# 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

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.  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
 [1] Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006 [2] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [3] Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 [4] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [5] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [6] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [7] Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011 [8] Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 [9] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [10] Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 [11] John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026 [12] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [13] Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014 [14] Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53 [15] Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141 [16] Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881 [17] Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko, Renée Lampe. Modeling the pressure distribution in a spatially averaged cerebral capillary network. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021016 [18] Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 [19] Brandy Rapatski, James Yorke. Modeling HIV outbreaks: The male to female prevalence ratio in the core population. Mathematical Biosciences & Engineering, 2009, 6 (1) : 135-143. doi: 10.3934/mbe.2009.6.135 [20] Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195

2019 Impact Factor: 1.053

## Metrics

• PDF downloads (154)
• HTML views (0)
• Cited by (75)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]