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Social Choice and Welfare

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09-12-2015 | Original Paper

Multilevel multidimensional consistent aggregators

Author: Mihir Bhattacharya

Published in: Social Choice and Welfare | Issue 4/2016

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Abstract

This paper examines the structure of consistent, multidimensional, multilevel aggregators in two distinct models- one, where the set of evaluations is the unit interval and the other, where it is finite. In the first model, we characterize a class of separable rules called component-wise \(\alpha \)-median rules. In the finite model, separability is no longer guaranteed. In addition to consistency, stronger notions of unanimity and anonymity are required to characterize a class of separable rules called bipartite rules.

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Katz (1998) and Samuels and Snyder (2001) provide empirical evidence of gerrymandering in different electoral systems and countries respectively.

Alternatively, voters submit a ‘vote’ for an alternative which is aggregated into a single alternative.

Macé (2013) provides another model of aggregation over evaluations.

The literature on fuzzy set aggregation deals with various versions of the consistency axiom in the one-dimensional model. Dubois and Prade (1985) and Marichal (2009) provide extensive surveys of this literature.

Perote-Peña (2005), Bervoets and Merlin (2012) and Plott (1973) analyze models that are similar in spirit to ours with related notions of consistency.

Recall that Chambers (2008) uses a different notion of representative consistency.

There is a fairly extensive literature on the aggregation of sets of issues/alternatives—see for instance, Barberà et al. (1991), Plott (1973), Goodin and List (2006) and Kasher and Rubinstein (1997).

See List and Puppe (2009) for a comprehensive survey of this literature.

intBox(.) denotes the interior of Box(.).

\(A^{C}\) is the complement of set A.

Recall that \(\mathbf{0}=(0,0,\ldots ,0) \in {A}\) and \(\mathbf{1} = (1,1,\ldots ,1) \in {A}.\)

Recall that component anonymity only states that the aggregator is invariant only over the issue for which the opinions are permuted.

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Title: Multilevel multidimensional consistent aggregators
Author: Mihir Bhattacharya
Publication date: 09-12-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 4/2016
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-015-0938-8

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