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Social Choice and Welfare

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01-04-2015

Multidimensional welfare rankings under weight imprecision: a social choice perspective

Author: Stergios Athanassoglou

Published in: Social Choice and Welfare | Issue 4/2015

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Abstract

Ranking alternatives based on multidimensional welfare indices depends, sometimes critically, on how the different dimensions of welfare are weighted. In this paper, a theoretical framework is presented that yields a set of consensus rankings in the presence of such weight imprecision. The main idea is to consider a vector of weights as an imaginary voter submitting preferences over alternatives. With this voting construct in mind, the well-known Kemeny rule from social choice theory is introduced as a means of aggregating the preferences of many plausible choices of weights, suitably weighted by the importance attached to them. The axiomatic characterization of Kemeny’s rule due to Young and Levenglick (1978) and Young (1988) extends to the present context. An analytic solution is derived for an interesting special case of the model corresponding to generalized weighted means and the \(\epsilon \)-contamination framework of Bayesian statistics. The model is applied to the ARWU index of Shanghai University. Graph-theoretic insights are shown to facilitate computation significantly.

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While conceptually straightforward, the application of Tarjan’s famous circuit-detection algorithm to facilitate the computation of Kemeny optimal rankings has not, to the best of my knowledge, been previously pursued in the literature.

For simplicity, I choose the normalized values to lie in [0,1], though any other interval \([x_{\min }, x_{\max }]\) would also work.

For example, in the case of the HDI, \(f\) could be set in the following manner: ask each country \(c\) to provide its importance function \(f_c\) on \(\Delta ^2\) and then set \(f=\sum _{c=1}^C f_c\) (where \(C\) is the total number of countries).

In the case of discrete \(f\), the integrals in Eqs. (3)–(4) would become summations over the relevant subset of \(\Delta ^{I-1}\) and all of the results naturally extend.

To be clear, in this paper I am using the term partial ranking to refer to a weak ordering: a reflexive, transitive and complete binary relation over the set of alternatives.

In the literature it is also sometimes referred to as the “maximum likelihood” or the “Condorcet” rule. As Young noted in Young (1988), the Marquis de Condorcet was the first to propose this way of deciding an election, even if he did not work out the formal details.

This definition of Kendall-\(\tau \) distance in the presence of partial rankings is consistent with Ailon (2007) and van Zuylen and Williamson (2009). It is appropriate in cases such as the paper’s, where partial input rankings are allowed but the output is constrained to be a full ranking.

Hence, note that while input rankings are allowed to be partial, \(K\) still has to be a set of full rankings (Ailon 2007; van Zuylen and Williamson 2009).

A skew symmetric matrix \(\varvec{Y}\) is a square matrix satisfying \(\varvec{Y}=-\varvec{Y}'\).

In what follows I will suppress dependence of the results on \(\varvec{\bar{w}}\) to avoid cumbersome notation.

While straightforward, I am unaware of other papers that make this connection to facilitate the computation or approximation of Kemeny-optimal rankings. In the numerical example, I use a Matlab implementation of Tarjan’s algorithm due to David F. Gleich, which can be found at http://www.mathworks.it/matlabcentral/fileexchange/24134-gaimc-graph-algorithms-in-matlab-code/content/gaimc/scomponents.m.

Here \(\varvec{e_k}\) denotes the corresponding standard basis vector in \(\mathfrak {R}^{I-1}\).

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Title: Multidimensional welfare rankings under weight imprecision: a social choice perspective
Author: Stergios Athanassoglou
Publication date: 01-04-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 4/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0858-z

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