Distributed Event-Triggered Nash Equilibrium Seeking in Noncooperative Duopoly Games

Core Concepts

The core message of this article is to propose a novel approach for locally stable convergence to Nash equilibrium in duopoly noncooperative games based on a distributed event-triggered control scheme.

Abstract

The article presents a novel approach for achieving Nash equilibrium in duopoly noncooperative games using a distributed event-triggered control scheme. The key highlights and insights are:

The proposed approach employs extremum seeking, with sinusoidal perturbation signals, to estimate the gradient (first derivative) of unknown quadratic payoff functions in a model-free fashion.

The event-triggered methodology is integrated with the noncooperative game setting, which is the first instance of such an integration. Each player evaluates independently the deviation between the current state variable and its last broadcasted value to update the player action.

The stability analysis is carried out using time-scaling technique, Lyapunov's direct method and averaging theory for discontinuous systems. The size of the ultimate small residual sets around the Nash equilibrium is quantified.

Numerical simulations are provided to illustrate the effectiveness of the proposed approach and its advantages over periodic sampled-data control methods.

Stats

The article does not contain any explicit numerical data or metrics to support the key logics. The analysis is primarily theoretical, with the simulation results providing qualitative insights.

Quotes

The article does not contain any striking quotes that directly support the key logics.

Key Insights Distilled From

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

by Vict... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07287.pdf

Deeper Inquiries

How can the proposed event-triggered Nash equilibrium seeking approach be extended to handle more complex game structures, such as multiplayer games or games with non-quadratic payoff functions

The proposed event-triggered Nash equilibrium seeking approach can be extended to handle more complex game structures by incorporating advanced game theory concepts and control strategies. For multiplayer games, the event-triggered control scheme can be adapted to consider interactions among multiple players by introducing communication protocols that allow players to update their strategies based on the actions of others. This can involve developing sophisticated triggering conditions that take into account the actions and payoffs of all players in the game.
In the case of games with non-quadratic payoff functions, the event-triggered approach can be enhanced by utilizing more advanced estimation techniques to approximate the gradients of the payoff functions. This may involve employing machine learning algorithms or reinforcement learning methods to adaptively estimate the payoff gradients in real-time. By incorporating these advanced techniques, the event-triggered control scheme can effectively handle the complexities introduced by non-quadratic payoff functions in games.

What are the potential limitations or drawbacks of the event-triggered control scheme compared to traditional periodic control methods in the context of noncooperative games

While the event-triggered control scheme offers several advantages over traditional periodic control methods, such as reduced communication and computation costs, there are potential limitations and drawbacks to consider in the context of noncooperative games. One limitation is the complexity of designing appropriate triggering conditions that ensure stability and convergence to Nash equilibrium in dynamic game environments. The design of triggering mechanisms that balance the trade-off between control performance and communication overhead can be challenging, especially in scenarios with multiple players and non-linear payoff functions.
Another drawback of the event-triggered control scheme is the potential for increased system complexity and implementation challenges. The integration of event-triggered strategies into existing control systems may require significant modifications to the system architecture and communication protocols. Additionally, the non-periodic nature of event-triggered updates can introduce uncertainties and delays in the control process, which may impact the overall system performance in noncooperative games.

The article focuses on duopoly games, but many real-world scenarios involve oligopolistic or monopolistic market structures. How can the proposed methodology be adapted to address these more general market settings

To adapt the proposed methodology to address more general market settings, such as oligopolistic or monopolistic structures, several modifications and enhancements can be made. In oligopolistic markets with a few dominant players, the event-triggered Nash equilibrium seeking approach can be extended to incorporate strategic interactions and competition dynamics among the key players. This may involve developing more sophisticated triggering conditions that consider the strategic behavior of the dominant firms and their impact on market outcomes.
In monopolistic market settings, where a single firm controls the market, the event-triggered control scheme can be tailored to address the unique challenges of monopolies, such as market power and pricing strategies. By incorporating game-theoretic concepts specific to monopolistic competition, the methodology can be adapted to analyze the strategic decisions of the monopolist and their effects on market equilibrium. This may involve designing specialized triggering mechanisms that account for the monopolist's pricing behavior and market dominance.
Overall, by customizing the event-triggered Nash equilibrium seeking approach to suit the characteristics of oligopolistic and monopolistic market structures, the methodology can provide valuable insights into strategic decision-making and equilibrium outcomes in a broader range of market environments.