Primal-Dual Formulation in Mean Field Games

• Primal-dual formulation is a measure-theoretic framework that couples a primal optimization with a dual maximization, ensuring value matching via occupation measures.
• It converts continuous-time control and mean field games into linear programs by leveraging stochastic process constraints and Hamilton-Jacobi-Bellman (HJB) subsolutions.
• Strong duality under classical regularity regimes enables precise characterization of Nash equilibria even in nonconvex and nonunique solution settings.

A primal-dual formulation is a measure-theoretic or variational framework that simultaneously expresses an optimization problem (the "primal") and a concave or maximization problem (the "dual"), with value-matching and feasibility conditions linking their solutions. In continuous-time control and game-theoretic contexts, primal-dual formulations enable both the characterization of optimal controls/policies and the precise analytical description of all equilibria (e.g., Nash equilibria in continuous-time mean field games). The technical core is often an equivalence between the original closed-loop control problem and a linear program over occupation measures, together with an abstract dual based on subsolutions to a corresponding Hamilton-Jacobi-Bellman (HJB) equation, resulting in strong duality under minimal regularity assumptions (Guo et al., 2 Mar 2025).

1. Measure-Theoretic Primal Formulation for Control and MFGs

Consider a continuous-time controlled diffusion process for a representative agent with state space $\R^d$, action space $A$ (Polish metric), and finite time horizon $[0,T]$. Given a deterministic mean-field flow $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$ and initial measure $\rho$, the agent selects a measurable, relaxed Markov policy $\gamma: [0,T]\times\R^d \to P(A)$. The state dynamics follow

$$X_t = X_0 + \int_0^t b^{\bm\mu,\gamma}(s,X_s)ds + \int_0^t \sigma^{\bm\mu,\gamma}(s,X_s) dW_s$$

for appropriate drift/diffusion $b,\sigma$ and $d$-dimensional Brownian motion $W$.

The occupation measure approach replaces stochastic process optimization with a linear program on $A$0, with $A$1 representing the law of $A$2 and $A$3 encoding time–state–action occupancy. The key linear "martingale-constraint" for smooth test functions $A$4 is: $A$5 where $A$6 denotes the controlled generator.

The corresponding primal LP is: $A$7 for bounded measurable running/terminal costs $A$8.

An equivalence theorem (Thm 3.5) establishes that the value of the occupation-measure LP matches the original closed-loop stochastic control value: $A$9 with a precise correspondence between optimal policies and optimal occupation measures via disintegration and superposition principles.

2. Dual Formulation: HJB Subsolutions and Abstract Duality

Duality is achieved by constructing the adjoint operator $[0,T]$0 acting on smooth test functions $[0,T]$1, leading to dual feasibility conditions:

• Terminal cost upper bound: $[0,T]$2
• Pointwise subsolution constraint: $[0,T]$3

This yields the dual maximization problem over smooth subsolutions of the formal HJB equation: $[0,T]$4 where

$[0,T]$5

The formal HJB equation for the value function $[0,T]$6 takes the shape: $[0,T]$7 Weak duality ($[0,T]$8) is immediate by pairing primal-feasible and dual-feasible elements.

3. Strong Duality and Regularity Regimes

Strong duality ($[0,T]$9) holds when the HJB admits a classical ($\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$0) solution $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$1. Under such circumstances, there exists a measurable selector $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$2 implementing the minimizer in the HJB, and the deterministic feedback policy $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$3 produces a weak solution to the SDE, saturating the dual constraints.

Elliptic/parabolic regularity arguments differentiate two regimes:

• Semilinear HJB (uncontrolled $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$4) admit classical solutions via Schauder estimates.
• Fully nonlinear HJB (controlled $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$5) with Lipschitz data yield classical solutions via Evans–Krylov theory.

In both, Itô’s formula verifies equality: $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$6 The proof structure involves measurable selection, construction of occupation measures, and demonstration of vanishing duality gap.

4. Primal-Dual Characterization of Nash Equilibria

A Nash equilibrium (NE) for MFGs is a triple $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$7 such that, given flow $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$8, $\bm\mu = (\mu_t)_{t\in[0,T]} \in P(\R^d)^{[0,T]}$9 solves the representative control problem and is consistent, $\rho$0.

The primal-dual system characterizing NE (Thm 4.12) is:

• Primal feasibility: $\rho$1
• Dual feasibility: $\rho$2
• Value matching: $\rho$3
• Consistency: time–state marginal of $\rho$4 is $\rho$5

When strong duality holds at $\rho$6, any Nash equilibrium must satisfy this system. Every minimizer or subsolution (whether pure or mixed) is faithfully represented, and joint primal-dual feasibility identifies all NEs.

5. Absence of Convexity and Uniqueness Assumptions

Unlike conventional approaches requiring convexity in the Hamiltonian or uniqueness of the optimizer, this primal-dual framework only presupposes measurability and boundedness of the coefficients $\rho$7. It allows for nonconvex, nonunique minimizers, and remains robust even when the HJB equation lacks classical (or continuous) solutions. Subsolution-based dual feasibility permits NE construction on the support of the actual flow, not just globally.

6. Key Technical Ingredients and Proof Structure

Crucial lemmas include:

• Disintegration: every occupation measure $\rho$8 can be uniquely decomposed into a Markov kernel $\rho$9 and a time-marginal $\gamma: [0,T]\times\R^d \to P(A)$0.
• Superposition principle: continuous solutions to the time-marginal Fokker–Planck equation lift to martingale solutions of the SDE.
• LP Duality theory: abstract weak duality holds whenever primal and dual cones are nonempty.
• PDE estimates (Schauder, Evans–Krylov) guarantee existence of classical HJB solutions in the respective regimes.

These technical ingredients underwrite the full identification of Nash equilibria through matched primal-dual feasibility and value matching, closing both the analytical and measure-theoretic duality gap (Guo et al., 2 Mar 2025).

Table: Analytical Structures in Primal-Dual MFG Formulation

Component	Mathematical Object	Function/Property
Primal LP	Occupation measures $\gamma: [0,T]\times\R^d \to P(A)$1	Minimizes expected cost given flow
Dual Problem	Smooth subsolutions $\gamma: [0,T]\times\R^d \to P(A)$2	Maximizes initial value subject to HJB-type inequalities
Constraint	Martingale-constraint Eqn	Enforces valid controlled diffusion paths
NE Characterization	Triple $\gamma: [0,T]\times\R^d \to P(A)$3	Feasibility, value matching, law-consistency
Regularity regime	Classical HJB solution	Ensures strong duality, explicit feedback
Extension	Nonconvex/Nonunique regime	Subsolution-based feasibility, generalized NE

This primal-dual analytical framework has established itself as a rigorous and complete characterization tool in continuous-time MFGs, general stochastic control, and beyond, providing full identification of Nash equilibria without restrictive convexity or uniqueness conditions.