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Lie-Bracket Nash Equilibrium Seeking with Bounded Update Rates for Noncooperative Games

Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira, Miroslav Krstic, Tamer Basar

TL;DR

The paper tackles local convergence to Nash equilibrium in static quadratic noncooperative games using a model-free, bounded-update-rate extremum-seeking approach based on Lie-bracket averaging. By reformulating the dynamics into an input-affine, highly oscillatory system and applying Lie-bracket approximations, it derives a gradient-like averaged system that converges to a neighborhood of the Nash point under Lyapunov stability analysis. The main result shows exponential convergence to the Nash neighborhood with a residual that scales as $\mathcal{O}(1/\tilde{\omega})$, independent of perturbation amplitudes. A four-player oligopoly example validates the method, demonstrating distributed Nash-seeking without information sharing and improved payoffs, highlighting practical applicability in multi-agent, model-free settings.

Abstract

This paper proposes a novel approach for local convergence to Nash equilibrium in quadratic noncooperative games based on a distributed Lie-bracket extremum seeking control scheme. This is the first instance of noncooperative games being tackled in a model-free fashion integrated with the extremum seeking method of bounded update rates. In particular, the stability analysis is carried out using Lie-bracket approximation and Lyapunov's direct method. We quantify the size of the ultimate small residual sets around the Nash equilibrium and illustrate the theoretical results numerically on an example in an oligopoly setting.

Lie-Bracket Nash Equilibrium Seeking with Bounded Update Rates for Noncooperative Games

TL;DR

The paper tackles local convergence to Nash equilibrium in static quadratic noncooperative games using a model-free, bounded-update-rate extremum-seeking approach based on Lie-bracket averaging. By reformulating the dynamics into an input-affine, highly oscillatory system and applying Lie-bracket approximations, it derives a gradient-like averaged system that converges to a neighborhood of the Nash point under Lyapunov stability analysis. The main result shows exponential convergence to the Nash neighborhood with a residual that scales as , independent of perturbation amplitudes. A four-player oligopoly example validates the method, demonstrating distributed Nash-seeking without information sharing and improved payoffs, highlighting practical applicability in multi-agent, model-free settings.

Abstract

This paper proposes a novel approach for local convergence to Nash equilibrium in quadratic noncooperative games based on a distributed Lie-bracket extremum seeking control scheme. This is the first instance of noncooperative games being tackled in a model-free fashion integrated with the extremum seeking method of bounded update rates. In particular, the stability analysis is carried out using Lie-bracket approximation and Lyapunov's direct method. We quantify the size of the ultimate small residual sets around the Nash equilibrium and illustrate the theoretical results numerically on an example in an oligopoly setting.
Paper Structure (7 sections, 2 theorems, 57 equations, 3 figures)

This paper contains 7 sections, 2 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Nash Equilibrium Seeking with Bounded Update Rates
  4. Lie-Bracket Approximations
  5. Stability Analysis
  6. Simulation Results
  7. Conclusion

Key Result

Theorem 5.1

For the $N$-player quadratic noncooperative game in (eq:Ji_v1), consider the dynamics of the concatenated strategies employed by the players (eq:dthetaA/dt_v2). The Lie-bracket approximation is given by (eq:dtheta-/dt_v5). Under Assumptions assum:Nash, assum:SDD, and assum:w, for $\|\theta(0)-\theta where $\|\theta(t)-\theta^{\ast}\|=\sqrt{(\theta(t)-\theta^{\ast})^{\top}(\theta(t)-\theta^{\ast})}

Figures (3)

  • Figure 1: Block Diagram of the proposed Lie-bracket Nash equilibrium seeking with bounded update rates.
  • Figure 2: Players' actions, $\theta_i(t)$ .
  • Figure 3: Payoff functions, $J_i(t)$.

Theorems & Definitions (3)

  • Theorem 5.1
  • proof
  • Theorem 7.1