Dynamic Games and Applications

, Volume 4, Issue 2, pp 208–230 | Cite as

On the Rate of Convergence for the Mean-Field Approximation of Controlled Diffusions with Large Number of Players

  • Vassili N. Kolokoltsov
  • Marianna Troeva
  • Wei YangEmail author


In this paper, we investigate the mean field games of N agents who are weakly coupled via the empirical measures. The underlying dynamics of the representative agent is assumed to be a controlled nonlinear diffusion process with variable coefficients. We show that individual optimal strategies based on any solution of the main consistency equation for the backward-forward mean filed game model represent a 1/N-Nash equilibrium for approximating systems of N agents.


Nonlinear diffusion Kinetic equation Forward-backward system Dynamic law of large numbers Rates of convergence Tagged particle ϵ-Nash equilibrium 



The research of Vassili N. Kolokoltsov has been partially supported by IPI of the Russian Academy of Science, Russian Foundation for Basic Research (Grants No. 11-01-12026, No. 12-07-00115), and the Ministry of Education and Science of the Russian Federation (Grant No. 4402). The research of Marianna Troeva has been partially supported by the Ministry of Education and Science of the Russian Federation (Grant No. 4402).

The authors also would like to thank the referees for their valuable suggestions.


  1. 1.
    Achdou Y, Capuzzo-Dolcetta I (2010) Mean field games: numerical methods. SIAM J Numer Anal 48:1136–1162 CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Andersson D, Djehiche B (2011) A maximum principle for SDEs of mean-field type. Appl Math Optim 63:341–356 CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bailleul IF (2011) Sensitivity for the Smoluchowski equation. J Phys A, Math Theor 44(24):245004 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Belopol’skaya YaI (2001) Nonlinear equations in diffusion theory. Probability and statistics. Part 4. In: Zapiski nauchnogo seminara POMI, vol 278. POMI, St Petersburg, pp 15–35. English version: (2003) J Math Sci (New York) 118(6):5513–5524 Google Scholar
  5. 5.
    Belopol’skaya YaI (2005) A probabilistic approach to a solution of nonlinear parabolic equations. Theory Probab Appl 49(4):589–611 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Benaim M, Le Boudec J-Y (2008) A class of mean field interaction models for computer and communication systems. In: 6th international symposium on modeling and optimization in mobile, ad hoc, and wireless networks and workshops. doi: 10.1109/WIOPT.2008.4586140 Google Scholar
  7. 7.
    Benaim M, Weibull J (2003) Deterministic approximation of stochastic evolution in games. Econometrica 71(3):873–903 CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bogachev VI, Röckner M, Shaposhnikov SV (2009) Nonlinear evolution and transport equations for measures. Dokl Math 80(3):785–789 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bordenave C, McDonald D, Proutiere A (2007) A particle system in interaction with a rapidly varying environment: mean field limits and applications. arXiv:math/0701363v2. Accessed 12 January 2007
  10. 10.
    Buckdahn R, Djehiche B, Li J, Peng S (2009) Mean-field backward stochastic differential equations: a limit approach. Ann Probab 37(4):1524–1565 CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Cepeda E, Fournier N (2011) Smoluchowski’s equation: rate of convergence of the Marcus-Lushnikov process. Stoch Process Appl 121(6):1411–1444 CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Crisan D (2006) Particle approximations for a class of stochastic partial differential equations. Appl Math Optim 54(3):293–314 CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Del Moral P (2004) Feynman-Kac formulae. Genealogical and interacting particle systems with applications. Probability and its applications. Springer, New York CrossRefzbMATHGoogle Scholar
  14. 14.
    Ferrari PA (1996) Limit theorems for tagged particles. Disordered systems and statistical physics: rigorous results (Budapest, 1995). Markov Process Related Fields 2(1):17–40 Google Scholar
  15. 15.
    Gast N, Gaujal B (2009) A mean field approach for optimization in particle systems and applications. In: Proceedings of the fourth international ICST conference on performance evaluation methodologies and tools. doi: 10.4108/ICST.VALUETOOLS2009.7477 Google Scholar
  16. 16.
    Gomes DA, Mohr J, Souza RR (2010) Discrete time, finite state space mean field games. J Math Pures Appl 9(93):308–328 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Grigorescu I (1999) Uniqueness of the tagged particle process in a system with local interactions. Ann Probab 27(3):1268–1282 CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Guéant O, Lasry J-M, Lions P-L (2010) Mean field games and applications. Paris-Princeton lectures on mathematical finance. Springer, Berlin Google Scholar
  19. 19.
    Guérin H, Méléard S, Nualart E (2006) Estimates for the density of a nonlinear Landau process. J Funct Anal 238:649–677 CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Huang M (2010) Large-population LQG games involving a major player: the Nash certainty equivalence principle. SIAM J Control Optim 48:3318–3353 CrossRefzbMATHGoogle Scholar
  21. 21.
    Huang M, Caines PE, Malhamé RP (2003) Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. In: Proceedings of the 42nd IEEE conference on decision and control, Maui, Hawaii, pp 98–103 Google Scholar
  22. 22.
    Huang M, Malhamé RP, Caines PE (2006) Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6:221–252 zbMATHMathSciNetGoogle Scholar
  23. 23.
    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(9):1560–1571 CrossRefGoogle Scholar
  24. 24.
    Huang M, Caines PE, Malhamé RP (2010) The NCE (mean field) principle with locality dependent cost interactions. IEEE Trans Autom Control 55(12):2799–2805 CrossRefGoogle Scholar
  25. 25.
    Jourdain B, Roux R (2011) Convergence of a stochastic particle approximation for fractional scalar conservation laws. Stoch Process Appl 121(5):957–988 (English summary) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Kolokoltsov VN (2006) On the regularity of solutions to the spatially homogeneous Boltzmann equation with polynomially growing collision kernel. Adv Stud Contemp Math 12:9–38 zbMATHMathSciNetGoogle Scholar
  27. 27.
    Kolokoltsov VN (2007) Nonlinear Markov semigroups and interacting Lévy type processes. J Stat Phys 126(3):585–642 zbMATHMathSciNetGoogle Scholar
  28. 28.
    Kolokoltsov VN (2010) Nonlinear Markov processes and kinetic equations. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  29. 29.
    Kolokoltsov VN (2011) Nonlinear Lévy and nonlinear Feller processes: an analytic introduction. arXiv:1103.5591. Published in: Antoniouk AV, Melnik RV (eds) (2013) Mathematics and Life Sciences. De Gruyter, Berlin, pp 45–70
  30. 30.
    Kolokoltsov VN (2012) Nonlinear Markov games on a finite state space (mean-field and binary interactions). Int J Stat Probab 1(1):77–91. Canadian Center of Science and Education (Open access journal) CrossRefGoogle Scholar
  31. 31.
    Kolokoltsov VN, Malafeyev OA (2010) Understanding game theory. World Scientific, Singapore CrossRefzbMATHGoogle Scholar
  32. 32.
    Kolokoltsov VN, Yang W (2012) Sensitivity analysis for HJB equations with application to coupled backward-forward systems. Preprint. Optimization (to appear) Google Scholar
  33. 33.
    Kolokoltsov VN, Yang W (2013) On existence results of general kinetic equations with a path-dependent feature. Open J Optim 2(2):39–44 CrossRefGoogle Scholar
  34. 34.
    Kolokoltsov VN, Li J, Yang W (2012) Mean field games and nonlinear Markov processes. arXiv:1112.3744
  35. 35.
    Kunita H (1997) Stochastic flows and stochastic differential equations. Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  36. 36.
    Lachapelle A, Salomon J, Turinici G (2010) Computation of mean field equilibria in economics. Math Models Methods Appl Sci 20:567–588 CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Lasry J-M, Lions P-L (2006) Jeux à champ moyen. I. Le cas stationnaire. C R Math Acad Sci Paris 343(9):619–625 (French) [Mean field games. I. The stationary case] CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Le Boudec J-Y, McDonald D, Mundinger J (2007) A generic mean field convergence result for systems of interacting objects. In: QEST 2007. 4th international conference on quantitative evaluation of systems, pp 3–18 Google Scholar
  39. 39.
    Lions P-L (2012) Théorie des jeux à champs moyen et applications. Cours au Collège de France.
  40. 40.
    Man PLW, Norris JR, Bailleul I, Kraft M (2010) Coupling algorithms for calculating sensitivities of Smoluchowski’s coagulation equation. SIAM J Sci Comput 32(2):635–655 CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Olla S (2001) Central limit theorems for tagged particles and for diffusions in random environment. In: Milieux alleatoires, Panor. Synthéses, vol 12. Soc Math France, Paris, pp 75–100 Google Scholar
  42. 42.
    Osada H (2010) Tagged particle processes and their non-explosion criteria. J Math Soc Jpn 62(3):867–894 CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Piasecki J, Sadlej K (2003) Deterministic limit of tagged particle motion: effect of reflecting boundaries. Physica A 323(1–4):171–180 CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Vassili N. Kolokoltsov
    • 1
  • Marianna Troeva
    • 2
  • Wei Yang
    • 3
    Email author
  1. 1.Department of StatisticsUniversity of WarwickCoventryUK
  2. 2.North-Eastern Federal UniversityYakutskRussia
  3. 3.Department of Mathematics and StatisticsUniversity of StrathclydeGlasgowUK

Personalised recommendations