On non-uniqueness in mean field games
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- by Erhan Bayraktar and Xin Zhang PDF
- Proc. Amer. Math. Soc. 148 (2020), 4091-4106
Corrigendum: Proc. Amer. Math. Soc. 149 (2021), 1359-1360.
Abstract:
We analyze an $N+1$-player game and the corresponding mean field game with state space $\{0,1\}$. The transition rate of the $j$th player is the sum of his control $\alpha ^j$ plus a minimum jumping rate $\eta$. Instead of working under monotonicity conditions, here we consider an anti-monotone running cost. We show that the mean field game equation may have multiple solutions if $\eta < \frac {1}{2}$. We also prove that although multiple solutions exist, only the one coming from the entropy solution is charged (when $\eta =0$), and therefore resolve a conjecture of Hajek and Livesay.References
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Additional Information
- Erhan Bayraktar
- Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109-1043
- MR Author ID: 743030
- ORCID: 0000-0002-1926-4570
- Email: erhan@umich.edu
- Xin Zhang
- Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109-1043
- ORCID: 0000-0002-0036-5996
- Email: zxmars@umich.edu
- Received by editor(s): August 16, 2019
- Received by editor(s) in revised form: January 28, 2020
- Published electronically: May 11, 2020
- Additional Notes: This research was supported in part by the National Science Foundation under grants DMS-1613170.
- Communicated by: David Levin
- © Copyright 2020 The authors
- Journal: Proc. Amer. Math. Soc. 148 (2020), 4091-4106
- MSC (2010): Primary 60F99, 60J27, 60K35, 93E20
- DOI: https://doi.org/10.1090/proc/15046
- MathSciNet review: 4127851