Seeking Nash Equilibrium in Noncooperative Quadratic Games Under Delayed Information Exchange

Abstract

In this article, we study the seeking of Nash equilibrium (NE) in a noncooperative quadratic game where all agents exchange their delayed strategy information with their neighbors. To extend best response algorithms to the delayed information setting, an estimation mechanism for each agent to estimate the current strategy profile is designed. Based on the best response strategy to the estimations, the strategy profile dynamics of all agents is established, which is revealed to converge asymptotically to the NE when agents exchange multistep-delay information via the Lyapunov–Krasovskii functional approach. In the scenario, where agents exchange one-step-delay information, the exponential convergence of the strategy profile dynamics to the NE can be guaranteed by restricting the learning rate to less than an upper bound. Moreover, a lower bound on the learning rate for instability of the NE is proposed. Numerical simulations are provided for verifying the developed results.