  This paper considers the problem of randomly assigning a set of objects to a
set of agents based on the ordinal preferences of agents. We generalize the
well-known immediate acceptance algorithm to the afore-mentioned random
environments and define the probabilistic rank rule (PR rule). We introduce two
new axioms: sd-rank-fairness, and equal-rank envy-freeness. Sd-rank-fairness
implies sd-efficiency. Equal-rank envy-freeness implies equal treatment of
equals. Sd-rank-fairness and equal-rank envy-freeness are enough to
characterize the PR rule.
