Nash equilibrium flows in a congested pedestrian system
Mean-field games (Lasry–Lions, 2007) model rational agents in a congested crowd where each agent minimizes travel cost + congestion penalty. The density field ρ satisfies a Fokker–Planck equation while the value function u satisfies a Hamilton–Jacobi–Bellman equation. Optimal flows emerge as a Nash equilibrium of the infinite-player limit.