Mean-Field Game — McKean-Vlasov

N→∞ rational agents each minimize cost J = ∫(½u² + α·dist_to_mean + noise) dt. The population density m(x,t) evolves via Fokker-Planck; optimal control u* = -∂V/∂x via Hamilton-Jacobi-Bellman. Nash equilibrium = fixed point of this forward-backward system.

Agent Density m(x,t)

Value Function V(x,t)

Particle Trajectories

Interaction α (mean-seeking)

Noise σ

Control cost r

Target position

Mean: — | Variance: — | Iteration: —