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Generic Properties of First-Order Mean Field Games

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  • Alberto Bressan

    (Penn State University)

  • Khai T. Nguyen

    (North Carolina State University)

Abstract

We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of view of generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for “nearly all” mean field games (in the Baire category sense) the best reply map is single-valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies.

Suggested Citation

  • Alberto Bressan & Khai T. Nguyen, 2023. "Generic Properties of First-Order Mean Field Games," Dynamic Games and Applications, Springer, vol. 13(3), pages 750-782, September.
    • Handle: RePEc:spr:dyngam:v:13:y:2023:i:3:d:10.1007_s13235-022-00487-3
    DOI: 10.1007/s13235-022-00487-3
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    References listed on IDEAS

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    1. Erhan Bayraktar & Xin Zhang, 2019. "On non-uniqueness in mean field games," Papers 1908.06207, arXiv.org, revised Mar 2020.
    2. Seierstad, Atle & Sydsaeter, Knut, 1977. "Sufficient Conditions in Optimal Control Theory," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(2), pages 367-391, June.
    3. Diogo A. Gomes & Levon Nurbekyan & Mariana Prazeres, 2018. "One-Dimensional Stationary Mean-Field Games with Local Coupling," Dynamic Games and Applications, Springer, vol. 8(2), pages 315-351, June.
    4. Diogo Gomes & João Saúde, 2014. "Mean Field Games Models—A Brief Survey," Dynamic Games and Applications, Springer, vol. 4(2), pages 110-154, June.
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