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Linear-Quadratic Mean Field Games

Author

Listed:
  • A. Bensoussan

    (The University of Texas at Dallas
    City University of Hong Kong)

  • K. C. J. Sung

    (The University of Hong Kong)

  • S. C. P. Yam

    (The Chinese University of Hong Kong)

  • S. P. Yung

    (The University of Hong Kong)

Abstract

We provide a comprehensive study of a general class of linear-quadratic mean field games. We adopt the adjoint equation approach to investigate the unique existence of their equilibrium strategies. Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward–backward ordinary differential equation. For the one-dimensional case, we establish the unique existence of the equilibrium strategy. For a dimension greater than one, by applying the Banach fixed point theorem under a suitable norm, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls in the underlying dynamics and is always satisfied whenever the coefficients of the mean field term are vanished, and hence, our theories include the classical linear-quadratic stochastic control problems as special cases. As a by-product, we also establish a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations. Numerical examples of nonexistence of the equilibrium strategy will also be illustrated. Finally, a similar approach has been adopted to study the linear-quadratic mean field type stochastic control problems and their comparisons with mean field games.

Suggested Citation

  • A. Bensoussan & K. C. J. Sung & S. C. P. Yam & S. P. Yung, 2016. "Linear-Quadratic Mean Field Games," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 496-529, May.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:2:d:10.1007_s10957-015-0819-4
    DOI: 10.1007/s10957-015-0819-4
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    References listed on IDEAS

    as
    1. Hamidou Tembine & Quanyan Zhu & Tamer Basar, 2011. "Risk-sensitive mean field stochastic differential games," Post-Print hal-00643547, HAL.
    2. Alan Kirman, 1993. "Ants, Rationality, and Recruitment," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 108(1), pages 137-156.
    3. A. Bensoussan & J. Frehse, 2000. "Stochastic Games for N Players," Journal of Optimization Theory and Applications, Springer, vol. 105(3), pages 543-565, June.
    4. Olivier Guéant & Pierre Louis Lions & Jean-Michel Lasry, 2011. "Mean Field Games and Applications," Post-Print hal-01393103, HAL.
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    6. Buckdahn, Rainer & Li, Juan & Peng, Shige, 2009. "Mean-field backward stochastic differential equations and related partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3133-3154, October.
    7. Bensoussan, A. & Yam, S.C.P. & Zhang, Z., 2015. "Well-posedness of mean-field type forward–backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3327-3354.
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