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On the Approximability of the Yolk in the Spatial Model of Voting

Ran Hu, James P. Bailey

TL;DR

It is shown that for an even number of voters in a two-dimensional space that the yolk radius is at most twice the size of the LP yolk radius, and that for all other settings, the LP yolk can be arbitrarily small relative to the yolk.

Abstract

In the spatial model of voting, the yolk and LP (linear programming) yolk are important solution concepts for predicting outcomes for a committee of voters. McKelvey and Tovey showed that the LP yolk provides a lower bound approximation for the size of the yolk and there has been considerable debate on whether the LP yolk is a good approximation of the yolk. In this paper, we show that for an odd number of voters in a two-dimensional space that the yolk radius is at most twice the size of the LP yolk radius. However, we also show that (1) even in this setting, the LP yolk center can be arbitrarily far away from the yolk center (relative to the radius of the yolk) and (2) for all other settings (an even number of voters or in dimension $k\geq 3$) that the LP yolk can be arbitrarily small relative to the yolk. Thus, in general, the LP yolk can be an arbitrarily poor approximation of the yolk.

On the Approximability of the Yolk in the Spatial Model of Voting

TL;DR

It is shown that for an even number of voters in a two-dimensional space that the yolk radius is at most twice the size of the LP yolk radius, and that for all other settings, the LP yolk can be arbitrarily small relative to the yolk.

Abstract

In the spatial model of voting, the yolk and LP (linear programming) yolk are important solution concepts for predicting outcomes for a committee of voters. McKelvey and Tovey showed that the LP yolk provides a lower bound approximation for the size of the yolk and there has been considerable debate on whether the LP yolk is a good approximation of the yolk. In this paper, we show that for an odd number of voters in a two-dimensional space that the yolk radius is at most twice the size of the LP yolk radius. However, we also show that (1) even in this setting, the LP yolk center can be arbitrarily far away from the yolk center (relative to the radius of the yolk) and (2) for all other settings (an even number of voters or in dimension ) that the LP yolk can be arbitrarily small relative to the yolk. Thus, in general, the LP yolk can be an arbitrarily poor approximation of the yolk.

Paper Structure

This paper contains 8 sections, 12 theorems, 25 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Notation
  4. Relationship Between Yolks and LP Yolks: The Worst Case
  5. Properties of Yolks and LP Yolks
  6. Upper Bound on the Ratio Between the Yolk and LP Yolk in two dimensions
  7. Conclusions and Future Directions
  8. Additional Proofs

Key Result

Theorem 1

Let $I\subset \mathbb{R}^k$. The LP yolk radius can be arbitrarily small relative to the yolk radius when (1) $k\geq 3$ or (2) $k=2$ and $|I|$ is even.

Figures (13)

  • Figure 1: Examples of limiting median hyperplanes and non-limiting median hyperplanes in $2$-dimension space. The black points are voters, the dotted black circle is yolk and the blue circle is LP yolk. The hyperplane $\textbf{H} _1(a_1,b_1), \textbf{H} _2(a_2,b_2)$ and $\textbf{H} _4(a_4,b_4)$ are limiting median hyperplanes, and $\textbf{H} _3(a_3,b_3), \textbf{H} _5(a_5,b_5)$ are non-limiting median hyperplanes. $\textbf{H} _ 3(a_3,b_3)$ rotates $\delta$ with $i$ as the center, that is, $\textbf{H} _ 4(a_4,b_4)$.
  • Figure 2: A set of non-degenerate ideal points yielding an arbitrarily small LP Yolk.
  • Figure 3: Set of limiting median hyperplanes and the resulting LP Yolk.
  • Figure 4: Set of ideal points and the yolk yielding a ratio of $\frac{1}{2}+\epsilon$.
  • Figure 5: Set of limiting median lines and a ball that intersects them all. The measurements indicate the distance between the two balls and the height of the new ball that intersects all limiting median hyperplanes.
  • ...and 8 more figures

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 1
  • proof
  • Proposition 2: First observed in stone1992limiting
  • ...and 22 more