Pairwise ranking: Choice of method can produce arbitrarily different rank order

We examine three methods for ranking by pairwise comparison: PerronRank (Principal Eigenvector), HodgeRank and TropicalRank. We show 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.