Prescribed-Time Fully Distributed Nash Equilibrium Seeking Strategy in Networked Games

Cheng Qian and Lei Ding ,,

Corresponding author: Lei Ding.

Citation: C.Qian and L.Ding, “Prescribed-time fully distributed Nash equilibrium seeking strategy in networked games,”IEEE/CAA J.Autom.Sinica, vol.11, no.1, pp.261–263, Jan.2024.

The authors are with the Institute of Advanced Technology for Carbon Neutrality, College of Automation & the College of Artificial Intelligence,Nanjing University of Posts and Telecommunications, Nanjing 210023, China(e-mail: dinglei@njupt.edu.cn; 1021051827@njupt.edu.cn).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JAS.2023.123933

Dear Editor,

This letter is concerned with prescribed-time fully distributed Nash equilibrium seeking for networked games under directed graphs.An adaptive algorithm is proposed to ensure the convergence of all players to the Nash equilibrium without requiring any knowledge of global parameters.Moreover, it is theoretically proved that the convergence time of the proposed seeking strategy can be predefined based on practical requirements.Finally, a numerical example is presented to validate the effectiveness of the proposed method.

As a key concept in game theory, Nash equilibrium has found wide-ranging applications in practical engineering fields [1], [2].As a result, numerous research methods have been proposed for the design and analysis of distributed Nash equilibrium seeking strategies depending only on the local neighborhood information [3].For instance, a distributed Nash equilibrium seeking strategy for general games was proposed by combining a leader-follower consensus protocol and a gradient play [4].The problem of distributed Nash equilibrium seeking was investigated for networked systems with bounded control inputs [5].A distributed nonsmooth algorithm with a projected differential inclusion was proposed to solve the generalized Nash equilibrium seeking problem for multi-cluster games [6].In the presence of external disturbances, a robust distributed algorithm was proposed to drive all agents to reach the Nash equilibrium [7].

Note that, in all aforementioned results, the parameter design of distributed Nash equilibrium seeking strategies is always dependent on the global information such as the eigenvalues of Laplacian matrices, which may be difficult to obtain in practical implementations.As a result, there is a growing demand for the development of fully distributed strategies which can be effective in avoiding global information [8].In [9], adaptive approaches were utilized to achieve fully distributed Nash equilibrium seeking in networked games with undirected graph.Under directed graph, a fully distributed approach to finding the Nash equilibrium was presented for high-order players subject to actuator limitations [10].

While significant progress has been made in the fully distributed Nash equilibrium seeking, practical scenarios still pose some challenging issues in terms of algorithm convergence speeds.It should be mentioned that the distributed Nash equilibrium seeking strategies[9], [10] can only ensure the asymptotic convergence.However, in certain engineering applications, it may be required to realize the convergence of algorithms within a prescribed time rather than an infinity time [11].To meet this requirement, various distributed control strategies have been presented.For example, a distributed timevarying seeking strategy that utilizes a prescribed-time observer under undirected graphs was proposed to achieve the convergence within a set time [12].Utilizing the distributed motion-planning method and the gradient search, a class of prescribed-time distributed Nash equilibrium seeking algorithms have been developed for first and second order multi-agent systems [13].However, these distributed methods require global information of the game and thus are not fully distributed.Therefore, it is necessary and important to design a prescribed-time fully distributed Nash equilibrium seeking strategy under directed graphs, which is the motivation behind this letter.

Building on the analysis above, this letter makes the following contributions: 1) A new prescribed-time fully distributed Nash equilibrium seeking strategy under directed graphs is proposed by designing an adaptive algorithm to adjust the control parameter according to the consensus error; 2) Theoretical analysis is conducted to prove that the proposed strategy can make all agents convergent to the Nash equilibrium in a prescribed time, which is beneficial for practical engineering applications with specific time constraints.

wheremis a positive constant.

Remark 1: Assumption 1 is a solid base for guaranteeing the existence of gradient vectors with continuous differentiability.Assumption 2 indicates the strong/strict monotonicity of pseudo-gradient vectors, guaranteeing the existence and uniqueness of the Nash equilibrium.

Assumption 3: The communication topology is directed and strongly connected.

Main results: Note that in a networked game, each player’s action and objective function are only available for itself but not others.As a result, the traditional centralized algorithm is no longer applicable due to a lack of global information about all players’ action and objective functions.In order to solve this problem, it is supposed that each player can generate a local estimate on the players’ actionsx.clusions of Theorem 1 can be obtained.

Remark 2: It is noted that, due to the presence of communication networks, the algorithm (5)-(8) will be inevitably subject to a variety of communication constraints such as communication delays and data losses.As a result, it is an interesting topic to further consider the effects of such communication constraints on distributed prescribed-time Nash equilibrium seeking strategy, which will be investigated in our future research work.■

Remark 3: Different from the existing literature [4], [9], [10], [13],Theorem 1 shows clearly that the proposed distributed Nash equilibrium seeking strategy is fully distributed by utilizing a PI adaptive algorithm to avoid the global information under a directed graph, and also ensures the prescribed convergence timeTby introducingin the adaptive parameter (8).

Numericalexample:Consider a 5-player game in which each pla yer’sactionxi∈R2.Let

Fig.1.Strongly connected communication topology.

Fig.2.The trajectories of players’ actions.

Conclusion: This letter introduces a new prescribed-time fully distributed Nash equilibrium seeking strategy under directed graphs.The prescribed-time seeking strategy does not require any global information on communication topology and allows to set the convergence time in advance based on specific requirements.Future research will be focused on fully distributed Nash equilibrium seeking in presence of cyberattacks [14] and communication constraints[15].

Fig.3.The estimates y ij on players’ actions x for all i ,j ∈{1,2,,5}.

Acknowledgments: This work was supported by the National Natural Science Foundation of China (NSFC) (62073171) and the Natural Science Foundation of Jiangsu Province (BK20200744).