# Characterization of the row geometric mean ranking with a group consensus axiom

## TL;DR

This paper axiomatizes the row geometric mean ranking method, proving it is uniquely characterized by three properties including group consensus preservation, anonymity, and responsiveness, thus providing a solid theoretical foundation.

## Contribution

It introduces an axiomatic characterization of the row geometric mean ranking, highlighting its unique properties and group consensus invariance.

## Key findings

- Proves the uniqueness of the row geometric mean ranking under three axioms.
- Establishes group consensus invariance as a key property.
- Provides theoretical validation for the ranking method.

## Abstract

An axiomatic approach is applied to the problem of extracting a ranking of the alternatives from a pairwise comparison ratio matrix. The ordering induced by row geometric mean method is proved to be uniquely determined by three independent axioms, anonymity (independence of the labelling of alternatives), responsiveness (a kind of monotonicity property) and aggregation invariance, which requires the preservation of group consensus, that is, the pairwise ranking between two alternatives should remain unchanged if unanimous individual preferences are combined by geometric mean.