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Minimax Theory and its Applications 03 (2018), No. 2, 227--260
Copyright Heldermann Verlag 2018

The Variational Structure and Time-Periodic Solutions for Mean-Field Games Systems

Marco Cirant
Dip. di Matematica "Tullio Levi-Civita", UniversitÓ di Padova, via Trieste 63, 35121 Padova, Italy

Levon Nurbekyan
King Abdullah University of Science and Technology, CEMSE Division, Thuwal 23955-6900, Saudi Arabia

We observe that mean-field game (MFG) systems admit a two-player infinite-dimensional general-sum differential game formulation. We show that particular regimes of this game reduce to previously known variational principles. Furthermore, based on the game-perspective we derive new variational formulations for first-order MFG systems with congestion. Finally, we use these findings to prove the existence of time-periodic solutions for viscous MFG systems with a coupling that is not a non-decreasing function of density.

Keywords: Infinite-dimensional differential games, congestion problems, saddle-point formulation.

MSC: 35Q91, 35Q93, 35A01.

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