## 1 Introduction
Let \(\mathbf{\lambda}=\text{col}((\lambda_{i})_{i\in\mathcal{N}})\in\mathcal{R}^{Nm}\), \(\mathbf{L}=L\bigotimes I_{m}\in\mathcal{R}^{Nm\times Nm}\), where \(L\) is the Laplacian matrix and \(I_{m}\) is a unit matrix of order m. In order to solve for v-GNEs, it is necessary to add the equality constraint \(\mathbf{L}\mathbf{\lambda}=\mathbf{0}\), and the augmented Lagrangian function is \(\mathbb{L}_{i}(x_{i},x_{-i},\lambda_{i},z_{i})=\mathbb{J}_{i}(x_{i},x_{-i})+ \lambda_{i}^{T}(Ax-b)+\sum_{i=1}^{N}\sum_{j=1}^{N}z_{i}^{T}L_{ij}\lambda_{j}\), where \(z_{i}\in\mathcal{R}^{m}\) serves to satisfy the coupling constraints and coordinate the local multipliers \(\lambda_{i}\) to bring them to consensus. Let \(\mathbf{z}=\text{col}(z_{1},z_{2},...,z_{N})\in\mathcal{R}^{Nm}\), there have

\[\mathbb{L}_{i}(x_{i},x_{-i},\lambda_{i},z_{i})=\mathbb{J}_{i}(x_{i},x_{-i})+ \lambda_{i}^{T}(Ax-b)+\mathbf{z}^{T}\mathbf{L}\mathbf{\lambda},\quad i\in\mathcal{N}. \tag{5}\]

Suppose \((x_{i}^{*},x_{-i}^{*})\) is the optimal solution of (5) and it follows from the definition of the generalized Lagrangian function that \((x_{i}^{*},x_{-i}^{*})\) is also the variational GNE of the game (3), then \(x^{*}\) satisfies Formula (4). Let \(\mathbf{A}=\text{diag}(A_{i})_{i\in\mathcal{N}}\), \(\mathbf{b}=\text{col}(b_{i})_{i\in\mathcal{N}}\), when \(\lambda_{1}=\lambda_{2}=...=\lambda_{N}=\lambda^{*}\), the KKT condition for a v-GNE is

\[ \begin{cases}0\in\mathbb{F}(x)+\mathbb{G}(x)+A^{T}\mathbf{\lambda},\\ 0=\mathbf{L}\mathbf{\lambda},\\ 0\in\mathbf{b}+\mathbf{L}\mathbf{\lambda}+N_{\mathbb{R}_{\geq 0}^{Nm}}(\mathbf{\lambda})-\mathbf{ Ax}-\mathbf{Lz},\end{cases}  \tag{6}\]

where \(\mathbb{F}(x)=\text{col}(\
abla_{x_{i}}f_{i}(x_{i},x_{-i}))_{i\in\mathcal{N}}\), \(\mathbb{G}(x)=\text{col}(\partial_{x_{i}}g_{i}(x_{i}))_{i\in\mathcal{N}}\). ## 3 Distributed Algorithm

### Global Information Setting

In this subsection, the derivation of Algorithm 2 is given based on the assumption that the global decision set of the game model is public. Next, we need to use the KKT condition obtained in the previous section to derive two operators. Using these two operators and a positive definite matrix, we can apply the DR operator splitting method to obtain Algorithm 2. Equation (10) is the standard form of the DR operator splitting method. Let \(\mathbf{v}=[x^{T},z^{T},\lambda^{T}]^{T}\) and define \(\mathbb{A}\), \(\mathbb{B}\) as

\[\mathbb{A}:\mathbf{v}\longmapsto\left[ \begin{array}{c}\mathbb{F}(x)\\ \mathbf{0}\\ b\end{array} \right]+\frac{1}{2}\left[ \begin{array}{ccc}\mathbf{0}&\mathbf{0}&\mathbf{A} ^{T}\\ \mathbf{0}&\mathbf{0}&\mathbf{L}\\ -\mathbf{A}&-\mathbf{L}&\mathbf{0}\end{array} \right]\mathbf{v}, \tag{7}\]

\[\mathbb{B}:\mathbf{v}\longmapsto\left[ \begin{array}{c}\mathbb{G}(x)\\ \mathbf{0}\\ N_{\mathbb{R}_{\geq 0}^{Nm}}(\mathbf{\lambda})\end{array} \right]+\frac{1}{2}\left[ \begin{array}{ccc}\mathbf{0}&\mathbf{0}&\mathbf{A}^{T}\\ \mathbf{0}&\mathbf{0}&\mathbf{L}\\ -\mathbf{A}&-\mathbf{L}&\mathbf{0}\end{array} \right]\mathbf{v}, \tag{8}\]

then the KKT condition (6) is equal to \(0\in\mathbb{A}(\mathbf{v})+\mathbb{B}(\mathbf{v})\). Denote \(\Phi\) as

\[\Phi=\frac{1}{2}\left[ \begin{array}{ccc}\mathbf{0}&\mathbf{0}&-\mathbf{A}^{T}\\ \mathbf{0}&\mathbf{0}&-\mathbf{L}\\ -\mathbf{A}&-\mathbf{L}&\mathbf{0}\end{array} \right]+\left[ \begin{array}{ccc}\alpha^{-1} &\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\tau^{-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\gamma^{-1}\end{array} \right], \tag{9}\]

where \(\alpha=\text{diag}(\alpha_{i}\mathbf{I})\), \(\tau=\text{diag}(\tau_{i}\mathbf{I})\), \(\gamma=\text{diag}(\gamma_{i}\mathbf{I})\) are fixed step in the iteration, and \(r_{k}\) meets \(\sum_{k\in\mathcal{N}}r_{k}(1-r_{k})=\infty\), (\(0<r_{k}<1\)). Then we can design the Algorithm 2:\[