## 5 Numerical Experiments
### Performance of the Algorithm
Notice that, in this case, the Lipschitz estimates \(\Lambda_{1}\) and \(\Lambda_{2}\) coincide, since \(L=L_{t}\) and the algorithm based on such an estimate is much more efficient than the one based on \(\Lambda_{0}\)._  
**Example 2**. : _We now consider a linear-quadratic network generalized Nash equilibrium problem, as in the framework of Section 4.3. We assume that the player network is the same as Example 1. We set the function \(k(t)=t+5\), with \(t\in[0,15]\), \(\phi=0.1/\rho(G)\) or \(\phi=0.5/\rho(G)\), and \(K(t)\) defined as in (38) with \(C(t)=\min\{13+t,18\}\). 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 satisfies assumption (3) with constant \(M=1\). In order to find the unique variational equilibrium \(x(t)\) of the problem, we found the maximizer of the potential function (30) by means of the MATLAB function quadprog. As in Example 1, we run the algorithm to approximate the fifth component of the solution vector \(x(t)\)._  
_Table 6 shows the number of iterations performed by the three versions of the algorithm (based on the estimates \(\Lambda_{0}\), \(\Lambda_{1}\) and \(\Lambda_{2}\)) for different values of \(\phi\) and \(\epsilon_{tot}\). Results show again that the algorithm based on the estimate \(\Lambda_{2}\) outperforms the other two versions based on the estimates \(\Lambda_{0}\) and \(\Lambda_{1}\). Table 7 shows the convergence of the approximate mean value of \(x_{5}(t)\) for different values of \(\phi\)._ \begin{table}
\begin{tabular}{c c c c c c c} \hline \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 \(1\) & 2835 & 179 & 179 & 7985 & 343 & 343 \\ \(0.5\) & 5668 & 357 & 357 & 15,968 & 684 & 684 \\ \(0.2\) & 13,983 & 881 & 881 & 43,207 & 1701 & 1701 \\ \(0.1\) & 27,965 & 1760 & 1760 & 86,413 & 3401 & 3401 \\ \(0.05\) & 55,928 & 3519 & 3519 & 172,824 & 6800 & 6800 \\ \(0.02\) & 128,807 & 8111 & 8111 & 433,801 & 15,862 & 15,862 \\ \(0.01\) & 257,613 & 16,220 & 16,220 & 867,601 & 31,722 & 31,722 \\ \hline Lipschitz estimate & \(\Lambda_{0}=45.07\) & \(\Lambda_{1}=3.14\) & \(\Lambda_{2}=3.14\) & \(\Lambda_{0}=138.59\) & \(\Lambda_{1}=5.66\) & \(\Lambda_{2}=5.66\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Example 1 with a constant feasible region: number of iterations of the algorithm based on the Lipschitz estimate used. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\epsilon_{tot}\) & \(\phi=0.1/\rho(G)\) & \(\phi=0.5/\rho(G)\) \\ \hline \(1\) & 13.0189685363 & 15.2238272767 \\ \(0.5\) & 13.0189711251 & 15.2238276377 \\ \(0.2\) & 13.0189724192 & 15.2238277070 \\ \(0.1\) & 13.0189727295 & 15.2238277328 \\ \(0.05\) & 13.0189728137 & 15.2238277376 \\ \(0.02\) & 13.0189728309 & 15.2238277386 \\ \(0.01\) &  
mean value of the solution by modifying an algorithm previously derived to approximate black box Lipschitz functions. We then applied our results to two class of games played on networks, where we assumed that some terms of the players' utility functions as well as the constraint set could vary with time. In future work, we plan to investigate the case where also the adjacency matrix of the graph is time-dependent. [PERSON] and [PERSON] have contributed equally to this work. All authors have read and agreed to the published version of the manuscript. This research was partially supported by the research project \"Programma ricerca di ateneo UNICT 2020-22 linea 2-OMNIA\" of the University of Catania. This support is gratefully acknowledged. The data presented in this study are available in the article.