# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 5 Numerical Studies
### Numerical Results  
The experimental settings described above satisfy all assumptions presented in [21]. We selected the step size in Lemma 8 to fulfill all conditions required by Theorem 1. To compare the performance of Algorithm 1 and the algorithm in [21], we conducted experiments using the same random initial condition for both algorithms. As shown in Figures 2 and 3, Algorithm 1 proposed in this paper converges faster than the algorithm in [21], which is referred to as Algorithm 2 in the following text. Figure 2 compares the convergence of the two algorithms under the partial decision setting by plotting the relative error of their decisions. The results show that Algorithm 1 has smaller relative errors than Algorithm 2 (the algorithm in [21]) for the same number of iterations, indicating a faster convergence rate for Algorithm 1. Figure 3 illustrates the trajectory of the total cost of all agents in the market corresponding to Algorithm 1 and Algorithm 2 (the algorithm in [21]), where the total cost is generated by \(\sum\limits_{i=1}^{N}J_{i}(x_{i},\mathbf{x}_{-i})\). The trajectory eventually converges to the same minimum value, indi  
Figure 3: The total cost of all agents generated by Algorithm 1 and Algorithm 2 (the algorithm in [21]). Figure 2: Relative error \(\left\|x_{k}-x^{*}\right\|/\left\|x^{*}\right\|\) plot generated by Algorithm 1 and Algorithm 2 (the algorithm in [21]). cating the correctness and accuracy of Algorithm 1. Furthermore, the convergence of the trajectory represents the effectiveness of the algorithm in terms of optimizing the total cost of all agents in the market. In Figure 4, we can observe the trajectory of each agent's decision in the Cournot market game, which has been solved by Algorithm 1. As we can see from the graph, the decision trajectories of all agents converge, which is a strong indication that the GNE obtained by executing Algorithm 1 can effectively minimize the cost incurred by each agent. It is worth noting that the convergence of decision trajectories is a critical aspect of the game theory that is often used to evaluate the effectiveness of a given algorithm. Therefore, the observed convergence in Figure 4 is an encouraging sign that Algorithm 1 can be a viable approach for solving similar problems in the future. During the iterative process of Algorithm 1, the estimated value of each agent's decision for all other agents is first obtained according to the formula \(\mathbf{x}_{-i,k+1}^{i}=\frac{1}{1+\tau_{i}d_{i}}\left(\mathbf{x}_{-i,k}^{i}+\tau_{i} \sum_{j=1}^{N}w_{i,j}\mathbf{x}_{-i,k}^{j}\right)\). This formula is used to compute the value of \(\mathbf{x}_{-i,k+1}^{i}\). Subsequently, the estimated decision value is used for the iteration of \((\mathbf{x},\mathbf{z},\mathbf{\lambda})\), which is a mathematical notation used to represent the values of the decision variables. When the algorithm converges, i.e., when the GNE is obtained, the estimated decision value of each agent for \(x_{i}\) is equal to the actual decision value of \(x_{i}\), i.e., \(\mathbf{x}_{i}^{j}=\mathbf{x}_{i}^{j}\) for \(j=1,\ldots,N\). Figure 5a shows the trajectory of the standard deviation of each agent's estimated decision set for \(x_{i}\) with increasing iteration times, which is generated by \(\sqrt{\frac{1}{N}\sum\limits_{j=1}^{N}\left(\mathbf{x}_{i}^{j}-\sum\limits_{j=1}^{ N}\mathbf{x}_{i}^{j}\right)^{2}}\) for \(j=1,\ldots,N\). It can be seen that the standard deviation of the estimated decision set for each agent eventually converges to 0. This indicates that the results satisfy the condition that the estimated value of each agent's decision for \(x_{i}\) is equal to the actual decision value of \(x_{i}\). As shown in Figure 5b, taking agent 3 as an example, the trajectory plot of all agents' estimated decision values for agent 3 is displayed, where \(x_{3}^{3}\) represents the actual value of agent 3's decision. The trajectory plot eventually converges to the same value, visually confirming the accuracy of Algorithm 1.