This paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons T and convex but non-smooth Hamiltonian H, as well as for smooth H and T large enough. The phenomenon is analysed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In particular, a new result on uniqueness for the MFG PDEs with small data, e.g., small T, is proved. Some results are also extended to MFGs with two populations.
Accepté le :
DOI : 10.1051/cocv/2018026
Mots-clés : Mean Field Games, finite horizon, non-uniqueness of solutions, uniqueness of solutions, multipopulation MFG
@article{COCV_2019__25__A44_0,
author = {Bardi, Martino and Fischer, Markus},
title = {On non-uniqueness and uniqueness of solutions in finite-horizon {Mean} {Field} {Games}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018026},
mrnumber = {4009550},
zbl = {1437.91049},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018026/}
}
TY - JOUR AU - Bardi, Martino AU - Fischer, Markus TI - On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018026/ DO - 10.1051/cocv/2018026 LA - en ID - COCV_2019__25__A44_0 ER -
%0 Journal Article %A Bardi, Martino %A Fischer, Markus %T On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018026/ %R 10.1051/cocv/2018026 %G en %F COCV_2019__25__A44_0
Bardi, Martino; Fischer, Markus. On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 44. doi : 10.1051/cocv/2018026. http://www.numdam.org/articles/10.1051/cocv/2018026/
[1] , and , Mean Field Games models of segregation. Math. Model. Methods Appl. Sci. 27 (2017) 75–113. | DOI | MR | Zbl
[2] , Wellposedness of mean field games with common noise under a weak monotonicity condition. SIAM J. Control Optim. 54 (2016) 30–48. | DOI | MR | Zbl
[3] , Strong Solutions for Time-Dependent Mean Field Games with Non-Separable Hamiltonians. Preprint (2016). | arXiv | MR
[4] , Explicit solutions of some linear-quadratic mean field games, Netw. Heterog. Media 7 (2012) 243–261. | DOI | MR | Zbl
[5] and , Uniqueness of solutions in mean field games with several populations and Neumann conditions, in PDE Models for Multi-Agent Phenomena, edited by , , . Springer INdAM Series. Preprint (2017). | arXiv | MR
[6] and , Nonlinear elliptic systems and mean field games. NoDEA Nonlinear Differ. Equ. Appl. 23 (2016) 23–44. | DOI | MR | Zbl
[7] and , Linear-quadratic N-person and mean-field games with ergodic cost. SIAM J. Control Optim. 52 (2014) 3022–3052. | DOI | MR | Zbl
[8] , , and , Fokker-Planck-Kolmogorov Equations. Vol. 207 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2015) | MR
[9] and , Stable solutions in potential mean field game systems. NoDEA Nonlinear Differ. Equ. Appl. 25 (2018) 1–26 | DOI | MR | Zbl
[10] , Notes on Mean Field Games from P-L. Lions’ lectures at Collège de France (2010).
[11] , , and , The Master Equation and the Convergence Problem in Mean Field Games. Preprint (2015). | arXiv
[12] and , Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51 (2013) 2705–2734. | DOI | MR | Zbl
[13] , and , Control of McKean-Vlasov dynamics versus mean field games. Math. Financ. Econ. 7 (2013) 131–166. | DOI | MR | Zbl
[14] , and , Mean field games with common noise. Ann. Probab. 44 (2016) 3740–3803. | DOI | MR | Zbl
[15] , Multi-population mean field games systems with Neumann boundary conditions. J. Math. Pures Appl. 103 (2015) 1294–1315. | DOI | MR | Zbl
[16] , On the Existence of Oscillating Solutions in Non-Monotone Mean-Field Games. Preprint (2017). | arXiv | MR
[17] and , Time-Dependent Focusing Mean-Field Games: the Sub-Critical Case. J Dyn. Differ. Equ. 31 (2019) 49–79. | DOI | MR | Zbl
[18] Mean Field Games Systems of Partial Differential Equations: uniqueness of solutions. Master thesis, University of Padua (2018).
[19] , On the connection between symmetric N-player games and mean field games. Ann. Appl. Probab. 27 (2017) 757–810. | DOI | MR | Zbl
[20] and , Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2006). | MR | Zbl
[21] , Partial Differential Equations of Parabolic Type. Prentice-Hall Inc., Englewood Cliffs, N.J. (1964). | MR | Zbl
[22] , Stochastic Differential Equations and Applications, Vol. 1. Academic Press, New York (1975). | MR | Zbl
[23] , and , Continuous time finite state mean field games. Appl. Math. Optim. 68 (2013) 99–143. | DOI | MR | Zbl
[24] , and , Economic Models and Mean-Field Games Theory. IMPA Mathematical Publications, Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro (2015). | MR
[25] , and , One-dimensional stationary mean-field games with local coupling. Dyn. Games Appl. 8 (2018) 315–351. | DOI | MR | Zbl
[26] , , and , Regularity Theory for Mean-Field Game Systems. Springer (2016). | DOI | MR
[27] and , Mean field games models: a brief survey. Dyn. Games Appl. 4 (2014) 110–154. | DOI | MR | Zbl
[28] and , Existence and uniqueness of solutions for Bertrand and Cournot mean field games. Appl. Math. Optim. 77 (2018) 47–71. | DOI | MR | Zbl
[29] , A reference case for mean field games models. J. Math. Pures Appl. 92 (2009) 276–294. | DOI | MR | Zbl
[30] , and , Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, edited by , et al. Lecture Notes in Mathematics. Springer, Berlin (2011) 205–266. | MR | Zbl
[31] , and , Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ϵ-Nash equilibria. IEEE Trans. Automa. Control 52 (2007) 1560–1571. | DOI | MR | Zbl
[32] , and , An invariance principle in large population stochastic dynamic games. J. Syst. Sci. Complex. 20 (2007) 162–172. | DOI | MR | Zbl
[33] , and , Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 (2006) 221–251. | DOI | MR | Zbl
[34] and , A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14 (1977) 619–633. | MR | Zbl
[35] , Foundations of Modern Probability, 2nd edn. Probability and Its Applications. Springer, New York (2001). | MR | Zbl
[36] and , Brownian Motion and Stochastic Calculus, 2nd edn. Vol. 113 of Graduate Texts in Mathematics. Springer, New York (1991). | MR | Zbl
[37] , Mean field games via controlled martingale problems: existence of Markovian equilibria. Stoch. Process. Appl. 125 (2015) 2856–2894. | DOI | MR | Zbl
[38] , A general characterization of the mean field limit for stochastic differential games. Probab. Theory Relat. Fields 165 (2016) 581–648. | DOI | MR | Zbl
[39] and , Mean field games. Jpn. J. Math. 2 (2007) 229–260. | DOI | MR | Zbl
[40] , Lectures at Collège de France (2008–2009).
[41] and , Linear quadratic risk-sensitive and robust mean field games. IEEE Trans. Automa. Control 62 (2016) 1062–1077. | DOI | MR | Zbl
[42] , and , Comparison theorem and estimates for transition probability densities of diffusion processes. Probab. Theory Relat. Fields 127 (2003) 388–406. | DOI | MR | Zbl
[43] , Uniqueness for linear-quadratic mean field games with common noise. Dyn. Games Appl. 8 (2018) 199–210. | DOI | MR | Zbl
[44] , A Note on Nonconvex Mean Field Games. Preprint (2016). | arXiv | MR
[45] and , Mean field games for large-population multiagent systems with Markov jump parameters. SIAM J. Control Optim. 50 (2012) 2308–2334. | DOI | MR | Zbl
Cité par Sources :
