Balancing Efficiency and Complexity in Large Scale Multi-Agent Optimization: A Reconfigurable Mean Field Game Approach

This paper explores the trade-off between efficiency and complexity in mean-field game (MFG) theory for large-scale multi-agent systems (LS-MAS). While MFG reduces computational burden by avoiding the curse of dimensionality and achieving Nash equilibrium, its solutions often yield inefficient social cost compared to the centralized McKeanVlasov control. To address this, we propose an extended MFG (EMFG) framework that improves efficiency with minimal complexity increase by introducing a decomposed mean field term using probability density functions (PDFs). Agents are grouped based on terminal PDF constraints, and an actor-critic-decomposed mass (ACDM) algorithm is developed to solve the resulting forward-backward PDE system. Although the PDF decomposition enlarges the neural network, the algorithm effectively balances control efficiency and computational cost. We provide an induction-based proof to bound the inefficiency gap between EMFG and centralized control and establish Lyapunov stability for the convergence of ACDM.