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Fuzzy Analysis of Sentiment Terms for Topic Detection Process in Social Networks Read chapter Read first chapter

2018 | OriginalPaper | Chapter

Monotonicity of a Profile of Rankings with Ties

Authors : Raúl Pérez-Fernández, Irene Díaz, Susana Montes, Bernard De Baets

Published in: Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations

Publisher: Springer International Publishing

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Abstract

A common problem in social choice theory concerns the aggregation of the rankings expressed by several voters. Two different settings are often discussed depending on whether the aggregate is assumed to be a latent true ranking that voters try to identify or a compromise ranking that (partially) satisfies most of the voters. In a previous work, we introduced the notion of monotonicity of a profile of rankings and used it for statistically testing the existence of this latent true ranking. In this paper, we consider different extensions of this property to the case in which voters provide rankings with ties.

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previous chapter On Dissimilarity Measures at the Fuzzy Partition Level
next chapter Consistency Properties for Fuzzy Choice Functions: An Analysis with the Łukasiewicz t-norm
Footnotes
1
A reversal is a switch of consecutive elements in a ranking. The minimum number of reversals needed for changing a given ranking into another one is measured by the Kendall distance function [8].
 
2
For any ranking \(\succ \) on \(\mathscr {C}\), the restriction of the relation https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-91476-3_26/468293_1_En_26_IEq57_HTML.gif on \(\mathcal {L}^*(\mathscr {C})\) to \(\mathcal {L}(\mathscr {C})\) coincides with the relation https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-91476-3_26/468293_1_En_26_IEq60_HTML.gif on \(\mathcal {L}(\mathscr {C})\). Therefore, the use of the same notation is justified.
 
3
The pre-order closure of a relation R is the smallest reflexive and transitive relation containing R [9].
 
Literature
1.
Arrow, K.J.: Social Choice and Individual Values. Wiley, New York (1963)MATH
2.
Pérez-Fernández, R., Alonso, P., Díaz, I., Montes, S., De Baets, B.: Monotonicity as a tool for differentiating between truth and optimality in the aggregation of rankings. J. Math. Psychol. 77, 1–9 (2017)MathSciNetCrossRef
3.
Pérez-Fernández, R., Alonso, P., Díaz, I., Montes, S., De Baets, B.: Monotonicity-based consensus states for the monometric rationalisation of ranking rules and how they are affected by ties. Int. J. Approximate Reason. 91, 131–151 (2017)MathSciNetCrossRef
4.
Kemeny, J.G.: Mathematics without numbers. Daedalus 88(4), 577–591 (1959)
5.
Schulze, M.: A new monotonic, clone-independent, reversal symmetric, and condorcet-consistent single-winner election method. Soc. Choice Welfare 36, 267–303 (2011)MathSciNetCrossRef
6.
Borda, J.C.: Mémoire sur les Élections au Scrutin. Histoire de l’Académie Royale des Sciences, Paris (1781)
7.
Black, D.: Partial justification of the Borda count. Public Choice 28, 1–15 (1976)CrossRef
8.
Kendall, M.G.: A new measure of rank correlation. Biometrika 30, 81–93 (1938)CrossRef
9.
Foldes, S.: Fundamental Structures of Algebra and Discrete Mathematics. Wiley, New York (1994)CrossRef
Metadata
Title
Monotonicity of a Profile of Rankings with Ties
Authors
Raúl Pérez-Fernández
Irene Díaz
Susana Montes
Bernard De Baets
Copyright Year
2018
Book
Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations

Print ISBN: 978-3-319-91475-6

Electronic ISBN: 978-3-319-91476-3

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-91476-3_26

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