Buttazzo, G. ; Casas, E. ; de Teresa, L. ; Glowinski, R. ; Leugering, G. ; Trélat, E. ; Zhang, X.(éd.)
We propose and investigate a discrete-time mean field game model involving risk-averse agents, each of them controlling a linear dynamical system. The model under study is a coupled system of dynamic programming equations with a Kolmogorov equation. The agents’ risk aversion is modeled by composite risk measures. The existence of a solution to the coupled system is obtained with a fixed point approach. The corresponding feedback control allows to construct an approximate Nash equilibrium for a related dynamic game with finitely many players.
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Keywords: Risk measures, mean field games, approximate Nash equilibria, Cournot equilibria
@article{COCV_2021__27_1_A46_0,
author = {Fr\'ed\'eric Bonnans, J. and Lavigne, Pierre and Pfeiffer, Laurent},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Discrete-time mean field games with risk-averse agents},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021044},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021044/}
}
TY - JOUR AU - Frédéric Bonnans, J. AU - Lavigne, Pierre AU - Pfeiffer, Laurent ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Discrete-time mean field games with risk-averse agents JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021044/ DO - 10.1051/cocv/2021044 LA - en ID - COCV_2021__27_1_A46_0 ER -
%0 Journal Article %A Frédéric Bonnans, J. %A Lavigne, Pierre %A Pfeiffer, Laurent %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Discrete-time mean field games with risk-averse agents %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021044/ %R 10.1051/cocv/2021044 %G en %F COCV_2021__27_1_A46_0
Frédéric Bonnans, J.; Lavigne, Pierre; Pfeiffer, Laurent. Discrete-time mean field games with risk-averse agents. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 44. doi: 10.1051/cocv/2021044
[1] , , , and , Partial differential equation models in macroeconomics. Philo. Trans. R. Soc. A 372 (2014) 20130397.
[2] , and , An extended mean field game for storage in smart grids. J. Optim. Theory Appl. 2020 (2020) 1–27.
[3] , , and , Coherent measures of risk. Math. Finance 9 (1999) 203–228.
[4] Basel Committee on Banking Supervision, Messages from the academic literature on risk measurement for the tradingbook (2011).
[5] and , Convex analysis and monotone operator theory in Hilbert spaces. CMS Books in Mathematics. Springer-Verlag New York (2011).
[6] , , , , and , Transmit strategies for massive machine-type communications based on mean field games. In 2018 15th International Symposium on Wireless Communication Systems (ISWCS). IEEE (2018) 1–5.
[7] , and , Schauder estimates for a class of potential mean field games of controls. Appl. Math. Optim. 2019 (2019) 1–34.
[8] and , Mean field game of controls and an application to trade crowding. Math. Financial Econ. 12 (2018) 335–363.
[9] and , Composition of time-consistent dynamic monetary risk measures in discrete time. Int. J. Theor. Appl. Finance 14 (2011) 137–162.
[10] and , Stochastic finance: an introduction in discrete time. Walter de Gruyter (2011).
[11] and , On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields 162 (2015) 707–738.
[12] and , Existence and uniqueness of solutions for Bertrand and Cournot mean field games. Appl. Math. Optim. 2015 (2015) 1–25.
[13] , and , Nonlocal Bertrand and Cournot mean field games with general nonlinear demand schedule. Preprint (2020). | arXiv
[14] , and , Mean Field Games and Applications. Springer Berlin, Heidelberg (2011) 205–266.
[15] , and , Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 (2006) 221–252.
[16] , and , Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ε-nash equilibria. IEEE Trans. Autom. Control 52 (2007) 1560–1571.
[17] , On classical solutions to the mean field game system of controls. Preprint (2019). | arXiv
[18] and , Jeux à champ moyen. i–le cas stationnaire. Comp. Rendus Math. 343 (2006) 619–625.
[19] and , Jeux à champ moyen. ii–horizon fini et contrôle optimal. Comp. Rendus Mathématique 343 (2006) 679–684.
[20] and , Mean field games. Jpn. J. Math. 2 (2007) 229–260.
[21] and , Linear quadratic risk-sensitive and robust mean field games. IEEE Trans. Autom. Control 62 (2017) 1062–1077.
[22] , Optimality conditions in variational form for non-linear constrained stochastic control problems. Math. Control Related Fields 10 (2020) 493–526.
[23] , Risk-averse dynamic programming for Markov decision processes. Math. Program. 125 (2010) 235–261.
[24] and , Conditional risk mappings. Math. Oper. Res. 31 (2006) 544–561.
[25] , and , Markov–Nash equilibria in mean-field games with discounted cost. SIAM J. Control Optim. 56 (2018) 4256–4287.
[26] , Minimax and risk averse multistage stochastic programming. Eur. J. Oper. Res. 219 (2012) 719–726. Feature Clusters.
[27] , and , Risk-sensitive mean-field games. IEEE Trans. Autom. Control 59 (2014) 835–850.
[28] , Optimal transport: Old and New. Springer Verlag (2008).
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This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and by the FIME Lab (Laboratoire de Finance des Marchés de l’Energie), Paris.
