Skip to main content

Advertisement

Log in

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

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

Abstract

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.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Achdou Y, Capuzzo-Dolcetta I (2010) Mean field games: numerical methods. SIAM J Numer Anal 48:1136–1162

    Article  MATH  MathSciNet  Google Scholar 

  2. Andersson D, Djehiche B (2011) A maximum principle for SDEs of mean-field type. Appl Math Optim 63:341–356

    Article  MATH  MathSciNet  Google Scholar 

  3. Bailleul IF (2011) Sensitivity for the Smoluchowski equation. J Phys A, Math Theor 44(24):245004

    Article  MathSciNet  Google Scholar 

  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. Belopol’skaya YaI (2005) A probabilistic approach to a solution of nonlinear parabolic equations. Theory Probab Appl 49(4):589–611

    Article  MathSciNet  Google Scholar 

  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. Benaim M, Weibull J (2003) Deterministic approximation of stochastic evolution in games. Econometrica 71(3):873–903

    Article  MATH  MathSciNet  Google Scholar 

  8. Bogachev VI, Röckner M, Shaposhnikov SV (2009) Nonlinear evolution and transport equations for measures. Dokl Math 80(3):785–789

    Article  MATH  MathSciNet  Google Scholar 

  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. Buckdahn R, Djehiche B, Li J, Peng S (2009) Mean-field backward stochastic differential equations: a limit approach. Ann Probab 37(4):1524–1565

    Article  MATH  MathSciNet  Google Scholar 

  11. Cepeda E, Fournier N (2011) Smoluchowski’s equation: rate of convergence of the Marcus-Lushnikov process. Stoch Process Appl 121(6):1411–1444

    Article  MATH  MathSciNet  Google Scholar 

  12. Crisan D (2006) Particle approximations for a class of stochastic partial differential equations. Appl Math Optim 54(3):293–314

    Article  MATH  MathSciNet  Google Scholar 

  13. Del Moral P (2004) Feynman-Kac formulae. Genealogical and interacting particle systems with applications. Probability and its applications. Springer, New York

    Book  MATH  Google Scholar 

  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. 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. Gomes DA, Mohr J, Souza RR (2010) Discrete time, finite state space mean field games. J Math Pures Appl 9(93):308–328

    Article  MathSciNet  Google Scholar 

  17. Grigorescu I (1999) Uniqueness of the tagged particle process in a system with local interactions. Ann Probab 27(3):1268–1282

    Article  MATH  MathSciNet  Google Scholar 

  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. Guérin H, Méléard S, Nualart E (2006) Estimates for the density of a nonlinear Landau process. J Funct Anal 238:649–677

    Article  MATH  MathSciNet  Google Scholar 

  20. Huang M (2010) Large-population LQG games involving a major player: the Nash certainty equivalence principle. SIAM J Control Optim 48:3318–3353

    Article  MATH  Google Scholar 

  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. 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

    MATH  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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

    MATH  MathSciNet  Google Scholar 

  27. Kolokoltsov VN (2007) Nonlinear Markov semigroups and interacting Lévy type processes. J Stat Phys 126(3):585–642

    MATH  MathSciNet  Google Scholar 

  28. Kolokoltsov VN (2010) Nonlinear Markov processes and kinetic equations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  31. Kolokoltsov VN, Malafeyev OA (2010) Understanding game theory. World Scientific, Singapore

    Book  MATH  Google Scholar 

  32. Kolokoltsov VN, Yang W (2012) Sensitivity analysis for HJB equations with application to coupled backward-forward systems. Preprint. Optimization (to appear)

  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

    Article  Google Scholar 

  34. Kolokoltsov VN, Li J, Yang W (2012) Mean field games and nonlinear Markov processes. arXiv:1112.3744

  35. Kunita H (1997) Stochastic flows and stochastic differential equations. Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  36. Lachapelle A, Salomon J, Turinici G (2010) Computation of mean field equilibria in economics. Math Models Methods Appl Sci 20:567–588

    Article  MATH  MathSciNet  Google Scholar 

  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]

    Article  MATH  MathSciNet  Google Scholar 

  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. Lions P-L (2012) Théorie des jeux à champs moyen et applications. Cours au Collège de France. http://www.college-defrance.fr/default/EN/all/equ_der/cours_et_seminaires.htm

  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

    Article  MATH  MathSciNet  Google Scholar 

  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. Osada H (2010) Tagged particle processes and their non-explosion criteria. J Math Soc Jpn 62(3):867–894

    Article  MATH  MathSciNet  Google Scholar 

  43. Piasecki J, Sadlej K (2003) Deterministic limit of tagged particle motion: effect of reflecting boundaries. Physica A 323(1–4):171–180

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kolokoltsov, V.N., Troeva, M. & Yang, W. On the Rate of Convergence for the Mean-Field Approximation of Controlled Diffusions with Large Number of Players. Dyn Games Appl 4, 208–230 (2014). https://doi.org/10.1007/s13235-013-0095-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-013-0095-6

Keywords

Navigation