Optimal stopping with private information

Many economic situations are modeled as stopping problems. Examples are timing of market entry decisions, search, and irreversible investment. We analyze a principalagent problem where the principal and the agent have di erent preferences over stopping rules. The agent privately observes a signal that in uences his own and the principal's payo . Based on his observation the agent decides when to stop. In order to in uence the agents stopping decision the principal chooses a transfer that conditions only on the time the agent stopped. We derive a monotonicity condition such that all cut-o stopping rules can be implemented using such a transfer. The condition generalizes the single-crossing condition from static mechanism design to optimal stopping problems. We give conditions under which the transfer is unique and derive a closed form solution based on re ected processes. We prove that there always exists a cut-o rule that is optimal for the principal and, thus, can be implemented using a posted-price mechanism. An application of our result leads to a new purely stochastic representation formula for the value function in optimal stopping problems based on re ected processes. Finally, we apply the model to characterize the set of implementable policies in the context of job search and irreversible investment.