## 1 Introduction
Let \(\mathbf{O}_{\overline{\Lambda}}=2\mathbf{P}_{\overline{\Lambda}}-\mathbf{I}\) and \(\mathbf{O}_{\overline{\Xi}}=2\mathbf{P}_{\overline{\Xi}}-\mathbf{I}\), when \(r_{k}=0.5\), the iterative form of the Algorithm 1 can be transformed as (According to reference [20], when \(r_{k}=1\), it is the alternating direction method of Multipliers.) \[\mathbf{\omega}_{\mathbf{k}+1}=\frac{1}{2}\mathbf{\omega}_{\mathbf{k}}+\frac{1}{2}\mathbf{O}_{ \overline{\Xi}}\mathbf{O}_{\overline{\Lambda}}(\mathbf{\omega}_{\mathbf{k}}). \tag{21}\]

Figure 1: The figure shows the process by which all customers agree on the estimated value of GNE. The vertical coordinate is the variance, i.e.,\(Var(x_{i}^{[1]},...,x_{i}^{[N]}),i\in\{1,2,...N\}\). When Assumptions 1-4 hold, the operators \(\mathbb{A}\) and \(\mathbb{B}\) are both maximally monotone. Referring to the theory of monotone operators [21], it is easy to prove the convergence of Algorithm 2. Next, we focus on the convergence of Algorithm 1. Consider the real vector space \(\mathcal{R}^{n(N+|\xi|)+n(N+|\xi|)}\) and define an inner product operation \(\langle\mathbf{\omega},\mathbf{\omega}^{\prime}\rangle=\langle\overline{\Phi}\mathbf{ \omega},\mathbf{\omega}^{\prime}\rangle\) by which an inner product space \(\mathbb{Q}\) can be obtained. Lemma 1: _Assuming that the matrix \(\overline{\Phi}\) is positive definite, then \(\overline{\pi},\overline{\tau},\overline{\tau}\) must satisfy the following inequalities:_

\[\forall\delta>0,\quad \begin{cases}\overline{\pi}_{i}\leq(\delta+c\hat{L}_{i}+ \max_{j\in\{1,2\ldots n\}}\sum_{k=1}^{m}|[-R_{i}^{T}A_{i}^{T}]_{jk}|)^{-1},\\ \overline{\pi}_{i}\leq(\delta+2d_{i})^{-1},\\ \overline{\gamma}_{i}\leq(\delta+\max_{j\in\{1,2\ldots n\}}\sum_{k=1}^{n_{i} }|[-A_{i}R_{i}]_{jk}|+2d_{i})^{-1},\end{cases}  \tag{22}\]

_where \(d_{i}=\frac{1}{2}\sum_{k=1}^{m}|[L]_{ik}|\) denotes the degree of player \(i\). This is a result of a calculation by [PERSON] which states that a sufficient condition for a positive definite matrix is that all eigenvalues are greater than zero [22]._

Lemma 2: _Denoted \(\widetilde{\mathbb{B}}\) by \(\overline{\Phi}^{-1}\overline{\mathbb{B}}\), the operator \(\widetilde{\mathbb{B}}\) is maximally monotone on \(\mathbb{Q}\) when Assumptions 1-4 hold and \(\Phi\) satisfies Lemma 1. Moreover, \(\mathbf{O}_{\overline{\mathbb{B}}}\) is nonexpansive on \(\mathbb{Q}\)._

Proof: According to Definition 9.12 in [23] and Theorem 20.25 in [23], the operator \(\hat{\mathbb{G}}\) is maximally monotone when Assumptions 1 holds. Operator \(\overline{\mathbb{B}}\) consists of a maximally monotone operator and a symmetric matrix. Therefore, operator \(\overline{\mathbb{E}}\) is maximally monotone. Since the matrix fair is positive definite, by Proposition 20.24 in [23], we can conclude that \(\widetilde{\mathbb{B}}\) is also maximally monotone. Moreover, by Corollary 23.11 in [23], we can know that the inverse preconditioner \(\mathbf{O}_{\overline{\mathbb{B}}}\) is nonexpansive. Lemma 3: _Denoted \(\widetilde{\mathbb{A}}\) by \(\overline{\Phi}^{-1}\overline{\mathbb{A}}\), the operator \(\widetilde{\mathbb{A}}\) is maximally monotone on \(\mathbb{Q}\) when Assumptions 2-5 hold and \(\Phi\) satisfies Lemma 1. Moreover, \(\mathbf{O}_{\overline{\mathbb{A}}}\) is nonexpansive on \(\mathbb{Q}\)._

Proof: The proof is similar to that of Lemma 2. When \(\mathbf{R}^{T}\widehat{\mathbb{F}}+\frac{1}{2}c\mathbf{L}\) is a maximally monotone operator, the conclusion of the Lemma 3 can be obtained. Theorem 1: _Suppose \(\lambda^{*}\) satisfies (6) and that \(x^{*}\) is the \(v\)-GNE of Problem (3). When Lemma 3 is valid, the sequences \(\{\mathbf{\xi}_{k}\}\) and \(\{\lambda_{k}\}\) generated by Algorithm 1 satisfy \(\lim_{k\to\infty}\mathbf{\xi}_{k}=(1_{N}\otimes x^{*})\) and \(\lim_{k\to\infty}\mathbf{\lambda}_{k}=(1_{N}\otimes\lambda^{*})\)[21]._

Proof: Since \(\lambda^{*}\) and \(x^{*}\) are generalized Nash equilibrium solutions, \(col(x^{*},z^{*},\lambda^{*})=(\overline{\mathbb{A}}+\overline{\mathbb{B}})col (x^{*},z^{*},\lambda^{*})\) holds.