Partial order similarity based on mutual information

TL;DR

This paper introduces a novel similarity measure for partial orders based on adjusted mutual information, capturing the degree of match between rankings while considering the importance of disagreement positions.

Contribution

The authors propose a new similarity measure for partial orders using mutual information, with efficient computation and sensitivity to disagreement positions.

Findings

Similarity measure equals one for identical partial orders.

Low similarity indicates independent partial orders.

Computational complexity is polynomial, efficient for typical cases.

Abstract

Comparing the ranking of candidates by different voters is an important topic in social and information science with a high relevance from the point of view of practical applications. In general, ties and pairs of incomparable candidates may occur, thus, the alternative rankings are described by partial orders. Various distance measures between partial orders have already been introduced, where zero distance is corresponding to a perfect match between a pair of partial orders, and larger values signal greater differences. Here we take a different approach and propose a similarity measure based on adjusted mutual information. In general, the similarity value of unity is corresponding to exactly matching partial orders, while a low similarity is associated to a pair of independent partial orders. The time complexity of the computation of this similarity measure is $\mathcal{O}(\left|{\mathcal C}\right|^3)$ in the worst case, and $\mathcal{O}(\left|{\mathcal C}\right|^2\ln \left|{\mathcal C}\right|)$ in the typical case of partial orders corresponding to trees with constant branching number, where $\left|{\mathcal C}\right|$ denotes the number of candidates. An interesting feature of our approach is that the similarity measure is sensitive to the position of the disagreements in the ranking: Differences at the highly ranked candidates induce larger similarity drop compared to disagreements at the bottom candidates.