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We argue for the application of bibliometric indices to quantify thelong-term uncertainty of outcome in sports. The Euclidean index is proposed toreward quality over quantity, while the rectangle index can be an appropriatemeasure of core performance. Their differences are highlighted through anaxiomatic analysis and several examples. Our approach also requires a weightingscheme to compare different achievements. The methodology is illustrated bystudying the knockout stage of the UEFA Champions League in the 20 seasonsplayed between 2003 and 2023: club and country performances as well as threetypes of competitive balance are considered. Measuring competition at the levelof national associations is a novelty. All results are remarkably robustconcerning the bibliometric index and the assigned weights. Since theperformances of national associations are more stable than the results ofindividual clubs, it would be better to build the seeding in the UEFA ChampionsLeague group stage upon association coefficients adjusted for league finishingpositions rather than club coefficients.

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