Strategy-proof and fair assignment is wasteful

I prove there exists no assignment mechanism that is strategy-proof, non-wasteful and satisfies equal treatment of equals. When outside options may exist, this strengthens the impossibility result of Bogomolnaia and Moulin (2001) by weakening ordinal efficiency to non-wastefulness. My result solves an open question posed by Erdil (2014) and complements his results on the efficient frontier of random assignment mechanisms.