Strong Solutions to Submodular Mean Field Games with Common Noise and Related McKean-Vlasov FBSDES

Jodi Dianetti
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Jodi Dianetti: Center for Mathematical Economics, Bielefeld University

No 674, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

Abstract: This paper studies multidimensional mean field games with common noise and the related system of McKean-Vlasov forward-backward stochastic differential equations de- riving from the stochastic maximum principle. We first propose some structural conditions which are related to the submodularity of the underlying mean field game and are a sort of opposite version of the well known Lasry-Lions monotonicity. By reformulating the represen- tative player minimization problem via the stochastic maximum principle, the submodularity conditions allow to prove comparison principles for the forward-backward system, which cor- respond to the monotonicity of the best reply map. Building on this property, existence of strong solutions is shown via Tarski’s fixed point theorem, both for the mean field game and for the related McKean-Vlasov forward-backward system. In both cases, the set of solutions enjoys a lattice structure, with minimal and maximal solutions which can be constructed by iterating the best reply map or via the fictitious play algorithm.

Keywords: Mean field games with common noise; FBSDE; Mean field FBSDE with conditional law; Stochastic Maximum principle; Submodular cost function: Tarski's fixed point theorem; ficitious play (search for similar items in EconPapers)
Pages: 33
Date: 2023-01-09
New Economics Papers: this item is included in nep-gth
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