# Mean Field Games

• Peter E. Caines
• Minyi Huang
• Roland P. Malhamé
Reference work entry

## 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 chapter 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 ε-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.

## Keywords

Mean field games MFG MFG equilibria Mean field game equations MFG PDEs MFG analytic methods MFG probabilistic methods Major minor agent and common noise games Non-linear Markov processes MFG applications

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## Authors and Affiliations

• Peter E. Caines
• 1
Email author
• Minyi Huang
• 2
• Roland P. Malhamé
• 3
1. 1.Department of Electrical and Computer EngineeringMcGill UniversityMontrealCanada
2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
3. 3.Department of Electrical EngineeringÉcole Polytechnique de MontréalMontrealCanada