Top

Social Choice and Welfare

Published in:

23-07-2018 | Original Paper

Constitutionally consistent voting rules over single-peaked domains

Author: Mihir Bhattacharya

Published in: Social Choice and Welfare | Issue 2/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Production efficiency and profit taxation

next article Congruence relations on a choice space

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

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.

Arrow KJ, Sen A, Suzumura K (2010) Handbook of social choice & welfare, vol 2. Elsevier, Amsterdam

Austen-Smith D, Banks JS (2005) Positive political theory II: strategy and structure. University of Michigan Press, Ann ArborCrossRef

Barberà S, Beviá C (2002) Self-selection consistent functions. J Econ Theory 105:263–277CrossRef

Barberà S, Jackson MO (2004) Choosing how to choose: self-stable majority rules and constitutions. Q J Econ 119:1011–1048CrossRef

Barberà S, Sonnenschein H, Zhou L (1991) Voting by committees. Econometrica: J Econometric Soc 59(3):595–609CrossRef

Bhattacharya M (2016) Multilevel multidimensional consistent aggregators. Soc Choice Welf 46:839–861CrossRef

Binmore K (1975) An example in group preference. J Econ Theory 10:377–385CrossRef

Black D, Newing RA, McLean I, McMillan A, Monroe BL (1958) The theory of committees and elections. Springer, New York

Blais A (2008) To keep or to change first past the post?: the politics of electoral reform. OUP, OxfordCrossRef

Border KC, Jordan JS (1983) Straightforward elections, unanimity and phantom voters. Rev Econ Stud 50:153–170CrossRef

Chambers CP (2008) Consistent representative democracy. Games Econ Behav 62(2):348–363CrossRef

Chambers CP (2009) An axiomatic theory of political representation. J Econ Theory 144(1):375–389CrossRef

Chandrasekher M (2015) Dynamically consistent voting rules. J Econ Theory 160:175–187CrossRef

Downs A (1957) An economic theory of democracy. Harper, New York

Fung, LW, Fu, K-S (1975) An axiomatic approach to rational decision making in a fuzzy environment. In: Zadeh Lotfi A, Fu K-S, Tanaka K, Shimura M (eds) Fuzzy sets and their applications to cognitive and decision processes, pp 227–256.

Koray S (2000) Self-selective social choice functions verify arrow and Gibbard–Satterthwaite theorems. Econometrica 68:981–996CrossRef

Koray S, Slinko A (2008) Self-selective social choice functions. Soc Choice Welf 31:129–149CrossRef

Koray S, Unel B (2003) Characterization of self-selective social choice functions on the tops-only domain. Soc Choice Welf 20:495–507CrossRef

Lainé J, Ozkes AI, Sanver R (2016) Hyper-stable social welfare functions. Soc Choice Welf 46:157–182CrossRef

Moulin H (1980) On strategy-proofness and single peakedness. Public Choice 35:437–455CrossRef

Nermuth M (1994) Two-Stage Discrete Aggregation: The Ostrogorski Paradox and Related Phenomena. Soc Choice Welf 9:99–116

Rubinstein A (1980) Stability of decision systems under majority rule. J Econ Theory 23:150–159CrossRef

Ünel B (1999) Explorations of self-selective social choice functions. Doctoral dissertation, Bilkent University

Title: Constitutionally consistent voting rules over single-peaked domains
Author: Mihir Bhattacharya
Publication date: 23-07-2018
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2019
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-1145-1

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 2/2019

Production efficiency and profit taxation

The weighted-egalitarian Shapley values

Strategy-proof aggregation rules and single peakedness in bounded distributive lattices

Flat rate taxes and relative poverty measurement: a reconsideration

Congruence relations on a choice space

Manipulability of consular election rules