Nash equilibrium seeking in noncooperative games with time-varying delays and bias

In this paper, we study the Nash equilibrium seeking algorithm in noncooperative games with general quadratic payoff functions in the presence of time-varying delays and bias. We consider delays with large distinct known constant parts and a small time-varying uncertainty. Such delays may arise from the payoff measurement process. By employing the time-delay approach to averaging, we introduce an equivalent, approximation-free representation of the Nash equilibrium seeking estimation error system, which is a perturbation of a linear time-invariant delayed system. Given large delays, we first choose appropriate gains to guarantee stability of this linear system, and then, the practical semi-global stability analysis is provided by employing a variation of constants formula. Furthermore, the sampled-data Nash equilibrium seeking in the presence of large distinct known delays and time-varying bias is also presented. Numerical simulations validate the efficiency of the results.