Never stop or never start? Optimal stopping under a mixture of CPT and EUT preferences

We consider the problem of finding the best time to stop a diffusion process for an agent with a preference model that is a mixture of expected utility theory (EUT) and cumulative prospect theory (CPT). In view of time-inconsistency, we consider two types of agents: a naive agent, who is not aware of the time-inconsistency and thus re-plans at every instant, and a sophisticated agent, who is aware of the time-inconsistency and takes a so-called intra-personal equilibrium strategy by correctly anticipating her actions in the future. We show that under a wide range of the CPT preference parameter values, the naive agent will never stop. For a sophisticated agent, we use a different notion of intra-personal equilibrium from the one employed by Ebert and Strack (2018). We show that any two-threshold strategy, which is to stop when the diffusion process reaches either an upper threshold or a lower threshold, cannot be an intra-personal equilibrium if the agent overweights worst, unlikely outcomes disproportionally. We derive a sufficient and necessary condition for the strategy of stopping everywhere to be an intra-personal equilibrium and show that this condition does not hold and thus the sophisticated agent may choose to start the diffusion process for some commonly used probability weighting functions.