Minimax Theory and its Applications 03 (2018), No. 2, 261--284
Copyright Heldermann Verlag 2018
Displacement Convexity for First-Order Mean-Field Games
Diogo A. Gomes
King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia
diogo.gomes@kaust.edu.sa
Tommaso Seneci
King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia
tommaso.seneci@kaust.edu.sa
Here, we consider the planning problem for first-order mean-field games (MFG). When there is no coupling between players, MFG degenerate into optimal transport problems. Displacement convexity is a fundamental tool in optimal transport that often reveals hidden convexity of functionals and, thus, has numerous applications in the calculus of variations. We explore the similarities between the Benamou-Brenier formulation of optimal transport and MFG to extend displacement convexity methods to MFG. In particular, we identify a class of functions, that depend on solutions of MFG, that are convex in time and, thus, obtain new a priori bounds for solutions of MFG. A remarkable consequence is the log-convexity of Lq norms. This convexity gives bounds for the density of solutions of the planning problem and extends displacement convexity of Lq norms from optimal transport. Additionally, we prove the convexity of Lq norms for MFG with congestion.
Keywords: Mean field game, congestion, optimal transport, displacement convexity.
MSC: 91A13, 35Q91, 26B25.
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