Comparing votes and seats with a diagonal (dis-) proportionality measure, using the slope-diagonal deviation (SDD) with cosine, sine and sign

Colignatus, Thomas (2017): Comparing votes and seats with a diagonal (dis-) proportionality measure, using the slope-diagonal deviation (SDD) with cosine, sine and sign.

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 - with 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 w = v / V and z = s / S. Then k = Cos[w, z] is a symmetric measure of similarity of the two vectors, θ = ArcCos[k] is the angle between the two vectors, and Sin[θ] = Sqrt[1 – b p] is a measure of disproportionality along the diagonal in {w, z} space. The geometry that uses Sin appears to be less sensitive than voters, representatives and researchers are to disproportionalities. This likely relates to the Weber-Fechner law. A disproportionality measure with improved sensitivity for human judgement is 10 √Sin[θ]. This puts an emphasis on the first digits of a scale of 10, which can be seen as an inverse (Bart Simpson) report card. The suggested measure has a sound basis in the theory of voting and statistics. The measure of 10 √Sin[θ] satisfies the properties of a metric and may be called the slope-diagonal deviation (SDD) metric. The cosine is the geometric mean of the slope b of the regression through the origin of z given w and slope p of w given z. The sine uses the deviation of this mean from the diagonal. The paper provides (i) theoretical foundations, (ii) evaluation of the relevant literature in voting theory and statistics, (iii) example outcomes of both theoretical cases and the 2017 elections in Holland, France and the UK, and (iv) comparison to other disproportionality measures and scores on criteria. Using criteria that are accepted in the voting literature, SDD appears to be better than currently available measures. It is rather amazing that the measure has not been developed a long time ago and been used for long. My search in the textbooks and literature has its limits however. A confusing aspect of variables in the unit simplex is that "proportionality" concerns only the diagonal in the {w, z} scatter plot while generally (e.g. in non-normalised space) any line through the origin is proportional.

• Item Type: MPRA Paper
• Institution: Thomas Cool Consultancy & Econometrics
• Language: English
• Keywords: General Economics, Social Choice, Social Welfare, Election, Majority Rule, Parliament, Party System, Representation, Proportion, District, Voting, Seat, Metric, Euclid, Distance, Cosine, Sine, Gallagher, Loosemore-Hanby, Sainte-Laguë, Largest Remainder, Webster, Jefferson, Hamilton, Slope Diagonal Deviation, Correlation, Diagonal regression, Regression through the origin, Apportionment, Disproportionality, Equity, Inequality, Lorenz, Gini coefficient
• Subjects: A - General Economics and Teaching > A1 - General Economics > A10 - General; D - Microeconomics > D6 - Welfare Economics > D63 - Equity, Justice, Inequality, and Other Normative Criteria and Measurement; D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations; D - Microeconomics > D7 - Analysis of Collective Decision-Making > D72 - Political Processes: Rent-Seeking, Lobbying, Elections, Legislatures, and Voting Behavior

References

Colignatus is the name in science of Thomas Cool, econometrician and teacher of mathematics, Scheveningen, Holland, http://econpapers.repec.org/RAS/pco170.htm

Belov, D. and R.D. Armstrong (2011), "Distributions of the Kullback-Leibler divergence with applications", J.Math. Stat. Psychol., May, 64(Pt 2):291-309, https://www.ncbi.nlm.nih.gov/pubmed/21492134, res. report: https://www.lsac.org/docs/default-source/research-(lsac-resources)/rr-09-02.pdf

Caulfield, M.J. (2010), "Apportioning Representatives in the United States Congress", MAA Convergence, https://www.maa.org/press/periodicals/convergence/apportioning-representatives-in-the-united-states-congress-introduction

Colignatus, Th. (2006), “On the sample distribution of the adjusted coefficient of determination (R2Adj) in OLS”, http://library.wolfram.com/infocenter/MathSource/6269/

Colignatus, Th. (2007), "Correlation and regression in contingency tables. A measure of association or correlation in nominal data (contingency tables), using determinants", https://mpra.ub.uni-muenchen.de/3660/

Colignatus, Th. (2009, 2015), "Elegance with Substance", https://zenodo.org/record/291974

Colignatus, Th. (2010), "Single vote multiple seats elections. Didactics of district versus proportional representation, using the examples of the United Kingdom and The Netherlands", https://mpra.ub.uni-muenchen.de/22782/

Colignatus, Th. (2011), "Conquest of the Plane", https://zenodo.org/record/291972

Colignatus, Th. 2014), "Voting Theory for Democracy", Thomas Cool Consultancy & Econometrics, https://zenodo.org/record/291985

Colignatus, Th. (2017a), "The performance of four possible rules for selecting the Prime Minister after the Dutch Parliamentary elections of March 2017", https://mpra.ub.uni-muenchen.de/77616/

Colignatus, Th. (2017b), "Two conditions for the application of Lorenz curve and Gini coefficient to voting and allocated seats", https://mpra.ub.uni-muenchen.de/80297/

Dongen, S. van & A.J. Enright (2012), "Metric distances derived from cosine similarity and Pearson and Spearman correlations", https://arxiv.org/abs/1208.3145

Dumont, P. & J.-F. Caulier (2003), "The "effective number of relevant parties": How voting power improves Laakso-Taagepera's index", http://centres.fusl.ac.be/CEREC/document/2003/cerec2003_7.pdf

Draper, N.R. & Y. Yang (1997), "Generalization of the geometric mean functional relationship", Computational Statistics & Data Analysis, Volume 23, Issue 3, 9 January 1997, Pages 355-372; Preprint 1995 Technical report no 943, http://www.stat.wisc.edu/node/1470

Eisenhauer, J.G. (2003), "Regression through the origin", Teaching Statistics. Volume 25, Number 3, Autumn 2003, p76-80

Erb, I. and C. Notredame (2016), "How should we measure proportionality on relative gene expression data?", Theory Biosci. (2016) 135:21–36, https://link.springer.com/content/pdf/10.1007%2Fs12064-015-0220-8.pdf or https://link.springer.com/article/10.1007%2Fs12064-015-0220-8

Gallagher, M. (1991), "Proportionality, disproportionality and electoral systems", Electoral Studies, 10:1, 33-51, https://www.tcd.ie/Political_Science/draft/staff/michael_gallagher/ElectoralStudies1991.pdf

Gallagher, M. (2007), Electoral Systems Web Site, Dublin, Trinity College, http://www.tcd.ie/Political_Science/Staff/Michael.Gallagher/ElSystems/index.php

Gallagher, M. (2017), "Election indices dataset", accessed at 2017-07-31, http://www.tcd.ie/Political_Science/staff/michael_gallagher/ElSystems/Docts/ElectionIndices.pdf

Goldenberg, J. & S.D. Fisher (2017), "The Sainte-Laguë index of disproportionality and Dalton’s principle of transfers", Accepted for publication in Party Politics, March 2017,

Hill, I.D. (1997), "Measuring proportionality", Voting matters, 8, p7-8, http://www.mcdougall.org.uk/VM/VOL1/ISSUE01-23.pdf

Johnston, J. (1972), "Econometric methods", 2nd edition, McGraw-Hill

Karpov, A. (2008), "Measurement of disproportionality in proportional representation systems", Mathematical and Computer Modelling, Volume 48, Issues 9–10, November 2008, Pages 1421-1438, http://www.sciencedirect.com/science/article/pii/S0895717708001933

Kestelman, P. (2005), "Apportionment and Proportionality: A Measured View", Voting Matters, 20, p 12-22, http://www.votingmatters.org.uk/ISSUE20/I20P4.PDF

Koppel, M, and A. Diskin (2009), "Measuring disproportionality, volatility and malapportionment: axiomatization and solutions", Social Choice and Welfare, August, 33:281, https://www.researchgate.net/publication/225444815_Measuring_disproportionality_volatility_and_malapportionment_Axiomatization_and_solutions

Laakso, M. (1980), "Electoral Justice as a criterion for different systems of proportional representation", Scandinavian Political Studies, Bind 3 (New Series) (1980) 3, https://tidsskrift.dk/scandinavian_political_studies/article/view/32355/30159

Laakso, M., & R. Taagepera (2007), "Proportional representation in Scandinavia: Implications for Finland", Scandinavian Political Studies, https://www.researchgate.net/publication/230000636_Proportional_Representation_in_Scandinavia_Implications_for_Finland

Leznik, M. & C. Tofallis (2005), "Estimating Invariant Principal Components Using Diagonal Regression", http://researchprofiles.herts.ac.uk/portal/en/publications/estimating-invariant-principal-components-using-diagonal-regression(d3379080-bda2-4d76-ab8d-b557c7fabea1).html

Lovell, D. V. Pawlowsky-Glahn, J.J. Egozcue, S. Marguerat, J. Bähler (2015), "Proportionality: A Valid Alternative to Correlation for Relative Data", PLOS Computational Biology, http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1004075

Malkevitch, J. (2002), "Apportionment", AMS Feature Column, http://www.ams.org/samplings/feature-column/fcarc-apportion1

Malkevitch, J. (2017), "Pairwise Equity in Apportionment (2017)", https://www.york.cuny.edu/~malk/gametheory/tc-2017-pairwise-apportion.html

NCSS Statistical Software (undated), "Lin's concordance correlation coefficient", Chapter 301 of the documentation, https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Lins_Concordance_Correlation_Coefficient.pdf

Pawlowsky-Glahn, V., J.J. Egozcue, R. Meziat (2007), "The statistical analysis of compositional data: The Aitchison geometry", https://laboratoriomatematicas.uniandes.edu.co/cursocoda/04-Vera-geometry.pdf

Pukelsheim, F. (2014), "Proportional representation. Apportionment methods and their applications", Springer,

Renwick, A. (2015), "Electoral Disproportionality: What Is It and How Should We Measure It?", http://blogs.reading.ac.uk/readingpolitics/2015/06/29/electoral-disproportionality-what-is-it-and-how-should-we-measure-it/

Samuelson, P. (1942), " A Note on Alternative Regressions", Econometrica, Vol. 10, No. 1 (Jan., 1942), pp. 80-83, : http://www.jstor.org/stable/1907024

Taagepera, R. and M.S. Shugart (1989), "Seats and Votes", Yale

Taagepera, R. and M. Laakso (2006), "Proportionality Profiles of West European Electoral Systems", European Journal of Political Research 8(4):423 - 446 · May 2006, https://www.researchgate.net/publication/230041613_Proportionality_Profiles_of_West_European_Electoral_Systems

Taagepera, R. and B. Grofman (2003), "Mapping the indices of seats-votes disproportionality and inter-election volatility", Party Politics, 9(6), p659-677, http://escholarship.org/uc/item/0m9912ff#page-1

Tofallis, C. (2000), "Multiple Neutral Regression", Operational Research Paper 14, UHBS 2000:13, http://uhra.herts.ac.uk/bitstream/handle/2299/689/S7.pdf?sequence=1

Quinn, T. (2017), "An introduction to proportionality", https://cran.r-project.org/web/packages/propr/vignettes/a_introduction.html

Zand, M.S., J. Wang, S. Hulchey (2015), "Graphical Representation of Proximity Measures for Multidimensional Data. Classical and Metric Multidimensional Scaling", The Mathematica Journal 17, http://www.mathematica-journal.com/2015/09/graphical-representation-of-proximity-measures-for-multidimensional-data/