  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) as well as prescribed-time extremum seeking (ES)
and source seeking 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.
