Seeking Nash Equilibrium in Non-cooperative Quadratic Games Under Delayed Information Exchange

TL;DR

This paper develops a method for finding Nash equilibria in non-cooperative quadratic games with delayed information exchange, ensuring convergence through estimation mechanisms and Lyapunov techniques.

Contribution

It introduces a novel estimation-based best-response algorithm for delayed information settings and proves convergence properties under different delay scenarios.

Findings

Convergence to Nash equilibrium is guaranteed with multi-step delays using Lyapunov-Krasovskii functionals.

Exponential convergence is achieved with one-step delays under a learning rate restriction.

Numerical simulations confirm the theoretical convergence results.

Abstract

In this paper, we investigate the seeking of Nash equilibrium (NE) in a non-cooperative 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 multi-step-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.