In this paper we propose two new monotonicity conditions that could serve as sufficient conditions for uniqueness of Nash equilibria in mean field games. In this study we aim for unconditional uniqueness that is independent of the length of the time horizon, the regularity of the starting distribution of the agents, or the regularization effect of a non-degenerate idiosyncratic noise. Through a rich class of simple examples we show that these new conditions are not only in dichotomy with each other, but also with the two widely studied monotonicity conditions in the literature, the Lasry–Lions monotonicity and displacement monotonicity conditions.
Bibliographical noteKAUST Repository Item: Exported on 2023-07-18
Acknowledged KAUST grant number(s): ORA-2021-CRG10-4674.2
Acknowledgements: PJG acknowledges the support of the National Science Foundation through NSF Grants DMS-2045027 and DMS-1905449. We acknowledge the support of the Heilbronn Institute for Mathematical Research and the UKRI/EPSRC Additional Funding Programme for Mathematical Sciences through the focused research grant “The master equation in Mean Field Games”. ARM has also been partially supported by the EPSRC via the NIA with grant number EP/X020320/1 and by the King Abdullah University of Science and Technology Research Funding (KRF) under Award No. ORA-2021-CRG10-4674.2. Last but not least, we would like to thank the referee for their careful reading of our manuscript and for the important comments they gave.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.
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