We study random assignment economies with expected-utility agents, each of them eventually obtaining a single object. We focus attention on assignment problems that must respect object-invariant (or uniform) weak priorities such as seniority rights in student residence assignment. We propose the sequential pseudomarket mechanism: the set of agents is partitioned into ordered priority groups that are called in turns to participate in a pseudomarket for the remaining objects. SP is characterized by the concept of consistent weak ex-ante efficiency (CWEE), that is, weak ex-ante efficiency complemented by consistency to economy reduction. Moreover, it is shown that CWEE generically implies ex-ante efficiency.