Time-inconsistent stopping, myopic adjustment & equilibrium stability: with a mean-variance application

TL;DR

This paper develops a framework for time-inconsistent optimal stopping problems in Markov chains, introducing equilibrium concepts, stability analysis, and applying the theory to mean-variance and variance problems.

Contribution

It introduces a comprehensive equilibrium framework for time-inconsistent stopping problems, including existence, stability, and verification results, with applications to mean-variance models.

Findings

Equilibrium existence is established under fixed point arguments.

Multiple equilibria and non-uniqueness are generally possible.

Stability notions are developed and analyzed for equilibrium strategies.

Abstract

For a discrete time Markov chain and in line with Strotz' consistent planning we develop a framework for problems of optimal stopping that are time-inconsistent due to the consideration of a non-linear function of an expected reward. We consider pure and mixed stopping strategies and a (subgame perfect Nash) equilibrium. We provide different necessary and sufficient equilibrium conditions including a verification theorem. Using a fixed point argument we provide equilibrium existence results. We adapt and study the notion of the myopic adjustment process and introduce different kinds of equilibrium stability. We show that neither existence nor uniqueness of equilibria should generally be expected. The developed theory is applied to a mean-variance problem and a variance problem.