  Motivated by a problem of scheduling unit-length jobs with weak preferences
over time-slots, the random assignment problem (also called the house
allocation problem) is considered on a uniform preference domain. For the
subdomain in which preferences are strict except possibly for the class of
unacceptable objects, Bogomolnaia and Moulin characterized the probabilistic
serial mechanism as the only mechanism satisfying equal treatment of equals,
strategyproofness, and ordinal efficiency. The main result in this paper is
that the natural extension of the probabilistic serial mechanism to the domain
of weak, but uniform, preferences fails strategyproofness, but so does every
other mechanism that is ordinally efficient and treats equals equally. If
envy-free assignments are required, then any (probabilistic or deterministic)
mechanism that guarantees an ex post efficient outcome must fail even a weak
form of strategyproofness.
