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Optimal Control and Potential Games in the Mean Field

Felix Höfer, H. Mete Soner

TL;DR

The paper establishes a general bridge between mean-field control (MFC) and a potential mean-field game (MFG) of controls in a setting with non-Markovian dynamics, common noise, and jumps. By formulating an invariant value function and using a probabilistic coin-flip construction, it shows that minimizers of the MFC problem induce Nash equilibria of the associated MFG, including extensions to law-dependent dynamics. It also connects these mean-field constructs to Wasserstein gradient flows and provides a range of examples, such as price-interaction models, Kuramoto synchronization, and Cucker-Smale flocking, along with an explicit linear-quadratic case. An invariance principle ensures that the fundamental conclusions hold across different probabilistic representations, making the results robust for both stochastic and deterministic flow formulations. Collectively, the work broadens the applicability of potential mean-field games by accommodating jumps, common noise, and path-dependent interactions, with potential implications for economic, physical, and biological systems.

Abstract

We study a mean field optimal control problem with general non-Markovian dynamics, including both common noise and jumps. We show that its minimizers are Nash equilibria of an associated mean field game of controls. These types of games are necessarily potential, and the Nash equilibria derived as the minimizers of the control problem are closely connected to McKean-Vlasov equations of Langevin type. To illustrate the general theory, we present several examples, including a mean field game of controls with interactions through a price variable, and mean field Cucker-Smale Flocking and Kuramoto models. We also establish the invariance property of the value function, a key ingredient used in our proofs.

Optimal Control and Potential Games in the Mean Field

TL;DR

The paper establishes a general bridge between mean-field control (MFC) and a potential mean-field game (MFG) of controls in a setting with non-Markovian dynamics, common noise, and jumps. By formulating an invariant value function and using a probabilistic coin-flip construction, it shows that minimizers of the MFC problem induce Nash equilibria of the associated MFG, including extensions to law-dependent dynamics. It also connects these mean-field constructs to Wasserstein gradient flows and provides a range of examples, such as price-interaction models, Kuramoto synchronization, and Cucker-Smale flocking, along with an explicit linear-quadratic case. An invariance principle ensures that the fundamental conclusions hold across different probabilistic representations, making the results robust for both stochastic and deterministic flow formulations. Collectively, the work broadens the applicability of potential mean-field games by accommodating jumps, common noise, and path-dependent interactions, with potential implications for economic, physical, and biological systems.

Abstract

We study a mean field optimal control problem with general non-Markovian dynamics, including both common noise and jumps. We show that its minimizers are Nash equilibria of an associated mean field game of controls. These types of games are necessarily potential, and the Nash equilibria derived as the minimizers of the control problem are closely connected to McKean-Vlasov equations of Langevin type. To illustrate the general theory, we present several examples, including a mean field game of controls with interactions through a price variable, and mean field Cucker-Smale Flocking and Kuramoto models. We also establish the invariance property of the value function, a key ingredient used in our proofs.
Paper Structure (34 sections, 19 theorems, 166 equations)

This paper contains 34 sections, 19 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. MFC, MFG and Main Results
  3. Context and Earlier Results
  4. Notation and Setting
  5. Probabilistic Structures
  6. Controlled State Process
  7. Main Result
  8. The Mean Field Control Problem
  9. The Potential Mean Field Game of Controls
  10. Main Result
  11. Law-Dependent Dynamics
  12. Proof of Theorem \ref{['thm:pot_game_finite']}
  13. Preliminaries
  14. Completion of the Proof
  15. The Case of Law-Dependent Dynamics
  16. ...and 19 more sections

Key Result

Theorem 1

Under natural regularity and growth conditions on the coefficients, for any minimizer $\alpha^*$ of the MFC problem eq:MFC_cost1, the pair $(\alpha^*,\nu^*)$ with $\nu^*_t:={\mathcal{L}}((X^{\alpha^*}_{\cdot\land t}, \alpha^*_t) \,|\, W^0,N^0)$ for $t\in[0,T]$, is a Nash equilibrium of the MFG of co

Theorems & Definitions (43)

  • Theorem
  • Definition 2.1
  • Definition 2.2: Admissible controls
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Theorem 3.6
  • ...and 33 more