Optimal stopping with private information

Many economic situations are modeled as stopping problems. Examples include job search, timing of market entry decisions, irreversible investment or the pricing of American options. This paper analyzes optimal stopping as a mechanism design problem with transfers. We show that under a dynamic single crossing condition a stopping rule can be implemented by a transfer that only depends on the realized stopping decision if and only if it is a cut-off rule. We characterize the transfer implementing a given stopping rule using a novel technique based on constrained stochastic processes.As an application we prove that in any Markovian optimal stopping problem there exists a welfare maximizing mechanism that does not require any communication. We discuss revenue maximization for separable processes.