Sliding-Mode Nash Equilibrium Seeking for a Quadratic Duopoly Game

This paper introduces a new method to achieve stable convergence to Nash equilibrium in duopoly noncooperative games. Inspired by the recent fixed-time Nash Equilibrium seeking (NES) [30] as well as prescribed-time extremum seeking (ES) [34] and source seeking [33] schemes, our approach employs a distributed sliding mode control (SMC) scheme, integrating extremum seeking with sinusoidal perturbation signals to estimate the pseudogradients of quadratic payoff functions. Notably, this is the first attempt to address noncooperative games without relying on models, combining classical extremum seeking with relay components instead of proportional control laws. We prove finite-time convergence of the closed-loop average system to Nash equilibrium using stability analysis techniques such as time-scaling, Lyapunov's direct method, and averaging theory for discontinuous systems. Additionally, we quantify the size of residual sets around the Nash equilibrium and validate our theoretical results through simulations.