An application of incomplete pairwise comparison matrices for ranking top tennis players

Pairwise comparison is an important tool in multi-attribute decision making. Pairwise comparison matrices (PCM) have been applied for ranking criteria and for scoring alternatives according to a given criterion. Our paper presents a special application of incomplete PCMs: ranking of professional tennis players based on their results against each other. The selected 25 players have been on the top of the ATP rankings for a shorter or longer period in the last 40 years. Some of them have never met on the court. One of the aims of the paper is to provide ranking of the selected players, however, the analysis of incomplete pairwise comparison matrices is also in the focus. The eigenvector method and the logarithmic least squares method were used to calculate weights from incomplete PCMs. In our results the top three players of four decades were Nadal, Federer and Sampras. Some questions have been raised on the properties of incomplete PCMs and remains open for further investigation.Comment: 14 pages, 2 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

Incomplete pairwise comparison matrices: Ranking top women tennis players

The method of pairwise comparisons is frequently applied for ranking purposes. This article aims to rank top women tennis players based on their win/lose ratios. Incomplete pairwise comparison matrices (PCMs) were constructed from data obtained from the WTA (Women’s Tennis Association) homepage. The database contains head-to-head results from the period between 1973 and 2022 for 28 players who had the position No. 1 in the official rankings of WTA. The weight vector was calculated from the incomplete PCM with the logarithmic least squares method and the eigenvector method. The results are not surprising: Serena Williams, Steffi Graf, and Martina Navratilova stand in the first three positions, and Martina Hingis, Kim Clijsters, and Justine Henin follow them. We also tested the frequently used probability-based Bradley-Terry method and found high rank-correlation values. Using graph representations, the results gave us a deeper insight into the properties of incomplete PCMs. Special attention was given to the nontransitive triads. A data modification was necessary to remove ties in order to apply the commonly used tests. The results indicate that ordinally nontransitive triads are not significant in the data we analysed

On the coincidence of optimal completions for small pairwise comparison matrices with missing entries

Incomplete pairwise comparison matrices contain some missing judgements. A natural approach to estimate these values is provided by minimising a reasonable measure of inconsistency after unknown entries are replaced by variables. Two widely used inconsistency indices for this purpose are Saaty's inconsistency index and the geometric inconsistency index, which are closely related to the eigenvector and the logarithmic least squares priority deriving methods, respectively. The two measures are proven to imply the same optimal filling for incomplete pairwise comparison matrices up to order four but not necessarily for order at least five.Comment: 10 pages, 1 figur

Characterisation of the consistent completion of AHP comparison matrices using graph theory

[EN] Decision-making is frequently affected by uncertainty and/or incomplete information, which turn decision-making into a complex task. It is often the case that some of the actors involved in decision-making are not sufficiently familiar with all of the issues to make the appropriate decisions. In this paper, we are concerned about missing information. Specifically, we deal with the problem of consistently completing an analytic hierarchy process comparison matrix and make use of graph theory to characterize such a completion. The characterization includes the degree of freedom of the set of solutions and a linear manifold and, in particular, characterizes the uniqueness of the solution, a result already known in the literature, for which we provide a completely independent proof. Additionally, in the case of nonuniqueness, we reduce the problem to the solution of nonsingular linear systems. 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A lexicographically optimal completion for pairwise comparison matrices with missing entries

Estimating missing judgements is a key component in many multi-criteria decision making techniques, especially in the Analytic Hierarchy Process. Inspired by the Koczkodaj inconsistency index and a widely used solution concept of cooperative game theory called the nucleolus, the current study proposes a new algorithm for this purpose. In particular, the missing values are substituted by variables, and the inconsistency of the most inconsistent triad is reduced first, followed by the inconsistency of the second most inconsistent triad, and so on. The necessary and sufficient condition for the uniqueness of the suggested lexicographically optimal completion is proved to be a simple graph-theoretic notion: the undirected graph associated with the pairwise comparisons, where the edges represent the known elements, should be connected. Crucially, our method does not depend on an arbitrarily chosen measure of inconsistency as there exists essentially one reasonable triad inconsistency index.Comment: 17 pages, 2 figure

Egy életminőség-rangsor a hazautalások alapján

A hazautalások a vendégmunkások és az őket küldő országok közötti kapcsolat egyik fontos mérőszámát jelentik. Ez egyben számszerűsített mutatója lehet annak is, hogy a saját hazájukhoz képest mely országokat részesítik előnyben az emberek, így egy életminőség jellegű rangsort állíthatunk fel azok között. Az elemzéshez a Világbank adatait használtuk 2010-től 2015-ig, az adatbázis a nemzetközi munkabér, illetve a személyek közötti bilaterális utalásokat tartalmazza. A javasolt mérőszám független az országok méretétől, és figyelembe veszi a teljes hálózat felépítését, azt feltételezve, hogy minden egységnyi átutalás felfogható egy preferenciaként a két érintett ország között

The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices

Complete and incomplete additive/multiplicative pairwise comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well known methods is proved in this paper. The arithmetic (geometric) mean of weight vectors, calculated from all spanning trees, is proved to be optimal to the (logarithmic) least squares problem, not only for complete, as it was recently shown in Lundy, M., Siraj, S., Greco, S. (2017): The mathematical equivalence of the "spanning tree" and row geometric mean preference vectors and its implications for preference analysis, European Journal of Operational Research 257(1) 197-208, but for incomplete matrices as well. Unlike the complete case, where an explicit formula, namely the row arithmetic/geometric mean of matrix elements, exists for the (logarithmic) least squares problem, the incomplete case requires a completely different and new proof. Finally, Kirchhoff's laws for the calculation of potentials in electric circuits is connected to our results.Comment: 21 pages, 6 figure

Models for Paired Comparison Data: A Review with Emphasis on Dependent Data

Thurstonian and Bradley-Terry models are the most commonly applied models in the analysis of paired comparison data. Since their introduction, numerous developments have been proposed in different areas. This paper provides an updated overview of these extensions, including how to account for object- and subject-specific covariates and how to deal with ordinal paired comparison data. Special emphasis is given to models for dependent comparisons. Although these models are more realistic, their use is complicated by numerical difficulties. We therefore concentrate on implementation issues. In particular, a pairwise likelihood approach is explored for models for dependent paired comparison data, and a simulation study is carried out to compare the performance of maximum pairwise likelihood with other limited information estimation methods. The methodology is illustrated throughout using a real data set about university paired comparisons performed by students.Comment: Published in at http://dx.doi.org/10.1214/12-STS396 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org