We consider the problem of allotting shares of a task or good among ag
ents with single peaked preferences over their own shares. Previous ch
aracterizations have examined rules, such as the uniform rule, which s
atisfy various symmetry requirements. We consider the case where agent
s might begin with natural claims to minimal or maximal allotments, or
might be treated with different priorities. We provide characterizati
ons of the rules which are strategy-proof and efficient, but which may
treat individuals asymmetrically. (C) 1997 Academic Press.