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Social Choice and Welfare

Erschienen in:

12.02.2016

A spatial analogue of May’s Theorem

verfasst von: Richard Lee Brady, Christopher P. Chambers

Erschienen in: Social Choice and Welfare | Ausgabe 1/2016

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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.

Vorheriger Artikel A conjecture on the construction of orderings by Borda’s rule

Nächster Artikel The Condorcet paradox revisited

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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

Titel: A spatial analogue of May’s Theorem
verfasst von: Richard Lee Brady
Christopher P. Chambers
Publikationsdatum: 12.02.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 1/2016
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-016-0949-0

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