# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 1 Introduction
### Literature Review  
The research of GNE was initiated by the seminal works of [4; 5]. In recent years, the computation of distributed GNE in monotone games has received considerable interest. However, most existing studies rely on the assumption that every agent in the system has access to the decision information of other agents, i.e., by employing a central node coordinator to orchestrate the information processing. The initial approach was based on the variational inequality method [6]. The variational inequality framework for generalized Nash equilibrium problems established in [6] has been extended to more general settings, such as time-dependent or stochastic problems in [7]. Ref. [8] developed a Tikhonov-regularized primal-dual algorithm, and [9] devised a primal-dual gradient method, which can reduce the compensation and assumed that it can acquire the decisions of all adversaries that influence each cost. In ref. [10], a payoff-based GNE algorithm was introduced, which has a diminishing step size and converges to a class of convex games. Recently, an operator-splitting method has emerged as a very powerful design technique, which ensures global convergence and features a fixed step size and succinct convergence analysis. However, most GNE application scenarios are aggregative games, such as [3; 11; 12; 13; 14]. In refs. [12; 13], the algorithms are semi-decentralized, requiring a central node to disseminate the common multipliers and aggregate variables, thus resulting in a star topology. In ref. [14], this is relaxed by a projection gradient algorithm based on two-layer consensus, both of which are only suitable for aggregative games. In networked non-cooperative games, a distributed forward-backward operator-splitting algorithm for seeking the GNE has been proposed by [PERSON] and [PERSON] [15], based on the convex analysis framework of [PERSON] [16]. Under partial information, the algorithm requires only local objective function, feasible set, and block of affine constraints for each agent. It also introduces a local estimate of all agents' decisions. They further extended their algorithm to an asynchronous setting by using auxiliary variables associated with Graphology [17]. In this setting, each agent uses private data and delay information from their neighbors' latency information to iterate asynchronously. They proved the convergence and effectiveness of their algorithm under mild assumptions. Ref. [18] investigated a generalized Nash equilibrium problem in which players are modeled as nodes of a network and each player's utility function depends on its own and its neighbors' actions, and derived a variational decomposition of the game under a quadratic reference model with shared constraints, which is illustrated with numerical examples. [PERSON] et al. [19] has developed a novel algorithm that differs from the existing projective pseudo gradient dynamics in that it is fully distributed, one-layer, and uses a proximal best response with consensus terms. It can overcome the limitations of double-layer iterations or conservative step sizes of the gradient-based methods. It can also extend the applicability of the proximal point method by analyzing the restricted monotonicity property. [PERSON] [20] applied a preconditioned proximal point algorithm (PPPA) that decomposes the merit-seeking task into Nash equilibrium (NE) computation of regularised subgame sequences based on local information. The PPPA also performs distributed updates of multipliers and auxiliary variables, which require multiple communications among agents to solve the subgame. For games with general coupling costs and affine coupling constraints, [15; 21] adopt operator methods to perform distributed decentralized GNE optimization: forward-backward optimization algorithm for strongly monotone games [15] and preconditioned proximal optimization algorithm for monotone games [21].