Strategy-proof allocation with outside option

Strategy-proof mechanisms are widely used in market design. In an abstract allocation model in which outside options are available to agents, we prove two results that demonstrate the interplay between strategy-proofness and outside options. The two results treat deterministic allocation mechanisms and random allocation mechanisms in a unified way. The first result proves that, for individually rational and strategy-proof mechanisms, pinning down agents' probabilities of consuming outside options is equivalent to pinning down a mechanism. The second result presents a sufficient condition for two strategy-proof mechanisms to be welfare equivalent. We show the usefulness of the two results in several applications.