Stability of Equilibria in Time-inconsistent Stopping Problems

TL;DR

This paper studies how the optimal values in time-inconsistent stopping problems change with reward functions and transition kernels, revealing conditions for semi-continuity and continuity, and highlighting cases where these properties fail.

Contribution

It establishes semi-continuity of optimal values under local uniform convergence and total variation, and identifies conditions for full continuity in time-inconsistent stopping problems.

Findings

Optimal value is semi-continuous under local uniform convergence of $f$ and $Q$.

Continuity of optimal value may fail without uniform convergence.

Continuity holds under relaxed limits over $ ext{epsilon}$-equilibria with uniform convergence.

Abstract

We investigate the stability of equilibrium-induced optimal values with respect to (w.r.t.) reward functions $f$ and transition kernels $Q$ for time-inconsistent stopping problems under nonexponential discounting in discrete time. First, with locally uniform convergence of $f$ and $Q$ equipped with total variation distance, we show that the optimal value is semi-continuous w.r.t. $(f,Q)$. We provide examples showing that continuity may fail in general, and the convergence for $Q$ in total variation cannot be replaced by weak convergence. Next we show that with the uniform convergence of $f$ and $Q$, the optimal value is continuous w.r.t. $(f,Q)$ when we consider a relaxed limit over $\varepsilon$-equilibria. We also provide an example showing that for such continuity the uniform convergence of $(f,Q)$ cannot be replaced by locally uniform convergence.