Electronic Communications in Probability

A rank-based mean field game in the strong formulation

Erhan Bayraktar and Yuchong Zhang

Full-text: Open access


We discuss a natural game of competition and solve the corresponding mean field game with common noise when agents’ rewards are rank-dependent. We use this solution to provide an approximate Nash equilibrium for the finite player game and obtain the rate of convergence.

Article information

Electron. Commun. Probab. Volume 21 (2016), paper no. 72, 12 pp.

Received: 26 March 2016
Accepted: 6 October 2016
First available in Project Euclid: 19 October 2016

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60H 91A

mean field games competition common noise rank-dependent interaction non-local interaction strong formulation

Creative Commons Attribution 4.0 International License.


Bayraktar, Erhan; Zhang, Yuchong. A rank-based mean field game in the strong formulation. Electron. Commun. Probab. 21 (2016), paper no. 72, 12 pp. doi:10.1214/16-ECP24. https://projecteuclid.org/euclid.ecp/1476841494.

Export citation


  • [1] René Carmona and François Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim. 51 (2013), no. 4, 2705–2734.
  • [2] René Carmona, François Delarue, and Daniel Lacker, Mean field games with common noise, to appear in Annals of Probability.
  • [3] René Carmona and Daniel Lacker, A probabilistic weak formulation of mean field games and applications, Ann. Appl. Probab. 25 (2015), no. 3, 1189–1231.
  • [4] Olivier Guéant, Jean-Michel Lasry, and Pierre-Louis Lions, Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., vol. 2003, Springer, Berlin, 2011, pp. 205–266.
  • [5] Edwin Hewitt, Integration by parts for Stieltjes integrals, Amer. Math. Monthly 67 (1960), 419–423.
  • [6] Minyi Huang, Roland P. Malhamé, and Peter E. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst. 6 (2006), no. 3, 221–251.
  • [7] Daniel Lacker and Kevin Webster, Translation invariant mean field games with common noise, Electron. Commun. Probab. 20 (2015), no. 42, 13.
  • [8] Jean-Michel Lasry and Pierre-Louis Lions, Mean field games, Jpn. J. Math. 2 (2007), no. 1, 229–260.
  • [9] Cédric Villani, Optimal transport: old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009.