One-Dimensional Forward–Forward Mean-Field Games

Diogo A. Gomes, Levon Nurbekyan, Marc Sedjro

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


While the general theory for the terminal-initial value problem for mean-field games (MFGs) has achieved a substantial progress, the corresponding forward–forward problem is still poorly understood—even in the one-dimensional setting. Here, we consider one-dimensional forward–forward MFGs, study the existence of solutions and their long-time convergence. First, we discuss the relation between these models and systems of conservation laws. In particular, we identify new conserved quantities and study some qualitative properties of these systems. Next, we introduce a class of wave-like equations that are equivalent to forward–forward MFGs, and we derive a novel formulation as a system of conservation laws. For first-order logarithmic forward–forward MFG, we establish the existence of a global solution. Then, we consider a class of explicit solutions and show the existence of shocks. Finally, we examine parabolic forward–forward MFGs and establish the long-time convergence of the solutions.
Original languageEnglish (US)
Pages (from-to)619-642
Number of pages24
JournalApplied Mathematics & Optimization
Issue number3
StatePublished - Nov 1 2016

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The authors were supported by KAUST baseline and start-up funds.


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