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Mean Field Games and Mean Field Type Control Theory

  • Book
  • © 2013

Overview

Authors:
  1. Alain Bensoussan

    1. Department of Systems Engineering and Engineering Management, University of Texas at Dallas Naveen Jindal School of Management, Richardson, Texas, USA, and City University of Hong Kong, Kowloon, Hong Kong SAR

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  2. Jens Frehse

    1. Universitat Bonn Institut für Angewandte Mathematik, Bonn, Germany

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  3. Phillip Yam

    1. Department of Statistics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR

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  • This is the first contribution of this work to present a unified approach for both problems, using either HJB and FP coupled equations or stochastic maximum principle
  • Subject matter is interesting and has received much recent attention
  • Gives an overview of the mean-field game and mean-field type control problems

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Introduction

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 1-5

  3. General Presentation of Mean Field Control Problems

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 7-9

  4. The Mean Field Games

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 11-14

  5. The Mean Field Type Control Problems

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 15-29

  6. Approximation of Nash Games with a Large Number of Players

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 31-43

  7. Linear Quadratic Models

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 45-57

  8. Stationary Problems

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 59-66

  9. Different Populations

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 67-87

  10. Nash Differential Games with Mean Field Effect

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 89-97

  11. Analytic Techniques

    • Alain Bensoussan, Jens Frehse, Phillip Yam
    Pages 99-124

  12. Back Matter

    Pages 125-128
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Keywords

About this book

​Mean field games and Mean field type control introduce new problems in Control Theory. The terminology “games” may be confusing. In fact they are control problems, in the sense that one is interested in a single decision maker, whom we can call the representative agent. However, these problems are not standard, since both the evolution of the state and the objective functional is influenced but terms which are not directly related to the state or the control of the decision maker. They are however, indirectly related to him, in the sense that they model a very large community of agents similar to the representative agent. All the agents behave similarly and impact the representative agent. However, because of the large number an aggregation effect takes place. The interesting consequence is that the impact of the community can be modeled by a mean field term, but when this is done, the problem is reduced to a control problem. ​

Reviews

From the book reviews:

“This brief monograph is a presentation of some aspects of MFG theory and of a related class of problems, called mean field type control, where the state of the control system is driven by a McKean-Vlasov equation, namely, a stochastic differential equation where the drift depends also on the distribution of the state itself. … I believe the book should be mandatory reading for all mathematicians and engineers working in the field of MFG and related topics.” (Martino Bardi, Mathematical Reviews, November, 2014)

Authors and Affiliations

  • Department of Systems Engineering and Engineering Management, University of Texas at Dallas Naveen Jindal School of Management, Richardson, Texas, USA, and City University of Hong Kong, Kowloon, Hong Kong SAR

    Alain Bensoussan

  • Universitat Bonn Institut für Angewandte Mathematik, Bonn, Germany

    Jens Frehse

  • Department of Statistics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR

    Phillip Yam

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