Mean Field Control and Mean Field Game Models with Several Populations

In this paper, 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.

[1]  Martino Bardi,et al.  Uniqueness of solutions in Mean Field Games with several populations and Neumann conditions , 2018 .

[2]  R. Carmona,et al.  Control of McKean–Vlasov dynamics versus mean field games , 2012, 1210.5771.

[3]  P. Caines,et al.  Large-population cost-coupled LQG problems: generalizations to non-uniform individuals , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[4]  Yves Achdou,et al.  Mean Field Games for Modeling Crowd Motion , 2018, Computational Methods in Applied Sciences.

[5]  Daniel Andersson,et al.  A Maximum Principle for SDEs of Mean-Field Type , 2011 .

[6]  Alain Bensoussan,et al.  Linear-Quadratic Mean Field Games , 2014, Journal of Optimization Theory and Applications.

[7]  Minyi Huang,et al.  Nash Certainty Equivalence in Large Population Stochastic Dynamic Games: Connections with the Physics of Interacting Particle Systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[8]  Olivier Guéant,et al.  Mean Field Games and Applications , 2011 .

[9]  Alain Bensoussan,et al.  The Master equation in mean field theory , 2014, 1404.4150.

[10]  Marco Cirant,et al.  Multi-population Mean Field Games systems with Neumann boundary conditions , 2015 .

[11]  Marco Cirant,et al.  Bifurcation and segregation in quadratic two-populations mean field games systems , 2015, 1511.09343.

[12]  Ronnie Sircar,et al.  Bertrand and Cournot Mean Field Games , 2015 .

[13]  A. Lachapelle,et al.  COMPUTATION OF MEAN FIELD EQUILIBRIA IN ECONOMICS , 2010 .

[14]  P. Caines,et al.  Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[15]  Josselin Garnier,et al.  Large Deviations for a Mean Field Model of Systemic Risk , 2012, SIAM J. Financial Math..

[16]  M. Bardi,et al.  Mean field games models of segregation , 2016, 1607.04453.

[17]  Minyi Huang,et al.  Large-Population Cost-Coupled LQG Problems With Nonuniform Agents: Individual-Mass Behavior and Decentralized $\varepsilon$-Nash Equilibria , 2007, IEEE Transactions on Automatic Control.

[18]  Laurent Pfeiffer.  Risk-averse Merton’s Portfolio Problem , 2016 .

[19]  A. Bensoussan,et al.  O C ] 2 0 N ov 2 01 5 Mean-field-game model for Botnet defense in Cybersecurity , 2018 .

[20]  Francois Delarue,et al.  The Master Equation for Large Population Equilibriums , 2014, 1404.4694.

[21]  O. Pironneau,et al.  Dynamic programming for mean-field type control , 2014 .

[22]  Peter E. Caines,et al.  An Invariance Principle in Large Population Stochastic Dynamic Games , 2007, J. Syst. Sci. Complex..

[23]  A. Bensoussan,et al.  Mean Field Games and Mean Field Type Control Theory , 2013 .

[24]  Ronnie Sircar,et al.  Fracking, Renewables, and Mean Field Games , 2017, SIAM Rev..

[25]  Ermal Feleqi.  The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players , 2013, Dyn. Games Appl..

[26]  Marie-Therese Wolfram,et al.  Socio-economic applications of finite state mean field games , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[27]  Pierre-Louis Lions,et al.  Partial differential equation models in macroeconomics , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  A. Bensoussan,et al.  On The Interpretation Of The Master Equation , 2015, 1503.07754.

[29]  R. Carmona,et al.  A probabilistic weak formulation of mean field games and applications , 2013, 1307.1152.

[30]  Marie-Therese Wolfram,et al.  On a mean field game approach modeling congestion and aversion in pedestrian crowds , 2011 .

[31]  Hamidou Tembine,et al.  Mean-Field-Type Games in Engineering , 2016, ArXiv.

[32]  R. Carmona,et al.  Mean Field Games and Systemic Risk , 2013, 1308.2172.

[33]  Boualem Djehiche,et al.  Mean-Field Type Modeling of Nonlocal Crowd Aversion in Pedestrian Crowd Dynamics , 2017, SIAM J. Control. Optim..

[34]  Diogo A. Gomes,et al.  Regularity Theory for Mean-Field Game Systems , 2016 .

[35]  Hamidou Tembine,et al.  Risk-Sensitive Mean-Field Type Control under Partial Observation , 2014, ArXiv.

[36]  Pierre-Louis Lions,et al.  Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis , 2013, 1305.6323.

[37]  Hamidou Tembine,et al.  A stochastic maximum principle for risk-sensitive mean-field-type control , 2014, CDC.

[38]  Peter E. Caines,et al.  Social Optima in Mean Field LQG Control: Centralized and Decentralized Strategies , 2012, IEEE Transactions on Automatic Control.

[39]  Aditya Mahajan,et al.  Team-optimal solution of finite number of mean-field coupled LQG subsystems , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[40]  Huyên Pham,et al.  Dynamic Programming for Optimal Control of Stochastic McKean-Vlasov Dynamics , 2016, SIAM J. Control. Optim..

[41]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[42]  Huyen Pham,et al.  Discrete Time McKean–Vlasov Control Problem: A Dynamic Programming Approach , 2015, Applied Mathematics & Optimization.