Finite-to-mean-field convergence rate and approximation quality for SMFG at practical constellation sizes

Establish explicit finite-to-mean-field convergence rates and quantify the approximation error of the mean field equilibrium for the Stackelberg Mean Field Game formulation of distributed data offloading in ultra-dense LEO satellite networks, specifically at practical constellation sizes on the order of M ≈ 5000 satellites, to characterize how well the finite-player Nash equilibria approximate the mean field equilibrium in this model.

Background

The paper critiques prior work that models LEO satellite energy management and offloading using a Stackelberg Mean Field Game (SMFG) formulation. While that approach provides a game-theoretic framework, the authors note that it lacks an analysis of how finite-player equilibria converge to the mean field equilibrium as the population size increases, especially at scales relevant to mega-constellations.

This gap is impactful because operators need quantitative guarantees that solutions designed under mean field assumptions remain accurate when deployed in constellations with thousands of satellites. Without explicit convergence rates or bounds on equilibrium approximation error, the practical fidelity of SMFG-based strategies remains unassessed at operational scales. The authors address this issue for their own HBAG framework by proving an O(M^{-1/4}) rate, but they identify the absence of analogous results for SMFG as an unresolved issue.