General stopping behaviors of naïve and non-committed sophisticated agents, with application to probability distortion

We consider the problem of stopping a diffusion process with a payoff functional that renders the problem time-inconsistent. We study stopping decisions of naıve agents who reoptimize continuously in time, as well as equilibrium strategies of sophisticated agents who anticipate but lack control over their future selves' behaviors. When the state process is one dimensional and the payoff functional satisfies some regularity conditions, we prove that any equilibrium can be obtained as a fixed point of an operator. This operator represents strategic reasoning that takes the future selves' behaviors into account. We then apply the general results to the case when the agents distort probability and the diffusion process is a geometric Brownian motion. The problem is inherently time-inconsistent as the level of distortion of a same event changes over time. We show how the strategic reasoning may turn a na¨ıve agent into a sophisticated one. Moreover, we derive stopping strategies of the two types of agent for various parameter specifications of the problem, illustrating rich behaviors beyond the extreme ones such as "neverstopping" or "never-starting".