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Social Choice and Welfare

Erschienen in:

23.07.2018 | Original Paper

Constitutionally consistent voting rules over single-peaked domains

verfasst von: Mihir Bhattacharya

Erschienen in: Social Choice and Welfare | Ausgabe 2/2019

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Abstract

Constitutional consistency requires that the voting rule produce the same outcome at any vote profile as the one it produces at its induced vote profile for any given set of voting rules (or constitution) consisting of the voting rule itself. We consider this type of consistency in two voting models with single-peaked preferences, one with a finite set of alternatives and the other, when the set of alternatives is the interval [0, 1]. We show that cumulative-threshold rules are the only unanimous, anonymous and constitutionally consistent voting rules. These rules assign monotone decreasing (increasing) thresholds to each alternative and pick the minimum (maximum) alternative from the range of the vote profile that receives more cumulative votes (votes received by all the alternatives smaller (or greater) than itself) than the threshold assigned to it. This class of rules consists of the min, max and median rules. The addition of continuity leads to the characterization of k-median rules in the interval voting model.

Vorheriger Artikel Production efficiency and profit taxation

Nächster Artikel Congruence relations on a choice space

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Rubinstein (1980) and Binmore (1975) first studied stability of decision systems and group preferences respectively.

See Blais (2008) for a study of electoral reform pertaining to changes in voting rules in the UK and Canada.

With some exceptions- Barberà and Beviá (2002) and Ünel (1999) obtain some positive results in the political economy and the social choice theory setting respectively.

A preference \(\succeq \) defined over the set of alternatives X is single-peaked if there exists a ‘peak’ \(x^{*}\) and an ordering \((\ge )\) over X such that (i) \([y<x\le x^{*}]\Rightarrow [x \succ y] \) and (ii) \([ x^{*}\ge x>y ] \Rightarrow [x \succ y]\) for all \(x,y\in X\).

This is proved in the “Appendix”.

We define a median rule as the rule which picks the \(\frac{n}{2}^{th}\)(\(\frac{n+1}{2}^{th}\)) lowest alternative in the vote profile if n is even(odd).

Barberà and Beviá (2002) studies a notion of self-selection consistency which pertains to decision-making among groups.

The range of the vote profile v is the set containing alternatives greater than or equal to the smallest alternative in the vote profile which are also less than or equal to the greatest alternative in the vote profile according to a pre-defined ordering. Ünel (1999) makes the observation that positional social choice functions are self-selective in the single-peaked domain.

See Nermuth (1994), Chambers (2008, 2009) and Bhattacharya (2016).

See Sect. 2 for formal definitions.

Note that if we consider tops-only and unanimous social choice functions instead of voting rules all the results in the paper can be proved with respect to social choice functions. I am thankful to an anonymous reviewer for these observations.

A preference \(\succeq \) is weakly-ordered if it satisfies: (i) Reflexivity: For all \(x\in X\), \(x\succeq x\) (ii) Completeness: For all \(x,y\in X\) either \(x\succeq y\) or \(y\succeq x\) (iii) Transitivity: For all \(x,y,z \in X\) if \(x\succeq y\) and \(y\succeq z\) then \(x\succeq z\).

We show later that if voting rules satisfy constitutional consistency then they will be immune to manipulative voting.

When X is finite we abuse notation slightly by denoting the range as \([\underline{v}, \overline{v}]\) i.e. the set of alternatives greater than or equal to \(\underline{v}\) and less than or equal to \(\overline{v}\).

We prove this in the “Appendix”.

Note that \(\alpha \)-median rules are similar to generalized median rules. The latter are defined for a fixed population while the former are defined for a variable size population.

I thank an anonymous reviewer for the shorter proof.

We relax this assumption later.

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Titel: Constitutionally consistent voting rules over single-peaked domains
verfasst von: Mihir Bhattacharya
Publikationsdatum: 23.07.2018
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 2/2019
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-1145-1

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