Equilibrium concepts for time-inconsistent stopping problems in continuous time

TL;DR

This paper introduces a new strong equilibrium concept for time-inconsistent stopping problems in continuous time, providing a more accurate subgame perfect equilibrium and a method to construct optimal equilibria.

Contribution

It defines the strong equilibrium, compares it with existing notions, and offers a new iteration method to construct and prove the existence of optimal mild equilibria.

Findings

Optimal mild equilibrium is always a strong equilibrium under certain conditions.

A new iteration method is developed for constructing optimal mild equilibria.

Existence of optimal mild equilibrium is established.

Abstract

A \emph{new} notion of equilibrium, which we call \emph{strong equilibrium}, is introduced for time-inconsistent stopping problems in continuous time. Compared to the existing notions introduced in ArXiv: 1502.03998 and ArXiv: 1709.05181, which in this paper are called \emph{mild equilibrium} and \emph{weak equilibrium} respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous-time Markov chain and the discount function is log sub-additive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.