General Optimal Stopping without Time Consistency
Hanqing Jin, Yanzhao Yang
TL;DR
This work tackles dynamic optimal stopping under time-inconsistent preferences by introducing a structure-free notion of sophisticated stopping that integrates forward and backward viewpoints. The authors develop a backward-iteration mechanism built around a time-mapping ${F}$ to identify approachable times and a delimiting set to obtain a minimal delimiting time ${\rho^{*}}$, proving ${\tau_{*}=\rho^{*}}$ as the sophisticated time. They illustrate the method with finite-horizon BIS comparisons, a counterexample showing limits of the naive limit ${\rho^{(\infty)}}$, and a detailed non-exponential discounting analysis in continuous time, including a concrete 1D Brownian example yielding ${\rho^{(\infty)}}={\tau_{a^{*}}}$. The results provide a practical, general framework for equilibrium stopping without time consistency, with applications to non-exponential discounting and broader dynamic decision problems. The paper demonstrates both the potential and limitations of backward iteration and clarifies how sophisticated stopping differs from classical BIS in a time-inconsistent setting.
Abstract
In this paper, we propose a new framework for solving a general dynamic optimal stopping problem without time consistency. A sophisticated solution is proposed and is well-defined for any time setting with general flows of objectives. A backward iteration is proposed to find the solution. The iteration works with an additional condition, which holds in interesting cases including the time inconsistency arising from non-exponential discounting. Even if the iteration does not work, the equilibrium solution can still be studied by a forward definition.