Distances Between Partial Preference Orderings

TL;DR

This paper introduces two methods for measuring the distance between partial preference orderings, one combinatorial and one based on belief functions, with the latter being more efficient for high-dimensional problems.

Contribution

It presents a novel belief function-based approach to compute distances between partial preference orderings, overcoming combinatorial complexity issues.

Findings

The combinatorial method accurately computes distances but is inefficient for high dimensions.

The belief function approach effectively models missing information in partial preferences.

Simple examples demonstrate the practical application of both methods.

Abstract

This paper proposes to establish the distance between partial preference orderings based on two very different approaches. The first approach corresponds to the brute force method based on combinatorics. It generates all possible complete preference orderings compatible with the partial preference orderings and calculates the Frobenius distance between all fully compatible preference orderings. Unfortunately, this first method is not very efficient in solving high-dimensional problems because of its big combinatorial complexity. That is why we propose to circumvent this problem by using a second approach based on belief functions, which can adequately model the missing information of partial preference orderings. This second approach to the calculation of distance does not suffer from combinatorial complexity limitation. We show through simple examples how these two theoretical methods work.