## 1 Introduction
As shown in Figure 4, it can be found that all three algorithms mentioned above minimize the cost function, and in comparison, using Algorithm 1 leads to the fastest decrease in the cost function. In order to clearly show how the algorithm works, the decreasing cost function of the 15 participants is shown in Figure 5, and it can be found that the cost function of each participant converges to the minimum value continuously and smoothly. Figure 2: The black arrows start at the client end and end at the router chosen by that client, and the clients at either end of the yellow arc are neighbors. Figure 4: The figure shows how the sum of the costs of all customers changes under the action of the three algorithms. The ordinate is \(\sum_{i=1}^{i=N}[f_{i}(x_{i},x_{-i})+g_{i}(x_{i})]\). Figure 5: The figure shows how the private cost per customer changes under the action of Algorithm 1. The ordinate is \(f_{i}(x_{i},x_{-i})+g_{i}(x_{i}),i\in\{1,...N\}\). \[=\langle\hat{\mathbf{x}}^{x}+\hat{\mathbf{x}}^{y}-\hat{\mathbf{x}}^{*},\mathbf{R} ^{T}(\widehat{\mathbb{F}}(\hat{\mathbf{x}})-\widehat{\mathbb{F}}(\hat{\mathbf{x}}^{*} ))\rangle\]0 The blue, red, and yellow curves indicate the convergence process of Algorithms 1 and 2, and DA algorithm, respectively. At the end of this section, the process of reaching agreement on the estimated variables \(\hat{\mathbf{x}}\) of the global decision set \(x\) by the participants \(i\in\mathcal{N}\) during the iteration of the algorithm is shown in \[=\langle\hat{\mathbf{x}}^{x}+\hat{\mathbf{x}}^{y}-\hat{\mathbf{x}}^{*},\mathbf{R} ^{T}(\widehat{\mathbb{F}}(\hat{\mathbf{x}})-\widehat{\mathbb{F}}(\hat{\mathbf{x}}^{*} ))\rangle\]1 As mentioned in this paper, the algorithm under local information elevates the decision vector latitude of the participants, and each participant has not only their own local decision \(x_{i}=x_{i}^{[i]}\in\mathcal{R}\), but also an estimate of the other participants' decisions \(x_{-i}^{[i]}=\text{col}(...,x_{i-1}^{[i]},x_{i+1}^{[i]},...,x_{N}^{[i]})\), where \(x_{i}^{[i]}\) denotes the estimate of participant \(i\) on participant \(j\). Relatively, for each participant \(i\), there exist N estimates of him, and we define this set as \(\mathbf{E}\), i.e., \(\mathbf{E}=\{x_{i}^{[1]},...,x_{i}^{[N]}\}\), let k be the number of iterations, if \(\forall i\in\{1,...,N\},\lim_{k\rightarrow\infty}Var(\mathbf{E})=0\), then it is proved that all decision makers reach agreement on the estimate of the global decision set, which is equivalent to knowing the global decision set. ## 6 Summary and Prospect

In this paper, we study algorithms for finding the GNE of noncooperative game problems for two different situations. In the case of global decision information, each participant can use the global decision set to minimize the cost function. In the case of partial decision information, participants need to estimate the global decision set and perform local optimization by estimating variables, while consensus dynamics with the Laplacian matrix ensures that the local estimations will reach the same GNE. This paper devotes an entire section to the derivation of Algorithms 1 and 2. In addition, we derive the step size condition for the convergence of the algorithm using the Gershgorin Circle Theorem and analyze the convergence of the algorithm under mild assumptions. Finally, we simulated both algorithms and compared them with the FB algorithm. This paper assumes that the objective function of each player is continuously differentiable, but in practical problems, there may be nonsmooth or nonconvex situations. Therefore, how to generalize the algorithm proposed in this paper to more general cases is a worthwhile research question. This paper only considers undirected connected graphs as the communication topology among players, but in real networks, there may be directed graphs or disconnected graphs. Therefore, how to analyze and design distributed algorithms that apply to these complex topological structures is a challenging problem.