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We examine three methods for ranking by pairwise comparison: PrincipalEigenvector, HodgeRank and Tropical Eigenvector. It is shown that the choice ofmethod can produce arbitrarily different rank order.To be precise, for any twoof 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 bythe two methods are the prescribed ones. We discuss the implications of thisresult in practice, study the geometry of the methods, and state some openproblems.

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