Nash Equilibrium Seeking with Arbitrarily Delayed Player Actions

In this paper, we propose a non-model based strategy for locally stable convergence to Nash equilibrium in a quadratic noncooperative (duopoly) game with arbitrarily delayed player actions. In our noncooperative scenario, the players have access only to their own payoff values, again with delay. The proposed approach is based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal perturbation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown locally quadratic functions. Indeed, this is the first contribution which considers extremum seeking for noncooperative games in the presence of delays. In order to compensate distinct delays in the inputs of the two players, we have employed boundary control via predictor feedback with averaging-based estimates. We apply a small-gain analysis for the resulting Input-to-State Stable hyperbolic PDEODE loop as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the time delays, in order to obtain local convergence results to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and corroborate the theoretical results numerically on an example of a two-player game with delays.