Part XII: Mean Field Games
The \(N\to\infty\) limit of rational agents — Hamilton-Jacobi-Bellman optimal control coupled to Fokker-Planck density evolution, recovering urban economics endogenously and connecting to CCD equilibrium.
Part Overview
Mean field games provide the ultimate bridge between individual rationality and collective dynamics. The HJB equation governs each agent's optimal control, while the Fokker-Planck equation tracks the population density. Their coupling — backward in time for value, forward for density — yields a self-consistent equilibrium. Cole-Hopf linearization reveals a hidden Schrödinger structure. Applied to cities, MFG recovers Wardrop equilibrium, prices of anarchy, and the Achdou-Capuzzo Dolcetta numerical scheme.
Key Topics
- • HJB equation
- • Cole-Hopf \(\to\) Schrödinger equation
- • MFG system (HJB+FP)
- • Stationary MFG
- • Price of Anarchy
- • MFG on networks
- • Wardrop equilibrium recovery
- • Achdou-Capuzzo Dolcetta scheme
3 chapters | Optimal control meets crowd dynamics | From individual rationality to collective equilibrium
Chapters
Chapter 1: HJB Equation
The Hamilton-Jacobi-Bellman equation for optimal control of urban agents. Dynamic programming principle, viscosity solutions, and the Cole-Hopf transformation \(u = -\epsilon \log \psi\) that linearizes HJB into a Schrödinger equation.
Chapter 2: MFG System
The coupled HJB-Fokker-Planck system: value function \(u\) evolves backward, density \(m\) evolves forward. Stationary MFG, existence and uniqueness via monotonicity (Lasry-Lions), Price of Anarchy bounds, and MFG on networks recovering Wardrop equilibrium.
Chapter 3: MFG Numerics
The Achdou-Capuzzo Dolcetta finite difference scheme for MFG systems. Monotone discretization preserving the forward-backward structure, Newton iteration for the coupled system, and convergence to urban equilibrium density profiles.