Stability of Equilibria in Time-Inconsistent Stopping Problems
Erhan Bayraktar, Zhenhua Wang, Zhou Zhou · SIAM Journal on Control and Optimization · 2023
Abstract. We investigate the stability of equilibrium-induced optimal values with respect to reward functions [Formula: see text] and transition kernels [Formula: see text] for time-inconsistent stopping problems under nonexponential discounting in discrete time. First, with locally uniform convergence of [Formula: see text] and [Formula: see text] equipped with total variation distance, we show that the optimal value is semicontinuous with respect to [Formula: see text]. We provide examples showing that continuity may fail in general, and the convergence for [Formula: see text] in total variation cannot be replaced by weak convergence. Next we show that, with the uniform convergence of [Formula: see text] and [Formula: see text], the optimal value is continuous with respect to [Formula: see text] when we consider a relaxed limit over [Formula: see text]-equilibria. We also provide an example showing that for such continuity the uniform convergence of [Formula: see text] cannot be replaced by locally uniform convergence.