[Paper Review] Decentralized Convergence to Nash Equilibria in Constrained Mean Field Control.

Sergio Grammatico, Francesca Parise+2 more|arXiv (Cornell University)|Oct 16, 2014

Distributed Control Multi-Agent SystemsComputer Science10 references13 citations

TL;DR

This paper proposes decentralized iterative methods for mean field control in large populations of heterogeneous agents subject to convex constraints, showing convergence to a Nash equilibrium as population size increases. It extends mean field control theory to constrained settings, demonstrating applicability to linear quadratic control and electric vehicle charging problems.

ABSTRACT

This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and affected by the aggregate behavior of the overall population. For such large-scale systems, the theory of “mean field” games and control has been successfully applied in various scientific disciplines. While the existing mean field control literature is limited to unconstrained problems, we formulate mean field problems in the presence of heterogeneous convex constraints at the level of individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several iterative solution methods and show that, even in the presence of constraints, the mean field solution gets arbitrarily close to a mean field Nash equilibrium as the population size grows. We apply our methods to the constrained linear quadratic mean field control problem and to the constrained mean field charging control problem for large populations of plug-in electric vehicles.

Motivation & Objective

• To address the gap in mean field control theory for constrained systems with heterogeneous agents.
• To develop iterative, decentralized algorithms that converge to a Nash equilibrium despite individual constraints.
• To extend mean field control to real-world problems involving state and control constraints, such as electric vehicle charging.
• To rigorously analyze convergence properties of the proposed methods under convex constraints.

Proposed method

• Formulates a constrained mean field control problem with individual agents having heterogeneous convex constraints on states and controls.
• Introduces iterative, decentralized algorithms where each agent updates its strategy based on the population's mean field behavior.
• Uses a dual decomposition approach to handle convex constraints, enabling distributed computation.
• Applies the method to two case studies: linear quadratic control and plug-in electric vehicle charging.
• Employs a variational inequality framework to analyze convergence to Nash equilibrium.
• Proves that as population size grows, the mean field solution approaches a Nash equilibrium even under constraints.

Experimental results

Research questions

• RQ1 Can decentralized mean field control be extended to systems with heterogeneous convex constraints on individual agents?
• RQ2 Do iterative, decentralized algorithms converge to a Nash equilibrium when constraints are present?
• RQ3 How does the presence of constraints affect the convergence rate and stability of mean field control?
• RQ4 Can the proposed method be effectively applied to practical problems like electric vehicle charging with state and control limits?
• RQ5 What theoretical guarantees can be provided for convergence to Nash equilibrium in constrained mean field control?

Key findings

• The proposed decentralized algorithms converge to a mean field Nash equilibrium as the population size tends to infinity, even under heterogeneous convex constraints.
• The method successfully extends mean field control theory beyond unconstrained settings to include state and control constraints.
• For the linear quadratic mean field control problem, the algorithm achieves near-optimal performance under constraints.
• In the electric vehicle charging application, the method ensures that all vehicles satisfy individual charging constraints while minimizing system-wide cost.
• Theoretical analysis confirms convergence via a variational inequality framework, establishing the existence of a Nash equilibrium in the limit.
• Numerical results demonstrate robustness and scalability of the method in large-scale settings with diverse agent behaviors.