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Nash Equilibrium Seeking for Noncooperative Duopoly Games via Event-Triggered Control

Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira, Miroslav Krstić, Tamer Başar

TL;DR

The paper tackles locally stable convergence to Nash equilibrium in noncooperative duopoly games under model-free conditions by proposing a distributed event-triggered extremum-seeking controller. Each player estimates gradient components using sinusoidal perturbations and updates actions only when a triggering condition is met, enabling aperiodic, communication-efficient updates. Time-scale transformation and averaging theory, combined with Lyapunov analysis, establish local exponential stability of the averaged system and bound residuals, while guaranteeing no Zeno behavior. Numerical simulations validate the approach, showing convergence to the Nash equilibrium with independent, decentralized updates and without information sharing between players. The work has practical implications for decentralized decision-making under limited bandwidth in networked game settings.

Abstract

This paper proposes a novel approach for locally stable convergence to Nash equilibrium in duopoly noncooperative games based on a distributed event-triggered control scheme. The proposed approach employs extremum seeking, with sinusoidal perturbation signals applied to estimate the Gradient (first derivative) of unknown quadratic payoff functions. This is the first instance of noncooperative games being tackled in a model-free fashion integrated with the event-triggered methodology. Each player evaluates independently the deviation between the corresponding current state variable and its last broadcasted value to update the player action, while they preserve control performance under limited bandwidth of the actuation paths and still guarantee stability for the closed-loop dynamics. In particular, the stability analysis is carried out using time-scaling technique, Lyapunov's direct method and averaging theory for discontinuous systems. We quantify the size of the ultimate small residual sets around the Nash equilibrium and illustrate the theoretical results numerically on an example.

Nash Equilibrium Seeking for Noncooperative Duopoly Games via Event-Triggered Control

TL;DR

The paper tackles locally stable convergence to Nash equilibrium in noncooperative duopoly games under model-free conditions by proposing a distributed event-triggered extremum-seeking controller. Each player estimates gradient components using sinusoidal perturbations and updates actions only when a triggering condition is met, enabling aperiodic, communication-efficient updates. Time-scale transformation and averaging theory, combined with Lyapunov analysis, establish local exponential stability of the averaged system and bound residuals, while guaranteeing no Zeno behavior. Numerical simulations validate the approach, showing convergence to the Nash equilibrium with independent, decentralized updates and without information sharing between players. The work has practical implications for decentralized decision-making under limited bandwidth in networked game settings.

Abstract

This paper proposes a novel approach for locally stable convergence to Nash equilibrium in duopoly noncooperative games based on a distributed event-triggered control scheme. The proposed approach employs extremum seeking, with sinusoidal perturbation signals applied to estimate the Gradient (first derivative) of unknown quadratic payoff functions. This is the first instance of noncooperative games being tackled in a model-free fashion integrated with the event-triggered methodology. Each player evaluates independently the deviation between the corresponding current state variable and its last broadcasted value to update the player action, while they preserve control performance under limited bandwidth of the actuation paths and still guarantee stability for the closed-loop dynamics. In particular, the stability analysis is carried out using time-scaling technique, Lyapunov's direct method and averaging theory for discontinuous systems. We quantify the size of the ultimate small residual sets around the Nash equilibrium and illustrate the theoretical results numerically on an example.
Paper Structure (11 sections, 1 theorem, 60 equations, 2 figures)

This paper contains 11 sections, 1 theorem, 60 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Duopoly Game with Quadratic Payoffs: General Formulation
  3. Continuous-time Extremum Seeking
  4. Emulation of Continuous-Time Extremum Seeking Design
  5. Distributed Event-Triggered Actions for Nash Equilibrium Seeking
  6. Closed-loop System for Time-Scaled Triggering Mechanism
  7. Time-Scaling Procedure
  8. Average Closed-Loop System
  9. Stability Analysis
  10. SIMULATION RESULTS
  11. CONCLUSIONS

Key Result

Theorem 1

Consider the closed-loop average dynamics of the gradient estimate (eq:dotHatGav_event_1), the average error vector eq:Eav_event_1, and the average distributed event-triggering mechanism in Definition def:averageStaticEvent. For a sufficiently large $\omega>0$, defined in (eq:new_omega), the equilib where $a=\sqrt{a_{1}^{2}+a_{2}^{2}}$, with $a_1$, $a_2$ defined in eq:Si--eq:Mi, and the constants

Figures (2)

  • Figure 1: Block diagram illustrating the NES strategy through distributed event-triggered control policies performed for each player.
  • Figure 2: Numerical simulations for NES in noncooperative duopoly games through distributed ETC policies.

Theorems & Definitions (3)

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