From time-inconsistency to time-consistency for optimal stopping problems

For optimal stopping problems with time-inconsistent preference, we measure the inherent level of time-inconsistency by taking the time required to turn the naive strategies into the sophisticated ones. In particular, when in a repeated experiment, the naive agent can observe her actual sequence of actions that is inconsistent with what she has planned at the initial time, and she would then choose her immediate action based on the observations of her later actual behavior. The procedure is repeated until her actual sequence of actions is consistent with her plan at any time. We show that for the preference value of cumulative prospect theory, in which the time-inconsistency is due to the probability distortion, the higher the degree of probability distortion, the more severe the level of time-inconsistency, and the more time required to turn the naive strategies into the sophisticated ones.