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Continuous-time mean field games: a primal-dual characterization

Xin Guo, Anran Hu, Jiacheng Zhang, Yufei Zhang

TL;DR

This work introduces a primal-dual framework for continuous-time mean field games with measurable coefficients, establishing an LP over occupation measures for the representative player's problem and a dual problem that maximizes smooth subsolutions of the associated HJB equation. Strong duality between the LP and its dual yields a complete characterization of all Nash equilibria, without requiring convexity of the Hamiltonian or uniqueness of its optimizer, and remains valid even when the HJB lacks a classical or continuous solution. The analysis leverages the superposition principle to connect measure-valued LP solutions to weak solutions of the controlled state dynamics, and hinges on regularity results for semilinear and fully nonlinear PDEs to guarantee dual solvability in broad settings. Compared to HJB-FP and BSDE approaches, the primal-dual formulation provides a versatile verification mechanism for NE and a pathway to compute multiple equilibria, with potential numerical algorithms built on joint primal-dual optimization.

Abstract

This paper establishes a primal-dual formulation for continuous-time mean field games (MFGs) and provides a complete analytical characterization of the set of all Nash equilibria (NEs). We first show that for any given mean field flow, the representative player's control problem with {\it measurable coefficients} is equivalent to a linear program over the space of occupation measures. We then establish the dual formulation of this linear program as a maximization problem over smooth subsolutions of the associated Hamilton-Jacobi-Bellman (HJB) equation, which plays a fundamental role in characterizing NEs of MFGs. Finally, a complete characterization of \emph{all NEs for MFGs} is established by the strong duality between the linear program and its dual problem. This strong duality is obtained by studying the solvability of the dual problem, and in particular through analyzing the regularity of the associated HJB equation. Compared with existing approaches for MFGs, the primal-dual formulation and its NE characterization do not require the convexity of the associated Hamiltonian or the uniqueness of its optimizer, and remain applicable when the HJB equation lacks classical or even continuous solutions.

Continuous-time mean field games: a primal-dual characterization

TL;DR

This work introduces a primal-dual framework for continuous-time mean field games with measurable coefficients, establishing an LP over occupation measures for the representative player's problem and a dual problem that maximizes smooth subsolutions of the associated HJB equation. Strong duality between the LP and its dual yields a complete characterization of all Nash equilibria, without requiring convexity of the Hamiltonian or uniqueness of its optimizer, and remains valid even when the HJB lacks a classical or continuous solution. The analysis leverages the superposition principle to connect measure-valued LP solutions to weak solutions of the controlled state dynamics, and hinges on regularity results for semilinear and fully nonlinear PDEs to guarantee dual solvability in broad settings. Compared to HJB-FP and BSDE approaches, the primal-dual formulation provides a versatile verification mechanism for NE and a pathway to compute multiple equilibria, with potential numerical algorithms built on joint primal-dual optimization.

Abstract

This paper establishes a primal-dual formulation for continuous-time mean field games (MFGs) and provides a complete analytical characterization of the set of all Nash equilibria (NEs). We first show that for any given mean field flow, the representative player's control problem with {\it measurable coefficients} is equivalent to a linear program over the space of occupation measures. We then establish the dual formulation of this linear program as a maximization problem over smooth subsolutions of the associated Hamilton-Jacobi-Bellman (HJB) equation, which plays a fundamental role in characterizing NEs of MFGs. Finally, a complete characterization of \emph{all NEs for MFGs} is established by the strong duality between the linear program and its dual problem. This strong duality is obtained by studying the solvability of the dual problem, and in particular through analyzing the regularity of the associated HJB equation. Compared with existing approaches for MFGs, the primal-dual formulation and its NE characterization do not require the convexity of the associated Hamiltonian or the uniqueness of its optimizer, and remain applicable when the HJB equation lacks classical or even continuous solutions.

Paper Structure

This paper contains 26 sections, 17 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Stochastic control and linear programming.
  3. MFGs and NE.
  4. MFGs and linear programming.
  5. Our work.
  6. Primal-dual vs. existing approaches for MFGs.
  7. Our techniques and most related works.
  8. Notation.
  9. Mathematical setup of MFGs
  10. Representative player's problem.
  11. MFG and its NE.
  12. Primal formulation of representative player's problem
  13. The LP.
  14. Equivalence between \ref{['controlled_cl_diffusion']} and \ref{['Linear_programming_diffusion']}.
  15. Dual formulation and verification of NEs
  16. ...and 11 more sections

Key Result

Proposition 3.1

Suppose (H.assm:diffusion_state) holds. Let $\bm \mu \in \mathcal{P}({\mathbb R}^d|{\mathbb T})$ and let $(\bm X,\gamma)\in \mathcal{A}_{\rm cl}(\bm \mu)$ be defined on a probability space $(\Omega, \mathcal{F},\mathbb{P})$. Define $\nu\in \mathcal{M}_+({\mathbb R}^d)$ and $\xi\in \mathcal{M}_+({\ma Then $(\nu,\xi)\in \mathcal{D}_P(\bm \mu)$ and $J^{\bm \mu}_{\rm cl}(\bm X, \gamma)=J^{\bm \mu}_P(

Theorems & Definitions (42)

  • Definition 2.1
  • Proposition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Remark 3.1
  • Theorem 3.5
  • Remark 3.2
  • Lemma 4.1
  • Proposition 4.2
  • ...and 32 more