## 1 Introduction
* The theoretical results in this paper are verified by numerical experiments and the advantages of Algorithm 1 are shown by comparison with the existing method. ```
Initialize: For each player \(i\in\mathcal{N}\), let \(x_{i,0}\in\mathcal{R}^{n_{i}}\), \(z_{i,0}\in\mathcal{R}^{m}\) and \(\lambda_{i,0}\in\mathcal{R}^{m}\) for\(k=1,2,3,...\)do \(\tilde{x}_{i,k}=\arg\min_{x}(\frac{1}{2}\|x-(x_{i,k}-\alpha_{i}A_{i}^{T}\lambda _{i,k})\|^{2}+\alpha_{i}f_{i}(x_{i},x_{-i}))\) \(\tilde{z}_{i,k}=\tau_{i}\sum_{j=1}^{N}w_{ij}(\lambda_{j,k}-\lambda_{i,k})+ \tilde{z}_{i,k}\) \(\tilde{\lambda}_{i,k}=\lambda_{i,k}-\frac{1}{2}\gamma_{i}A_{i}x_{i,k}-\frac{1 }{2}\gamma_{i}\sum_{j=1}^{N}w_{ij}(z_{i,k}-z_{j,k})-\gamma_{i}b_{i}+\gamma_{i} A_{i}\tilde{x}_{i,k}+\gamma_{i}\sum_{j=1}^{N}w_{ij}(\tilde{z}_{i,k}-\tilde{z}_{j,k})\) \(\tilde{x}_{i,k}=2\tilde{x}_{i,k}-x_{i,k}\) \(\tilde{z}_{i,k}=2\tilde{z}_{i,k}-z_{i,k}\) \(\tilde{\lambda}_{i,k}=2\tilde{\lambda}_{i,k}-\lambda_{i,k}\) \(\tilde{\pi}_{i,k}=prox_{a_{i}g_{i}}(\tilde{\lambda}_{i,k}-\frac{1}{2}\alpha_{i }A_{i}^{T}\tilde{\lambda}_{i,k})\) \(\tilde{\pi}_{i,k}=\frac{1}{2}\gamma_{i}\sum_{j=1}^{N}w_{ij}(\tilde{\gamma}_{i,j }-\tilde{\lambda}_{i,k})+\tilde{z}_{i,k}\) \(\tilde{\pi}_{i,k}=proj_{\pi_{0}^{m}}(\tilde{\lambda}_{i,k}-\frac{1}{2}\gamma_{ i}A_{i}\tilde{\pi}_{i,k}-\frac{1}{2}\gamma_{i}\sum_{j=1}^{N}w_{ij}(\tilde{z}_{i,k}- \tilde{z}_{j,k})+\gamma_{i}A_{i}\tilde{\pi}_{i,k}+\gamma_{i}\sum_{j=1}^{N}w_{ ij}(\tilde{z}_{i,k}-\tilde{z}_{j,k}))\) \(x_{i,k+1}=x_{i,k}+2r_{k}\overline{x}_{i,k}-2r_{k}\widetilde{x}_{i,k}\) \(z_{i,k+1}=z_{i,k}+2r_{k}\overline{z}_{i,k}-2r_{k}\overline{z}_{i,k}\) \(\lambda_{i,k+1}=\lambda_{i,k}+2r_{k}\overline{\lambda}_{i,k}-2r_{k}\overline{ \lambda}_{i,k}\) endfor
```

**Return:** The sequence \((x_{i,k})_{k=1}^{\infty}\) will eventually approximate the optimal solution. ```

**Algorithm 2** GNE seeking under global-decision

In this paper, Section 2 derives the KKT conditions of the variational problem of the original game using dual analysis method; Section 3 proposes distributed Algorithms 1 and 2 based on DR; Section 4 analyzes the convergence of the two algorithms; Section 5 presents the simulation results of these two algorithms; and Section 6 summarizes the whole paper and looks forward to the future. ## 2 Preliminary Acquaintance

### Graph Theory

The graph \(\mathcal{G}\) is mainly represented by two sets, namely the set of vertices \(\mathcal{N}\) and the set of edges \(\varepsilon\), i.e., \(\mathcal{G}=\{\mathcal{N},\varepsilon\}\), where \(\mathcal{N}=\{1,...,N\}\) and \(\varepsilon\subset\mathcal{N}\times\mathcal{N}\). A player \(i\) and a player \(j\) are said to be neighbors of \(i\) if they can exchange information, and this relationship is represented in the connectivity graph as \((i,j)\in\varepsilon\). Moreover, the set of neighbors of player \(i\) is \(\mathcal{N}_{-i}=\{j|(i,j)\in\varepsilon\}\). A path in a graph \(\mathcal{G}\) is an interleaved sequence of vertices in \(\mathcal{N}\) and edges in \(\varepsilon\), where the end point of each edge is the start point of the next edge. If \(\forall i,j\in\mathcal{N},(i,j)\in\varepsilon\), then the graph \(\mathcal{G}\) can be called a connected graph. Define \(\mathcal{R}\) as the set of real numbers then the adjacency matrix of the graph \(\mathcal{G}\) is \(W=[w_{ij}]\in\mathcal{R}^{N\times N}\), where \(\mathcal{R}\) denotes the set of real numbers, and whether the value of \(w_{ij}\) is greater than zero indicates whether there exists a path between \(i\) and \(j\). Let \(d_{i}=\sum_{j=1}^{N}w_{ij}\), and the weighted degree matrix of the adjacency matrix \(W\) is \(WD=\text{diag}(d_{i})_{i\in\mathcal{N}}\). Then \(L=WD-W\) is the Laplace matrix of the graph \(\mathcal{G}\). Suppose \(\mathcal{G}\) is a connected graph and is undirected, i.e., \(W=W^{T}\); then \(L\) has a simple eigenvalue \(s_{1}=0\) and all other eigenvalues are greater than zero. Arrange all eigenvalues in descending order as: \(s_{N}>\)\(s_{N-1}>\)... \(>\)\(s_{2}>\)0.