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    This study develops a distributed observer strategy for multiplayer systems to achieve generalized Nash equilibrium (GNE). The method ensures players reach GNE even with limited information and complex system dynamics.

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    Area of Science:

    • Control Systems Engineering
    • Game Theory
    • Distributed Systems

    Background:

    • Generalized Nash equilibrium (GNE) seeking is crucial for decentralized decision-making in multiplayer systems.
    • Existing methods often require global information or strong communication topologies.
    • Second-integrator systems present unique challenges due to their dynamics and constraints.

    Purpose of the Study:

    • To develop a distributed observer-based strategy for GNE seeking in second-integrator multiplayer systems.
    • To address scenarios with limited information (e.g., velocity unavailability) and complex system dynamics (e.g., Euler-Lagrange, integrator chains).
    • To ensure asymptotic convergence to GNE under directed communication topologies.

    Main Methods:

    • Synthesis of a distributed observer using a composite variable for state estimation.
    • Application of nonsmooth analysis to prove observer accuracy under strongly connected topologies.
    • Leveraging convex theory for asymptotic convergence analysis.
    • Development of extension strategies including auxiliary dynamics, adaptation techniques, and coordinate transformations for various system types.

    Main Results:

    • The proposed distributed observer enables accurate estimation of other players' variables.
    • All decision variables converge to the generalized Nash equilibrium (GNE).
    • Effective GNE seeking strategies are presented for velocity-free, nonlinear Euler-Lagrange, and integrator chain systems.
    • Simulation results validate the theoretical findings.

    Conclusions:

    • The distributed observer-based approach is effective for GNE seeking in second-integrator systems.
    • The developed strategies are robust to information limitations and system complexities.
    • The framework provides a unified approach for distributed GNE seeking in various multiplayer systems.