Agreement in Circular Societies

Hardin, Christopher S.

doi:10.4169/000298910x474970

Cited by 6 publications

(12 citation statements)

References 1 publication

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“…1. There is some literature on circular societies (namely societies whose approval sets are circular arcs) [11,35]. Hardin's result [11] is that in a circular (q, p)-agreeable society S we have a(S) ≥ q−1 p , and this bound is tight.…”

Section: Final Remarksmentioning

confidence: 99%

“…There is some literature on circular societies (namely societies whose approval sets are circular arcs) [11,35]. Hardin's result [11] is that in a circular (q, p)-agreeable society S we have a(S) ≥ q−1 p , and this bound is tight. As for piercing numbers of such families, it is easy to see that an upper bound on the piercing numbers of a families of circular arcs with the (p, q)-property is the upper bound on the piercing numbers of families of intervals satisfying the same property, plus 1.…”

Section: Final Remarksmentioning

confidence: 99%

“…The political spectrum is often modeled as a line, with conservative candidates on the right and liberal candidates on the left. In other settings, a spectrum could be multi-dimensional [10], or circular [11], etc.…”

Section: Introductionmentioning

confidence: 99%

“…Since then there have been a number of other results of this type (e.g. [13,11,10,14]) that consider other hypotheses on approval sets and other geometric spaces for the political spectrum X. These results are all in some sense variants of Helly's Theorem.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Piercing numbers in approval voting

Zerbib

2019

Mathematical Social Sciences

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We survey a host of results from discrete geometry that have bearing on the analysis of geometric models of approval voting. Such models view the political spectrum as a geometric space, with geometric constraints on voter preferences. Results on piercing numbers then have a natural interpretation in voting theory, and we survey their implications for various classes of geometric constraints on voter approval sets.

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Section: Final Remarksmentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Piercing numbers in approval voting

Zerbib

2019

Mathematical Social Sciences

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“…examined agreement in linear societies, in which voters may approve of an interval of options chosen from a linear spectrum, and higher-dimensional analogues [1]. Recognizing that a circular model is often more appropriate, Hardin studied approval voting in circular societies [2]. Furthermore, Klawe et.…”

Section: Introductionmentioning

confidence: 99%

Approval Voting in Product Societies

Mazur

Sondjaja

Wright

et al. 2017

The American Mathematical Monthly

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Abstract. In approval voting, individuals vote for all platforms that they find acceptable. In this situation it is natural to ask: When is agreement possible? What conditions guarantee that some fraction of the voters agree on even a single platform? Berg et. al. found such conditions when voters are asked to make a decision on a single issue that can be represented on a linear spectrum. In particular, they showed that if two out of every three voters agree on a platform, there is a platform that is acceptable to a majority of the voters. Hardin developed an analogous result when the issue can be represented on a circular spectrum. We examine scenarios in which voters must make two decisions simultaneously. For example, if voters must decide on the day of the week to hold a meeting and the length of the meeting, then the space of possible options forms a cylindrical spectrum. Previous results do not apply to these multi-dimensional voting societies because a voter's preference on one issue often impacts their preference on another. We present a general lower bound on agreement in a two-dimensional voting society, and then examine specific results for societies whose spectra are cylinders and tori.

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Double-interval societies

Klawe¹,

Nyman²,

Scott³

et al. 2014

The Mathematics of Decisions, Elections, and Games

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Abstract. Consider a society of voters, each of whom specify an approval set over a linear political spectrum. We examine double-interval societies, in which each person's approval set is represented by two disjoint closed intervals, and study this situation where the approval sets are pairwise-intersecting: every pair of voters has a point in the intersection of their approval sets. The approval ratio for a society is, loosely speaking, the popularity of the most popular position on the spectrum. We study the question: what is the minimal guaranteed approval ratio for such a society? We provide a lower bound for the approval ratio, and examine a family of societies that have rather low approval ratios. These societies arise from double-n strings: arrangements of n symbols in which each symbol appears exactly twice.

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Agreement in Circular Societies

Cited by 6 publications

References 1 publication

Piercing numbers in approval voting

Piercing numbers in approval voting

Approval Voting in Product Societies

Double-interval societies

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