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Distributed Event-Triggered Nash Equilibrium Seeking for Noncooperative Games

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

TL;DR

This work addresses decentralized Nash equilibrium seeking for $N$-player noncooperative games with unknown quadratic payoffs. It introduces distributed event-triggered extremum seeking using sinusoidal perturbations to estimate pseudo-gradients and a time-scale/averaging analysis to handle discontinuities, proving local exponential stability and Zeno avoidance. Theoretical results show convergence to a neighborhood of the NE with a residual bound $O\left(a+\frac{1}{\omega}\right)$, and simulations in an oligopoly example validate aperiodic triggering and NE attainment without payoff-model information sharing. The approach enables bandwidth-efficient, scalable NES in networked game settings and lays groundwork for broader applications in distributed control under communication constraints.

Abstract

We propose locally convergent Nash equilibrium seeking algorithms for $N$-player noncooperative games, which use distributed event-triggered pseudo-gradient estimates. The proposed approach employs sinusoidal perturbations to estimate the pseudo-gradients of unknown quadratic payoff functions. This is the first instance of noncooperative games being tackled in a model-free fashion with event-triggered extremum seeking. Each player evaluates independently the deviation between the corresponding current pseudo-gradient estimate and its last broadcasted value from the event-triggering mechanism to tune individually the player action, while they preserve collectively the closed-loop stability/convergence. We guarantee Zeno behavior avoidance by establishing a minimum dwell-time to avoid infinitely fast switching. In particular, the stability analysis is carried out using Lyapunov's method and averaging for systems with discontinuous right-hand sides. We quantify the size of the ultimate small residual sets around the Nash equilibrium and illustrate the theoretical results numerically on an oligopoly setting.

Distributed Event-Triggered Nash Equilibrium Seeking for Noncooperative Games

TL;DR

This work addresses decentralized Nash equilibrium seeking for -player noncooperative games with unknown quadratic payoffs. It introduces distributed event-triggered extremum seeking using sinusoidal perturbations to estimate pseudo-gradients and a time-scale/averaging analysis to handle discontinuities, proving local exponential stability and Zeno avoidance. Theoretical results show convergence to a neighborhood of the NE with a residual bound , and simulations in an oligopoly example validate aperiodic triggering and NE attainment without payoff-model information sharing. The approach enables bandwidth-efficient, scalable NES in networked game settings and lays groundwork for broader applications in distributed control under communication constraints.

Abstract

We propose locally convergent Nash equilibrium seeking algorithms for -player noncooperative games, which use distributed event-triggered pseudo-gradient estimates. The proposed approach employs sinusoidal perturbations to estimate the pseudo-gradients of unknown quadratic payoff functions. This is the first instance of noncooperative games being tackled in a model-free fashion with event-triggered extremum seeking. Each player evaluates independently the deviation between the corresponding current pseudo-gradient estimate and its last broadcasted value from the event-triggering mechanism to tune individually the player action, while they preserve collectively the closed-loop stability/convergence. We guarantee Zeno behavior avoidance by establishing a minimum dwell-time to avoid infinitely fast switching. In particular, the stability analysis is carried out using Lyapunov's method and averaging for systems with discontinuous right-hand sides. We quantify the size of the ultimate small residual sets around the Nash equilibrium and illustrate the theoretical results numerically on an oligopoly setting.
Paper Structure (12 sections, 3 theorems, 86 equations, 3 figures)

This paper contains 12 sections, 3 theorems, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $N$-Player Game with Quadratic Payoffs: General Formulation
  3. Continuous-time Nash Equilibrium Seeking
  4. Emulation of the Continuous-Time Nash Equilibrium Seeking Design
  5. Distributed Event-Triggered Tuning for Nash Equilibrium Seeking
  6. Closed-Loop System for Time-Scaled Triggering Mechanism
  7. Rescaling of Time
  8. Average Closed-Loop System
  9. Stability Analysis
  10. Simulation Results
  11. Conclusions
  12. Average of Discontinuous Systems

Key Result

Theorem 1

Consider the closed-loop average dynamics of the pseudo-gradient estimate (eq:dotHatGav_event_1), the average error vector (eq:Eav_event_1), the average static event-triggering mechanism in Definition def:averageStaticEvent, and Assumptions assumption3 and assum:SDD. For $\omega>0$ sufficiently larg where $a=\sqrt{\sum_{i=1}^{n}a_{i}^{2}}$, with $a_i$ defined in eq:Si and the constants $m$ and $M_

Figures (3)

  • Figure 1: Block diagram of the NES strategy through distributed event-triggered tuning policies of Definition \ref{['def:staticEvent']} performed for each player.
  • Figure 2: Event-triggered Nash equilibrium seeking system.
  • Figure 3: Block diagram of the complete closed-loop system.

Theorems & Definitions (5)

  • Definition 1: NES Static-Triggering Condition
  • Definition 2: Average Static-Triggering Condition
  • Theorem 1
  • Theorem 2
  • Theorem 3