Abstract
We introduce an abstract framework for the study of general mean field games and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set valued map expressing the admissible trajectories for the microscopical agents. The techniques used can be applied to consider a broad class of dependence between the trajectories of the single agent and the state of the system. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2008)
Aubin, J.-P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264. Springer, Berlin (1984)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis, Modern Birkhäuser Classics. Birkhäuser, Boston (2009). Reprint of the 1990 edition
Billingsley, P.: Convergence of Probability Measures, Second. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1999)
Bonnet, B., Frankowska, H.: Differential inclusions in Wasserstein spaces: the Cauchy-Lipschitz framework. J. Differ. Equ. (2020)
Cannarsa, P., Capuani, R.: Existence and uniqueness for mean field games with state constraints, PDE models for multiagent phenomena. In: Springer INdAM Ser., vol. 28, pp. 49–71. Springer, Cham (2018)
Cannarsa, P., Capuani, R., Cardaliaguet, P.: \(C^{1,1}\)-smoothness of constrained solutions in the calculus of variations with application to mean field games. Math. Eng. 1(1), 174–203 (2019). https://doi.org/10.3934/Mine.2018.1.174
Cannarsa, P., Capuani, R., Cardaliaguet, P.: Mean field games with state constraints: from mild to pointwise solutions of the PDE system. Calc. Var. Partial Differ. Equ. 60(3), 108 (2021). https://doi.org/10.1007/s00526-021-01936-4
Capuani, R., Marigonda, A.: Constrained mean field games equilibria as fixed point of random lifting of set-valued maps. IFAC-PapersOnLine 55(30), 180–185 (2022). https://doi.org/10.1016/j.ifacol.2022.11.049
Capuani, R., Marigonda, A., Mogentale, M.: Random lifting of set valued maps, large-scale scientific computing. In: Lecture Notes in Comput. Sci., vol. 13127, pp. 297–305. Springer, Cham (2022)
Himmelberg, C.J.: Measurable relations. Fundam. Math. 87, 53–72 (1975)
Hiriart-Urruty, J.-B.: Images of connected sets by semicontinuous multifunctions. J. Math. Anal. Appl. 111(2), 407–422 (1985)
Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006)
Huang, M., Caines, P.E., Malhamé, R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized \(\epsilon \)-Nash equilibria. IEEE Trans. Autom. Control 52(9), 1560–1571 (2007)
Jimenez, C., Marigonda, A., Quincampoix, M.: Dynamical systems and Hamilton-Jacobi-Bellman equations on the Wasserstein space and their \(L^{2}\) representations. SIMA (2021). To appear
Kirr, E., Petrusel, A.: Continuous dependence on parameters of the fixed points set for some set valued operators. Discuss. Math., Differ. Incl. 17(1–2), 29–41 (1997)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006). (French, with English and French summaries)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006). (French, with English and French summaries)
Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
Lawson, J.D.: Embeddings of compact convex sets and locally compact cones. Pac. J. Math. 66(2), 443–453 (1976)
Lim, T.-C.: On fixed point stability for set valued contractive mappings with applications to generalized differential equations. J. Math. Anal. Appl. 110, 436–441 (1985)
Marigonda, A., Quincampoix, M.: Mayer control problem with probabilistic uncertainty on initial positions. J. Differ. Equ. 264(5), 3212–3252 (2018). https://doi.org/10.1016/j.jde.2017.11.014
Nadler, S.B. Jr.: Multi-valued contraction mappings. Pac. J. Math. 30, 475–488 (1969)
Pata, V.: Fixed Point Theorems and Applications. Unitext, vol. 116. Springer, Cham (2019)
Roberts, J.W.: The embedding of compact convex sets in locally convex spaces. Can. J. Math. 30(3), 449–454 (1978). https://doi.org/10.4153/CJM-1978-038-0
Funding
Open access funding provided by Università degli Studi di Verona within the CRUI-CARE Agreement. Rossana Capuani has received funding from the Italian Ministry of University and Research according to D.M. 1062/2021 PON Ricerca e Innovazione 2014-2020, Asse IV Istruzione e ricerca per il recupero - Azione IV.4 - Dottorati e contratti di ricerca su tematiche dell’innovazione, Azione IV.6 Contratti di ricerca su tematiche Green.
Antonio Marigonda has been partially supported by INdAM-GNAMPA Project 2023 GNAMPA CUP_E53C22001930001 entitled “Mean field games methods for mobility and sustainable development”.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Giovanni Colombo
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Capuani, R., Marigonda, A. & Ricciardi, M. Random Lift of Set Valued Maps and Applications to Multiagent Dynamics. Set-Valued Var. Anal 31, 28 (2023). https://doi.org/10.1007/s11228-023-00693-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11228-023-00693-0