Stability of Equilibria in Time-inconsistent Stopping Problems

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 ε-equilibria. We also provide an example showing that for such continuity the uniform convergence of (f, Q) cannot be replaced by locally uniform convergence.