## 1 Introduction
\begin{split}&\tilde{\mathbf{v}}=(Id+\Phi^{-1}\mathbb{A})^{-1}(\tilde{ \mathbf{v}}_{k}),\\ &\tilde{\mathbf{v}}_{k}=2\tilde{\mathbf{v}}-\mathbf{v}_{k},\\ &\overline{\mathbf{v}}_{k}=(Id+\Phi^{-1}\mathbb{B})^{-1}(\tilde{\mathbf{ v}}_{k}),\\ &\mathbf{v}_{k+1}=\mathbf{v}_{k}+2\tau_{k}(\overline{\mathbf{v}}_{k}-\tilde{\mathbf{v}}).\end{split}  \tag{10}\]

### Partial Information Setting

In this subsection, players are reluctant to disclose their decisions due to privacy concerns, i.e., player \(i\) is not aware of other players' decisions \(x_{-i}\). In this case, each player can only find the GNE by exchanging local information with their trusted neighbors through the connectivity graph \(\mathcal{G}\) and minimizing their objective function with the obtained information. Next, we will obtain an equivalent form of the variational problem under certain assumptions. By following a similar process as in the previous subsection, we can derive Algorithm 1. **Assumption 4**: \(\mathcal{G}\) _is undirected and connected and it has no self-loops._

Define \(\mathbf{x}^{[i]}\) to denote the estimate of agent \(i\) for GNE \(x^{*}\), where \(\mathbf{x}^{[i]}=\text{col}(x_{j}^{[i]})_{j\in\mathcal{N}}\) and \(\mathbf{x}_{-i}^{[i]}=\text{col}(x_{j}^{[i]})_{j\in\mathcal{N},j\
eq i}\). We next investigated under what conditions the player can use the estimator \(\mathbf{x}^{[i]}\) to minimize the objective function and why this can be done. **Remark 1**: \(x_{j}^{[i]}\) _represent the state estimate of \(i\) versus \(j\), and \(x_{i}^{[i]}=x_{i}\). When \(\forall(i,j)\), \(\mathbf{x}^{[i]}=\mathbf{x}^{[i]}\), all the agents reach an agreement._

The variational problem of the original game problem (3) can be written as the following optimization problem:

\[ \begin{cases}\operatorname*{arg\,min}_{x\in\mathcal{R}^{n}}&\langle\mathbb{F} (x^{*}),x\rangle+g(x),\\ \text{s.t.}&\mathbf{Ax}\leq\mathbf{b},\\ &\mathbf{L\lambda}=0,\end{cases}  \tag{11}\]

where \(\mathbb{F}(x)=\text{col}(\
abla_{x_{i}}f_{i}(x_{i},x_{-i}))_{i\in\mathcal{N}}\), \(g(x)=\sum_{i=1}^{N}(g_{i}(x_{i}))\), \(\mathbf{A}=\text{diag}(A_{1},A_{2},...,A_{N})\), \(\mathbf{b}=\text{col}(b_{1},...,b_{N})\). In order that the decision variable \(x_{i}\) and the estimation variable \(x_{-i}^{[i]}\) of agent \(i\) can be selected from \(\mathbf{x}^{[i]}\), define two linear mappings: let

\[R_{i}=\left[ \begin{array}{cc}\mathbf{0}_{n_{i}\times n_{<i}}&I_{n_{i}}&\mathbf{0}_{n _{i}\times n_{>i}}\end{array} \right], \tag{12}\]

\[S_{i}=\left[ \begin{array}{cc}I_{n_{<i}}&\mathbf{0}_{n_{<i}\times n_{i}}&\mathbf{0}_ {n_{<i}\times n_{>i}}\\ \mathbf{0}_{n_{>i}\times n_{<i}}&\mathbf{0}_{n_{>i}\times n_{i}}&I_{n_{>i}}\end{array}  \right], \tag{13}\]

where \(n_{<i}=\sum_{j<i,j\in\mathcal{N}}n_{j}\), \(n_{<i}=\sum_{j>i,j\in\mathcal{N}}n_{j}\), i.e.,\(R_{i}\in\mathcal{R}^{n_{i}\times n}:x\longmapsto x_{i}\) and \(S_{i}\in\mathcal{R}^{(n-n_{i})\times n}:x\longmapsto x_{-i}\). Hence \(R_{i}\mathbf{x}^{[i]}=x_{i}^{[i]}=x_{i}\) and \(S_{i}\mathbf{x}^{[i]}=\mathbf{x}_{-i}^{[i]}\). Furthermore, these two operators satisfy the following formulas:

\[ \begin{split}& R_{i}S_{i}^{T}=\mathbf{0}_{n_{i}\times(n-n_{i})},\\ & R_{i}R_{i}^{T}=I_{n_{i}},\\ & S_{i}S_{i}^{T}=I_{(n-n_{i})\times(n-n_{i})},\\ & S_{i}^{T}S_{i}+R_{i}R_{i}^{T}=I_{n}.\end{split}  \tag{14}\]Define \(\mathbf{\hat{x}}=\text{col}(\mathbf{x}^{[i]})_{i=1}^{N}\), \(\hat{\mathbb{F}}(\mathbf{\hat{x}}^{*})=\text{col}(\
abla_{x_{i}}f_{i}(x_{i},\mathbf{x}_{ -i}^{[i]*})_{i\in\mathcal{N}}\), if \(\forall(i,j)\in\varepsilon,\mathbf{x}^{[i]}=\mathbf{x}^{[j]}\), then \(\hat{\mathbb{F}}(\mathbf{\hat{x}}^{*})=\mathbb{F}(\mathbf{x}^{*})\). With \(R_{\mathbf{x}}^{[i]}=x_{i}\) and \(\mathbf{R}=\text{diag}(R_{i})\), the stacking decision \(x\) is equivalent to \(\mathbf{R}\mathbf{\hat{x}}\). Since \(\forall(i,j)\in\varepsilon,\mathbf{x}^{[i]}=\mathbf{x}^{[j]}\Leftrightarrow(L \bigotimes I_{n})\mathbf{\hat{x}}=\mathbf{0}\), let \(\mathbf{\hat{L}}=(L\bigotimes I_{n})\in\mathcal{R}^{Nn\times Nn}\), then (11) is equivalent to

\[