We consider the problem of allotting shares of a task or good among agents with single peaked preferences over their own shares. Previous characterizations have examined rules, such as the uniform rule, which satisfy various symmetry requirements. We consider the case where agents might begin with natural claims to minimal or maximal allotments, or might be treated with different priorities. We provide characterizations of the rules which are strategy-proof and efficient, but which may treat individuals asymmetrically.Journal of Economic LiteratureClassification Numbers: D71, D78, D63. © 1997 Academic Press.