# A characterization of the Logarithmic Least Squares Method

## TL;DR

This paper provides an axiomatic characterization of the Logarithmic Least Squares Method, establishing it as the unique approach satisfying correctness in consistent cases and invariance under triad transformations.

## Contribution

It offers the first axiomatic foundation for the Logarithmic Least Squares Method, clarifying its unique properties in deriving preference vectors from pairwise comparison matrices.

## Key findings

- Proves the method is the only one satisfying the specified axioms.
- Shows the method reproduces the inducing vector for consistent matrices.
- Demonstrates invariance to transformations on triads.

## Abstract

We provide an axiomatic characterization of the Logarithmic Least Squares Method (sometimes called row geometric mean), used for deriving a preference vector from a pairwise comparison matrix. This procedure is shown to be the only one satisfying two properties, correctness in the consistent case, which requires the reproduction of the inducing vector for any consistent matrix, and invariance to a specific transformation on a triad, that is, the weight vector is not influenced by an arbitrary multiplication of matrix elements along a 3-cycle by a positive scalar.