Skip to main content
SpringerLink
Log in

The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

This note contains a detailed derivation of the equations of the recent mean field games theory (abbr. MFG), developed by M. Huang, P.E. Caines, and R.P. Malhamé on one hand and by J.-M. Lasry and P.-L. Lions on the other, associated with a class of stochastic differential games, where the players belong to several populations, each of which consisting of a large number of similar and indistinguishable individuals, in the context of periodic diffusions and long-time-average (or ergodic) costs. After introducing a system of N Hamilton–Jacobi–Bellman (abbr. HJB) and N Kolmogorov–Fokker–Planck (abbr. KFP) equations for an N-player game belonging to such a class of games, the system of MFG equations (consisting of as many HJB equations, and of as many KFP equations as is the number of populations) is derived by letting the number of the members of each population go to infinity. For the sake of clarity and for reader’s convenience, the case of a single population of players, as formulated in the work of J.-M. Lasry and P.-L. Lions, is presented first. The note slightly improves the results in this case too, by dealing with more general dynamics and costs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We make the standard convention that \(\prod_{j=l+1}^{N}dm(x^{j}) =1\) for l=N and \(\prod_{j=1}^{l-1}{dm_{j}^{N}(x^{j}) } =1\) for l=1.

References

  1. Bardi M, Feleqi E Nonlinear elliptic systems and mean field games. Preprint

  2. Cardaliaguet P (2010) Notes on mean field games (from P.-L. Lions’ lectures at Collège de France)

  3. Guéant O, Lasry J-M, Lions P-L (2011) Mean field games and applications. In: Carmona RA et al. (eds) Paris–Princeton lectures on mathematical finance 2010. Lecture notes in math, vol 2003. Springer, Berlin, pp 205–266

    Chapter  Google Scholar 

  4. Hewitt E, Savage LJ (1955) Symmetric measures on Cartesian products. Trans Am Math Soc 80:470–501

    Article  MathSciNet  MATH  Google Scholar 

  5. Huang M, Caines PE, Malhame RP (2003) Individual and mass behaviour in large population stochastic wireless power control problems. In: Centralized and Nash equilibrium solutions, IEEE conference on decision and control, HI, USA, December 2003, pp 98–103

    Google Scholar 

  6. Huang M, Caines PE, Malhamé RP (2006) Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6:221–251

    MathSciNet  MATH  Google Scholar 

  7. Huang M, Caines PE, Malhamé RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ϵ-Nash equilibria. IEEE Trans Autom Control 52:1560–1571

    Article  Google Scholar 

  8. Lasry J-M, Lions P-L (2006) Jeux à champ moyen. I. Le cas stationnaire. C R Acad Sci Paris 343:619–625

    Article  MathSciNet  MATH  Google Scholar 

  9. Lasry J-M, Lions P-L (2006) Jeux à champ moyen. II. Horizon fini et contrôle optimal. C R Acad Sci Paris 343:679–684

    Article  MathSciNet  MATH  Google Scholar 

  10. Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2(1):229–260

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35131, Padova (PD), Italy

    Ermal Feleqi

Authors
  1. Ermal Feleqi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ermal Feleqi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Feleqi, E. The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players. Dyn Games Appl 3, 523–536 (2013). https://doi.org/10.1007/s13235-013-0088-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-013-0088-5

Keywords

Search

Navigation