Rational agents navigate a crowd — Nash equilibrium from a PDE system, not individual optimization
Density field ρ(x,y)
Value function u(x,y)
Mean Field Game: Each rational agent minimizes cost J = ∫[½|v|² + α·ρ(x,t)]dt to reach goal (right side). The optimal control satisfies the Hamilton-Jacobi-Bellman equation coupled with Fokker-Planck for density evolution. At Nash equilibrium, no agent can unilaterally improve — the crowd self-organizes into efficient lanes despite individual optimization. Introduced by Lasry-Lions 2006 and Huang-Malhamé-Caines 2006.