June  2012, 7(2): 243-261. doi: 10.3934/nhm.2012.7.243

Explicit solutions of some linear-quadratic mean field games

1. 

Dipartimento di Matematica, Università di Padova, via Trieste, 63; I-35121 Padova, Italy

Received  November 2011 Revised  March 2012 Published  June 2012

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.
Citation: Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243
References:
[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.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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

[5]

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

[6]

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.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

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

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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. Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

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.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

[24]

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

show all references

References:
[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.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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

[5]

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

[6]

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.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

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

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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. Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

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.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

[24]

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

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