An extremum seeking-based approach for Nash equilibrium seeking in N-cluster noncooperative games

Abstract

This paper considers Nash equilibrium seeking for N-cluster noncooperative games in which the explicit expressions of the agents’ local objective functions are not available to the agents. As an alternative, the output values of the agents’ local objective functions are supposed to be measurable. An extremum seeker is designed to achieve the Nash equilibrium seeking for the N-cluster noncooperative games. The design of the extremum seeker is based on a dynamic average consensus protocol and the modulation of the sinusoidal dither signals. Through Lyapunov stability analysis, the convergence results are analytically studied. Local and nonlocal convergence results are derived under local and global characterizations of the Nash equilibrium for the N-cluster noncooperative games, respectively. Compared with our previous works on Nash equilibrium seeking for the N-cluster noncooperative games, the main advantages and characteristics of the proposed method are threefold. Firstly, the proposed method broadens the applicability of the existing Nash equilibrium seeking strategies to circumstances in which the agents cannot access the explicit expressions of their local objective functions. Moreover, the proposed seeking strategy performs as a unified strategy that solves the noncooperative games and the social cost minimization problems without utilizing explicit model information. Secondly, the proposed seeking strategy requires each agent to update fewer auxiliary variables compared with the existing methods and hence reduces the communication and computation costs. Thirdly, different from most of the existing works that adopt singular perturbation to analyze the stability of extremum seekers, this paper establishes the convergence results via Lyapunov stability analysis, which is novel and provides some new insights on the analysis of extremum seekers. A numerical example is provided to verify the effectiveness of the proposed method.