Time-Inconsistent Stopping Under Distorted Probabilities

In this chapter we study stopping strategies in the presence of distorted probabilities, in both discrete and continuous time. Probability distortion is a salient ingredient for a number of important models in behavioral economics, including cumulative prospect theory (Kahneman and Tversky (Econometrica 47:263–291, 1979), Tversky and Kahneman (Journal of Risk and Uncertainty 5:297–323, 1992)) and rank-dependent utility (Quiggin (Journal of Economic Behavior & Organization 3:323–343, 1982), Schmeidler (Econometrica 57:571–587, 1989)). Contrary to the expected utility theory, in the prospect theory model, economic agents do not weight outcomes by their objective probabilities but rather by transformed probabilities. These transformed probabilities (or decision weights) allow the model to capture economic behavior observed in experimental settings showing that people tend to overweight small probabilities and underweight large probabilities. Similarly, rank-dependent expected utility overweighs unlikely extreme outcomes. Importantly, in a dynamic context probability weighting makes the decision maker’s problem inherently time inconsistent. Mathematically, the reward functional with probability distortion involves the so-called Choquet integral (Choquet (Annales de l’Institut Fourier, 5:131–295, 1954)), instead of the conventional expectation.