The Fundamental Theorem of Barzilai does not hold

Abstract

The extensively cited work of Barzilai, J. (1997): Deriving weights from pairwise comparison matrices, published in Journal of the Operational Research Society, 48(12), 1226-1232., derives the geometric mean method from two simple axioms. This note reveals that the central result of the paper called the Fundamental Theorem by the author does not hold as there exists at least one further method satisfying both requirements.

Proposition 1 does not hold as the column mean method does not meet independence of relative measurements due to the normalisation of the weights