On the convergence of closed-loop nash equilibria to the mean field game limit

Annals of Applied Probability

Volumn 30, Issue 4, 2020, Pages 1693-1761

  Lacker, Daniel ^a  

^a Och Spine at New York Presbyterian Hospitals   (United States)

Author keywords

Closed loop controls; McKean Vlasov equations; Mean field games; Relaxed controls; Stochastic differential games

Indexed keywords

EID: 85090312609     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/19-AAP1541     Document Type: Article

References (67)

1. BAFICO, R. and BALDI, P. (1981/82). Small random perturbations of Peano phenomena. Stochastics 6 279-292. MR0665404 https://doi.org/10.1080/17442508208833208
2. BAYRAKTAR, E. and COHEN, A. (2018). Analysis of a finite state many player game using its master equation. SIAM J. Control Optim. 56 3538-3568. MR3860894 https://doi.org/10.1137/17M113887X
3. BEN AROUS, G. and ZEITOUNI, O. (1999). Increasing propagation of chaos for mean field models. Ann. Inst. Henri Poincaré Probab. Stat. 35 85-102. MR1669916 https://doi.org/10.1016/S0246-0203(99) 80006-5
4. BERGIN, J. and BERNHARDT, D. (1992). Anonymous sequential games with aggregate uncertainty. J. Math. Econom. 21 543-562. MR1195676 https://doi.org/10.1016/0304-4068(92)90026-4
5. BERTSEKAS, D. and SHREVE, S. (1996). Stochastic Optimal Control: The Discrete Time Case. Athena Scientific.
6. BORKAR, V. S. and GHOSH, M. K. (1992). Stochastic differential games: Occupation measure based approach. J. Optim. Theory Appl. 73 359-385. MR1163266 https://doi.org/10.1007/BF00940187
7. BRÉMAUD, P. and YOR, M. (1978). Changes of filtrations and of probability measures. Z. Wahrsch. Verw. Gebiete 45 269-295. MR0511775 https://doi.org/10.1007/BF00537538
8. BRUNICK, G. and SHREVE, S. (2013). Mimicking an Itô process by a solution of a stochastic differential equation. Ann. Appl. Probab. 23 1584-1628. MR3098443 https://doi.org/10.1214/12-aap881
9. CARDALIAGUET, P.., CIRANT, M. and PORRETTA, A. (2018). Remarks on Nash equilibria in mean field game models with a major player. arXiv preprint arXiv:1811.02811.
10. CARDALIAGUET, P. (2017). The convergence problem in mean field games with local coupling. Appl. Math. Optim. 76 177-215. MR3679342 https://doi.org/10.1007/s00245-017-9434-0
11. CARDALIAGUET, P., DELARUE, F., LASRY, J.-M. and LIONS, P.-L. (2019). The Master Equation and the Convergence Problem in Mean Field Games. Annals of Mathematics Studies 201. Princeton Univ. Press, Princeton, NJ. MR3967062
12. CARDALIAGUET, P. and RAINER, C. (2009). Stochastic differential games with asymmetric information. Appl. Math. Optim. 59 1-36. MR2465706 https://doi.org/10.1007/s00245-008-9042-0
13. CARMONA, R. (2016). Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications. Financial Mathematics 1. SIAM, Philadelphia, PA. MR3629171
14. CARMONA, R., CERENZIA, M. and PALMER, A. Z. (2018). The Dyson game. arXiv preprint arXiv:1808.02464.
15. CARMONA, R. and DELARUE, F. (2013). Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51 2705-2734. MR3072222 https://doi.org/10.1137/120883499
16. CARMONA, R. and DELARUE, F. (2017). Probabilistic Theory of Mean Field Games: Vol. I, Mean Field FBSDEs, Control, and Games. Stochastic Analysis and Applications. Springer Verlag.
17. CARMONA, R. and DELARUE, F. (2018). Probabilistic Theory of Mean Field Games with Applications. II: Mean Field Games with Common Noise and Master Equations. Probability Theory and Stochastic Modelling 84. Springer, Cham. MR3753660
18. CARMONA, R., DELARUE, F. and LACKER, D. (2016). Mean field games with common noise. Ann. Probab. 44 3740-3803. MR3572323 https://doi.org/10.1214/15-AOP1060
19. CARMONA, R., FOUQUE, J.-P. and SUN, L.-H. (2015). Mean field games and systemic risk. Commun. Math. Sci. 13 911-933. MR3325083 https://doi.org/10.4310/CMS.2015.v13.n4.a4
20. CECCHIN, A., DAI PRA, P., FISCHER, M. and PELINO, G. (2019). On the convergence problem in mean field games: A two state model without uniqueness. SIAM J. Control Optim. 57 2443-2466. MR3981375 https://doi.org/10.1137/18M1222454
21. CECCHIN, A. and PELINO, G. (2019). Convergence, fluctuations and large deviations for finite state mean field games via the master equation. Stochastic Process. Appl. 129 4510-4555. MR4013871 https://doi.org/10.1016/j.spa.2018.12.002
22. CHASSAGNEUX, J.-F., CRISAN, D. and DELARUE, F. (2014). A probabilistic approach to classical solutions of the master equation for large population equilibria. arXiv preprint arXiv:1411.3009.
23. DELARUE, F. (2017). Mean field games: A toy model on an Erdös-Renyi graph. In Journées MAS 2016 de la SMAI-Phénomènes Complexes et Hétérogènes. ESAIM Proc. Surveys 60 1-26. EDP Sci., Les Ulis. MR3772471
24. DELARUE, F. and FOGUEN TCHUENDOM, R. (2020). Selection of equilibria in a linear quadratic mean-field game. Stochastic Process. Appl. 130 1000-1040. MR4046528 https://doi.org/10.1016/j.spa.2019.04.005
25. DELARUE, F., LACKER, D. and RAMANAN, K. (2020). From the master equation to mean field game limit theory: Large deviations and concentration of measure. Ann. Probab. 48 211-263. MR4079435 http://dx.doi.org/10.1214/19-AOP1359
26. DELARUE, F., LACKER, D. and RAMANAN, K. (2019). From the master equation to mean field game limit theory: A central limit theorem. Electron. J. Probab. 24 Paper No. 51, 54. MR3954791 https://doi.org/10.1214/19-EJP298
27. EL KAROUI, N. and MÉLÉARD, S. (1990). Martingale measures and stochastic calculus. Probab. Theory Related Fields 84 83-101. MR1027822 https://doi.org/10.1007/BF01288560
28. EL KAROUI, N., NGUYEN, D. and JEANBLANC-PICQUÉ, M. (1987). Compactification methods in the control of degenerate diffusions: Existence of an optimal control. Stochastics 20 169-219. MR0878312 https://doi.org/10.1080/17442508708833443
29. EL KAROUI, N., NGUYEN, D. H. and JEANBLANC-PICQUÉ, M. (1988). Existence of an optimal Markovian filter for the control under partial observations. SIAM J. Control Optim. 26 1025-1061. MR0957652 https://doi.org/10.1137/0326057
30. FELEQI, E. (2013). The derivation of ergodic mean field game equations for several populations of players. Dyn. Games Appl. 3 523-536. MR3127148 https://doi.org/10.1007/s13235-013-0088-5
31. FIGALLI, A. (2008). Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254 109-153. MR2375067 https://doi.org/10.1016/j.jfa.2007.09.020
32. FILIPPOV, A. F. (1962). On certain questions in the theory of optimal control. J. SIAM Control Ser. A 1 76-84. MR0149985
33. FISCHER, M. (2017). On the connection between symmetric N-player games and mean field games. Ann. Appl. Probab. 27 757-810. MR3655853 https://doi.org/10.1214/16-AAP1215
34. FLEMING, W. H. (1977). Generalized solutions in optimal stochastic control. In Differential Games and Control Theory, II (Proc. 2nd Conf., Univ. Rhode Island, Kingston, R.I., 1976) Lecture Notes in Pure and Appl. Math. 30 147-165. MR0688675
35. FLEMING, W. H. and NISIO, M. (1984). On stochastic relaxed control for partially observed diffusions. Nagoya Math. J. 93 71-108. MR0738919 https://doi.org/10.1017/S0027763000020742
36. FRIEDMAN, A. (1972). Stochastic differential games. J. Differential Equations 11 79-108. MR0292528 https://doi.org/10.1016/0022-0396(72)90082-4
37. FUDENBERG, D. and LEVINE, D. (1986). Limit games and limit equilibria. J. Econom. Theory 38 261-279. MR0841698 https://doi.org/10.1016/0022-0531(86)90118-3
38. FUDENBERG, D. and LEVINE, D. (2009). Open-loop and closed-loop equilibria in dynamic games with many players. In A Long-Run Collaboration on Long-Run Games 41-58. World Scientific.
39. GANGBO, W. and SWIECH, A. (2015). Existence of a solution to an equation arising from the theory of mean field games. J. Differential Equations 259 6573-6643. MR3397332 https://doi.org/10.1016/j.jde.2015.08.001
40. GÄRTNER, J. (1988). On the McKean-Vlasov limit for interacting diffusions. Math. Nachr. 137 197-248. MR0968996 https://doi.org/10.1002/mana.19881370116
41. GYÖNGY, I. (1986). Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theory Related Fields 71 501-516. MR0833267 https://doi.org/10.1007/BF00699039
42. HAMADENE, S. and LEPELTIER, J.-P. (1995). Zero-sum stochastic differential games and backward equations. Systems Control Lett. 24 259-263. MR1321134 https://doi.org/10.1016/0167-6911(94)00011-J
43. HAUSSMANN, U. G. and LEPELTIER, J.-P. (1990). On the existence of optimal controls. SIAM J. Control Optim. 28 851-902. MR1051628 https://doi.org/10.1137/0328049
44. HUANG, M., CAINES, P. E. and MALHAMÉ, R. P. (2007). Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized Nash equilibria. IEEE Trans. Automat. Control 52 1560-1571. MR2352434 https://doi.org/10.1109/TAC.2007.904450
45. HUANG, M., MALHAMÉ, R. P. and CAINES, P. E. (2006). Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 221-251. MR2346927
46. JABIN, P.-E. and WANG, Z. (2016). Mean field limit and propagation of chaos for Vlasov systems with bounded forces. J. Funct. Anal. 271 3588-3627. MR3558251 https://doi.org/10.1016/j.jfa.2016.09.014 [47] JACOD, J. and MÉMIN, J. (1981). Weak and strong solutions of stochastic differential equations: Existence and stability. In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Math. 851 169-212. Springer, Berlin. MR0620991
47. KALLENBERG, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). Springer, New York. MR1876169 https://doi.org/10.1007/978-1-4757-4015-8
48. KRYLOV, N. V. and RÖCKNER, M. (2005). Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 154-196. MR2117951 https://doi.org/10.1007/ s00440-004-0361-z
49. LACKER, D. (2015). Mean field games via controlled martingale problems: Existence of Markovian equilibria. Stochastic Process. Appl. 125 2856-2894. MR3332857 https://doi.org/10.1016/j.spa.2015.02.006
50. LACKER, D. (2016). A general characterization of the mean field limit for stochastic differential games. Probab. Theory Related Fields 165 581-648. MR3520014 https://doi.org/10.1007/s00440-015-0641-9
51. LACKER, D. (2018). On a strong form of propagation of chaos for McKean-Vlasov equations. Electron. Commun. Probab. 23 Paper No. 45, 11. MR3841406 https://doi.org/10.1214/18-ECP150
52. LASRY, J.-M. and LIONS, P.-L. (2006). Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343 619-625. MR2269875 https://doi.org/10.1016/j.crma.2006.09.019
53. LASRY, J.-M. and LIONS, P.-L. (2006). Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343 679-684. MR2271747 https://doi.org/10.1016/j.crma.2006.09.018
54. LASRY, J.-M. and LIONS, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229-260. MR2295621 https://doi.org/10.1007/s11537-007-0657-8
55. NEUFELD, A. and NUTZ, M. (2014). Measurability of semimartingale characteristics with respect to the probability law. Stochastic Process. Appl. 124 3819-3845. MR3249357 https://doi.org/10.1016/j.spa.2014.07.006
56. NUTZ, M., SAN MARTIN, J. and TAN, X. (2018). Convergence to the mean field game limit: A case study. arXiv preprint arXiv:1806.00817.
57. OELSCHLÄGER, K. (1984). A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12 458-479. MR0735849
58. PROTTER, P. E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. Springer, Berlin. Version 2.1, Corrected third printing. MR2273672 https://doi.org/10.1007/978-3-662-10061-5
59. ROXIN, E. (1962). The existence of optimal controls. Michigan Math. J. 9 109-119. MR0136844
60. ROYDEN, H. L. (1968). Real Analysis, 2nd ed. Macmillan. MR0151555
61. STROOCK, D. W. and VARADHAN, S. R. S. (2006). Multidimensional Diffusion Processes. Classics in Mathematics. Springer, Berlin. Reprint of the 1997 edition. MR2190038
62. SZNITMAN, A.-S. (1991). Topics in propagation of chaos. In École D'Été de Probabilités de Saint-Flour XIX-1989. Lecture Notes in Math. 1464 165-251. Springer, Berlin. MR1108185 https://doi.org/10.1007/BFb0085169
63. TREVISAN, D. (2013). Zero noise limits using local times. Electron. Commun. Probab. 18 no. 31, 7. MR3056068 https://doi.org/10.1214/ECP.v18-2587
64. TREVISAN, D. (2016). Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron. J. Probab. 21 Paper No. 22, 41. MR3485364 https://doi.org/10.1214/ 16-EJP4453
65. VERETENNIKOV, A. Y. (1981). On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR, Sb. 39 387.
66. VILLANI, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI. MR1964483 https://doi.org/10.1090/gsm/058
67. WALSH, J. B. (1986). An introduction to stochastic partial differential equations. In École D'été de Probabilités de Saint-Flour, XIV-1984. Lecture Notes in Math. 1180 265-439. Springer, Berlin. MR0876085 https://doi.org/10.1007/BFb0074920