# American Institute of Mathematical Sciences

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, Università 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. 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. doi: 10.1137/090758477. Google Scholar [3] O. Alvarez and M. Bardi, Ergodic problems in differential games, in, Advances in Dynamic Game Theory, 9 (2007), 131. Google Scholar [4] O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations,, Mem. Amer. Math. Soc., 204 (2010). Google Scholar [5] R. J. Aumann, Markets with a continuum of traders,, Econometrica, 32 (1964), 39. 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, (1997). Google Scholar [7] T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory,", Second edition, (1995). Google Scholar [8] A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications,", Applied Mathematical Sciences, 151 (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, (2005). Google Scholar [11] W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", 2nd edition, 25 (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. Google Scholar [13] O. Guéant, "Mean Field Games and Applications to Economics,", Ph.D. Thesis, (2009). Google Scholar [14] O. Guéant, A reference case for mean field games models,, J. Math. Pures Appl. (9), 92 (2009), 276. Google Scholar [15] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in, 2003 (2011), 205. 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 (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, (2003), 98. 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. 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. 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. 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. 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. 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. 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. Google Scholar

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##### 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. 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. doi: 10.1137/090758477. Google Scholar [3] O. Alvarez and M. Bardi, Ergodic problems in differential games, in, Advances in Dynamic Game Theory, 9 (2007), 131. Google Scholar [4] O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations,, Mem. Amer. Math. Soc., 204 (2010). Google Scholar [5] R. J. Aumann, Markets with a continuum of traders,, Econometrica, 32 (1964), 39. 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, (1997). Google Scholar [7] T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory,", Second edition, (1995). Google Scholar [8] A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications,", Applied Mathematical Sciences, 151 (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, (2005). Google Scholar [11] W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", 2nd edition, 25 (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. Google Scholar [13] O. Guéant, "Mean Field Games and Applications to Economics,", Ph.D. Thesis, (2009). Google Scholar [14] O. Guéant, A reference case for mean field games models,, J. Math. Pures Appl. (9), 92 (2009), 276. Google Scholar [15] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in, 2003 (2011), 205. 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 (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, (2003), 98. 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. 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. 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. 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. 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. 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. 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. Google Scholar
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