Ranked Realities: Is the Squared Kemeny Rule the Key to Fairer Averages?

The core problem the Squared Kemeny rule aims to solve is the unfairness inherent in traditional rank aggregation methods, like Kemeny's rule, which often prioritize majority opinions at the expense of minority preferences. It addresses the need for a more balanced and representative aggregate ranking when combining multiple ranking criteria, ensuring that each input ranking exerts a proportional influence on the final outcome. This is particularly relevant in scenarios where different criteria should be weighted fairly, such as in hotel searches or university ratings, where a single dominant factor shouldn't overshadow others.

The primary difference lies in how they handle input rankings. Kemeny's rule seeks to minimize the total distance to all input rankings, which often favors the majority. This can lead to a situation where a single dominant criterion or viewpoint disproportionately influences the outcome. The Squared Kemeny rule, on the other hand, minimizes the squared swap distances. This approach ensures that each input ranking, regardless of its weight, exerts a proportional influence on the final outcome. This means that a hotel with excellent reviews can compensate for a slightly higher price, creating a more balanced and representative ranking.

The Squared Kemeny rule offers several key advantages. Firstly, it provides proportionality, meaning input rankings influence the output ranking based on their assigned weights. Secondly, it is responsive to changes in its input, ensuring that all criteria are taken into account. Finally, it has an axiomatic characterization based on neutrality, reinforcement, continuity, and a proportionality axiom, solidifying its role as a well-founded rank aggregation method. This contrasts with methods like Kemeny's rule, which can be less sensitive to nuances in the input data.

In real-world scenarios, the Squared Kemeny rule can lead to fairer and more representative rankings. For example, in hotel searches, it allows hotels to compensate for a lower price ranking by having high review scores and a good location. This contrasts with Kemeny's rule, which might simply output the cheapest hotel, regardless of other factors. Similarly, in university ratings, the Squared Kemeny rule can ensure that various factors, such as research output, student satisfaction, and faculty quality, are considered proportionally, providing a more holistic and balanced assessment of a university's standing. The core implication is a shift away from winner-takes-all approaches to a system that values diverse perspectives.

One major challenge with the Squared Kemeny rule is its computational complexity; it's been shown to be NP-complete. This means finding the exact solution can be computationally expensive for large datasets. However, approximation algorithms exist to mitigate this issue. Despite this, the Squared Kemeny rule holds significant potential, especially where majoritarian rules are undesirable. It opens the door to exploring the meaning of proportional rules and their applications in modern digital markets and future work exploring the topic of proportional rank aggregation, where fairness and the proportional influence of different ranking criteria are paramount.