A Class of Mean-Field Games with Optimal Stopping and its Inverse Problem

This paper revisits the well-studied optimal stopping problem but within the large-population framework. In particular, two classes of optimal stopping problems are formulated by taking into account the relative performance criteria. It is remarkable the relative performance criteria, also understood by the Joneses preference, habit formation utility, or relative wealth concern in economics and finance, plays an important role in explaining various decision behaviors such as price bubbles. By introducing such criteria in large-population setting, a given agent can compare his individual stopping rule with the average behaviors of its cohort. The associated mean-field games are formulated in order to derive the decentralized stopping rules. The related consistency conditions are characterized via some coupled equation system and the asymptotic Nash equilibrium properties are also verified. In addition, some inverse mean-field optimal stopping problem is also introduced and discussed.