We explore the consequences of weakening the notion of incentive
compatibility from strategy-proofness to ordinal Bayesian incentive
compatibility (OBIC) in the random assignment model. If the common prior of the
agents is a uniform prior, then a large class of random mechanisms are OBIC
with respect to this prior -- this includes the probabilistic serial mechanism.
We then introduce a robust version of OBIC: a mechanism is locally robust OBIC
if it is OBIC with respect all independent priors in some neighborhood of a
given independent prior. We show that every locally robust OBIC mechanism
satisfying a mild property called elementary monotonicity is strategy-proof.
This leads to a strengthening of the impossibility result in Bogomolnaia and
Moulin (2001): if there are at least four agents, there is no locally robust
OBIC and ordinally efficient mechanism satisfying equal treatment of equals