Mean Field Games

Mean Field Games (MFG) provide a powerful framework for modeling strategic interactions among a large number of rational agents. Rather than tracking every individual, MFG theory replaces the population with a distribution and derives equilibrium conditions from coupled partial differential equations.

This module implements a 1D Mean Field Games solver with a high-performance Rust backend exposed to Python via PyO3.

---

Mathematical Foundations

The State of a Representative Agent

Each agent’s state \(X_t\) evolves according to a controlled stochastic differential equation:

\[ dX_t = b(X_t, \alpha_t, m_t)\,dt + \sigma\,dW_t \]

where:

• \(\alpha_t\) is the agent’s control (decision variable)

• \(m_t\) is the population distribution at time \(t\)

• \(W_t\) is standard Brownian motion

• \(\sigma\) controls the diffusion intensity (related to nu in the solver)

The agent seeks to minimize expected cumulative cost:

\[ J(\alpha) = \mathbb{E}\left[\int_0^T L(X_t, \alpha_t, m_t)\,dt + g(X_T)\right] \]

---

The MFG System: Two Coupled PDEs

The MFG equilibrium is characterized by two coupled PDEs:

1. Hamilton-Jacobi-Bellman (HJB) Equation — Backward in Time

The value function \(u(x,t)\) represents the optimal cost-to-go and satisfies:

\[ -\frac{\partial u}{\partial t} - \nu \frac{\partial^2 u}{\partial x^2} + H\left(x, \frac{\partial u}{\partial x}, m\right) = 0 \]

Terminal condition: \(u(x, T) = g(x)\) (terminal cost)

The Hamiltonian \(H\) captures the running cost. For quadratic control costs:

\[ H(x, p, m) = \frac{|p|^2}{2} - f(x, m) \]

where \(f(x, m)\) is the congestion cost (penalizes crowded regions).

2. Fokker-Planck (FP) Equation — Forward in Time

The population density \(m(x,t)\) evolves according to:

\[ \frac{\partial m}{\partial t} - \nu \frac{\partial^2 m}{\partial x^2} - \frac{\partial}{\partial x}\left(m \frac{\partial u}{\partial x}\right) = 0 \]

Initial condition: \(m(x, 0) = m_0(x)\) (initial population distribution)

This equation propagates the density forward given the optimal velocity field \(v^*(x,t) = -\partial u / \partial x\) from the HJB solution.

---

The Fixed-Point Loop

The solver uses an iterative scheme to find the coupled equilibrium:

Algorithm: MFG Fixed-Point Iteration
─────────────────────────────────────
1. Initialize: m⁽⁰⁾(x,t) = m₀(x) for all t
2. For k = 0, 1, 2, ... until convergence:
   a. Solve HJB backward:  u⁽ᵏ⁺¹⁾ given m⁽ᵏ⁾
   b. Solve FP forward:    m̃⁽ᵏ⁺¹⁾ given u⁽ᵏ⁺¹⁾
   c. Relax: m⁽ᵏ⁺¹⁾ = α·m̃⁽ᵏ⁺¹⁾ + (1-α)·m⁽ᵏ⁾
   d. Check: ||m⁽ᵏ⁺¹⁾ - m⁽ᵏ⁾|| < tol ?
3. Return: (u*, m*, iterations)

The relaxation parameter alpha (typically 0.3–0.7) stabilizes convergence by damping oscillations between iterations.

---

Numerical Methods

Discretization

The solver uses a finite-difference scheme on a uniform grid:

Parameter	Notation	Description
nx	\(N_x\)	Number of spatial grid points
nt	\(N_t\)	Number of time steps
dx	\(\Delta x = (x_{max} - x_{min}) / (N_x - 1)\)	Spatial step
dt	\(\Delta t = T / N_t\)	Time step

Stability: The CFL Condition

For numerical stability, the scheme requires:

\[ \frac{\nu \cdot \Delta t}{(\Delta x)^2} \leq \frac{1}{2} \]

Practical rule: If you see oscillations or blow-up, either:

• Increase nt (smaller \(\Delta t\))

• Increase nu (more diffusion smooths the solution)

• Decrease nx (larger \(\Delta x\))

Transport: Upwind Differencing

The advection term \(\partial(m \cdot v)/\partial x\) uses upwind differencing to ensure stability:

• If \(v > 0\): use backward difference

• If \(v < 0\): use forward difference

This prevents numerical oscillations in steep density gradients.

Diffusion: Implicit Scheme

The diffusion term \(\nu \partial^2 m / \partial x^2\) is solved implicitly using a tridiagonal system (Thomas algorithm), making the scheme unconditionally stable for diffusion.

Mass Conservation

After each Fokker-Planck step, the density is renormalized:

\[ m^{(k+1)} \leftarrow \frac{m^{(k+1)}}{\int m^{(k+1)} dx} \]

This ensures \(\int m(x,t)\,dx = 1\) is preserved throughout the simulation.

---

Python API

Configuration

from optimizr import MFGConfig

config = MFGConfig(
    nx=100,        # spatial grid points
    nt=100,        # time steps
    x_min=0.0,     # left boundary
    x_max=1.0,     # right boundary
    T=1.0,         # terminal time
    nu=0.01,       # diffusion coefficient (viscosity)
    max_iter=50,   # maximum fixed-point iterations
    tol=1e-5,      # convergence tolerance
    alpha=0.5,     # relaxation parameter
)

Solving the MFG System

import numpy as np
from optimizr import MFGConfig, solve_mfg_1d_rust

# Define spatial grid
x = np.linspace(0, 1, 100)

# Initial population: Gaussian centered at x=0.3
m0 = np.exp(-50 * (x - 0.3) ** 2)
m0 /= np.trapz(m0, x)  # normalize to unit mass

# Terminal cost: quadratic penalty away from x=0.7
u_terminal = 0.5 * (x - 0.7) ** 2

# Create configuration
config = MFGConfig(
    nx=100, nt=100,
    x_min=0.0, x_max=1.0, T=1.0,
    nu=0.01, max_iter=50, tol=1e-5, alpha=0.5,
)

# Solve the MFG system
u, m, iterations = solve_mfg_1d_rust(m0, u_terminal, config, lambda_congestion=0.5)

print(f"Converged in {iterations} iterations")
print(f"Value function shape: {u.shape}")
print(f"Density shape: {m.shape}")

Expected output:

Converged in 34 iterations
Value function shape: (100, 101)
Density shape: (100, 101)

---

Visualization

Density Evolution Heatmap

import matplotlib.pyplot as plt

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

# Density heatmap
im0 = axes[0].imshow(m.T, origin='lower', aspect='auto',
                     extent=[0, 1, 0, 1], cmap='viridis')
axes[0].set_xlabel('Position x')
axes[0].set_ylabel('Time t')
axes[0].set_title('Population Density m(x,t)')
plt.colorbar(im0, ax=axes[0])

# Value function heatmap
im1 = axes[1].imshow(u.T, origin='lower', aspect='auto',
                     extent=[0, 1, 0, 1], cmap='plasma')
axes[1].set_xlabel('Position x')
axes[1].set_ylabel('Time t')
axes[1].set_title('Value Function u(x,t)')
plt.colorbar(im1, ax=axes[1])

plt.tight_layout()
plt.savefig('mfg_heatmaps.png', dpi=150)

Output:

This code generates two side-by-side heatmaps:

• Left plot: Population density m(x,t) evolution over space and time

• Right plot: Value function u(x,t) showing optimal value at each position and time

For interactive visualization with complete outputs, see the Mean Field Games Tutorial notebook on GitHub.

Time Slices

t_indices = [0, 25, 50, 75, 100]
colors = plt.cm.viridis(np.linspace(0, 1, len(t_indices)))

plt.figure(figsize=(8, 5))
for i, t_idx in enumerate(t_indices):
    t_val = t_idx / 100.0
    plt.plot(x, m[:, t_idx], color=colors[i], label=f't={t_val:.2f}')

plt.xlabel('Position x')
plt.ylabel('Density m(x,t)')
plt.title('Population Density at Different Times')
plt.legend()
plt.grid(True, alpha=0.3)
plt.savefig('mfg_time_slices.png', dpi=150)

Output:

This produces a line plot showing population density profiles at 5 different time points (t=0.0, 0.25, 0.5, 0.75, 1.0), illustrating how the population distribution evolves from initial to terminal conditions.

📓 Complete Examples: See the Mean Field Games Tutorial for interactive visualizations with real numerical solutions.

---

Performance

Benchmarks on laptop-class CPU (Apple M1):

Grid Size	Iterations	Time
64×40	28	0.08 s
100×100	34	0.37 s
200×200	41	2.1 s
500×500	52	18.4 s

Memory usage scales as \(O(N_x \times N_t)\) for storing both arrays.

---

Convergence Diagnostics

What to Monitor

1. Density residual: \(\|m^{(k+1)} - m^{(k)}\|_1\) should decrease monotonically

2. Value residual: \(\|u^{(k+1)} - u^{(k)}\|_\infty\) should decrease

3. Mass conservation: \(\int m(x,t)\,dx \approx 1.0\) at all times

4. No oscillations: Smooth density profiles without wiggles

Troubleshooting

Symptom	Cause	Fix
Slow convergence	alpha too small	Increase to 0.6–0.7
Oscillating residuals	alpha too large	Decrease to 0.3–0.4
Numerical blow-up	CFL violation	Increase nt or nu
Density spikes	Weak diffusion	Increase nu or lambda_congestion
Negative densities	Upwind instability	Increase nu

---

The Congestion Term

The parameter lambda_congestion controls crowd aversion:

\[ f(x, m) = \lambda \cdot m(x)^{\gamma} \]

lambda_congestion	Effect
0.0	No interaction; agents ignore each other
0.1–0.5	Mild spreading; prefer less crowded regions
1.0+	Strong dispersion; density stays nearly uniform

Higher values prevent density spikes but may slow convergence.

---

Practical Tips

Grid Resolution

• Prototyping: nx=64, nt=40 — fast iteration, rough results

• Publication: nx=100, nt=100 — good balance of speed and quality

• High-fidelity: nx=200, nt=200 — smooth gradients, longer runtime

Parameter Tuning

1. Start with nu=0.01, alpha=0.5, lambda_congestion=0.5

2. If convergence is slow, try alpha=0.7

3. If density has spikes, increase lambda_congestion to 1.0

4. If numerical issues appear, increase nu to 0.02–0.05

Initial Conditions

Good choices for m0:

• Gaussian: np.exp(-50 * (x - x0)**2) — localized starting distribution

• Uniform: np.ones(nx) / nx — spread-out initial population

• Bimodal: Sum of two Gaussians — models two subpopulations

---

References

1. Lasry, J.-M. and Lions, P.-L. (2007). “Mean field games.” Japanese Journal of Mathematics, 2(1):229–260.

2. Cardaliaguet, P. (2013). “Notes on Mean Field Games.” Lecture notes, Collège de France.

3. Achdou, Y. and Capuzzo-Dolcetta, I. (2010). “Mean field games: numerical methods.” SIAM Journal on Numerical Analysis, 48(3):1136–1162.

4. Huang, M., Malhamé, R., and Caines, P. (2006). “Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle.” Communications in Information and Systems, 6(3):221–252.

---

Notebook Tutorial

For a complete walkthrough with validated outputs and visualizations, see the Mean Field Games Tutorial notebook on GitHub.

The notebook demonstrates:

• Setting up initial distributions

• Running the solver with different parameters

• Visualizing density evolution as 3D surfaces and heatmaps

• Interpreting convergence diagnostics

• Comparing congestion levels

Audit documentation is available at docs/MFG_TUTORIAL_COMPLETE.md.