Skip to main content
SpringerLink
Log in

Social Optima of Backward Linear-Quadratic-Gaussian Mean-Field Teams

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

This paper studies a class of stochastic linear-quadratic-Gaussian (LQG) dynamic optimization problems involving a large number of weakly-coupled heterogeneous agents. By “heterogeneous,” we mean agents are endowed with different types of parameters thus they are not statistically identical. Specifically, discrete-type heterogeneous agents are considered here which are more practical than homogeneous-type agents, and at the same time, more tractable than continuum-type heterogeneous agents. Unlike well-studied mean-field-game, these agents formalize a team with cooperation to minimize some social cost functional. Moreover, unlike standard social optima literature, the state here evolves by some backward stochastic differential equation (BSDE) in which the terminal instead initial condition is specified. Accordingly, the related social cost is represented by some recursive functional for which the initial state is considered. Applying a backward version of person-by-person optimality, we construct an auxiliary control problem for each agent based on decentralized information. The decentralized social strategy is derived by a class of new consistency condition (CC) systems, which are mean-field-type forward-backward stochastic differential equations (FBSDEs). The well-posedness of such consistency condition system is obtained via Riccati decoupling method. The related asymptotic social optimality is also verified.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Abou-Kandil, H., Freiling, G., Ionescu, V., Jank, G.: Matrix Riccati Equation in Control and Systems Theory. Birkhäuser, Basel (2003)

    Book  Google Scholar 

  2. Albi, G., Choi, Y.P., Fornasier, M., Kalise, D.: Mean field control hierarchy. Appl. Math. Optim. 76, 93–135 (2017)

    Article  MathSciNet  Google Scholar 

  3. Arabneydi, J., Mahajan, A.: Team-optimal solution of finite number of mean-field coupled LQG subsystems. In: Proceedings of the 2015 54th IEEE Conference on Decision and Control (CDC). IEEE, (2015), pp. 5308–5313

  4. Bensoussan, A., Sung, K., Yam, S., Yung, S.: Linear-quadratic mean field games. J. Optim. Theory Appl. 169, 496–529 (2016)

    Article  MathSciNet  Google Scholar 

  5. Bismut, J.: An introductory approach to duality in optimal stochastic control. SIAM Rev. 20, 62–78 (1978)

    Article  MathSciNet  Google Scholar 

  6. Buckdahn, R., Li, J., Peng, S.: Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor agents. SIAM J. Control Optim. 52, 451–492 (2014)

    Article  MathSciNet  Google Scholar 

  7. Carmona, R., Delarue, F.: Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51, 2705–2734 (2013)

    Article  MathSciNet  Google Scholar 

  8. Chen, Y., Bušić, A., Meyn, S.P.: State estimation for the individual and the population in mean field control with application to demand dispatch. IEEE Trans. Autom. Control 62, 1138–1149 (2016)

    Article  MathSciNet  Google Scholar 

  9. de Waal, P.R., van Schuppen, J.H.: A class of team problems with discrete action spaces: optimality conditions based on multimodularity. SIAM J. Control Optim. 38, 875–892 (2000)

    Article  MathSciNet  Google Scholar 

  10. Duffie, D., Epstein, L.: Stochastic differential utility. Econometrica 60, 353–394 (1992)

    Article  MathSciNet  Google Scholar 

  11. El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997)

    Article  MathSciNet  Google Scholar 

  12. El Karoui, N., Peng, S., Quenez, M.C.: A dynamic maximum principle for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 11, 664–693 (2001)

    Article  MathSciNet  Google Scholar 

  13. Hu, Y., Huang, J., Li, X.: Linear quadratic mean field game with control input constraint. ESAIM Control Optim. Calc. Var. 24, 901–919 (2018)

    Article  MathSciNet  Google Scholar 

  14. Hu, Y., Huang, J., Nie, T.: Linear-quadratic-Gaussian mixed mean-field games with heterogeneous input constraints. SIAM J. Control Optim. 56, 2835–2877 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Huang, M., Nguyen, S.: Linear-Quadratic Mean Field Social Optimization with a Major Player. arXiv preprint arXiv:1904.03346, (2019)

  17. Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6, 221–252 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Huang, M., Caines, P.E., Malhamé, R.P.: Large-population cost-coupled LQG problems with non-uniform agents: individual-mass behavior and decentralized \(\varepsilon \)-Nash equilibria. IEEE Trans. Autom. Control 52, 1560–1571 (2007)

    Article  Google Scholar 

  19. Huang, J., Wang, G., Xiong, J.: A maximum principle for partial information backward stochastic control problems with application. SIAM J. Control Optim. 48, 2106–2117 (2009)

    Article  MathSciNet  Google Scholar 

  20. Huang, M., Caines, P.E., Malhamé, R.P.: Social optima in mean field LQG control: centralized and decentralized strategies. IEEE Trans. Autom. Control 57, 1736–1751 (2012)

    Article  MathSciNet  Google Scholar 

  21. Huang, J., Wang, S., Wu, Z.: Backward mean-field linear-quadratic-Gaussian (LQG) games: full and partial information. IEEE Trans. Autom. Control 61, 3784–3796 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Li, X., Sun, J., Xiong, J.: Linear quadratic optimal control problems for mean-field backward stochastic differential equations. Appl. Math. Optim. (2016). https://doi.org/10.1007/S00245-017-9464-7

    Article  Google Scholar 

  24. Lim, A.E., Zhou, X.Y.: Linear-quadratic control of backward stochastic differential equations. SIAM J. Control Optim. 40, 450–474 (2001)

    Article  MathSciNet  Google Scholar 

  25. Long, N.V.: A Survey of Dynamic Games in Economics. World Scientific, Singapore (2010)

    Book  Google Scholar 

  26. Ma, J., Yong, J.: Forward-backward stochastic differential equations and their applications. Springer, New York (1999)

    MATH  Google Scholar 

  27. Nourian, M., Caines, P.E.: \(\epsilon \)-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM J. Control Optim. 51, 3302–3331 (2013)

    Article  MathSciNet  Google Scholar 

  28. Nuno, G., Moll, B.: Social optima in economies with heterogeneous agents. Rev. Econ. Dynam. 28, 150–180 (2018)

    Article  Google Scholar 

  29. Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)

    Article  MathSciNet  Google Scholar 

  30. Pardoux, E., Tang, S.: Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Relat. Fields 114, 123–150 (1999)

    Article  MathSciNet  Google Scholar 

  31. Peng, S.: Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim. 27, 125–144 (1993)

    Article  MathSciNet  Google Scholar 

  32. Piccoli, B., Rossi, F., Trélat, E.: Control to flocking of the kinetic Cucker–Smale Model. SIAM J. Math. Anal. 47, 4685–4719 (2015)

    Article  MathSciNet  Google Scholar 

  33. Sun, J., Yong, J.: Linear quadratic stochastic differential games: open-loop and closed-loop saddle points. SIAM J. Control Optim. 52, 4082–4121 (2014)

    Article  MathSciNet  Google Scholar 

  34. Wang, G., Wu, Z.: The maximum principle for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control 54, 1230–1242 (2009)

    Article  MathSciNet  Google Scholar 

  35. Wang, B., Zhang, J.: Social optima in mean field linear-quadratic-Gaussian models with Markov jump parameters. SIAM J. Control Optim. 55, 429–456 (2017)

    Article  MathSciNet  Google Scholar 

  36. Xu, W.: A maximum principle for optimal control for a class of controlled systems. J. Austral. Math. Soc. Ser. B 38, 172–181 (1996)

    Article  MathSciNet  Google Scholar 

  37. Yeung, D.W., Petrosjan, L.A.: Cooperative Stochastic Differential Games. Springer, New York (2006)

    Google Scholar 

  38. Yong, J.: Linear forward–backward stochastic differential equations with random coefficients. Probab. Theory Relat. Fields 135, 53–83 (2006)

    Article  MathSciNet  Google Scholar 

  39. Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees and the Associate Editor for careful reading and constructive comments. This work was supported by the National Natural Science Foundation of China (No. 12001317, 12001320), Shandong Provincial Natural Science Foundation (No. ZR2020QA019), RGC 153005/14P, 153275/16P, P0030808, P0008686, P0031044, QILU Young Scholars Program of Shandong University and the Fundamental Research Funds of Shandong University (11030072064071). The authors also acknowledge the financial support from PolyU-SDU joint research center.

Author information

Authors and Affiliations

  1. Zhongtai Securities Institute for Financial Studies, Shandong University, Shandong, 250100, Jinan, China

    Xinwei Feng

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

    Jianhui Huang

  3. School of Management, Shandong University, Shandong, 250100, Jinan, China

    Shujun Wang

Authors
  1. Xinwei Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianhui Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shujun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shujun Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, X., Huang, J. & Wang, S. Social Optima of Backward Linear-Quadratic-Gaussian Mean-Field Teams. Appl Math Optim 84 (Suppl 1), 651–694 (2021). https://doi.org/10.1007/s00245-021-09782-8

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-021-09782-8

Keywords

Search

Navigation