We argue for the application of bibliometric indices to quantify 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 16 seasons played between
2003 and 2019: 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. Inequality has not
increased among the elite clubs and between the national associations, however,
it has changed within some countries. 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.Comment: 23 pages, 9 figures, 7 table
