  For optimal stopping problems with time-inconsistent preference, we measure
the inherent level of time-inconsistency by taking the time needed 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 which are
inconsistent with what she has planned at the initial time, she then chooses
her immediate action based on the observations on her later actual behavior.
The procedure is repeated until her actual sequence of actions are 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.
