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Social Choice and Welfare
Published in: Social Choice and Welfare 2/2022

01-09-2021 | Original Paper

Generalized medians and a political center

Author: Tasos Kalandrakis

Published in: Social Choice and Welfare | Issue 2/2022

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Abstract

Building on properties of the median in one dimension under simple majority rule, I propose two generalizations for multi-dimensional environments and general voting rules. A P-ball (for pivotal) is a smallest radius ball such that, for every pair of alternatives, if all its members prefer one alternative over another then all members of some winning coalition share that preference (and there does not exist a winning coalition all the members of which have the opposite strict preference). A W-ball (for winning) is a smallest radius ball such that, if all members of some winning coalition prefer one alternative over another, then there exists a member of that ball that shares that preference. P-balls and W-balls coincide with the unicameral yolk (McKelvey in Am J Polit Sci 30(2):283–314, 1986; Ferejohn et al., in Soc Choice Welf 1:45–67, 1984) under simple majority rule. Using these constructs, I generalize and sharpen McKelvey’s (Am J Polit Sci 30(2):283–314, 1986) circular bounds on the set of alternatives socially preferred to any alternative, and bound the core and the uncovered set for general voting rules. I study comparative statics on the effect of changes in the voting rule.
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Appendix
Available only for authorised users
Footnotes
1
Of course, there is a vast literature on the broader question of social preference aggregation that proposes different solution concepts in such environments. A review of this literature is outside the scope of this study, and I direct the reader to, e.g., Austen-Smith and Banks (1999) for a comprehensive review.
 
2
See Appendix B.
 
3
The special case when only a finite subset of committee members in \({\mathbb {R}}^D\) have voting rights is captured by the general assignment of voting weights that I detail in the next paragraph.
 
4
Of course, whether a model (finite or infinite) provides a good approximation depends on the details of the mode of application of the model and the purpose of the approximation. For instance, Tovey (2010) provides examples where use of the infinite models may or may not provide a good approximation.
 
5
Note that the restriction that \(m_\ell \in (0,1)\) is not consequential in this finite setting. In particular, it does not preclude the unanimity rule because we can raise \(m_\ell\) above the sum of voting weights of all except the committee member with the smallest voting weight. Unanimity is precluded in the infinite committee setting by the assumption \(m_\ell \in (0,1).\)
 
6
The definition appears most succinctly in Koehler (1990), page 238, as (emphasis in the original): “Define a point-specific yolk with respect to a proposal z as a circle of minimum radius which intersects every majority line in the subset of majority lines such that z is located in the intersection of their minority sides.”
 
7
Also, the optimization problems herein are implemented without distinction to limiting and non-limiting “majority” lines, neither does the additional classification of hyperplanes to a- and b-type in Koehler (1990) feature in this study.
 
8
See also Stone and Tovey (1992); Koehler (1992); McKelvey and Tovey (2010).
 
9
See also Penn (2006); Duggan (2013).
 
10
For example, see the discussion in Martin et al. (2016), page 516, for McKelvey’s original bound on the uncovered set.
 
11
Practically here I assume that no more than D points fall on the same hyperplane.
 
12
Because Tovey allows an even sized committee (N even), he also admits the possibility of two distinct parallel median hyperplanes. Indeed, there would be an infinity of median hyperplanes in between any such pair of parallel median hyperplanes. My definition of pivotal hyperplanes more generally isolates the only two relevant ones because once these two extreme ones intersect with a candidate solution, all remaining ones in between also do.
 
13
Indeed, if (\(DW_y\)) yields a negative solution, its absolute value provides a rough gauge of the size of the core set which is non-empty around y.
 
Literature
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Metadata
Title
Generalized medians and a political center
Author
Tasos Kalandrakis
Publication date
01-09-2021
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 2/2022
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-021-01359-2

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