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

A páros összehasonlításokon alapuló rangsorolás módszertani és alkalmazási kérdései = Methodological and applicational issues of paired comparison based ranking

A páros összehasonlításokkal történő rangsorolás egyaránt felmerül a döntéselmélet, a preferenciák modellezése, a társadalmi választások elmélete, a tudománymetria, a statisztika, a pszichológia, vagy a sport területén. Ilyen esetekben gyakran nincs lehetőség az alternatívák egyetlen, objektív skálán történő értékelésére, csak azok egymással való összevetésére. Ez három, részben összefüggő kérdést vet fel. Az első a vizsgált gyakorlati probléma matematikai reprezentációja, a második az így keletkező feladat megoldása, a harmadik a kapott eredmény értelmezése. Értekezésünk az első kettőre fókuszál, bár a 7. fejezetben szereplő alkalmazásban az utóbbira is kitérünk. ____ Paired comparison based ranking problems are given by a tournament matrix representing the performance of some objects against each other. They arise in many different fields like social choice theory (Chebotarev and Shamis, 1998), sports (Landau, 1895, 1914; Zermelo, 1929) or psychology (Thurstone, 1927). The usual goal is to determine a winner (possibly a set of winners) or a complete ranking for the objects. There were some attempts to link the two areas (i.e. Bouyssou (2004)), however, they achieved a limited success. We will deal only with the latter issue, allowing for different preference intensities (including ties), incomplete and multiple comparisons among the objects. The ranking includes three areas: representation of the practical problem as a mathematical model, its solution, and interpretation of the results. The third issue strongly depends on the actual application, therefore it is not addressed in the thesis, however, it will appear in Chapter 7

Additive and multiplicative properties of scoring methods for preference aggregation

The paper reviews some additive and multiplicative properties of ranking procedures used for generalized tournaments with missing values and multiple comparisons. The methods analysed are the score, generalised row sum and least squares as well as fair bets and its variants. It is argued that generalised row sum should be applied not with a fixed parameter, but a variable one proportional to the number of known comparisons. It is shown that a natural additive property has strong links to independence of irrelevant matches, an axiom judged unfavourable when players have different opponents

Paired Comparisons Analysis: An Axiomatic Approach to Rankings in Tournaments

In this paper we present an axiomatic analysis of several ranking methods for tournaments. We find that two of them exhibit a very good behaviour with respect to the set of properties under consideration. One of them is the maximum likelihood ranking, the most common method in statistics and psychology. The other one is a new ranking method introduced in this paper: recursive Buchholz. One of the most widely studied methods in social choice, the fair bets ranking, also performs quite well, but fails to satisfy some arguably important properties.Tournament;ranking;paired comparisons;fair bets;maximum likelihood

Rangsorolás páros összehasonlításokkal. Kiegészítések a felvételizői preferencia-sorrendek módszertanához (Paired comparisons ranking. A supplement to the methodology of application-based preference ordering)

A Közgazdasági Szemle márciusi számában Telcs és szerzőtársai [2013] a felvételizők preferenciái alapján új megközelítést javasolt a felsőoktatási intézmények rangsorolására. Az alábbi írás új szempontokat biztosít ezen alapötlet gyakorlati megvalósításához. Megmutatja, hogy az alkalmazott modell ekvivalens az alternatívák egy aggregált páros összehasonlítási mátrix révén végzett rangsorolásával, ami rávilágít a szerzők kiinduló hipotéziseinek vitatható pontjaira. A szerző röviden áttekinti a hasonló feladatok megoldására javasolt módszereket, különös tekintettel azok axiomatikus megalapozására, majd megvizsgálja a Telcs és szerzőtársai [2013] által alkalmazott eljárásokat. Végül említést tesz egy hasonló megközelítéssel élő cikkről, és megfogalmaz néhány, a vizsgálat továbbfejlesztésére vonatkozó javaslatot. _____ In the March issue of Közgazdasági Szemle, Telcs et al. suggested a new approach to university ranking through preference ordering of applicants. The paper proposes new aspects to the implementation of this idea. It is shown that the model of these is equivalent to the ranking of alternatives based on paired comparisons, which reveals the debatable points in their hypotheses. The author reviews briefly the methods proposed in the literature, focusing on their axiomatic properties, and thoroughly examines the procedures of Telcs et al. [2013]. The paper presents an article which applied a similar approach and suggests some improvements to it

How to choose the most appropriate centrality measure?

We propose a new method to select the most appropriate network centrality measure based on the user's opinion on how such a measure should work on a set of simple graphs. The method consists in: (1) forming a set $\cal F$ of candidate measures; (2) generating a sequence of sufficiently simple graphs that distinguish all measures in $\cal F$ on some pairs of nodes; (3) compiling a survey with questions on comparing the centrality of test nodes; (4) completing this survey, which provides a centrality measure consistent with all user responses. The developed algorithms make it possible to implement this approach for any finite set $\cal F$ of measures. This paper presents its realization for a set of 40 centrality measures. The proposed method called culling can be used for rapid analysis or combined with a normative approach by compiling a survey on the subset of measures that satisfy certain normative conditions (axioms). In the present study, the latter was done for the subsets determined by the Self-consistency or Bridge axioms.Comment: 26 pages, 1 table, 1 algorithm, 8 figure

On the influence of rankings

Ranking systems are becoming increasingly important in many areas, in the Web environment and academic life for instance. In a world with a tremendous amount of choices, rankings play the crucial role of influencing which objects are 'tasted' or selected. This selection generates a feedback when the ranking is based on citations, as is the case for the widely used invariant method. The selection affects new stated opinions (citations), which will, in turn, affect next ranking. The purpose of this paper is to investigate this feedback in the context of journals by studying some simple but reasonable dynamics. Our main interest is on the long run behavior of the process and how it depends on the preferences, in particular on their diversity. We show that multiple long run behavior may arise due to strong self enforcing mechanisms at work with the invariant method. These effects are not present in a simple search model in which individuals are influenced by the cites of the papers they first read.ranking, scoring, invariant method, search