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Minimax Theory and its Applications 03 (2018), No. 2, 173--209
Copyright Heldermann Verlag 2018

Mean Field Control and Mean Field Game Models with Several Populations

Alain Bensoussan
International Center for Decision and Risk Analysis, Jindal School of Management, University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, U.S.A.

Tao Huang
Dept. of Mathematics, Wayne State University, 4841 Cass Avenue, Detroit, MI 48201, U.S.A.

Mathieu Laurière
Department of Operations Research and Financial Engineering, Princeton University, Charlton Street, Princeton, NJ 08544, U.S.A.

We investigate the interaction of two populations with a large number of indistinguishable agents. The problem consists in two levels: the interaction between agents of a same population, and the interaction between the two populations. In the spirit of mean field type control (MFC) problems and mean field games (MFG), each population is approximated by a continuum of infinitesimal agents. We define four different problems in a general context and interpret them in the framework of MFC or MFG. By calculus of variations, we derive formally in each case the adjoint equations for the necessary conditions of optimality. Importantly, we find that in the case of a competition between two coalitions, one needs to rely on a system of master equations in order to describe the equilibrium. Examples are provided, in particular linear-quadratic models for which we obtain systems of ODEs that can be related to Riccati equations.

Keywords: Mean field type control problems, mean field games, master equation, Hamilton-Jacobi-Bellman equation, linear quadratic problems.

MSC: 35Q91, 35Q93, 91A13; 35A01, 35B20, 35K40

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