## 1 Introduction
Substituting \(\mathbb{A}=\overline{\mathbb{A}}\), \(\overline{\mathbb{B}}=\overline{\mathbb{B}}\), \(\Phi=\overline{\Phi}\) and \(\mathbf{v}=\mathbf{\omega}\) into (10), the distributed algorithm under the local information setting, i.e., Algorithm 1 can be derived. In Algorithm 1, both the local decision \(x_{i}\) and the local estimate \(x_{-i}^{[i]}\) are updated by the Laplace matrix, which causes the global estimate \(x^{[i]}\) of agent \(i\) get closer to its neighbors, and the gradient descent causes the local cost function to decrease. The range of values of the step size parameter \(\alpha_{i}\), \(\tau_{i}\), \(\tau_{i}\) depends on the positive definiteness of the matrix \(\overline{\Phi}\), which can be found analytically using the Gershgorin. Note that the update step in the algorithm \(r_{k}\) needs to meet two conditions according to the requirements of the Douglas-Rachford method, i.e., \(0<r_{k}<1\) and \(\sum_{k=1}^{\infty}r_{k}(1-r_{k})=\infty\). **Remark 2**. : _In the pseudo-code, the input parameters are \(x_{i,0}^{[i]}\), \(x_{-i,0}^{[i]}\), \(z_{i,0}\) and \(\lambda_{i,0}\), where \((x_{i,0}^{[i]},\mathbf{x}_{-i,0}^{[i]})\) is player i's estimate of the global decision set at iteration 0 and the output parameter is \((x_{j,k}^{[i]},\mathbf{x}_{-j,k}^{[i]})\), which is player i's final estimate of the global decision set._

Observing the output parameters \((x_{j,k}^{[i]},\mathbf{x}_{-j,k}^{[i]})\) after the algorithm iterations are completed reveals that eventually, all players have the same estimate of the global decision set, i.e., \(\forall i,j\in\mathcal{N},\quad(x_{j,k}^{[i]},x_{-i,k}^{[i]})=(x_{j,k}^{[i]},x_{ -j,k}^{[i]})\). As shown in Figure 1 in Section 3, the variance of the estimates of all players converges to zero during the iterations of the algorithm. ## 4 Convergence Analysis

In this section, two assumptions are proposed to analyze the convergence of Algorithms 1 and 2. First, the range of values of the step size \(\overline{\alpha},\overline{\tau},\overline{\tau}\) can be obtained according to Gershgorin, and then the convergence of the two algorithms can be proven based on two mild assumptions. **Assumption 5**. :
1. _The set of Nash equilibria of the variational problem of the original problem is nonempty_
2. _and_ \(\mathbf{R}^{T}\bar{\mathbb{F}}+\frac{1}{2}c\hat{\mathbf{L}}\) _is maximally monotone operator._

**Assumption 6**. :
1. _The pseudo-gradient operator_ \(\mathbb{F}\) _is strongly monotone and Lipschitz continuous, i.e., there exist_ \(\theta_{1},\theta_{2}>0\)_, satisfying_ \(\forall x,x^{\prime}\in\mathcal{R}^{n}\)_,_ \(\frac{\langle x-x^{\prime},\mathbb{F}(x)-\mathbb{F}(x^{\prime})\rangle}{\|x- x^{\prime}\|^{2}}\geq\theta_{1}\) _and_ \(\frac{\|\mathbb{F}(x)-\mathbb{F}(x^{\prime})\|}{\|x-x^{\prime}\|}\leq\theta_{2}\)_._
2. _The operator_ \(\mathbf{R}^{T}\bar{\mathbb{F}}\) _is Lipschitz continuous, i.e., there exist_ \(\theta_{3}>0\)_, satisfying_ \(\forall\mathbf{\hat{x}},\mathbf{\hat{x}}^{\prime}\in(R)^{nN}\)_,_ \(\frac{\|\mathbb{F}(\mathbf{\hat{x}})-\mathbb{F}(\mathbf{\hat{x}}^{\prime})\|}{\|\mathbf{ \hat{x}}-\mathbb{F}^{\prime}\|}\leq\theta_{2}\)_._

Note that if and only if \(\mathbf{0}\in\text{Zer}(\overline{\Phi}^{-1}\overline{\mathbb{A}}+\overline{\Phi }^{-1}\overline{\mathbb{B}})\), exists \(\mathbf{0}\in\text{Zer}(\overline{\Lambda}+\overline{\mathbb{B}})\). Define \(\mathbf{P}_{\overline{\Lambda}}:=(I+\overline{\Phi}^{-1}\overline{\mathbb{A}})^{-1}\) and \(\mathbf{P}_{\overline{\Xi}}:=(I+\overline{\Phi}^{-1}\overline{\mathbb{B}})^{-1}\), iterative process can be expressed as \(\mathbf{\omega}_{\mathbf{k}+1}=\mathbf{\omega}_{\mathbf{k}}+2r_{k}(\mathbf{P}_{\overline{\Lambda} }\mathbf{P}_{\overline{\Xi}}-\mathbf{P}_{\overline{\Xi}}-\mathbf{P}_{\overline{\Lambda}}) (\mathbf{\omega}_{\mathbf{k}})\).