  We examine three methods for ranking by pairwise comparison: Principal
Eigenvector, HodgeRank and Tropical Eigenvector. It is shown that the choice of
method can produce arbitrarily different rank order.To be precise, for any two
of the three methods, and for any pair of rankings of at least four items,
there exists a comparison matrix for the items such that the rankings found by
the two methods are the prescribed ones. We discuss the implications of this
result in practice, study the geometry of the methods, and state some open
problems.
