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

László Csató, Dóra Gréta Petróczy

TL;DR

This study proposes using bibliometric indices to quantify long-term uncertainty of knockout-stage outcomes in sports, applying the Euclidean index $E(\mathbf{x})$ and the rectangle index $R(\mathbf{x})$ within a rolling five-year window to UEFA Champions League data. By computing performance shares and the Herfindahl-Hirschman index, the authors compare club and national association performances and analyze three forms of competitive balance. The main contributions are an axiomatic comparison of $E$ and $R$, demonstration of robustness to index and weight choices, and novel insights into association-level seeding versus club-based seeding in the Champions League. The work has practical implications for UEFA policies and provides a flexible framework for assessing competition in knockout formats across sports.

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.

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

TL;DR

This study proposes using bibliometric indices to quantify long-term uncertainty of knockout-stage outcomes in sports, applying the Euclidean index and the rectangle index within a rolling five-year window to UEFA Champions League data. By computing performance shares and the Herfindahl-Hirschman index, the authors compare club and national association performances and analyze three forms of competitive balance. The main contributions are an axiomatic comparison of and , demonstration of robustness to index and weight choices, and novel insights into association-level seeding versus club-based seeding in the Champions League. The work has practical implications for UEFA policies and provides a flexible framework for assessing competition in knockout formats across sports.

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.

Paper Structure

This paper contains 8 sections, 2 theorems, 8 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology: bibliometric indices
  3. Data and implementation
  4. Results
  5. Related literature
  6. The performances of clubs and national associations
  7. Competitive balance
  8. Concluding remarks

Key Result

Proposition 1

The share of a competitor's achievement in the total achievement under the Euclidean and the rectangle indices does not change if the scores are multiplied by the same positive number $c > 0$.

Figures (9)

  • Figure 1: The rectangle index does not satisfy independence
  • Figure 2: The rectangle index does not satisfy depth relevance
  • Figure 3: The rectangle index does not satisfy directional consistency
  • Figure 4: Club performances in five-year periods between 2003 and 2023 The year on the x-axis indicates the finishing year of the last season in the period, e.g. 2018 corresponds to the seasons from 2013/14 to 2017/18.
  • Figure 5: Country performances in five-year periods between 2003 and 2023 The year on the x-axis indicates the finishing year of the last season in the period, e.g. 2018 corresponds to the seasons from 2013/14 to 2017/18.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 3.1
  • Example 3.2
  • Example 3.3