Set values for mean field games

İşeri̇, Meli̇h; Zhang, Jianfeng

doi:10.1090/tran/9255

Set values for mean field games
HTML articles powered by AMS MathViewer

by Meli̇h İşeri̇ and Jianfeng Zhang;

Trans. Amer. Math. Soc. 377 (2024), 7117-7174

DOI: https://doi.org/10.1090/tran/9255

Published electronically: August 16, 2024

HTML | PDF | Request permission

Abstract:

In this paper we study mean field games with possibly multiple mean field equilibria. Instead of focusing on the individual equilibria, we propose to study the set of values over all possible equilibria, which we call the set value of the mean field game. When the mean field equilibrium is unique, typically under certain monotonicity conditions, our set value reduces to the singleton of the standard value function which solves the master equation. The set value is by nature unique, and we shall establish two crucial properties: (i) the dynamic programming principle, also called time consistency; and (ii) the convergence of the set values of the corresponding $N$-player games, which can be viewed as a type of stability result. To our best knowledge, this is the first work in the literature which studies the dynamic value of mean field games without requiring the uniqueness of mean field equilibria. We emphasize that the set value is very sensitive to the type of the admissible controls. In particular, for the convergence one has to restrict to corresponding types of equilibria for the N-player game and for the mean field game. We shall illustrate this point by investigating three cases, two in finite state space models and the other in a continuous time model with controlled diffusions.

References

Dilip Abreu, David Pearce, and Ennio Stacchetti, Toward a theory of discounted repeated games with imperfect monitoring, Econometrica 58 (1990), no. 5, 1041–1063. MR 1079409, DOI 10.2307/2938299
Martino Bardi and Markus Fischer, On non-uniqueness and uniqueness of solutions in finite-horizon mean field games, ESAIM Control Optim. Calc. Var. 25 (2019), Paper No. 44, 33. MR 4009550, DOI 10.1051/cocv/2018026
Erhan Bayraktar and Asaf Cohen, Analysis of a finite state many player game using its master equation, SIAM J. Control Optim. 56 (2018), no. 5, 3538–3568. MR 3860894, DOI 10.1137/17M113887X
Erhan Bayraktar, Alekos Cecchin, Asaf Cohen, and François Delarue, Finite state mean field games with Wright-Fisher common noise, J. Math. Pures Appl. (9) 147 (2021), 98–162 (English, with English and French summaries). MR 4213680, DOI 10.1016/j.matpur.2021.01.003
Erhan Bayraktar, Alekos Cecchin, Asaf Cohen, and François Delarue, Finite state mean field games with Wright-Fisher common noise as limits of $N$-player weighted games, Math. Oper. Res. 47 (2022), no. 4, 2840–2890. MR 4515487, DOI 10.1287/moor.2021.1230
Alain Bensoussan, Jens Frehse, and Phillip S. C. Yam, Mean Field Games and Mean Field Type Control Theory, Springer Verlag, New York, 2013
Minyi Huang, Roland P. Malhamé, and Peter E. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst. 6 (2006), no. 3, 221–251. MR 2346927, DOI 10.4310/CIS.2006.v6.n3.a5
Pierre Cardaliaguet, Notes on mean field games, Lectures by P. L. Lions, Collège de France, 2010
P. Cardaliaguet, The convergence problem in mean field games with local coupling, Appl. Math. Optim. 76 (2017), no. 1, 177–215. MR 3679342, DOI 10.1007/s00245-017-9434-0
Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions, The master equation and the convergence problem in mean field games, Annals of Mathematics Studies, vol. 201, Princeton University Press, Princeton, NJ, 2019. MR 3967062, DOI 10.2307/j.ctvckq7qf
Guilherme Carmona, Nash Equilibria of Games with a Continuum of Players, Game Theory and Information, 0412009, University Library of Munich, Germany, 2004.
René Carmona and François Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim. 51 (2013), no. 4, 2705–2734. MR 3072222, DOI 10.1137/120883499
René Carmona and François Delarue, Probabilistic theory of mean field games with applications. I, Probability Theory and Stochastic Modelling, vol. 83, Springer, Cham, 2018. Mean field FBSDEs, control, and games. MR 3752669
René Carmona and François Delarue, Probabilistic theory of mean field games with applications. II, Probability Theory and Stochastic Modelling, vol. 84, Springer, Cham, 2018. Mean field games with common noise and master equations. MR 3753660
Alekos Cecchin, Paolo Dai Pra, Markus Fischer, and Guglielmo Pelino, On the convergence problem in mean field games: a two state model without uniqueness, SIAM J. Control Optim. 57 (2019), no. 4, 2443–2466. MR 3981375, DOI 10.1137/18M1222454
Alekos Cecchin and François Delarue, Selection by vanishing common noise for potential finite state mean field games, Comm. Partial Differential Equations 47 (2022), no. 1, 89–168. MR 4373880, DOI 10.1080/03605302.2021.1955256
Alekos Cecchin and Guglielmo Pelino, Convergence, fluctuations and large deviations for finite state mean field games via the master equation, Stochastic Process. Appl. 129 (2019), no. 11, 4510–4555. MR 4013871, DOI 10.1016/j.spa.2018.12.002
François Delarue, Restoring uniqueness to mean-field games by randomizing the equilibria, Stoch. Partial Differ. Equ. Anal. Comput. 7 (2019), no. 4, 598–678. MR 4022284, DOI 10.1007/s40072-019-00135-9
François Delarue and Rinel Foguen Tchuendom, Selection of equilibria in a linear quadratic mean-field game, Stochastic Process. Appl. 130 (2020), no. 2, 1000–1040. MR 4046528, DOI 10.1016/j.spa.2019.04.005
François Delarue, Daniel Lacker, and Kavita Ramanan, From the master equation to mean field game limit theory: large deviations and concentration of measure, Ann. Probab. 48 (2020), no. 1, 211–263. MR 4079435, DOI 10.1214/19-AOP1359
François Delarue, Daniel Lacker, and Kavita Ramanan, From the master equation to mean field game limit theory: a central limit theorem, Electron. J. Probab. 24 (2019), Paper No. 51, 54. MR 3954791, DOI 10.1214/19-EJP298
Mao Fabrice Djete, Large population games with interactions through controls and common noise: convergence results and equivalence between open-loop and closed-loop controls, ESAIM: COCV, 29 (2023), 39. \PrintDOI{10.1051/cocv/2023005}.
Zachary Feinstein, Continuity and sensitivity analysis of parameterized nash games, Econ Theory Bull (2022). \PrintDOI{10.1007/s40505-022-00228-0}.
Zachary Feinstein, Birgit Rudloff, and Jianfeng Zhang, Dynamic set values for nonzero-sum games with multiple equilibriums, Math. Oper. Res. 47 (2022), no. 1, 616–642. MR 4403768, DOI 10.1287/moor.2021.1143
Ermal Feleqi, The derivation of ergodic mean field game equations for several populations of players, Dyn. Games Appl. 3 (2013), no. 4, 523–536. MR 3127148, DOI 10.1007/s13235-013-0088-5
Markus Fischer, On the connection between symmetric $N$-player games and mean field games, Ann. Appl. Probab. 27 (2017), no. 2, 757–810. MR 3655853, DOI 10.1214/16-AAP1215
Markus Fischer and Francisco J. Silva, On the asymptotic nature of first order mean field games, Appl. Math. Optim. 84 (2021), no. 2, 2327–2357. MR 4304905, DOI 10.1007/s00245-020-09711-1
Rinel Foguen Tchuendom, Uniqueness for linear-quadratic mean field games with common noise, Dyn. Games Appl. 8 (2018), no. 1, 199–210. MR 3764697, DOI 10.1007/s13235-016-0200-8
Wilfrid Gangbo and Alpár R. Mészáros, Global well-posedness of master equations for deterministic displacement convex potential mean field games, Comm. Pure Appl. Math. 75 (2022), no. 12, 2685–2801. MR 4509653, DOI 10.1002/cpa.22069
Melih İşeri and Jianfeng Zhang, Set valued Hamilton-Jacobi-Bellman equations, Preprint, arXiv:2311.05727.
Daniel Lacker, A general characterization of the mean field limit for stochastic differential games, Probab. Theory Related Fields 165 (2016), no. 3-4, 581–648. MR 3520014, DOI 10.1007/s00440-015-0641-9
Daniel Lacker, On the convergence of closed-loop Nash equilibria to the mean field game limit, Ann. Appl. Probab. 30 (2020), no. 4, 1693–1761. MR 4133381, DOI 10.1214/19-AAP1541
Daniel Lacker and Luc Le Flem, Closed-loop convergence for mean field games with common noise, Ann. Appl. Probab. 33 (2023), no. 4, 2681–2733. MR 4612653, DOI 10.1214/22-aap1876
Jean-Michel Lasry and Pierre-Louis Lions, Mean field games, Jpn. J. Math. 2 (2007), no. 1, 229–260. MR 2295621, DOI 10.1007/s11537-007-0657-8
Mathieu Laurière and Ludovic Tangpi, Convergence of large population games to mean field games with interaction through the controls, SIAM J. Math. Anal. 54 (2022), no. 3, 3535–3574. MR 4438040, DOI 10.1137/22M1469328
Pierre-Louis Lions, Cours au Collège de France, www.college-de-france.fr.
Jin Ma, Jianfeng Zhang, and Zhaoyu Zhang, Deep learning methods for set valued PDEs, working paper.
Chenchen Mou and Jianfeng Zhang, Wellposedness of second order master equations for mean field games with nonsmooth data, Memoirs of the AMS, Accepted, arXiv:1903.09907.
Marcel Nutz, Jaime San Martin, and Xiaowei Tan, Convergence to the mean field game limit: a case study, Ann. Appl. Probab. 30 (2020), no. 1, 259–286. MR 4068311, DOI 10.1214/19-AAP1501
Triet Pham and Jianfeng Zhang, Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation, SIAM J. Control Optim. 52 (2014), no. 4, 2090–2121. MR 3227460, DOI 10.1137/120894907
Dylan Possamaï and Ludovic Tangpi, Non-asymptotic convergence rates for mean-field games: weak formulation and McKean-Vlasov BSDEs, Preprint, arXiv:2105.00484.
Yuliy Sannikov, Games with imperfectly observable actions in continuous time, Econometrica 75 (2007), no. 5, 1285–1329. MR 2347347, DOI 10.1111/j.1468-0262.2007.00795.x
Jianfeng Zhang, Backward stochastic differential equations, Probability Theory and Stochastic Modelling, vol. 86, Springer, New York, 2017. From linear to fully nonlinear theory. MR 3699487, DOI 10.1007/978-1-4939-7256-2