Bibliometric indices as a measure of performance and competitive balance in the knockout stage of the UEFA Champions League

TL;DR

This paper applies bibliometric indices to evaluate long-term performance and competitive balance in UEFA Champions League knockout stages, introducing new metrics and a weighting scheme for more stable national association assessments.

Contribution

It introduces the Euclidean and rectangle indices for sports performance measurement and applies them to UEFA data, including national associations, which is a novel approach.

Findings

Bibliometric indices are robust for measuring sports performance.

National association performance is more stable than club results.

Adjusted association coefficients could improve UEFA seeding processes.

Abstract

We argue for the application of bibliometric indices to quantify the long-term uncertainty of outcome in sports. The Euclidean index is proposed to reward quality over quantity, while the rectangle index can be an appropriate measure of core performance. Their differences are highlighted through an axiomatic analysis and several examples. Our approach also requires a weighting scheme to compare different achievements. The methodology is illustrated by studying the knockout stage of the UEFA Champions League in the 20 seasons played between 2003 and 2023: club and country performances as well as three types of competitive balance are considered. Measuring competition at the level of national associations is a novelty. All results are remarkably robust concerning the bibliometric index and the assigned weights. Since the performances of national associations are more stable than the results of individual clubs, it would be better to build the seeding in the UEFA Champions League group stage upon association coefficients adjusted for league finishing positions rather than club coefficients.