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In continuous time a mean field game is typically composed by a [[Hamilton–Jacobi–Bellman equation]] that describes the [[optimal control]] problem of an individual and a [[Fokker–Planck equation]] that describes the dynamics of the aggregate distribution of agents. Under fairly general assumptions it can be proved that a class of mean field games is the limit as <math>N\rightarrow\infty</math> of a ''N-player'' [[Nash equilibrium]].<ref>{{Cite web|url=https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf|title=Notes on Mean Field Games|last=Cardaliaguet|first=Pierre|date=September 27, 2013|website=|archive-url=|archive-date=|dead-url=|access-date=}}</ref>
In continuous time a mean field game is typically composed by a [[Hamilton–Jacobi–Bellman equation]] that describes the [[optimal control]] problem of an individual and a [[Fokker–Planck equation]] that describes the dynamics of the aggregate distribution of agents. Under fairly general assumptions it can be proved that a class of mean field games is the limit as <math>N\rightarrow\infty</math> of a ''N-player'' [[Nash equilibrium]].<ref>{{Cite web|url=https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf|title=Notes on Mean Field Games|last=Cardaliaguet|first=Pierre|date=September 27, 2013|website=|archive-url=|archive-date=|dead-url=|access-date=}}</ref>


A related concept to that of mean-field games is "mean-field-type control".{{cn|date=December 2018}} In this case a [[social planner]] controls a distribution of states and chooses a control strategy. The solution to a mean-field-type control problem can typically be expressed as dual Hamilton–Jacobi–Bellman equation coupled with Kolmogorov equation.{{cn|date=December 2018}} Mean-field-type game theory<ref>{{Cite web|url=https://www.sciencedirect.com/science/article/pii/S000510981500271X?via%3Dihub|title=ScienceDirect|website=www.sciencedirect.com|doi=10.1016/j.automatica.2015.06.036|access-date=2019-03-29}}</ref><ref>{{Cite journal|last=Djehiche|first=Boualem|last2=Tcheukam|first2=Alain|last3=Tembine|first3=Hamidou|date=2017-09-27|title=Mean-Field-Type Games in Engineering|url=http://www.aimspress.com/article/10.3934/ElectrEng.2017.1.18|journal=ElectrEng 2017, Vol. 1, Pages 18-73|language=en|doi=10.3934/ElectrEng.2017.1.18}}</ref><ref>{{Cite journal|last=Tembine|first=Hamidou|date=2017-12-06|title=Mean-field-type games|url=http://www.aimspress.com/Math/2017/4/706|journal=Math 2017, Vol. 2, Pages 706-735|language=en|doi=10.3934/Math.2017.4.706}}</ref><ref>{{Cite journal|last=Tembine|first=H.|last2=Duncan|first2=Tyrone E.|date=2018|title=Linear–Quadratic Mean-Field-Type Games: A Direct Method|url=https://www.mdpi.com/2073-4336/9/1/7|journal=Games|language=en|volume=9|issue=1|pages=7|doi=10.3390/g9010007|via=}}</ref> is the multi-agent generalization of mean-field-type control<ref>{{Cite journal|last=Andersson|first=Daniel|last2=Djehiche|first2=Boualem|date=2011-06-01|title=A Maximum Principle for SDEs of Mean-Field Type|url=https://doi.org/10.1007/s00245-010-9123-8|journal=Applied Mathematics & Optimization|language=en|volume=63|issue=3|pages=341–356|doi=10.1007/s00245-010-9123-8|issn=1432-0606}}</ref>.
A related concept to that of mean-field games is "mean-field-type control". In this case a [[social planner]] controls a distribution of states and chooses a control strategy. The solution to a mean-field-type control problem can typically be expressed as dual adjoint Hamilton–Jacobi–Bellman equation coupled with [[Fokker–Planck equation|Kolmogorov equation]]. Mean-field-type game theory<ref>{{Cite web|url=https://www.sciencedirect.com/science/article/pii/S000510981500271X?via%3Dihub|title=ScienceDirect|website=www.sciencedirect.com|doi=10.1016/j.automatica.2015.06.036|access-date=2019-03-29}}</ref><ref>{{Cite journal|last=Djehiche|first=Boualem|last2=Tcheukam|first2=Alain|last3=Tembine|first3=Hamidou|date=2017-09-27|title=Mean-Field-Type Games in Engineering|url=http://www.aimspress.com/article/10.3934/ElectrEng.2017.1.18|journal=ElectrEng 2017, Vol. 1, Pages 18-73|language=en|doi=10.3934/ElectrEng.2017.1.18}}</ref><ref>{{Cite journal|last=Tembine|first=Hamidou|date=2017-12-06|title=Mean-field-type games|url=http://www.aimspress.com/Math/2017/4/706|journal=Math 2017, Vol. 2, Pages 706-735|language=en|doi=10.3934/Math.2017.4.706}}</ref><ref>{{Cite journal|last=Tembine|first=H.|last2=Duncan|first2=Tyrone E.|date=2018|title=Linear–Quadratic Mean-Field-Type Games: A Direct Method|url=https://www.mdpi.com/2073-4336/9/1/7|journal=Games|language=en|volume=9|issue=1|pages=7|doi=10.3390/g9010007|via=}}</ref> is the multi-agent generalization of the single-agent mean-field-type control<ref>{{Cite journal|last=Andersson|first=Daniel|last2=Djehiche|first2=Boualem|date=2011-06-01|title=A Maximum Principle for SDEs of Mean-Field Type|url=https://doi.org/10.1007/s00245-010-9123-8|journal=Applied Mathematics & Optimization|language=en|volume=63|issue=3|pages=341–356|doi=10.1007/s00245-010-9123-8|issn=1432-0606}}</ref><ref>{{Cite book|url=https://www.springer.com/gp/book/9781461485070|title=Mean Field Games and Mean Field Type Control Theory|last=Bensoussan|first=Alain|last2=Frehse|first2=Jens|last3=Yam|first3=Phillip|date=2013|publisher=Springer-Verlag|isbn=9781461485070|series=SpringerBriefs in Mathematics|location=New York|language=en}}</ref>.


==See also==
==See also==

Revision as of 17:58, 29 March 2019

Mean field game theory is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal,[1] in the engineering literature by Peter E. Caines and his co-workers[2][3] and independently and around the same time by mathematicians Jean-Michel Lasry [fr] and Pierre-Louis Lions.[4][5][6][7]

Use of the term 'mean field' is inspired by mean field theory in physics which considers the behaviour of systems of large numbers of particles where individual particles have negligible impact upon the system.

In continuous time a mean field game is typically composed by a Hamilton–Jacobi–Bellman equation that describes the optimal control problem of an individual and a Fokker–Planck equation that describes the dynamics of the aggregate distribution of agents. Under fairly general assumptions it can be proved that a class of mean field games is the limit as of a N-player Nash equilibrium.[8]

A related concept to that of mean-field games is "mean-field-type control". In this case a social planner controls a distribution of states and chooses a control strategy. The solution to a mean-field-type control problem can typically be expressed as dual adjoint Hamilton–Jacobi–Bellman equation coupled with Kolmogorov equation. Mean-field-type game theory[9][10][11][12] is the multi-agent generalization of the single-agent mean-field-type control[13][14].

See also

References

  1. ^ Jovanovic, Boyan; Rosenthal, Robert W. (1988). "Anonymous Sequential Games". Journal of Mathematical Economics. 17 (1): 77–87. doi:10.1016/0304-4068(88)90029-8.
  2. ^ Huang, M. Y.; Malhame, R. P.; Caines, P. E. (2006). "Large Population Stochastic Dynamic Games: Closed-Loop McKean–Vlasov Systems and the Nash Certainty Equivalence Principle". Communications in Information and Systems. 6 (3): 221–252. doi:10.4310/CIS.2006.v6.n3.a5. Zbl 1136.91349.
  3. ^ Nourian, M.; Caines, P. E. (2013). "ε–Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents". SIAM Journal on Control and Optimization. 51 (4): 3302–3331. arXiv:1209.5684. doi:10.1137/120889496.
  4. ^ Lions, P. L.; Lasry, J. M. (2007). "Large investor trading impacts on volatility". Annales de l'Institut Henri Poincaré C. 24 (2): 311. Bibcode:2007AIHPC..24..311L. doi:10.1016/j.anihpc.2005.12.006.
  5. ^ Lasry, J. M.; Lions, P. L. (2007). "Mean field games". Japanese Journal of Mathematics. 2: 229–260. doi:10.1007/s11537-007-0657-8.
  6. ^ Lasry, J. M.; Lions, P. L. (2006). "Jeux à champ moyen. II – Horizon fini et contrôle optimal". Comptes Rendus Mathématique. 343 (10): 679. doi:10.1016/j.crma.2006.09.018.
  7. ^ Lasry, J. M.; Lions, P. L. (2006). "Jeux à champ moyen. I – Le cas stationnaire". Comptes Rendus Mathématique. 343 (9): 619. doi:10.1016/j.crma.2006.09.019.
  8. ^ Cardaliaguet, Pierre (September 27, 2013). "Notes on Mean Field Games" (PDF). {{cite web}}: Cite has empty unknown parameter: |dead-url= (help)
  9. ^ "ScienceDirect". www.sciencedirect.com. doi:10.1016/j.automatica.2015.06.036. Retrieved 2019-03-29.
  10. ^ Djehiche, Boualem; Tcheukam, Alain; Tembine, Hamidou (2017-09-27). "Mean-Field-Type Games in Engineering". ElectrEng 2017, Vol. 1, Pages 18-73. doi:10.3934/ElectrEng.2017.1.18.
  11. ^ Tembine, Hamidou (2017-12-06). "Mean-field-type games". Math 2017, Vol. 2, Pages 706-735. doi:10.3934/Math.2017.4.706.
  12. ^ Tembine, H.; Duncan, Tyrone E. (2018). "Linear–Quadratic Mean-Field-Type Games: A Direct Method". Games. 9 (1): 7. doi:10.3390/g9010007.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  13. ^ Andersson, Daniel; Djehiche, Boualem (2011-06-01). "A Maximum Principle for SDEs of Mean-Field Type". Applied Mathematics & Optimization. 63 (3): 341–356. doi:10.1007/s00245-010-9123-8. ISSN 1432-0606.
  14. ^ Bensoussan, Alain; Frehse, Jens; Yam, Phillip (2013). Mean Field Games and Mean Field Type Control Theory. SpringerBriefs in Mathematics. New York: Springer-Verlag. ISBN 9781461485070.

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