The Optimal Equilibrium for Time-Inconsistent Stopping Problems -- the Discrete-Time Case

TL;DR

This paper introduces an iterative method to identify the unique optimal equilibrium in infinite-horizon discrete-time stopping problems with non-exponential discounting, demonstrating the existence of a dominating equilibrium.

Contribution

It develops a novel iterative approach for finding subgame perfect Nash equilibria and proves the existence and uniqueness of an optimal equilibrium under decreasing impatience.

Findings

Existence of equilibrium via fixed-point iterations.

Uniqueness of the optimal equilibrium.

Optimal equilibrium yields larger value than others.

Abstract

We study an infinite-horizon discrete-time optimal stopping problem under non-exponential discounting. A new method, which we call the iterative approach, is developed to find subgame perfect Nash equilibria. When the discount function induces decreasing impatience, we establish the existence of an equilibrium through fixed-point iterations. Moreover, we show that there exists a unique optimal equilibrium, which generates larger value than any other equilibrium does at all times. To the best of our knowledge, this is the first time a dominating subgame perfect Nash equilibrium is shown to exist in the literature of time-inconsistency.