3426 \\
0.2 & 26,069 & 8554 & 3260 & 78,858 & 26,431 & 7995 \\
0.1 & 52,136 & 17,107 & 6518 & 157,715 & 52,860 & 15,989 \\
0.05 & 104,270 & 34,212 & 13,034 & 315,428 & 105,719 & 31,977 \\
0.02 & 246,321 & 92,250 & 30,788 & 815,228 & 249,976 & 85,721 \\
0.01 & 492,640 & 184,499 & 61,575 & \(>1\times 10^{6}\) & 499,950 & 171,441 \\ \hline
\begin{tabular}{c} Lipschitz \\ estimate \\ \end{tabular} & \(\Lambda_{0}=88.46\) & \(\Lambda_{1}=33.18\) & \(\Lambda_{2}=11.18\) & \(\Lambda_{0}=276.49\) & \(\Lambda_{1}=96.82\) & \(\Lambda_{2}=30.31\) \\ \end{tabular}
\end{table} 
Table 2: Example 1 with a moving feasible region: number of iterations of the algorithm based on the Lipschitz estimate used. Figure 2: Network topology of Example 1. _Moreover, we run the algorithm to solve the problem in the case the feasible region \(A(t)\) is constant, with \(U_{i}(t)=18\) for any \(t\) and \(i=1,\ldots,8\), and the map \(F\) is the same as before. Tables 4 and 5 show the number of iterations of the algorithm and the approximate mean value of \(x_{5}(t)\), respectively.