## 1 Introduction
** According to (21), \(\forall\hat{\mathbf{\omega}}^{*}\in Fix(\mathbf{O}_{\overline{\mathbb{B}}}\mathbf{O}_{ \widetilde{\Lambda}})\), we have

\[\|\hat{\mathbf{\omega}}^{\mathbf{k+1}}-\hat{\mathbf{\omega}}^{*}\|^{2}\] \[=\frac{1}{2}\|\hat{\mathbf{\omega}}^{\mathbf{k}}-\hat{\mathbf{\omega}}^{*}\|^ {2}+\frac{1}{2}\|\mathbf{O}_{\overline{\mathbb{B}}}\mathbf{O}_{\widetilde{\Lambda}} \hat{\mathbf{\omega}}^{\mathbf{k}}-\hat{\mathbf{\omega}}^{*}\|^{2}-\frac{1}{4}\|\mathbf{O}_{ \overline{\mathbb{B}}}\mathbf{O}_{\widetilde{\Lambda}}\hat{\mathbf{\omega}}^{\mathbf{k}}- \hat{\mathbf{\omega}}^{\mathbf{k}}\|^{2}\] \[\leq\frac{1}{2}\|\hat{\mathbf{\omega}}^{\mathbf{k}}-\hat{\mathbf{\omega}}^{*}\| ^{2}-\frac{1}{4}\|\mathbf{O}_{\overline{\mathbb{B}}}\mathbf{O}_{\widetilde{\Lambda}} \hat{\mathbf{\omega}}^{\mathbf{k}}-\hat{\mathbf{\omega}}^{\mathbf{k}}\|^{2}.\]

Since \(\overline{\mathbb{B}}\) is a maximal monotone operator and \(\overline{\Lambda}\) is a restricted monotone operator, the sequences \(\{\hat{\mathbf{\omega}}^{k}\}\) is Fejer monotone. Note that \(\sum_{k=1}^{\infty}\frac{1}{4}\|\mathbf{O}_{\overline{\mathbb{B}}}\mathbf{O}_{ \overline{\Lambda}}\hat{\mathbf{\omega}}^{\mathbf{k}}-\hat{\mathbf{\omega}}^{\mathbf{k}}\|^{2} \leq\|\hat{\mathbf{\omega}}^{\mathbf{0}}-\hat{\mathbf{\omega}}^{*}\|^{2}\), so \(\lim_{k\to\infty}|\mathbf{O}_{\overline{\mathbb{B}}}\mathbf{O}_{\overline{\Lambda}} \hat{\mathbf{\omega}}^{\mathbf{kthat each router can carry is finite, each client has a cost function \(f_{i}(x_{i},x_{-i})\) that depends on the decisions of all clients. Using the matrix \(A\), we can characterize the routers chosen by each client individually, and let \(x=\operatorname{col}(x_{1},x_{2}...x_{n})\), then \(Ax\) is the total supply vector to all routers given the action profile \(x\) of all clients. A constraint \(Ax<b\) is used to denote the bandwidth limit of these routers, where \(b=\operatorname{col}(b_{1},b_{2},...,b_{m})\in\mathcal{R}^{m}\). In addition, each client is restricted to exchange decisions only with its neighboring clients, and the adjacency relationship between clients is represented by a Laplace matrix. Then, the goal of each player in this model is to solve (3) in a situation where they can only communicate with adjacent clients. To satisfy Assumptions 1 and 2, we define function \(g_{i}(x_{i})=\eta_{i}(\sum_{j=1}^{n_{i}}[x_{i}]_{j})^{2}+\xi_{i}^{T}\), where \(x_{i},\eta_{i},\xi_{i}\in\mathcal{R}\). \(g_{i}(x_{i})\) is a strongly convex, Lipschitz continuous function. Define \(F:\mathcal{R}^{m}\to\mathcal{R}^{m}\) to map the amount paid by users connected to the router and the bandwidth they can allocate, then \(f_{i}(x_{i},x_{-i})=-F^{T}(Ax)A_{i}x_{i}\). Reference to economics [25], using a linear inverse demand function \(F_{j}(x)=\overline{F}_{j}-d_{j}[Ax]_{j}\) as the mapping \(F\). Denote gradient vector \(F(x)=\operatorname{col}(F_{1}(x),...,F_{m}(x))\), \(\overline{F}=\operatorname{col}(\overline{F_{1}},\overline{F_{2}},..., \overline{F_{m}})\), \(B=\operatorname{diag}\{d_{1},...,d_{m}\}\). Then \(F(x)=\overline{F}-BAx\) is the unit return vector function and the player's cost function can be written as \(g_{i}(x_{i})+\sum_{j=1}^{N}x_{j}^{T}A_{i}^{T}BA_{i}x_{i}-\overline{F}^{T}A_{i }x_{i}\). The experimental setup is described next, with 15 customers, i.e., \(N=15\), 4 routers, i.e., \(m=4\), and the total bandwidth that the 4 routers can carry being 9, 6, 12, and 8, i.e., \(b=\operatorname{col}(9,6,12,8)\). Clients 1, 3, 5, and 7 are connected to the first router; clients 2, 4, 6, and 8 are connected to the second router; clients 9, 10, and 13 are connected to the third router; and the remaining clients are connected to the fourth router, as shown in Figure 2.

As shown in Figure 3, Algorithms 1 and 2 and the DA algorithm based on the Doubly-Augmented operator splitting approach [13] all have good convergence results. Next, the cost functions of all participants are accumulated and gradually decrease with algorithm iterations.