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Explicit solutions of some linear-quadratic mean field games

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  • We consider $N$-person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Planck equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number $N$ of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [22]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.
    Mathematics Subject Classification: Primary: 91A13, 49N70; Secondary: 93E20, 91A06, 49N10.


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  • [1]

    Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Opt., 50 (2012), 77-109.doi: 10.1137/100790069.


    Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.doi: 10.1137/090758477.


    O. Alvarez and M. Bardi, Ergodic problems in differential games, in "Advances in Dynamic Game Theory," Ann. Internat. Soc. Dynam. Games, 9, Birkhäuser Boston, Boston, MA, (2007), 131-152.


    O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp.


    R. J. Aumann, Markets with a continuum of traders, Econometrica, 32 (1964), 39-50.doi: 10.2307/1913732.


    M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," With appendices by Maurizio Falcone and Pierpaolo Soravia, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997.


    T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory," Second edition, Academic Press, Ltd., London, 1995.


    A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications," Applied Mathematical Sciences, 151, Springer-Verlag, Berlin, 2002.


    P. Cardaliaguet, "Notes on Mean Field Games," from P.-L. Lions' lectures at Collège de France, 2010.


    J. C. Engwerda, "Linear Quadratic Dynamic Optimization and Differential Games," Wiley, Chichester, 2005.


    W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," 2nd edition, Stochastic Modelling and Applied Probability, 25, Springer, New York, 2006.


    D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308-328.


    O. Guéant, "Mean Field Games and Applications to Economics," Ph.D. Thesis, Université Paris-Dauphine, 2009.


    O. Guéant, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), 276-294.


    O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in "Paris-Princeton Lectures on Mathematical Finance 2010" (eds. R. A. Carmona, et al.), Lecture Notes in Math., 2003, Springer, Berlin, (2011), 205-266.


    R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980.


    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 Conference on Decision and Control," Maui, Hawaii, December, (2003), 98-103.


    M. Huang, P. E. Caines and R. P. Malhamé, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.


    M. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.doi: 10.1109/TAC.2007.904450.


    M. Huang, P. E. Caines and R. P. Malhamé, An invariance principle in large population stochastic dynamic games, J. Syst. Sci. Complex., 20 (2007), 162-172.doi: 10.1007/s11424-007-9015-4.


    A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588.doi: 10.1142/S0218202510004349.


    J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Acad. Sci. Paris, 343 (2006), 619-625.doi: 10.1016/j.crma.2006.09.019.


    J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Acad. Sci. Paris, 343 (2006), 679-684.doi: 10.1016/j.crma.2006.09.018.


    J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.

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