Extended mean field game theoretical optimal distributed control for large scale multi-agent systems: An efficiency-complexity tradeoff

Abstract

This paper investigates the tradeoff between optimal efficiency and computational complexity in the emerging mean-field game (MFG) theory and further develops a novel reconfigurable decomposition approach that can balance the efficiency-complexity of MFG theoretical optimal distributed control for large-scale multi-agent systems (LS-MAS). Generally, the MFG has the potential to overcome the “Curse of Dimensionality” in LS-MAS control by simplifying all agents' interactions into ones between individual agents and the collective average effects captured by the group's probability density function (PDF). However, the social cost associated with MFG Nash equilibria is generally inefficient compared to the centralized optimal cost associated with the McKean-Vlasov control problem. To enhance the efficiency of MFG theoretical control without significantly increasing complexity, a novel extended MFG (EMFG) is developed to efficiently balance the MFG efficiency and computational complexity through a decomposed mean field PDF term. Specifically, LS-MAS is divided into multiple groups based on desired terminal PDF constraints. Then, an actor-critic-decomposed mass (ACDM) algorithm is developed to attain the optimal control for LS-MAS by solving the coupled MFG forward-backward partial differential equation (PDE) system. While the PDF decomposition expands the neural network structure, it escalates computational complexity. However, the developed algorithm evaluates and balances LS-MAS optimal control efficiency and complexity. In addition, an induction-based proof is provided to demonstrate the reduction of the inefficiency bound between the optimal cost associated with McKean-Vlasov control and the social cost associated with the extended MFG equilibrium. After that, the Lyapunov stability analysis is presented to illustrate the convergence of the ACDM algorithm. Eventually, numerical simulations are provided to validate the proposed approach.