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Social Choice and Welfare

Erschienen in:

16.01.2017 | Original Paper

Social choice correspondences with infinitely many agents: serial dictatorship

verfasst von: Shino Takayama, Akira Yokotani

Erschienen in: Social Choice and Welfare | Ausgabe 3/2017

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Abstract

We study social choice correspondences (SCC) assigning a set of choices to each pair consisting of a nonempty subset of the set of alternatives and a weak preference profile. The SCC satisfies unanimity if when there is a weakly Pareto dominant alternative, the SCC selects this alternative. Stability requires that the SCC is unaffected by withdrawal of losing alternatives. Independence implies that the SCC selects the same outcome from a subset of the set of alternatives for two preference profiles that are the same on this set. We characterize the SCC satisfying the three axioms, when the set of alternatives is finite but includes more than three alternatives, and the set of agents can have any cardinality. We show that the SCC is a serial dictatorship à la Eraslan and McLennan (J Econ Theory 117:29–54, 2004) and that a serial dictatorship can include “invisible serial dictators” à la Kirman and Sondermann (J Econ Theory 5:267–277, 1972).

Vorheriger Artikel Strategic schools under the Boston mechanism revisited

Nächster Artikel Properties of multiwinner voting rules

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Cato (2013b) calls our unanimity condition strong Pareto.

Some serial dictatorship results are also found in Campbell and Kelly (2002) Example 3.12, 3.13 in pp. 51–52, Gevers (1979), Dutta et al. (2001) and Grether and Plott (1982).

The Appendix of Hurd and Loeb (1985) provides some important facts concerning ultrafilters. For discussions of ultrafilters in social choice, see Monjardet (1983).

Because we use their result in our proof of Theorem 1, after introducing technical terms, we state it as the Ultrafilter Lemma in Appendix I. In contrast with Campbell and Kelly (2000), we use a full domain of weak preferences, and then their Ultrafilter Lemma holds in our setting as proved in Appendix I.

See Footnote 1. In this article, we define strong Pareto of Cato (2013b) as unanimity.

For the classical works, Gibbard (1973); Satterthwaite (1975). Schmeidler and Sonnenschein (1974) also study the relationship between Arrow’s Impossibility Theorem and the Gibbard–Satterthwaite Theorem. For more recent works, Reny (2001); Eliaz (2004); Ubeda (2004); Vohra (2011). Many preceding works (Geanakoplos 2005; Yu 2012; Barberà 1980, 1983; Campbell and Kelly 2002, for a comprehensive survey) provide another proof of Arrow’s Impossibility Theorem in the case of finitely many agents. Technically, to prove our main theorem, we will use the result in Arrow (1959) that under our three axioms, the SCC generates a social welfare function.

These impossibility theorems include Arrow’s Impossibility Theorem (Arrow 1951), the Gibbard–Satterthwaite Theorem (Gibbard 1973; Satterthwaite 1975), the Impossibility theorem under strategic candidacy (Grether and Plott 1982; Dutta et al. 2001; Eraslan and McLennan 2004), and the characterization of game-theoretic solutions implementing only dictatorial social choices by Jackson and Srivastava (1996). It is also known that there is an interconnection between lexicographic preferences and both Arrow’s Impossibility Theorem and the Gibbard–Satterthwaite Theorem (see Fishburn 1975; Mitra and Sen 2014).

Ultrafilters are well-established objects both in mathematics and in social choice theory. For references, see Comfort and Negrepotis (1974) or Bell and Slomson (1969).

Here, we follow the terminology in Eraslan and McLennan (2004) and refer to \(\rho \) as a tie-breaking rule, even though \(\rho \) is not necessarily strict, and there still may exist multiple alternatives in \(\phi ^{\mathscr {U}, \rho }(X,R)\) after applying the “tie-breaking.” Man and Takayama (2013a) call \(\rho \) a tie-breaking preference.

We thank a referee for proposing the current (and more intuitive) definition to us.

In their paper, our stability is called Arrow’s Choice Axiom.

See Section 17 in Halmos (1974).

Hurd and Loeb (1985) provides a concise proof for this result, which is found as Theorem A.8 in page 221.

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Titel: Social choice correspondences with infinitely many agents: serial dictatorship
verfasst von: Shino Takayama
Akira Yokotani
Publikationsdatum: 16.01.2017
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 3/2017
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-017-1025-0

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