Fixed-Time Nash Equilibrium Seeking in Time-Varying Networks

• Published in: IEEE transactions on automatic control Vol. 68; no. 4; pp. 1954 - 1969
• Main Authors: Poveda, Jorge I., Krstic, Miroslav, Basar, Tamer
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-04-2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Aerodynamics; Algorithms; Asymptotic stability; Convergence; Cost function; Dynamic stability; Dynamics; Equilibrium; Extremum seeking; First order algorithms; Game theory; Games; Heuristic algorithms; Hybrid systems; Initial conditions; learning in games; Nash equilibria; noncooperative games; Numerical stability; Players
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3168527

In this article, we introduce first-order and zeroth-order Nash equilibrium seeking dynamics with fixed-time and practical fixed-time convergence certificates for noncooperative games having finitely many players. The first-order algorithms achieve exact convergence to the Nash equilibrium of the game in a finite time that can be additionally upper bounded by a constant that is independent of the initial conditions of the actions of the players. Moreover, these fixed-time bounds can be prescribed a priori by the system designer under an appropriate tuning of the parameters of the algorithms. When players have access only to measurements of their cost functions, we consider a class of distributed multitime scale zeroth-order model-free adaptive dynamics that achieve semiglobal practical fixed-time stability, qualitatively preserving the fixed-time bounds of the first-order dynamics as the time scale separation increases. Moreover, by leveraging the property of fixed-time input-to-state stability, further results are obtained for mixed games where some of the players implement different seeking dynamics. Fast and slow switching communication graphs are also incorporated using tools from hybrid systems. We consider potential games as well as general nonpotential strongly monotone games. Numerical examples illustrate our results.