The uncovered set in spatial voting games

Scott L. Feld; Bernard Grofman; Richard Hartly; Marc Kilgour; Nicholas Miller;

doi:10.1007/bf00126302

Computing the Yolk in Spatial Voting Games without Computing Median Lines

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v33i01.33012012 ◽

2019 ◽

Vol 33 ◽

pp. 2012-2019

Author(s):

Joachim Gudmundsson ◽

Sampson Wong

Keyword(s):

Upper Bound ◽

Linear Time ◽

Search Technique ◽

Important Concept ◽

Uncovered Set ◽

Spatial Voting ◽

Voting Games ◽

Parametric Search ◽

Linear Time Algorithms

The yolk is an important concept in spatial voting games: the yolk center generalises the equilibrium and the yolk radius bounds the uncovered set. We present near-linear time algorithms for computing the yolk in the plane. To the best of our knowledge our algorithm is the first that does not precompute median lines, and hence is able to break the best known upper bound of O(n4/3) on the number of limiting median lines. We avoid this requirement by carefully applying Megiddo’s parametric search technique, which is a powerful framework that could lead to faster algorithms for other spatial voting problems.

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In Search of the Uncovered Set

Political Analysis ◽

10.1093/pan/mpl007 ◽

2007 ◽

Vol 15 (1) ◽

pp. 21-45 ◽

Cited By ~ 22

Author(s):

Nicholas R. Miller

Keyword(s):

Computational Algorithm ◽

Voting Theory ◽

Legislative Action ◽

Uncovered Set ◽

Political Analysis ◽

Spatial Voting ◽

Voting Games ◽

General Relevance ◽

Straight Line ◽

Theoretical Considerations

This paper pursues a number of theoretical explorations and conjectures pertaining to the uncovered set in spatial voting games. It was stimulated by the article “The Uncovered Set and the Limits of Legislative Action” by W. T. Bianco, I. Jeliazkov, and I. Sened (2004, Political Analysis 12:256—78) that employed a grid-search computational algorithm for estimating the size, shape, and location of the uncovered set, and it has been greatly facilitated by access to the CyberSenate spatial voting software being developed by Joseph Godfrey. I bring to light theoretical considerations that account for important features of the Bianco, Jeliazkov, and Sened results (e.g., the straight-line boundaries of uncovered sets displayed in some of their figures, the “unexpectedly large” uncovered sets displayed in other figures, and the apparent sensitivity of the location of uncovered sets to small shifts in the relative sizes of party caucuses) and present theoretical insights of more general relevance to spatial voting theory.

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The Uncovered Set and the Limits of Legislative Action

Political Analysis ◽

10.1093/pan/mph018 ◽

2004 ◽

Vol 12 (3) ◽

pp. 256-276 ◽

Cited By ~ 30

Author(s):

William T. Bianco ◽

Ivan Jeliazkov ◽

Itai Sened

Keyword(s):

Social Choice ◽

Real World ◽

Solution Concept ◽

Collective Choice ◽

Simulation Technique ◽

Legislative Action ◽

Uncovered Set ◽

Spatial Voting ◽

Voting Games

We present a simulation technique for sorting out the size, shape, and location of the uncovered set to estimate the set of enactable outcomes in “real-world” social choice situations, such as the contemporary Congress. The uncovered set is a well-known but underexploited solution concept in the literature on spatial voting games and collective choice mechanisms. We explain this solution concept in nontechnical terms, submit some theoretical observations to improve our theoretical grasp of it, and provide a simulation technique that makes it possible to estimate this set and thus enable a series of tests of its empirical relevance.

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On β-Plurality Points in Spatial Voting Games

ACM Transactions on Algorithms ◽

10.1145/3459097 ◽

2021 ◽

Vol 17 (3) ◽

pp. 1-21

Author(s):

Boris Aronov ◽

Mark De Berg ◽

Joachim Gudmundsson ◽

Michael Horton

Keyword(s):

Spatial Voting ◽

Voting Games ◽

Equal Distance ◽

Alternative Proposal

Let V be a set of n points in mathcal R d , called voters . A point p ∈ mathcal R d is a plurality point for V when the following holds: For every q ∈ mathcal R d , the number of voters closer to p than to q is at least the number of voters closer to q than to p . Thus, in a vote where each v ∈ V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal p will not lose against any alternative proposal q . For most voter sets, a plurality point does not exist. We therefore introduce the concept of β-plurality points , which are defined similarly to regular plurality points, except that the distance of each voter to p (but not to q ) is scaled by a factor β , for some constant 0< β ⩽ 1. We investigate the existence and computation of β -plurality points and obtain the following results. • Define β * d := {β : any finite multiset V in mathcal R d admits a β-plurality point. We prove that β * d = √3/2, and that 1/√ d ⩽ β * d ⩽ √ 3/2 for all d ⩾ 3. • Define β ( p, V ) := sup {β : p is a β -plurality point for V }. Given a voter set V in mathcal R 2 , we provide an algorithm that runs in O ( n log n ) time and computes a point p such that β ( p , V ) ⩾ β * b . Moreover, for d ⩾ 2, we can compute a point p with β ( p , V ) ⩾ 1/√ d in O ( n ) time. • Define β ( V ) := sup { β : V admits a β -plurality point}. We present an algorithm that, given a voter set V in mathcal R d , computes an ((1-ɛ)ċ β ( V ))-plurality point in time O n 2 ɛ 3d-2 ċ log n ɛ d-1 ċ log 2 1ɛ).

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In quest of the banks set in spatial voting games

Social Choice and Welfare ◽

10.1007/s00355-012-0676-0 ◽

2012 ◽

Vol 41 (1) ◽

pp. 43-71 ◽

Cited By ~ 1

Author(s):

Scott L. Feld ◽

Joseph Godfrey ◽

Bernard Grofman

Keyword(s):

Spatial Voting ◽

Voting Games ◽

Banks Set

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The Core and the Stability of Group Choice in Spatial Voting Games

American Political Science Review ◽

10.2307/1958065 ◽

1988 ◽

Vol 82 (1) ◽

pp. 195-211 ◽

Cited By ~ 49

Author(s):

Norman Schofield ◽

Bernard Grofman ◽

Scott L. Feld

Keyword(s):

Political Stability ◽

Simple Majority ◽

Small Perturbations ◽

The Core ◽

Spatial Voting ◽

Voting Games ◽

Voting Game ◽

Stable Core ◽

The Stability ◽

Structurally Stable

The core of a voting game is the set of undominated outcomes, that is, those that once in place cannot be overturned. For spatial voting games, a core is structurally stable if it remains in existence even if there are small perturbations in the location of voter ideal points. While for simple majority rule a core will exist in games with more than one dimension only under extremely restrictive symmetry conditions, we show that, for certain supramajorities, a core must exist. We also provide conditions under which it is possible to construct a structurally stable core. If there are only a few dimensions, our results demonstrate the stability properties of such frequently used rules as two-thirds and three-fourths. We further explore the implications of our results for the nature of political stability by looking at outcomes in experimental spatial voting games and at Belgian cabinet formation in the late 1970s.

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