Skip to main content
SpringerLink
Log in

An Invariance Principle in Large Population Stochastic Dynamic Games

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

We study large population stochastic dynamic games where the so-called Nash certainty equivalence based control laws are implemented by the individual players. We first show a martingale property for the limiting control problem of a single agent and then perform averaging across the population; this procedure leads to a constant value for the martingale which shows an invariance property of the population behavior induced by the Nash strategies.

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

References

  1. R. P. Isaacs, Differential Games, John Wiley, 1965.

  2. T. Başar G.J. Olsder (1995) Dynamic Noncooperative Game Theory EditionNumber2 Academic Press London, UK

    Google Scholar 

  3. G.M. Erickson (1995) ArticleTitleDifferential game models of advertsing competition Europ. J. Oper. Res. 83 431–438 Occurrence Handle10.1016/0377-2217(94)00232-2

    Article  Google Scholar 

  4. E.J. Green (1984) ArticleTitleContinuum and finite-player noncooperative models of competition Econometrica 52 IssueID4 975–993 Occurrence Handle10.2307/1911194

    Article  Google Scholar 

  5. M. Huang, P. E. Caines, and R. P. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and Nash equilibrium solutions, in Proc. the 42nd IEEE Conf. Decision Contr., Maui, Hawaii, December 2003, 98–103.

  6. V.E. Lambson (1984) ArticleTitleSelf-enforcing collusion in large dynamic markets J. Econ. Theory 34 282–291 Occurrence Handle10.1016/0022-0531(84)90145-5

    Article  Google Scholar 

  7. M. Ali Khan and Y. Sun, Non-cooperative games with many players, in Handbook of Game Theory with Economic Applications (ed. by R. J. Aumann and S. Hart), North-Holland, 2002, 3.

  8. M. Huang R.P. Malhamé P.E. Caines (2005) Nash equilibria for large-population linear stochastic systems of weakly coupled agents E.K. Boukas R.P. Malhamé (Eds) Analysis, Control and Optimization of Complex Dynamic Systems Springer New York 215–252 Occurrence Handle10.1007/0-387-25477-3_9

    Chapter  Google Scholar 

  9. M. Huang, R. P. Malhamé, and P. E. Caines, Nash certainty equivalence in large population stochastic dynamic games: Connections with the physics of interacting particle systems, in Proc. the 45th IEEE Conference on Decision and Control, San Diego, CA, December 2006, 4921–4926.

  10. M. Huang, R. P. Malhamé, and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information and Control, 2007, to appear.

  11. M. Huang, P. E. Caines, and R. P. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behavior and decentralized ɛ-Nash equilibria, IEEE Transactions on Automatic Control, 2007, 52, to appear.

  12. R. Boel P. Varaiya (1977) ArticleTitleOptimal control of jump processes SIAM J. Control Optim. 15 IssueID1 92–119 Occurrence Handle10.1137/0315008

    Article  Google Scholar 

  13. M.H.A. Davis P. Varaiya (1973) ArticleTitleDynamic programming conditions for partially observable stochastic systems SIAM J. Control Optim. 11 226–261 Occurrence Handle10.1137/0311020

    Article  Google Scholar 

  14. D.A. Dawson J. Gärtner (1987) ArticleTitleLarge deviations from the McKean-Vlasov limit for weakly interacting diffusions Stochastics 20 247–308

    Google Scholar 

  15. A. S. Sznitman, Topics in propagation of chaos, in Ecole d’Eté de Probabilitiés de Saint-Flour XIX−1989, Lect. Notes Math. 1464, Springer-Verlag, Berlin, 1991, 165–252.

  16. A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge Univ. Press, 1992.

Download references

Author information

Authors and Affiliations

  1. Department of Information Engineering, Research School of Information Sciences and Engineering, The Australian National University, Canberra, ACT, 0200, Australia

    Minyi Huang

  2. Department of Electrical and Computer Engineering, McGill University, Montreal, QC, H3A 2A7, Canada

    Peter E. Caines

  3. Department of Electrical Engineering, École Polytechnique de Montréal, Montreal, QC, H3C 3A7, Canada

    Roland P. Malhamé

  4. GERAD, Montreal, Canada

    Peter E. Caines & Roland P. Malhamé

Authors
  1. Minyi Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter E. Caines
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roland P. Malhamé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minyi Huang.

Additional information

Dedicated to Professor Han-Fu Chen on the occasion of his 70th birthday.

This work was partially supported by the Australian Research Council (ARC) and National Sciences and Engineering Research Council of Canada (NSERC).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, M., Caines, P.E. & Malhamé, R.P. An Invariance Principle in Large Population Stochastic Dynamic Games. Jrl Syst Sci & Complex 20, 162–172 (2007). https://doi.org/10.1007/s11424-007-9015-4

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-007-9015-4

Keywords

Search

Navigation