SUBMODULAR MEAN FIELD GAMES: EXISTENCE AND APPROXIMATION OF SOLUTIONS

• 成果类型: Article
• 署名作者: Dianetti, Jodi; Ferrari, Giorgio; Fischer, Markus; Nendel, Max
• 署名单位: University of Bielefeld; University of Padua
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1655
• 发表日期: 2021
• 页码: 2538-2566
• 关键词: uniform integrability; dynamic-games; equilibrium; CONVERGENCE

摘要

We study mean field games with scalar Ito-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.