On time-inconsistent stopping problems and mixed strategy stopping times

TL;DR

This paper develops a game-theoretic framework for time-inconsistent stopping problems involving non-linear expected rewards, introducing mixed strategies and analyzing equilibrium conditions for diffusion processes.

Contribution

It introduces a novel mixed strategy stopping time framework with equilibrium analysis for time-inconsistent problems involving non-linear rewards.

Findings

Characterization of subgame perfect Nash equilibrium.

Existence and uniqueness results for equilibrium strategies.

Application to mean-variance and variance stopping problems.

Abstract

A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. A subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other necessary and sufficient equilibrium conditions including a smooth fit result are proved. Existence and uniqueness are investigated. A mean-variance and a variance problem are studied. The state process is a general one-dimensional It\^{o} diffusion.