Nash Equilibrium of 2-Agent Game with Quadratic Vector Payoff Functions and Its Stability

Abstract

It is common that each agent in a noncooperative system has multiple objectives but the stability property for Nash equilibria in such a game has seldomly been studied. To solve this, we illustrate the characterization of Nash equilibria in the noncooperative systems with quadratic vector payoff functions analytically. It turns out that the Nash equilibria of such systems can be characterized by a set. Depending on the parameters of the payoff functions, we investigate the property of the Nash equilibrium set and present some sufficient conditions where the set is compact and connected. Furthermore, we consider the pseudo-gradient dynamics for the agents and present a sufficient condition where the Nash equilibrium set is asymptotically stable. Several numerical examples are presented to illustrate our results.