# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 1 Introduction  
The problem of finding a generalized Nash equilibrium (GNE) for networked systems has gained significant interest recently owing to its applicability to multi-agent decision-making scenarios, such as demand-side management in intelligent grids [1], demand response in competitive markets [2], and electric vehicle charging [3]. In such systems, the agents aim to minimize cost functions under joint feasibility constraints with non-cooperative settings. They cannot reduce their local costs by unilaterally changing their decisions but by relying on the decisions of the other agents. The GNE is a self-executing outcome, and once computed, all agents execute it to achieve their own minimum cost. In these systems, each user has a local cost function to minimize, which is a function of their own and other users' decisions. Moreover, each user's feasible decision set is coupled with other users' decisions due to the limited network resources. A natural solution concept for such systems is the GNE, which captures the non-cooperative behavior of multiple interacting agents. A GNE is a vector of decisions such that no user can reduce their local cost by unilaterally deviating from it, given the decisions of other users. Hence, a GNE is a self-enforcing outcome that all agents will implement once computed, as it achieves their individual minimum costs.  
To compute GNE introspectively, most algorithms for finding Nash equilibrium require a central information center to collect and store all game information such as cost functions, coupling constraints, and feasible local sets in a traditional computing environment. However, this approach is not suitable for large-scale network games because it is difficult to observe the true decisions of all other players in this type of game. Additionally, having a central coordinator node may be impractical due to technical, geographical, or game-related reasons. In this case, each player needs to compute their local decision corresponding to GNE in a distributed way by utilizing their local objective function, local feasible set, and possibly local data related to the coupling constraints and communicating with their neighbors. Therefore, our goal is to develop a distributed algorithm that estimates the strategies of all other agents only by communicating with adjacent agents.  
This algorithm can achieve the goal of reflecting local decisions in the absence of global information and eventually reconstructing actual value.  
The proximal-point algorithm has been widely used in distributed computing of GNE problems. In our study, we demonstrate that it can achieve the goal of reflecting local decisions in the absence of global information while maintaining the distributed nature and convergence of the algorithm. That is to say, in the absence of a central coordinator, the generalized Nash equilibrium point is calculated by each participant using local information and communication with neighbors. This approach enables each player to compute a local decision corresponding to GNE by utilizing its local objective function, local feasible set, and possibly local data related to the coupling constraints and communicating with its neighbors. By continuously updating the proximal point, the proximal-point algorithm can achieve the convergence of restricted monotone and Lipschitz continuous pseudo-gradient games under appropriate fixed step size conditions. Our study proposes a new distributed GNE algorithm for games with shared biomimetic coupling constraints, which is based on the variational GNE approach and the proximal-point algorithm. Our results demonstrate the effectiveness of the proposed algorithm in achieving GNE in a distributed manner in the presence of shared constraints.