On the additivity of preference aggregation methods

The paper reviews some axioms of additivity concerning ranking methods used for generalized tournaments with possible missing values and multiple comparisons. It is shown that one of the most natural properties, called consistency, has strong links to independence of irrelevant comparisons, an axiom judged unfavourable when players have different opponents. Therefore some directions of weakening consistency are suggested, and several ranking methods, the score, generalized row sum and least squares as well as fair bets and its two variants (one of them entirely new) are analysed whether they satisfy the properties discussed. It turns out that least squares and generalized row sum with an appropriate parameter choice preserve the relative ranking of two objects if the ranking problems added have the same comparison structure.Comment: 24 pages, 9 figure

How to avoid ordinal violations in incomplete pairwise comparisons

Assume that some ordinal preferences can be represented by a weakly connected directed acyclic graph. The data are collected into an incomplete pairwise comparison matrix, the missing entries are estimated, and the priorities are derived from the optimally filled pairwise comparison matrix. Our paper studies whether these weights are consistent with the partial order given by the underlying graph. According to previous results from the literature, two popular procedures, the incomplete eigenvector and the incomplete logarithmic least squares methods fail to satisfy the required property. Here, it is shown that the recently introduced lexicographically optimal completion combined with any of these weighting methods avoids ordinal violation in the above setting. This finding provides a powerful argument for using the lexicographically optimal completion to determine the missing elements in an incomplete pairwise comparison matrix.Comment: 11 pages, 2 figure

Bibliometric indices as a measure of long-term competitive balance in knockout tournaments

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