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A Linearly Convergent Distributed Nash Equilibrium Seeking Algorithm for Aggregative Games Over Time-Varying

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    This study introduces a distributed algorithm to find Nash equilibrium in complex games with constraints. The method ensures convergence even with changing network conditions and constraints.

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    Area of Science:

    • Control Theory
    • Game Theory
    • Networked Systems

    Background:

    • Distributed computation of Nash equilibrium (NE) is challenging.
    • Time-varying unbalanced communication graphs introduce complexity.
    • Local closed convex set constraints require specialized handling.

    Purpose of the Study:

    • To develop a distributed discrete-time algorithm for computing Nash equilibrium.
    • To address challenges posed by time-varying unbalanced graphs and set constraints.
    • To analyze the convergence properties and step-size bounds of the proposed algorithm.

    Main Methods:

    • A novel distributed discrete-time NE seeking algorithm is proposed.
    • Combines average tracking and push-sum protocols for aggregate estimation.
    • Employs the method of feasible direction to manage set constraints.

    Main Results:

    • The algorithm achieves linear convergence, proven using the small gain theorem.
    • Explicit estimates for step-size upper bounds are derived.
    • Numerical simulations confirm the algorithm's effectiveness in a Nash-Cournot game.

    Conclusions:

    • The proposed distributed algorithm effectively computes Nash equilibrium under dynamic and constrained conditions.
    • The theoretical analysis provides guarantees on convergence and step-size selection.
    • The findings are validated through practical simulations, demonstrating real-world applicability.