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Social Choice and Welfare
Erschienen in: Social Choice and Welfare 1/2016

11.08.2015 | Original Paper

A note on the McKelvey uncovered set and Pareto optimality

verfasst von: Felix Brandt, Christian Geist, Paul Harrenstein

Erschienen in: Social Choice and Welfare | Ausgabe 1/2016

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Abstract

We consider the notion of Pareto optimality under the assumption that only the pairwise majority relation is known and show that the set of necessarily Pareto optimal alternatives coincides with the McKelvey uncovered set. As a consequence, the McKelvey uncovered set constitutes the coarsest Pareto optimal majoritarian social choice function. Moreover, every majority relation is induced by a preference profile in which the uncovered alternatives precisely coincide with the Pareto optimal ones. We furthermore discuss the structure of the McKelvey covering relation and the McKelvey uncovered set.
Vorheriger Artikel Fiscal policy and corruption
Nächster Artikel Preference exclusions for social rationality

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Fußnoten
1
Antisymmetry is not required for any of our results to hold. In fact, Theorem 1 is even stronger when also assuming antisymmetric individual preferences (since this only increases the difficulty of constructing a suitable preference profile).
 
2
Some authors call this strong Pareto optimality. In contrast, an alternative x would be weakly Pareto optimal if there is no alternative y with \(y\mathrel {P_i}x\) for all \(i\in N_{R}\). In the case of antisymmetric preferences, the two notions coincide.
 
3
Consider, for instance, the simple case of \(A=\{a,b,c\}\) and \(Q=\{(a,b),(a,c)\}\), which is easily seen not to be the covering relation for any preference profile.
 
4
Corollary 1 also entails an analogous weaker result for the special case of tournaments (i.e., antisymmetric majority relations \(R_\mathbf {Maj}\)), which was used as a Lemma by Brandt and Geist (2014).
 
5
The whole example was obtained from and proved minimal by an automated computer search based on the method developed by Brandt et al. (2014).
 
6
In fact, any tournament of size 7 can be realized by 3 agents.
 
7
This even holds when individual preferences are allowed to be weak orders.
 
8
The same counterexample also applies to the case of weak individual orders (i.e., without the assumption of antisymmetry of \(R_i\)). In this case there are 256 instead of 22 candidates for individual preferences relations, and additional constraints for the Pareto edges are required, which makes the resulting IP significantly larger, but still solvable within less than one second.
 
Literatur
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Metadaten
Titel
A note on the McKelvey uncovered set and Pareto optimality
verfasst von
Felix Brandt
Christian Geist
Paul Harrenstein
Publikationsdatum
11.08.2015
Verlag
Springer Berlin Heidelberg
Erschienen in
Social Choice and Welfare / Ausgabe 1/2016
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-015-0904-5

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