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Mean Field Games

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Handbook of Dynamic Game Theory

Abstract

Mean field game (MFG) theory studies the existence of Nash equilibria, together with the individual strategies which generate them, in games involving a large number of asymptotically negligible agents modeled by controlled stochastic dynamical systems. This is achieved by exploiting the relationship between the finite and corresponding infinite limit population problems. The solution to the infinite population problem is given by (i) the Hamilton-Jacobi-Bellman (HJB) equation of optimal control for a generic agent and (ii) the Fokker-Planck-Kolmogorov (FPK) equation for that agent, where these equations are linked by the probability distribution of the state of the generic agent, otherwise known as the system’s mean field. Moreover, (i) and (ii) have an equivalent expression in terms of the stochastic maximum principle together with a McKean-Vlasov stochastic differential equation, and yet a third characterization is in terms of the so-called master equation. The article first describes problem areas which motivate the development of MFG theory and then presents the theory’s basic mathematical formalization. The main results of MFG theory are then presented, namely the existence and uniqueness of infinite population Nash equilibiria, their approximating finite population ɛ-Nash equilibria, and the associated best response strategies. This is followed by a presentation of the three main mathematical methodologies for the derivation of the principal results of the theory. Next, the particular topics of major-minor agent MFG theory and the common noise problem are briefly described and then the final section concisely presents three application areas of MFG theory.

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, McGill University, 3480 University Street, H3A 2A7, Montreal, QC, Canada

    Peter E. Caines

  2. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, K1S xx, Ottawa, ON, Canada

    Minyi Huang

  3. Department of Electrical Engineering, École Polytechnique de Montréal, 2900 Edouard Montpetit, H3C 3A7, Montreal, QC, Canada

    Roland P. Malhamé

Authors
  1. Peter E. Caines
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  2. Minyi Huang
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  3. Roland P. Malhamé
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Corresponding author

Correspondence to Peter E. Caines .

Editor information

Editors and Affiliations

  1. Center for Advanced Study, University of Illinois Urbana-Champaign, Urbana, Illinois, USA

    Tamer Basar

  2. GERAD, HEC Montréal, Montreal, Québec, Canada

    Georges Zaccour

Section Editor information

  1. Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, 1308 West Main, 61801, Urbana, IL, USA

    Tamer Başar

  2. Département de sciences de la décision, GERAD, HEC Montréal, Montreal, QC, Canada

    Georges Zaccour

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Caines, P.E., Huang, M., Malhamé, R.P. (2017). Mean Field Games. In: Basar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-27335-8_7-1

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  • DOI: https://doi.org/10.1007/978-3-319-27335-8_7-1

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