Top

Social Choice and Welfare

Published in:

12-02-2016

A spatial analogue of May’s Theorem

Authors: Richard Lee Brady, Christopher P. Chambers

Published in: Social Choice and Welfare | Issue 1/2016

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

search-config

AI-assisted search

Off

Abstract

In a spatial model with Euclidean preferences, we establish that the geometric median satisfies Maskin monotonicity, anonymity, and neutrality. For three agents, it is the unique such rule.

previous article A conjecture on the construction of orderings by Borda’s rule

next article The Condorcet paradox revisited

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

Other works using the geometric median in economics or political science research include Cervone et al. (2012), Baranchuk and Dybvig (2009) and Chung and Duggan (2014). In particular, the latter work describes an interesting generalization of the concept to general convex preferences.

See, for example, Muller and Satterthwaite (1977), Dasgupta et al. (1979), Barbera and Peleg (1990) and Berga and Moreno (2009).

Since \(X=\mathbb {R}^d\), isometries are bijections that correspond to reflections, rotations, and translations.

The geometric median has a rich history in this special case and is sometimes referred to as the Fermat-Torricelli point of a triangle.

An interior angle is simply the angle formed by two adjacent sides of the triangle.

A median of a triangle is any of the line segments connecting a vertex to the midpoint of the opposite side of the triangle.

Note that \(c_1'\in con(\triangle _{c_1x_C^*c_2})\) will follow since it is assumed \(\triangle _{c_1'c_2'c_3'}\) has all angles less than or equal to \(120^{\circ }\) while \(\angle _{c_1x_C^*c_2}=120^{\circ }\).

A scalene triangle has all three interior angles of different measure.

Note that as \(c_3\rightarrow c_1\) we have \(\angle _{c_1c_2c_3}\rightarrow 0\) so finding such a \(c_3'\) is always possible by the Intermediate Value Theorem.

This is achieved by moving \(z_2\) and \(z_3\) in tandem towards \(\varphi (Z)\) until each side of the triangle is equal in length.

This follows by the midpoint collinear construction outlined previously.

See Brady and Chambers (2015) for a discussion and result using the dual solution.

Baranchuk N, Dybvig PH (2009) Consensus in diverse corporate boards. Rev Fin Stud 22(2):715–747CrossRef

Barbera S, Peleg B (1990) Strategy-proof voting schemes with continuous preferences. Soc Choice Welf 7(1):31–38CrossRef

Berga D, Moreno B (2009) Strategic requirements with indifference: single-peaked versus single-plateaued preferences. Soc Choice Welf 32(2):275–298CrossRef

Black D (1948) On the rationale of group decision-making. J Polit Econ 56(1):23–34CrossRef

Brady RL, Chambers CP (2015) Spatial implementation. Games Econ Behav 94(2015):200–205

Cervone DP, Dai R, Gnoutcheff D, Lanterman G, Mackenzie A, Morse A, Srivastava N, Zwicker WS (2012) Voting with rubber bands, weights, and strings. Math Soc Sci 64(1):11–27CrossRef

Chung H, Duggan J (2014) Directional equilibria, working paper

Coxeter H (1989) Introduction to geometry. Wiley, New York

Dasgupta P, Hammond P, Maskin E (1979) The implementation of social choice rules: Some general results on incentive compatibility. Rev Econ Stud 46:185–216CrossRef

Deimling K (2011) Nonlinear functional analysis. Courier Dover Publications, Mineola

Duggan J (2015) May’s Theorem in one dimension, working paper

Gini C, Galvani L (1929) Di talune estensioni dei concetti di media a caratteri qualitivi. Metron 8:3–209

Güler O (2010) Foundations of optimization, 258th edn. Springer, New YorkCrossRef

Haldane J (1948) Note on the median of a multivariate distribution. Biometrika 35(3–4):414–417CrossRef

Hotelling H (1929) Stability in competition. Econ J 39(153):41–57CrossRef

Maskin E (1999) Nash equilibrium and welfare optimality. Rev Econ Stud 66(1):23–38CrossRef

May KO (1952) A set of independent necessary and sufficient conditions for simple majority decision. Econometrica 20(4):680–684CrossRef

Muller E, Satterthwaite MA (1977) The equivalence of strong positive association and strategy-proofness. J Econ Theory 14(2):412–418CrossRef

Plott CR (1967) A notion of equilibrium and its possibility under majority rule. Am Econ Rev 57(4):787–806

Title: A spatial analogue of May’s Theorem
Authors: Richard Lee Brady
Christopher P. Chambers
Publication date: 12-02-2016
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 1/2016
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-016-0949-0

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 1/2016

Pairwise partition graphs and strategy-proof social choice in the exogenous indifference class model

The role of subjective beliefs in preferences for redistribution

The greatest unhappiness of the least number

Multidimensional political competition with non-common beliefs

On surplus-sharing in partnerships

Infinite-horizon social evaluation with variable population size