Discrete mean field games: Existence of equilibria and convergence

IF 1.1 Q3 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
J. Doncel, Nicolas Gast, B. Gaujal
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引用次数: 27

Abstract

We consider mean field games with discrete state spaces (called discrete mean field games in the following) and we analyze these games in continuous and discrete time, over finite as well as infinite time horizons. We prove the existence of a mean field equilibrium assuming continuity of the cost and of the drift. These conditions are more general than the existing papers studying finite state space mean field games. Besides, we also study the convergence of the equilibria of N -player games to mean field equilibria in our four settings. On the one hand, we define a class of strategies in which any sequence of equilibria of the finite games converges weakly to a mean field equilibrium when the number of players goes to infinity. On the other hand, we exhibit equilibria outside this class that do not converge to mean field equilibria and for which the value of the game does not converge. In discrete time this non- convergence phenomenon implies that the Folk theorem does not scale to the mean field limit.
离散平均场对策:平衡点的存在性和收敛性
我们考虑具有离散状态空间的平均场对策(以下称为离散平均场对策),并在连续和离散时间,有限和无限时间范围内分析这些对策。我们证明了假设代价和漂移连续的平均场平衡的存在性。这些条件比现有研究有限状态空间均场对策的论文更为普遍。此外,我们还研究了四种情况下N人对策的均衡收敛到平均场均衡的问题。一方面,我们定义了一类策略,当参与者数量趋于无穷时,有限对策的任意均衡序列弱收敛于平均场均衡。另一方面,我们展示了这类之外的均衡,它们不收敛于平均场均衡,也不收敛于博弈的价值。在离散时间下,这种不收敛现象意味着Folk定理不能扩展到平均场极限。
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来源期刊
Journal of Dynamics and Games
Journal of Dynamics and Games MATHEMATICS, INTERDISCIPLINARY APPLICATIONS-
CiteScore
2.00
自引率
0.00%
发文量
26
期刊介绍: The Journal of Dynamics and Games (JDG) is a pure and applied mathematical journal that publishes high quality peer-review and expository papers in all research areas of expertise of its editors. The main focus of JDG is in the interface of Dynamical Systems and Game Theory.
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