# Mean Field Game Theory

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, in the engineering literature by Peter E. Caines and his co-workers and independently and around the same time by mathematicians Jean-Michel Lasry  and Pierre-Louis Lions. 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. Read all..

## Explanation

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 ${\displaystyle N\to \infty }$ 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]

From Caines (2009), a relatively simple model of large-scale games is the linear-quadratic Gaussian model. The individual agent's dynamics are modeled as a stochastic differential equation

${\displaystyle dx_{i}=(a_{i}x_{i}+b_{i}u_{i})\,dt+\sigma _{i}\,dw_{i},\quad i=1,\dots ,N,}$

where ${\displaystyle x_{i}}$ is the state of the ${\displaystyle i}$-th agent, and ${\displaystyle u_{i}}$ is the control. The individual agent's cost is

${\displaystyle J_{i}(u_{i},\nu )=\mathbb {E} \left\{\int _{0}^{\infty }e^{-\rho t}\left[(x_{i}-\nu )^{2}+ru_{i}^{2}\right]\,dt\right\},\quad \nu =\Phi \left({\frac {1}{N}}\sum _{k\neq i}^{N}x_{k}+\eta \right).}$

The coupling between agents occurs in the cost function.

## 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. S2CID 36197045.
4. Lions, Pierre-Louis; Lasry, Jean-Michel (March 2007). "Large investor trading impacts on volatility". Annales de l'Institut Henri Poincaré C. 24 (2): 311–323. Bibcode:2007AIHPC..24..311L. doi:10.1016/j.anihpc.2005.12.006.
5. Lasry, Jean-Michel; Lions, Pierre-Louis (28 March 2007). "Mean field games". Japanese Journal of Mathematics. 2 (1): 229–260. doi:10.1007/s11537-007-0657-8. S2CID 1963678.
6. Lasry, Jean-Michel; Lions, Pierre-Louis (November 2006). "Jeux à champ moyen. II – Horizon fini et contrôle optimal" [Mean field games. II – Finite horizon and optimal control]. Comptes Rendus Mathématique (in French). 343 (10): 679–684. doi:10.1016/j.crma.2006.09.018.
7. Lasry, Jean-Michel; Lions, Pierre-Louis (November 2006). "Jeux à champ moyen. I – Le cas stationnaire" [Mean field games. I – The stationary case]. Comptes Rendus Mathématique (in French). 343 (9): 619–625. doi:10.1016/j.crma.2006.09.019.
8. Cardaliaguet, Pierre (September 27, 2013). "Notes on Mean Field Games" (PDF).
9. Tembine, Hamidou (September 2015). "Risk-sensitive mean-field-type games with Lp-norm drifts". Automatica. 59: 224–237. arXiv:1505.06280. doi:10.1016/j.automatica.2015.06.036. S2CID 8161026.
10. Djehiche, Boualem; Tcheukam, Alain; Tembine, Hamidou (2017). "Mean-Field-Type Games in Engineering". AIMS Electronics and Electrical Engineering. 1 (1): 18–73. arXiv:1605.03281. doi:10.3934/ElectrEng.2017.1.18. S2CID 16055840.
11. Tembine, Hamidou (2017). "Mean-field-type games". AIMS Mathematics. 2 (4): 706–735. doi:10.3934/Math.2017.4.706.
12. Duncan, Tyrone; Tembine, Hamidou (12 February 2018). "Linear–Quadratic Mean-Field-Type Games: A Direct Method". Games. 9 (1): 7. doi:10.3390/g9010007.
13. Andersson, Daniel; Djehiche, Boualem (30 October 2010). "A Maximum Principle for SDEs of Mean-Field Type". Applied Mathematics & Optimization. 63 (3): 341–356. doi:10.1007/s00245-010-9123-8. S2CID 121265168.
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.[page needed]