What Does a Nonlinear Fokker–Planck Equation Buy a Mean Field Game?

Mean field games almost always pair the agents’ Hamilton–Jacobi–Bellman equation with a Fokker–Planck equation linear in the density: the agents’ noises are independent, every interaction sits in the cost, and the agents never touch. This discussion paper asks what each kind of nonlinearity in the forward equation buys. In a stationary game in one dimension the crowd is an explicit statistics of the value function; the linear equation forces Boltzmann statistics, and without a crowd cost the game decouples. Noise, capacity and congestion each select another statistics and couple the game through the dynamics alone. Noise that grows with the density gives Tsallis statistics with edges and, past an exponent of 1.146, an empty gap; noise that falls with it gathers the crowd into one cluster until the branch folds at 0.219; a capacity gives a Fermi–Dirac plateau once the linear crowd would overflow the cap 1.7-fold; congestion gives the noise family in another game, whose optimum, 0.5 to 3 per cent better, needs a fee set by the crowd’s flow. A gradient pull buys nothing: it is a crowd cost in disguise. The open questions are where the computation sees what the theory cannot yet prove.