Displacement convexity for first-order mean-field games

Diogo A. Gomes, Tommaso Seneci

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)261-284
Number of pages24
JournalMinimax Theory and its Applications
Issue number2
StatePublished - 2018

Bibliographical note

Publisher Copyright:
© 2018, Heldermann Verlag.


  • Congestion
  • Displacement convexity
  • Mean field game
  • Optimal transport

ASJC Scopus subject areas

  • Analysis
  • Control and Optimization
  • Computational Mathematics


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