Stochastic Nash Equilibrium Seeking for Games with General Nonlinear Payoffs

Abstract

AbstractIn this paper, we propose a multi-input stochastic extremum seeking algorithm to solve Nash equilibrium seeking for a noncooperative game whose N players seek to maximize their general nonlinear payoff functions. Our algorithm is a non-model based approach for asymptotic, locally stable attainment of the Nash equilibria in probability. Different from classical game theory algorithms, where each player employs the knowledge of the functional form of his payoff and the knowledge of the other players's actions, a player employing our algorithm measures only his own payoff values, without knowing the functional form of his or other players’ payoff functions. For the general nonlinear payoffs, the convergence is not perfect but is biased in proportion to the third derivatives of the payoff functions and is dependent on the intensity of stochastic perturbation. We quantify the size of these residual biases. Compared to the deterministic extremum seeking with sinusoidal perturbation signals, the advantage of our algorithm lies in that each player independently employs his seeking strategy by choosing an ergodic stochastic probing signal and there is no requirement on different frequencies for different players.