Nash equilibrium in continuous crowd dynamics
Mean-Field Games (Lasry-Lions 2006) model Nash equilibria for large populations of rational agents. Two coupled PDEs: the Hamilton-Jacobi-Bellman equation ∂ₜu + ν∆u − |∇u|²/2 = f(m) for the value function u, and the Fokker-Planck equation ∂ₜm − ν∆m − div(m∇u) = 0 for the agent density m. Agents route from left to right minimizing travel time plus congestion cost f(m) = α·m.