Equilibria of Time-inconsistent Stopping for One-dimensional Diffusion Processes

TL;DR

This paper analyzes three types of equilibria in time-inconsistent stopping problems for one-dimensional diffusions, providing characterizations, relationships, and examples to clarify their properties.

Contribution

It offers necessary and sufficient conditions for weak equilibria, links between mild, weak, and strong equilibria, and illustrative examples highlighting their differences.

Findings

Characterization of weak equilibria with smooth-fit condition

Optimal mild equilibrium is also weak and sometimes strong

Examples demonstrating the non-equivalence of equilibrium types

Abstract

We consider three equilibrium concepts proposed in the literature for time-inconsistent stopping problems, including mild equilibria, weak equilibria and strong equilibria. The discount function is assumed to be log sub-additive and the underlying process is one-dimensional diffusion. We first provide necessary and sufficient conditions for the characterization of weak equilibria. The smooth-fit condition is obtained as a by-product. Next, based on the characterization of weak equilibria, we show that an optimal mild equilibrium is also weak. Then we provide conditions under which a weak equilibrium is strong. We further show that an optimal mild equilibrium is also strong under a certain condition. Finally, we provide several examples including one shows a weak equilibrium may not be strong, and another one shows a strong equilibrium may not be optimal mild.