Deriving weights from pairwise comparison matrices

We study the problem of deriving weights from pairwise comparisons. Properties of acceptable solutions are analysed, including immunity to rank reversals, independence of description of the problem, independence of scale inversion, left-right eigenvector asymmetry, uniqueness, independence of order of operations and inter-level consistency, preservation of the algebraic structure of the problem, extensibility to the additive case, related optimization models and related error measures. We derive the geometric mean from two simple axioms and establish that it is the only acceptable solution to this problem.