Variational Mean Field Game System

• Variational Mean Field Game System is a reformulation of coupled Hamilton–Jacobi and Fokker–Planck equations into a unified Euler–Lagrange framework via an introduced potential function.
• This approach simplifies analysis by encoding mass conservation and flow in the potential, which facilitates proofs of existence, uniqueness, and robust computability of solutions.
• The framework extends naturally to congestion models and crowd dynamics, employing variational inequality and monotonicity techniques for handling complex, nonlinear scenarios.

A variational mean field game system in one spatial dimension recasts the coupled Hamilton–Jacobi and Fokker–Planck (continuity) equations as the Euler–Lagrange equations of an underlying variational problem, using the introduction of a potential function. This approach exploits the structure of the continuity equation to reduce the complexity of the problem and, for a range of models, establishes existence, uniqueness, and computability of solutions under broad conditions.

1. Variational Recasting of the Mean Field Game System

The canonical mean field game (MFG) planning system in one dimension consists of two coupled partial differential equations: $$\begin{cases} -u_t - \lambda u_{xx} + H(u_x) + V(x) = g(m), \ m_t - \lambda m_{xx} - \partial_x (H'(u_x)m) = 0, \end{cases}$$ defined on $(0,T)\times \mathbb{T}$, where $u$ is the value function, $m$ is the density, $H$ the Hamiltonian, $V$ a potential, and $g$ the mean field coupling.

The variational approach leverages Poincaré’s Lemma to "integrate" the continuity equation for $m$ by introducing a potential function $\varphi(t,x)$ such that: $$\begin{cases} m = \varphi_x + 1, \ \lambda m_x + H'(u_x) m = \varphi_t + q(t), \end{cases}$$ where $q(t)$ is a time-dependent normalization.

This reformulation yields a variational problem over $(\varphi, q)$, with the Lagrangian density defined via the Legendre transform $L$ of $H$: $$L_0(z, y) = \begin{cases} L\left( \frac{z}{y} \right) y & \text{if } y > 0, \ +\infty & \text{if } y=0,\, z \neq 0, \ 0 & \text{if } y=0,\, z = 0. \end{cases}$$ The action to be minimized is: $$\int_0^T\!\!\int_\mathbb{T} \left[ L_0(\varphi_t + q - \lambda \varphi_{xx}, \varphi_x + 1) - V(x)\varphi_x + G(\varphi_x + 1) \right] dx\,dt,$$ subject to boundary and normalization constraints on $\varphi$.

The Euler–Lagrange equations precisely encode the original MFG system after transforming variables.

2. The Role of the Potential Function

The potential $\varphi$ represents a cumulative distribution or quantile function underlying the mass density $m$, with $m = \varphi_x + 1$. It transforms the continuity (Fokker–Planck) equation into a constraint on derivatives of $\varphi$, thus encoding the mass evolution and flux directly.

This reparameterization allows the entire planning MFG to be formulated as a single variational problem. The potential method therefore:

• Encodes the mass conservation and flow,
• Bypasses the need to enforce the continuity equation directly,
• Enables solution by calculus of variations methods,
• Simplifies analytical and computational treatment, especially in lower dimensions.

Once $\varphi$ has been determined as the minimizer, the corresponding MFG variables are reconstructed by: $$m(t, x) = \varphi_x(t, x) + 1, \quad u(t, x) = \int_0^x L'\left( \frac{\varphi_t(t, \tau) + q(t) - \lambda \varphi_{xx}(t, \tau)}{\varphi_x(t, \tau) + 1} \right) d\tau.$$

3. Existence and Uniqueness of Solutions

Existence is established via the direct method in the calculus of variations:

• Coercivity: Ensured by growth conditions $G(z) \geq C|z|^\gamma - C$, $L(w) \geq C|w|^\beta$.
• Lower Semicontinuity: Proved in suitable Sobolev spaces for the action functional.
• Compactness: The set of admissible potentials is convex, closed, weakly sequentially compact.

Strong convexity of the problem (in the case of strictly convex $G$), together with standard arguments, ensures uniqueness: any two minimizers must coincide due to the convexity of the functional.

The method establishes:

• Existence and uniqueness for the full MFG planning system under reasonable regularity and convexity conditions,
• Well-posedness even in degenerate or nonlinear settings (subject to the structure of the variational inequality for nonconvex congestion problems).

4. Extension to Congestion Models and Nonlinear Variational Inequalities

For first-order mean field games with congestion, the system takes the form: $$\begin{cases} -u_t + \frac{u_x^2}{2m^\alpha} = m^\mu, \ m_t - (u_x m^{1-\alpha})_x = 0, \end{cases}$$ with $m > 0$.

Introducing the potential yields: $$\begin{cases} m = \varphi_x + 1, \ u_x = (\varphi_x + 1)^{\alpha-1}(\varphi_t + q), \end{cases}$$ which, for certain parameter regimes, does not correspond to a convex variational problem but rather to a monotone operator system. The solution concept is then formulated via variational inequalities, searching for weak solutions such that: $$\langle A[\psi, \varpi], [\psi, \varpi] - [\varphi, q] \rangle \geq 0$$ for all admissible test functions, where $A$ encodes the system as an operator in function space. Existence follows via monotonicity and compactness arguments, utilizing regularization and Schaefer’s fixed point theorem.

5. Application to the One-Dimensional Hughes' Model

The variational/potential method is applied to the one-dimensional Hughes’ model for crowd motion: $$\begin{cases} -\rho_t + [\rho f^2(\rho) \Psi_x]_x = 0, \ f(\rho)|\Psi_x| = 1. \end{cases}$$ Introducing a potential $\varphi$ such that $\rho = \varphi_x$, the system can be rewritten as a Hamilton–Jacobi equation. For $f(\rho) = 1 - \rho$, the potential satisfies: $\varphi_t = \varphi_x (1 - \varphi_x),$ which may be explicitly solved via the Hopf–Lax formula: $$\varphi(t, x) = \min_y \left\{ t L\left( \frac{x-y}{t} \right) + \int_{-\infty}^y \rho_0(\tau)d\tau \right\},$$ where $L$ is the Legendre transform of the Hamiltonian $H$.

This demonstrates the versatility of the variational approach for modeling crowd dynamics, congestion, and MFG planning in low dimensions, allowing for analytic solutions and robust qualitative analysis.

Summary Table

Aspect	Main Content/Formulation
Variational formulation	Minimize integral action over $(\varphi, q)$ using $L_0$, with constraints on potential
Potential mechanism	Integrates continuity equation, reduces to single variational entity
Existence/uniqueness	Deducted via direct method, strong convexity, compactness, lower semicontinuity
Extension to congestion	System becomes monotone operator problem; solved as variational inequality (weak solution)
Application to Hughes' model	Reduces to Hamilton–Jacobi equation (with explicit solutions), illustrating unifying applicability

Theoretical and Practical Implications

• The potential (variational) approach systematically simplifies MFG planning problems in one dimension by eliminating the need for explicit continuity enforcement.
• Convex optimization and monotonicity techniques become directly applicable, providing strong theoretical guarantees and efficient algorithms.
• It extends naturally to more complex crowd models and congestion effects, with well-developed variational inequality tools providing existence and uniqueness even when convexity fails.

This framework enhances both the analytic tractability and computational viability of MFG models, establishing a foundational methodology for further developments in mean field game theory, particularly in one spatial dimension.