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December 2021 Submodular mean field games: Existence and approximation of solutions
Jodi Dianetti, Giorgio Ferrari, Markus Fischer, Max Nendel
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Ann. Appl. Probab. 31(6): 2538-2566 (December 2021). DOI: 10.1214/20-AAP1655


We study mean field games with scalar Itô-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. First, it allows us to prove existence of solutions via an application of Tarski’s fixed point theorem, covering cases with discontinuous dependence on the measure variable. Second, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist minimal and maximal solutions. Third, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. The mean field game is first defined over ordinary stochastic controls, then extended to relaxed controls. Our approach also allows us to prove existence of a strong solution for a class of submodular mean field games with common noise, where the representative player at equilibrium interacts with the (conditional) mean of its state’s distribution.

Funding Statement

Financial support through the German Research Foundation via CRC 1283 is gratefully acknowledged.


We thank two anonymous Referees for their pertinent and helpful comments.


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Jodi Dianetti. Giorgio Ferrari. Markus Fischer. Max Nendel. "Submodular mean field games: Existence and approximation of solutions." Ann. Appl. Probab. 31 (6) 2538 - 2566, December 2021.


Received: 1 July 2019; Revised: 1 September 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/20-AAP1655

Primary: 06B23 , 49J45 , 91A15 , 93E20

Keywords: complete lattice , first order stochastic dominance , Mean field games , submodular cost function , Tarski’s fixed point theorem

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.31 • No. 6 • December 2021
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