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

Abstract

Mean field games almost always pair the agents' Hamilton-Jacobi-Bellman equation with a Fokker-Planck equation that is linear in the density: the agents' noises are independent and they interact only through their costs, so they never touch. This discussion paper asks what a game gains when the forward equation is nonlinear in the density, case by case. In a stationary game the crowd is an explicit statistics of the value function. The linear equation forces Boltzmann statistics, and with no crowd cost the game decouples. Density-dependent noise gives Tsallis statistics with an edge and an empty gap, or, when noise falls with the density, a crowd that abandons its preferred places for one cluster; a cost of motion that diverges at capacity gives Fermi-Dirac statistics with a plateau; congestion gives the same family as noise with a different value function, and its planner's optimum is reached by a fee set by the crowd's local flow. A gradient physical pull toward the crowd buys nothing in a stationary game: it is a crowd cost in disguise, and a pull theta herds exactly as a conformity cost theta squared. Equilibria computed on one double-well baseline, each checked against an independent calculation, show every signature, and the open questions are where the computation sees what the theory cannot yet prove. Disclaimer. Research and education, not advice: see zamrik.com/disclaimer.