A note on the assignment problem with uniform preferences

Motivated by a problem of scheduling unit-length jobs with weak preferences over time-slots, the random assignment problem is considered on a uniform preference domain. It is shown that the natural extension of the probabilistic serial mechanism to the domain of weak, but uniform, preferences fails strategy-proofness, but so does every other mechanism that is ordinally efficient and treats equals equally. If envy-free assignments are required, any ex-post efficient (probabilistic or deterministic) mechanism must fail even a weak form of strategy-proofness.