# Measures of policy distance and inequality / disproportionality of votes and seats

Measures of policy distance and inequality / disproportionality of votes and seats

Didactics and routines using Mathematica

Thomas Colignatus

http://thomascool.eu

February 2 2018

http://thomascool.eu

February 2 2018

Abstract

Let v be a vector of votes for parties and s a vector of their seats gained in the House of Commons or the House of Representatives. We use a single zero for the lumped category of “Other”, of the wasted vote, for parties that got votes but no seats. Let V = 1’v be total turnout and S = 1’s the total number of seats, and let w = v / V and z = s / S be the perunages (or per ten or percentages). Let d[w, z] be the inequality / disproportionality of votes and seats. This can be the angle between the vectors(AID) and the sine-diagonal (SDID) measure based upon this. Parties can also be scored with policy vector p, using a “left-to-right” policy scale [0, 10]. A common voter-legislative distance is the weighted average a = p’ (z - w). With AID d[w, z] the present paper looks into the properties of d[p w, p z]. The latter term for variable w and z given p works as a disproportionality measure, and for variable p given w and z works as policy congruence. We can define an angular policy distance (APD) pd[w, z, p] that employs this d[p w, p z] properly. The APD is much more sensitive than the weighted average, but Sqrt[Abs[a]] has remarkably similar behaviour.

Keywords

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

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

### Contents

Contents

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

Start (evaluate this subsection for the initialisation packages)

## 1. Introduction

1. Introduction

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

2. Inequality / disproportionality measures of votes and seats

## 3. Before and after (half) elections

3. Before and after (half) elections

## 4. Properties and first distinctions

4. Properties and first distinctions

## 5. Basics of the Angular Policy Distance (APD)

5. Basics of the Angular Policy Distance (APD)

## 6. Properties and application of the APD

6. Properties and application of the APD

## 7. Focusing on d[p w, p z] (APT)

7. Focusing on d[p w, p z] (APT)

## 8. The weighted average policy distance (WA)

8. The weighted average policy distance (WA)

## 9. Relation of the WA to ALHID and APD

9. Relation of the WA to ALHID and APD

## 10. Conclusion

10. Conclusion

## Appendix A. Using d[w, p] and d[z, p]

Appendix A. Using d[w, p] and d[z, p]

## Appendix B. d[p w, p z] for ALHID

Appendix B. d[p w, p z] for ALHID

## Appendix C. Combining WA with the RPM

Appendix C. Combining WA with the RPM

## Appendix D. ||p - p’|| on top of d[w, z]

Appendix D. ||p - p’|| on top of d[w, z]

## References

References

Colignatus is the name in science of Thomas Cool, econometrician and teacher of mathematics, Scheveningen, Holland.

Colignatus, Th. (2017a), “Comparing votes and seats with cosine, sine and sign, with attention for the slope and enhanced sensitivity to disproportionality”, https://mpra.ub.uni-muenchen.de/81389/

Colignatus, Th. (2017b), “One woman, one vote. Though not in the USA, UK and France”, https://mpra.ub.uni-muenchen.de/82513/

Colignatus, Th. (2017c), “Statistics, slope, cosine, sine, sign, significance and R-squared”, https://boycottholland.wordpress.com/2017/10/21/statistics-slope-cosine-sine-sign-significance-and-r-squared/

Colignatus, Th. (2018), “Comparing the Aitchison distance and the angular distance for use as inequality or disproportionality measures for votes and seats”, https://www.wolframcloud.com/objects/thomas-cool/Voting/2018-01-18-Aitchison.nb

Golder, M. & Stramski, J. (2010), “Ideological congruence and electoral institutions”, American Journal of Political Science, 54(1), 90-106, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.606.8475&rep=rep1&type=pdf

Stokman, F., Knoop, J. van der & Oosten, R. C. H. van (2013), “Modeling collective decision making”, in Nee, V., Snijders, T. A. B. & Wittek, R. (eds.) (2013), “Handbook of Rational Choice Social Research”, Stanford University Press, p. 151-182. See also https://www.rug.nl/staff/f.n.stokman, while Dutch readers may look at http://decide.nl