N rational agents minimizing cost coupled through population density
Entropy: — | Mean speed: —
Mean-field game: each agent minimizes ∫[½|u|² + αF(m)]dt + terminal cost, where m is population density.
Equilibrium couples HJB (backward) with Fokker-Planck (forward). Agents shown as dots; heatmap shows density field m(x,t).
High α → strong crowding aversion (agents spread); low α → agents cluster at potential minimum.