In his lectures at College de France, P.L. Lions introduced the concept of Master equation, see [8] for Mean Field Games. It is introduced in a heuristic fashion, from the prospective as a system of partial differential equations, that the equation is associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in [3], composed of a formalism of derivations. The interest of this equation is that it contains interesting particular cases, which can be studied directly, in particular the system of HJB-FP (Hamilton-Jacobi-Bellman, Fokker-Planck) equations obtained as the limit of the finite Nash equilibrium game, when the trajectories are independent, see [6]. Usually, in mean field theory, one can bypass the large Nash equilibrium, by introducing the concept of representative agent, whose action is influenced by a distribution of similar agents, and obtains directly the system of HJB-FP equations of interest, see for instance [1]. Apparently, there is no such approach for the Master equation. We show here that it is possible. We first do it for the Mean Field type control problem, for which we interpret completely the Master equation. For the Mean Field Games itself, we solve a related problem, and obtain again the Master equation.
RésuméDans son cours au Collège de France, P.L. Lions a inroduit le concept d'équation maitresse, pour les jeux à champ moyen [8]. Ceci a été fait d'une manière heuristique, à partir du système d'équations aux dérivées partielles associé à un équilibre de Nash, pour un nombre fini, mais grand, de joueurs. La méthode repose sur un formalisme de dérivations. L'intérêt de cette équation maitresse est qu'elle contient des cas particuliers intéressants, qui peuvent être étudiés dicretement, en particulier le système des équations HJB-
