Agents: 500
Time: 0.00
Congestion: 0.00
Avg cost: 0.00
Mean Field Games (Lasry-Lions 2006; Huang-Malhamé-Caines 2006) model strategic interactions among a continuum of indistinguishable rational agents. Each agent minimizes an individual cost depending on their state and the aggregate distribution m(x,t).
−∂_t u − ν Δu + ½|∇u|² = f(m) (HJB backward)
∂_t m − ν Δm − div(m ∇u) = 0 (Fokker-Planck forward)
The Hamilton-Jacobi-Bellman equation gives optimal control u, while the Fokker-Planck equation governs the evolution of the crowd density m. These two PDEs are coupled: the optimal strategy depends on congestion f(m), creating a fixed-point problem. Applications: pedestrian flow, traffic, financial markets, epidemics.