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The theory of Mean Field Game of Controls considers a class of mean field games where the interaction is through the joint distribution of the state and control. It is well known that, for standard mean field games, certain monotonicity conditions are crucial to guarantee the uniqueness of mean field equilibria and then the global wellposedness for master equations. In the literature the monotonicity condition could be the Lasry–Lions monotonicity, the displacement monotonicity, or the anti-monotonicity conditions. In this paper, we investigate these three types of monotonicity conditions for Mean Field Games of Controls and show their propagation along the solutions to the master equations with common noises. In particular, we extend the displacement monotonicity to semi-monotonicity, whose propagation result is new even for standard mean field games. This is the first step towards the global wellposedness theory for master equations of Mean Field Games of Controls.
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This content will become publicly available on September 9, 2026
Mean field game master equations with anti-monotonicity conditions
It is well known that the monotonicity condition, either in Lasry–Lions sense or in displacement sense, is crucial for the global well-posedness of mean field game master equations, as well as for the uniqueness of mean field equilibria and solutions to mean field game systems. In the literature, the monotonicity conditions are always taken in a fixed direction. In this paper, we propose a new type of monotonicity condition in the opposite direction, which we call the anti-monotonicity condition, and establish the global well-posedness for mean field game master equations with non-separable Hamiltonians. Our anti-monotonicity condition allows our data to violate both the Lasry–Lions monotonicity and the displacement monotonicity conditions.
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- PAR ID:
- 10638681
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- Volume:
- 27
- Issue:
- 11
- ISSN:
- 1435-9855
- Page Range / eLocation ID:
- 4469 to 4499
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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