Filling Positions Without Transfers: Screening on Outside Options

A designer offers vertically-differentiated positions to agents in the absence of transfers. Agents have private outside options and may reject their offers ex-post. The designer has preferences over the quantity of agents who accept each position. We show that under a general condition on the distribution of outside options, an optimal mechanism for the designer offers all agents an identical lottery, and we characterize this mechanism. When our condition does not hold, the optimal mechanism may require screening agents by offering a menu of distinct lotteries. Our results follow from a decomposition of agents' participation probabilities in any feasible mechanism.