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On some mean field games and master equations through the lens of conservation laws

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Abstract

In this manuscript we derive a new nonlinear transport equation written on the space of probability measures that allows to study a class of deterministic mean field games and master equations, where the interaction of the agents happens only at the terminal time. The point of view via this transport equation has two important consequences. First, this equation reveals a new monotonicity condition that is sufficient both for the uniqueness of MFG Nash equilibria and for the global in time well-posedness of master equations. Interestingly, this condition is in general in dichotomy with both the Lasry–Lions and displacement monotonicity conditions, studied so far in the literature. Second, in the absence of monotonicity, the conservative form of the transport equation can be used to define weak entropy solutions to the master equation. We construct several concrete examples to demonstrate that MFG Nash equilibria, whether or not they actually exist, may not be selected by the entropy solutions of the master equation.

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Acknowledgements

Both authors are grateful to Alberto Bressan and Elio Marconi for the discussions about the structure of entropy solutions for scalar conservation laws in the case of non-convex flux functions. 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 a 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.

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  1. Department of Mathematics, Baylor University, Sid Richardson Building 1410 S. 4th Street, Waco, TX, 76706, USA

    P. Jameson Graber

  2. Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK

    Alpár R. Mészáros

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Graber, P.J., Mészáros, A.R. On some mean field games and master equations through the lens of conservation laws. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02859-z

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