Abstract
We consider stochastic differential games with \(N\) nearly identical players, linear-Gaussian dynamics, and infinite horizon discounted quadratic cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of \(N\) Hamilton–Jacobi–Bellman and \(N\) Kolmogorov–Fokker–Planck partial differential equations, proving that for small discount factors quadratic-Gaussian solutions exist and are unique. Then, we prove the convergence of such solutions to the unique quadratic-Gaussian solution of the pair of Mean Field equations. We also discuss some singular limits, such as vanishing discount, vanishing noise, and cheap control.
Similar content being viewed by others
Notes
Observe that the value \(\bar{\ell }\) depends on the fixed \(N\ge \bar{N}\), so that it is in fact \(\bar{\ell }=\bar{\ell }_N\). However, the number of players is being kept fixed throughout the rest of this step of the proof, so we can drop the the dependency in the notation without risk of confusion.
References
Achdou Y, Capuzzo-Dolcetta I (2010) Mean field games: numerical methods. SIAM J Numer Anal 48(3):1136–1162
Bardi M (2012) Explicit solutions of some linear-quadratic mean field games. Netw Heterog Media 7:243–261
Bardi M, Priuli FS (2014) Linear-quadratic \(N\)-person and mean-field games with ergodic cost. SIAM J Control Optim 52(5):3022–3052
Basar T, Olsder GJ (1995) Dynamic noncooperative game theory, 2nd edn. Academic Press, London
Bensoussan A, Frehse J (1995) Ergodic Bellman systems for stochastic games in arbitrary dimension. Proc R Soc London Ser A 449:65–77
Bensoussan A, Frehse J (2002) Regularity results for nonlinear elliptic systems and applications. Springer, Berlin
Bensoussan A, Sung KCJ, Yam SCP, Yung SP (2014) Linear-quadratic mean field games. ar**v:1404.5741
Engwerda JC (2005) LQ dynamic optimization and differential games. John Wiley, Hoboken, NJ
Gomes DA, Mohr J, Souza RR (2010) Discrete time, finite state space mean field games. J Math Pures Appl (9) 93(3):308–328
Guéant O, Lasry J-M, Lions P-L (2011) Mean field games and applications. In Carmona RA et al (eds) Paris-Princeton Lectures on mathematical finance 2010. Springer, Berlin, pp 205–266
Huang M, Caines PE, Malhamé RP (2003) Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. In Proceedings of the 42nd IEEE conference on decision and control, Maui, pp 98–103
Huang M, Caines PE, Malhamé RP (2006) Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6(3):221–251
Huang M, Caines PE, Malhamé RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized \(\epsilon \)-Nash equilibria. IEEE Trans Automat Control 52(9):1560–1571
Kolokoltsov VN, Troeva M, Yang W (2014) On the rate of convergence for the mean-field approximation of controlled diffusions with large number of players. Dyn Games Appl 4(2):208-230
Lachapelle A, Wolfram M-T (2011) On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transport Res B 45(10):1572–1589
Lancaster P, Rodman L (1995) Algebraic Riccati equations. Oxford University Press, New York
Lasry J-M, Lions P-L (2006) Jeux à champ moyen. I. Le cas stationnaire. CR Acad Sci Paris 343:619–625
Lasry J-M, Lions P-L (2006) Jeux à champ moyen. II. Horizon fini et contrôle optimal. CR Acad Sci Paris 343:679–684
Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2:229–260
Li T, Zhang J-F (2008) Asymptotically optimal decentralized control for large population stochastic multiagent systems. IEEE Trans Automat Control 53(7):1643–1660
Nourian M, Caines PE, Malhamé RP, Huang M (2013) Nash, social and centralized solutions to consensus problems via mean field control theory. IEEE Trans Automat Control 58(3):639–653
Priuli FS (2014) First order mean field games in pedestrian dynamics. ar**v:1402.7296
Serre D (2002) Matrices: theory and applications, 2nd edn. Springer, New York
Acknowledgments
The author was partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Priuli, F.S. Linear-Quadratic \(N\)-Person and Mean-Field Games: Infinite Horizon Games with Discounted Cost and Singular Limits. Dyn Games Appl 5, 397–419 (2015). https://doi.org/10.1007/s13235-014-0129-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13235-014-0129-8