  We study functions that produce a ranking of $n$ individuals from $n$ such
rankings and are impartial in the sense that the position of an individual in
the output ranking does not depend on the input ranking submitted by that
individual. When $n \geq 4$, two properties concerning the quality of the
output in relation to the input can be achieved in addition to impartiality:
individual full rank, which requires that each individual can appear in any
position of the output ranking; and monotonicity, which requires that an
individual cannot move down in the output ranking if it moves up in an input
ranking. When $n \geq 5$, monotonicity can be dropped to strengthen individual
full rank to weak unanimity, requiring that a ranking submitted by every
individual must be chosen as the output ranking. Mechanisms achieving these
results can be implemented in polynomial time. Both results are best possible
in terms of their dependence on $n$. The second result cannot be strengthened
further to a notion of unanimity that requires agreement on pairwise
comparisons to be preserved.
