Strategy-proof allocation with outside option

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