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  "body": "\n## Summary\n\nA mean-field game is a differential game with a very large number of small, identical players.\nEach player reacts only to the distribution of all the others.\nIn the limit of infinitely many players, a Nash equilibrium is described by a coupled system of two partial differential equations.\nA [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] backward in time gives the optimal strategy of one player.\nA [[Fokker–Planck equation|FVQWR7NS00]] forward in time gives the evolution of the distribution of players.\nThe theory was introduced independently by Jean-Michel Lasry and Pierre-Louis Lions and by Minyi Huang, Roland Malhamé and Peter Caines.\n\n## Explanations\n\n### L1 human\n\nThink of a crowd leaving a stadium: each person chooses a path to the exit, and the best path depends on where everyone else goes.\nNo single person changes the crowd, but the crowd shapes every choice.\nAn equilibrium is a crowd movement in which nobody can do better by changing their own path, given the movement of the crowd.\nMean-field games compute such equilibria by treating the crowd as a continuous density.\nThe same idea models traders in a market, drivers in traffic and firms competing for a resource.\n",
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