Online Appendix for Partial Strategyproofness: Relaxing Strategy- proofness for the Random Assignment Problem∗

Proof. Since deterministic mechanisms are just special cases of random mechanisms, Theorem 1 applies: A deterministic mechanism f is strategyproof if and only if it is swap monotonic, upper invariant, and lower invariant. Thus, strategyproofness implies swap monotonicity (i.e., sufficiency in Proposition 5). For necessity, observe that swap monotonicity implies upper and lower invariance for deterministic mechanisms: If a swap (say from Pi : a ¡ b to P 1 i : b ¡ a) affects an agent’s assignment, then the assignment must change strictly for the two objects a and b that are swapped. But under a deterministic mechanism, this change can only be from 0 to 1 or from 1 to 0. The only possible changes are therefore the ones where an agent receives a with certainty if it reports Pi : a ¡ b and receives b with certainty if she reports P 1 i : b ¡ a. ∗Department of Informatics, University of Zurich, Switzerland. Email: mennle, [email protected]. For the main paper, see www.ifi.uzh.ch/ce/publications/PSP.pdf, and for updates of this Online Appendix see www.ifi.uzh.ch/ce/publications/PSP-OAPP.pdf.