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Linear Quadratic Mean Field Games: Decentralized O(1/N)-Nash Equilibria

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Abstract

This paper studies an asymptotic solvability problem for linear quadratic (LQ) mean field games with controlled diffusions and indefinite weights for the state and control in the costs. The authors employ a rescaling approach to derive a low dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The rescaling technique is further used for performance estimates, establishing an O(1/N)-Nash equilibrium for the obtained decentralized strategies.

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References

  1. Huang M, Malhamé R P, and Caines P E, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inform. Systems, 2006, 6(3): 221–252.

    Article  MathSciNet  Google Scholar 

  2. Lasry J M and Lions P L, Mean field games, Japan. J. Math., 2007, 2(1): 229–260.

    Article  MathSciNet  Google Scholar 

  3. Bauso D, Tembine H, and Basar T, Opinion dynamics in social networks through mean-field games, SIAM J. Control Optim., 2016, 54(6): 3225–3257.

    Article  MathSciNet  Google Scholar 

  4. Carmona R, Fouque J P, and Sun L H, Mean field games and systemic risk, Communications in Mathematical Sciences, 2015, 13: 911–933.

    Article  MathSciNet  Google Scholar 

  5. Chan P and Sircar R, Fracking, renewables, and mean field games, SIAM Review, 2017, 59(3): 588–615.

    Article  MathSciNet  Google Scholar 

  6. De Paola A, Angeli D, and Strbac G, Distributed control of micro-storage devices with mean field games, IEEE Transactions on Smart Grid, 2016, 7(2): 1119–1127.

    Google Scholar 

  7. Huang X, Jaimungal S, and Nourian M, Mean-field game strategies for optimal execution, Applied Mathematical Finance, 2019, 26(2): 153–185.

    Article  MathSciNet  Google Scholar 

  8. Lachapelle A and Wolfram M T, On a mean field game approach modeling congestion and aversion in pedestrian crowds, Transportation Research Part B: Methodological, 2011, 45(10): 1572–1589.

    Article  Google Scholar 

  9. Lacker D and Zariphopoulou T, Mean field and n-agent games for optimal investment under relative performance criteria, Math. Finance, 2019, 29(4): 1003–1038.

    Article  MathSciNet  Google Scholar 

  10. Laguzet L and Turinici G, Individual vaccination as Nash equilibrium in an SIR model with application to the 2009–2010 influenza A (H1N1) epidemic in France, Bulletin of Mathematical Biology, 2015, 77(10): 1955–1984.

    Article  MathSciNet  Google Scholar 

  11. Li Z, Reppen A M, and Sircar R, A mean field games model for cryptocurrency mining, arXiv: 1912.01952, 2019.

  12. Ma Z, Callaway D S, and Hiskens I A, Decentralized charging control of large populations of plug-in electric vehicles, IEEE Transactions on Control Systems Technology, 2013, 21(1): 67–78.

    Article  Google Scholar 

  13. Salhab R, Malhamé R P, and Ny J L, A dynamic game model of collective choice in multiagent systems, IEEE Trans. Autom. Control, 2018, 63(3): 768–782.

    Article  MathSciNet  Google Scholar 

  14. Swiecicki I, Gobron T, and Ullmo D, Schrödinger approach to mean field games, Physical Review Letters, 2016, 116(12): 128701.

    Article  Google Scholar 

  15. Wang B and Huang M, Mean field production output control with sticky prices: Nash and social solutions, Automatica, 2019, 100: 90–98.

    Article  MathSciNet  Google Scholar 

  16. Yin H, Mehta P G, Meyn S P, et al., Synchronization of coupled oscillators is a game, IEEE Trans. Autom. Control, 2012, 57(4): 920–935.

    Article  MathSciNet  Google Scholar 

  17. Bensoussan A, Frehse J, and Yam S C P, Mean Field Games and Mean Field Type Control Theory, Springer, New York, 2013.

    Book  Google Scholar 

  18. Caines P E, Huang M, and Malhamé R P, Mean field games, Eds. by Başar T and Zaccour G, Handbook of Dynamic Game Theory, Springer, Berlin, 2017, 345–372.

    Google Scholar 

  19. Cardaliaguet P, Notes on Mean Field Games, University of Paris, Dauphine, 2013.

    MATH  Google Scholar 

  20. Carmona R and Delarue F, Probabilistic Theory of Mean Field Games with Applications I–II, Springer, Cham, Switzerland, 2018.

    Book  Google Scholar 

  21. Bensoussan A, Sung K C J, Yam S C P, et al., Linear-quadratic mean-field games, J. Optim. Theory Appl., 2016, 169(2): 496–529.

    Article  MathSciNet  Google Scholar 

  22. Huang J, Wang S, and Wu Z, Backward mean-field linear-quadratic-gaussian (LQG) games: Full and partial information, IEEE Trans. Autom. Control, 2016, 61(12): 3784–3796.

    Article  MathSciNet  Google Scholar 

  23. Huang M, Caines P E, and Malhamé R P, Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behavior and decentralized ε-Nash equilibria, IEEE Trans. Autom. Control, 2007, 52(9): 1560–1571.

    Article  Google Scholar 

  24. Li T and Zhang J F, Asymptotically optimal decentralized control for large population stochastic multiagent systems, IEEE Trans. Autom. Control, 2008, 53(7): 1643–1660.

    Article  MathSciNet  Google Scholar 

  25. Wang B C and Zhang J F, Mean field games for large-population multiagent systems with Markov jump parameters, SIAM J. Control Optim., 2012, 50(4): 2308–2334.

    Article  MathSciNet  Google Scholar 

  26. Bardi M and Priuli F S, Linear-quadratic N-person and mean-field games with ergodic cost, SIAM J. Control Optim., 2014, 52(5): 3022–3052.

    Article  MathSciNet  Google Scholar 

  27. Moon J and Başar T, Linear quadratic risk-sensitive and robust mean field games, IEEE Trans. Autom. Control, 2017, 62(3): 1062–1077.

    Article  MathSciNet  Google Scholar 

  28. Huang J and Huang M, Robust mean field linear-quadratic-Gaussian games with unknown L2-disturbance, SIAM J. Control Optim., 2017, 55(5): 2811–2840.

    Article  MathSciNet  Google Scholar 

  29. Tchuendom R F, Uniqueness for linear-quadratic mean field games with common noise, Dyn. Games Appl., 2018, 8(1): 199–210.

    Article  MathSciNet  Google Scholar 

  30. Huang M, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM J. Control Optim., 2010, 48(5): 3318–3353.

    Article  MathSciNet  Google Scholar 

  31. Nguyen S L and Huang M, Linear-quadratic-Gaussian mixed games with continuum-parametrized minor players, SIAM J. Control Optim., 2012, 50(5): 2907–2937.

    Article  MathSciNet  Google Scholar 

  32. Bensoussan A, Chau M H M, Lai Y, et al., Linear-quadratic mean field Stackelberg games with state and control delays, SIAM J. Control Optim., 2017, 55(4): 2748–2781.

    Article  MathSciNet  Google Scholar 

  33. Caines P E and Kizikale A C, ε-Nash equilibria for partially observed LQG mean field games with a major player, IEEE Trans. Autom. Control, 2017, 62(7): 3225–3234.

    Article  MathSciNet  Google Scholar 

  34. Huang M and Zhou M, Linear quadratic mean field games: Asymptotic solvability and relation to the fixed point approach, IEEE Trans. Autom. Control, 2020, 65(4): 1397–1412.

    Article  MathSciNet  Google Scholar 

  35. Ma Y and Huang M, Linear quadratic mean field games with a major player: The multi-scale approach, Automatica, 2020, 113(3): 108774.

    Article  MathSciNet  Google Scholar 

  36. Huang M and Yang X, Linear quadratic mean field social optimization: Asymptotic solvability and decentralized control, Appl. Math. Optim., 2021, (accepted).

  37. Nourian M and Caines P E, ε-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents, SIAM J. Control Optim., 2013, 51(4): 3302–3331.

    Article  MathSciNet  Google Scholar 

  38. Basna R, Hilbert A, and Kolokoltsov V, An epsilon-Nash equilibrium for non-linear Markov games of mean-field-type on finite spaces, Communications on Stochastic Analysis, 2014, 8(4): 449–468.

    Article  MathSciNet  Google Scholar 

  39. Cardaliaguet P, Delarue F, Lasry J M, et al., The master equation and the convergence problem in mean field games, arXiv: 1509.02505, 2015.

  40. Huang M and Zhou M, Linear quadratic mean field games — Part I: The asymptotic solvability problem, Proc. 23rd Internat. Symp. Math. Theory Networks and Systems, Hong Kong, China, July, 2018, 489–495.

  41. Huang M and Yang X, Linear quadratic mean field games: Decentralized O(1/N)-Nash equilibria, arXiv: 2107.09168, 2021.

  42. Yosida K, Functional Analysis, Springer-Verlag, Berlin, 6th edition, 1980.

    MATH  Google Scholar 

  43. Pham H, Continuous-Time Stochastic Control and Optimization with Financial Applications, Springer Science & Business Media, Berlin, 2009.

    Book  Google Scholar 

  44. Yong J and Zhou X Y, Stochastic Controls: Hamiltonian Systems and HJB Equations, SpringerVerlag, New York, 1999.

    Book  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Minyi Huang & Xuwei Yang

Authors
  1. Minyi Huang
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  2. Xuwei Yang
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Correspondence to Minyi Huang or Xuwei Yang.

Additional information

This research was supported by Natural Sciences and Engineering Research Council (NSERC) of Canada.

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Huang, M., Yang, X. Linear Quadratic Mean Field Games: Decentralized O(1/N)-Nash Equilibria. J Syst Sci Complex 34, 2003–2035 (2021). https://doi.org/10.1007/s11424-021-1266-y

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  • DOI: https://doi.org/10.1007/s11424-021-1266-y

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