# Comparing the Aitchison distance and the angular distance for use as inequality or disproportionality measures for votes and seats

Comparing the Aitchison distance and the angular distance for use as inequality or disproportionality measures for votes and seats

Didactics and routines using Mathematica

Thomas Colignatus

http://thomascool.eu

January 18 & February 6 2018

http://thomascool.eu

January 18 & February 6 2018

Abstract

Votes and seats satisfy only two of seven criteria for application of the Aitchison distance. Vectors of votes and seats, say for elections for political parties the House of Representatives, can be normalised to 1 or 100%, and then have the outward appearance of compositional data. The Aitchison geometry and distance for compositional data then might be considered for votes and seats too. However, there is an essential zero when a party gets votes but doesn’t gain a seat, and a zero gives an undefined logratio. In geology, changing from weights to volumes affects the percentages but not the Aitchison distance. For votes and seats there are no different scales or densities per party component however, and thus reportioning (perturbation) would be improper. Another key issue is subcompositional dominance. For votes {10, 20, 70} and seats {20, 10, 70} it is essential that we consider three parties. For a disproportionality measure we would value it positively that there is a match on 70. The Aitchison distance compares the ratios {10, 20, 70} / {20, 10, 70} = {1/2, 2, 1} with {20, 10, 70} / {20, 10, 70} = {1, 1, 1}, and then neglects a ratio equal to 1, since Log[1] = 0. In this case it essentially compares the subcompositions, i.e. votes {10, 20} and seats {20, 10}, rescales to {1/3, 2/3} and {2/3, 1/3}, and finds high disproportionality. This means that it essentially looks at a two party outcome instead of a three party outcome. It follows that votes and seats are better served by another distance measure. Suggested is the angular distance and the Sine-Diagonal Inequality / Disproportionality (SDID) measure based upon this. Users may of course apply both the angular and the Aitchison measures while being aware of the crucial differences in properties.

Keywords

Votes, Seats, Electoral System, Distance, Disproportionality, Aitchison Geometry, Angular Distance, Sine-Diagonal Inequality / Disproportionality, Loosemore-Hanby, Gallagher, Descriptive Statistics, Education, Reportion

MSC2010

00A69 General applied mathematics

28A75 Measure and integration. Length, area, volume, other geometric measure theory

62J20 Statistics. Diagnotics

97M70 Mathematics education. Behavioral and social sciences

28A75 Measure and integration. Length, area, volume, other geometric measure theory

62J20 Statistics. Diagnotics

97M70 Mathematics education. Behavioral and social sciences

JEL

A100 General Economics: General

D710 Social Choice; Clubs; Committees; Associations,

D720 Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

D630 Equity, Justice, Inequality, and Other Normative Criteria and Measurement

D710 Social Choice; Clubs; Committees; Associations,

D720 Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

D630 Equity, Justice, Inequality, and Other Normative Criteria and Measurement

Cloud

This paper is also available at (updated version with link unchanged):

https://www.wolframcloud.com/objects/thomas-cool/Voting/2018-01-18-Aitchison.nb

I thank Stephen Wolfram for making this analysis possible in the Wolfram Cloud. I thank professors Vera Pawlowsky and Josep Antoni Martín-Fernández of UdG for essential comments. All errors remain mine.

### Contents

Contents

### Start (evaluate this subsection for the initialisation packages)

Start (evaluate this subsection for the initialisation packages)

## 1. Introduction

1. Introduction

## 2. Euclidean improduct, norm, distance and angle

2. Euclidean improduct, norm, distance and angle

## 3. Theoretical requirements for a norm

3. Theoretical requirements for a norm

## 4. Inequality / disproportionality measures of votes and seats

4. Inequality / disproportionality measures of votes and seats

## 5. Compositional data

5. Compositional data

## 6. Log Closure Plus One

6. Log Closure Plus One

## 7. The Aitchison geometry

7. The Aitchison geometry

## 8. Aitchison geometry and voting data

8. Aitchison geometry and voting data

## 9. Scoring parties on a policy scale

9. Scoring parties on a policy scale

## 10. References in the voting literature to Aitchison’s analysis

10. References in the voting literature to Aitchison’s analysis

## 11. Conclusion

11. Conclusion

## Appendix A. Formal properties of the Aitchison distance

Appendix A. Formal properties of the Aitchison distance

## Appendix B. Approximation to the angle

Appendix B. Approximation to the angle

## Appendix C. Votes and seats in Catalunya December 2017

Appendix C. Votes and seats in Catalunya December 2017

## Appendix D. Aitchison distance and Webster / Sainte-Laguë

Appendix D. Aitchison distance and Webster / Sainte-Laguë

## Links

Links

## References

References