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General Stopping Behaviors of Naive 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 naive 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 naive
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 "never-stopping" or
"never-starting"
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