Finagle's Law and the Finagle Point, a New Solution Concept for Two-Candidate Competition in Spatial Voting Games Without a Core

The Shapley-Owen value (SOV, Owen and Shapley 1989, Optimal location of candidates in ideological space. International Journal of Game Theory 125–42), a generalization of the Shapley-Shubik value applicable to spatial voting games, is an important concept in that it takes us away from a priori concepts of power to notions of power that are directly tied to the ideological proximity of actors. SOVs can also be used to locate the spatial analogue to the Copeland winner, the strong point, the point with smallest win-set, which is a plausible solution concept for games without cores. However, for spatial voting games with many voters, until recently, it was too computationally difficult to calculate SOVs, and thus, it was impossible to find the strong point analytically. After reviewing the properties of the SOV, such as the result proven by Shapley and Owen that size of win sets increases with the square of distance as we move away from the strong point along any ray, we offer a computer algorithm for computing SOVs that can readily find such values even for legislatures the size of the U.S. House of Representatives or the Russian Duma. We use these values to identify the strong point and show its location with respect to the uncovered set, for several of the U.S. congresses analyzed in Bianco, Jeliazkov, and Sened (2004, The limits of legislative actions: Determining the set of enactable outcomes given legislators preferences. Political Analysis 12:256–76) and for several sessions of the Russian Duma. We then look at many of the experimental committee voting games previously analyzed by Bianco et al. (2006, A theory waiting to be discovered and used: A reanalysis of canonical experiments on majority-rule decision making. Journal of Politics 68:838–51) and show how outcomes in these games tend to be points with small win sets located near to the strong point. We also consider how SOVs can be applied to a lobbying game in a committee of the U.S. Senate.

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Stability and Centrality of Legislative Choice in the Spatial Context

American Political Science Review ◽

10.2307/1961966 ◽

1987 ◽

Vol 81 (2) ◽

pp. 539-553 ◽

Cited By ~ 37

Author(s):

Bernard Grofman ◽

Guillermo Owen ◽

Nicholas Noviello ◽

Amihai Glazer

Keyword(s):

Infinite Number ◽

Majority Rule ◽

Solution Concept ◽

Spatial Context ◽

Core Outcome ◽

Legislative Voting ◽

Spatial Voting ◽

Voting Games ◽

Legislative Choice ◽

Copeland Winner

Majority-rule spatial voting games lacking a core still always present a “near-core” outcome, more commonly known as the Copeland winner. This is the alternative that defeats or ties the greatest number of alternatives in the space. Previous research has not tested the Copeland winner as a solution concept for spatial voting games without a core, lacking a way to calculate where the Copeland winner was with an infinite number of alternatives. We provide a straightforward algorithm to find the Copeland winner and show that it corresponds well to experimental outcomes in an important set of experimental legislative voting games. We also provide an intuitive motivation for why legislative outcomes in the spatial context may be expected to lie close to the Copeland winner. Finally, we show a connection between the Copeland winner and the Shapley value and provide a simple but powerful algorithm to calculate the Copeland scores of all points in the space in terms of the (modified) power values of each of the voters and their locations in the space.

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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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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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ON THE CHACTERISTIC NUMBERS OF VOTING GAMES

International Game Theory Review ◽

10.1142/s0219198906001156 ◽

2006 ◽

Vol 08 (04) ◽

pp. 643-654 ◽

Cited By ~ 3

Author(s):

MATHIEU MARTIN ◽

VINCENT MERLIN

Keyword(s):

Upper Bound ◽

Solution Concept ◽

Voting Games ◽

Voting Game ◽

The Stability ◽

The One

This paper deals with the non-emptiness of the stability set for any proper voting game. We present an upper bound on the number of alternatives which guarantees the non emptiness of this solution concept. We show that this bound is greater than or equal to the one given by Le Breton and Salles (1990) for quota games.

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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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