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Social Choice and Welfare

Erschienen in:

02.11.2015 | Original Paper

A characterization of the n-agent Pareto dominance relation

verfasst von: Shaofang Qi

Erschienen in: Social Choice and Welfare | Ausgabe 3/2016

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Abstract

The Pareto dominance relation of a preference profile is (the asymmetric part of) a partial order. For any integer n, there exists an n-agent preference profile that generates a given Pareto dominance relation if and only if the dimension of the corresponding partial order is less than or equal to n; we provide a general characterization of when this is the case.

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Gibson and Powers identify majority rule with the plurality choice function restricted to the family of all 2-element subsets and extend McGarvey’s theorem to the case where the plurality choice function is restricted to the set of all l-element subsets, where \(1<l<n\) and n is the number of alternatives. Their result is also a corollary of one of Saari (1989)’s results on voting dictionaries.

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The inverse problems for both rules become trivial if we allow the number of individuals to vary.

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The maximal-element rationalizability problem in Bossert et al. (2005) is also related to such a multi-criteria procedure. Additionally, finding a characterization for the END\(_{n}\)-rationalizability (efficient and non-deteriorating rationalizability) of a choice function for a given integer n introduced by Bossert and Sprumont (2003) is also closely related to the inverse problem we study here.

With an additional structure, an aggregating map f that denotes how the agent aggregates various criteria, the multi-criteria procedure can be incorporated as an intermediate step in different individual decision-making models including choices with status quo bias or reference-dependence choices. See, for instance, Masatlioglu and Ok (2005); Ambrus and Rozen 2015); Ok et al. 2015).

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See Demuynck (2014) for additional results on the computational complexity of inverse Pareto problems.

Note that completeness implies reflexivity. Some authors define completeness only for any two distinct options.

To see this, note that \(\left( T_{Q\cup P^{d}}\right) ^{d}=T_{\left( Q\cup P^{d}\right) ^{d}}=T_{Q^{d}\cup P}\). Since \(T_{Q^{d}\cup P}\) is a partial order, given that Q is a partial-conjugate of P, \(T_{Q\cup P^{d}}\) is also a partial order.

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Titel: A characterization of the n-agent Pareto dominance relation
verfasst von: Shaofang Qi
Publikationsdatum: 02.11.2015
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 3/2016
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-015-0934-z

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