The Fundamental Theorem of Barzilai does not hold

TL;DR

This paper critically examines Barzilai's 1997 theorem, revealing that its main result, the Fundamental Theorem, is invalid because an alternative method also satisfies the same axioms.

Contribution

It identifies a flaw in Barzilai's theorem by demonstrating the existence of an additional method fulfilling the same axioms, challenging the theorem's uniqueness.

Findings

The Fundamental Theorem does not hold as previously claimed.

An alternative method satisfies the same axioms as Barzilai's method.

The result questions the validity of the original theorem's assumptions.

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.