On the additivity of preference aggregation methods

TL;DR

This paper examines axioms of additivity in ranking methods for generalized tournaments, analyzing how properties like consistency relate to independence of irrelevant comparisons and evaluating several methods' adherence to these axioms.

Contribution

It introduces weakened consistency axioms and analyzes multiple ranking methods, including a new variant of fair bets, for their compliance with these properties.

Findings

Least squares and generalized row sum preserve relative rankings under certain conditions.

Some ranking methods satisfy weakened additivity axioms.

A new variant of fair bets is proposed and analyzed.

Abstract

The paper reviews some axioms of additivity concerning ranking methods used for generalized tournaments with possible missing values and multiple comparisons. It is shown that one of the most natural properties, called consistency, has strong links to independence of irrelevant comparisons, an axiom judged unfavourable when players have different opponents. Therefore some directions of weakening consistency are suggested, and several ranking methods, the score, generalized row sum and least squares as well as fair bets and its two variants (one of them entirely new) are analysed whether they satisfy the properties discussed. It turns out that least squares and generalized row sum with an appropriate parameter choice preserve the relative ranking of two objects if the ranking problems added have the same comparison structure.