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    This study introduces novel distributed algorithms for solving complex aggregative games. These methods efficiently find generalized Nash equilibria (GNE) in both differentiable and nonsmooth scenarios, demonstrating practical effectiveness.

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    Area of Science:

    • Game Theory
    • Distributed Systems
    • Optimization

    Background:

    • Aggregative games involve player costs dependent on collective decisions.
    • Existing methods struggle with nonsmooth cost functions and complex constraints.
    • Finding generalized Nash equilibria (GNE) is crucial for strategic decision-making.

    Purpose of the Study:

    • To develop distributed algorithms for finding variational GNE in aggregative games.
    • To address both differentiable and nonsmooth cost function scenarios.
    • To analyze the convergence properties of the proposed algorithms.

    Main Methods:

    • For differentiable games: Gradient descent and dynamic average consensus.
    • For nonsmooth games: Continuous-time distributed projection-based algorithms using differential inclusions.
    • Convergence analysis via Lyapunov stability theory and singular perturbation analysis.

    Main Results:

    • Exponential convergence to variational GNE for differentiable games.
    • Convergence to variational GNE for nonsmooth games.
    • Simulation examples validate the proposed methods' effectiveness.

    Conclusions:

    • The developed distributed algorithms effectively compute variational GNE for aggregative games.
    • The methods are robust, handling both smooth and nonsmooth cost functions.
    • This research advances the solution of complex strategic decision-making problems in distributed settings.