Colignatus, Thomas (2017): Comparing votes and seats with cosine, sine and sign, with attention for the slope and enhanced sensitivity to disproportionality.
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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 w = v / V and z = s / S the perunages (often percentages). There are slopes b and p from the regressions through the origin (RTO) z = b w + e and w = p z + ε. Then k = Cos[v, s] = Cos[w, z] = Sqrt[b p]. The geometric mean slope is a symmetric measure of similarity of the two vectors. θ = ArcCos[k] is the angle between the vectors. Thus Sin[v, s] = Sin[w, z] = Sin[θ] = Sqrt[1 – b p] is metric and a measure of disproportionality in general. Geometry appears to be less sensitive to disproportionalities than voters, representatives and researchers tend to be. This likely relates to the WeberFechner law. Covariance gives a sign for majority switches. A disproportionality measure with enhanced sensitivity for human judgement is the sine diagonal disproportionality SDD = sign 10 √Sin[v, s]. This puts an emphasis on the first digits of a scale of 10, which can be seen as an inverse (Bart Simpson) report card. What does disproportionality measure ? The unit of account can be either the party or the individual representative. This distinguishes between the party average and the party marginal candidate. The difference z – w is often treated as a level, and Webster / SainteLaguë (WSL) uses the relative expression z / w – 1. For the party marginal candidate z – w already is relative, with the unit of account of the individual representative in the denominator. The Hamilton Largest Remainder (HLR) apportionment has the representative as the unit of account. The "Representative Largest Remainder" (RLR) uses a 0.5 natural quota. 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.
Item Type:  MPRA Paper 

Institution:  Thomas Cool Consultancy & Econometrics 
Original Title:  Comparing votes and seats with cosine, sine and sign, with attention for the slope and enhanced sensitivity to disproportionality 
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, LoosemoreHanby, SainteLaguë, Largest Remainder, Webster, Jefferson, Hamilton, Sine Diagonal Disproportionality, 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 DecisionMaking > D71  Social Choice ; Clubs ; Committees ; Associations D  Microeconomics > D7  Analysis of Collective DecisionMaking > D72  Political Processes: RentSeeking, Lobbying, Elections, Legislatures, and Voting Behavior 
Item ID:  81389 
Depositing User:  Thomas Colignatus 
Date Deposited:  16 Sep 2017 14:08 
Last Modified:  27 Sep 2019 05:30 
References:  Colignatus is the name in science of Thomas Cool, econometrician and teacher of mathematics, Scheveningen, Holland, http://econpapers.repec.org/RAS/pco170.htm References in the text to Wikipedia refer to it as a portal and no source. Balinksi, M. and H.P. Young (1976), "Criteria for proportional representation", IIASA Research Report December, RR76020, http://pure.iiasa.ac.at/525/1/RR76020.pdf Balinksi, M. and H.P. Young (1980), "The Webster method of apportionment", Proceedings of the National Academy of Sciences 77(1):14, February, https://www.researchgate.net/publication/7186641_The_Webster_method_of_apportionment Belov, D. and R.D. Armstrong (2011), "Distributions of the KullbackLeibler divergence with applications", J.Math. Stat. Psychol., May, 64(Pt 2):291309, https://www.ncbi.nlm.nih.gov/pubmed/21492134, res. report: https://www.lsac.org/docs/defaultsource/research(lsacresources)/rr0902.pdf Beumer, M. (2010), "Apportionment in theory and practice", MSc Thesis in logic, UvA, http://www.illc.uva.nl/Research/Publications/Reports/MoL201007.text.pdf Borg, I. & P.J.F. Groenen (2005), "Modern Multidimensional Scaling", Springer, http://www.springer.com/gp/book/9780387251509 Carey, J. M., & Hix, S. (2009), "The Electoral Sweet Spot: LowMagnitude Proportional Electoral Systems", Working Paper (draft, colour graphs), http://www.lse.ac.uk/government/research/resgroups/PSPE/pdf/PSPE_WP1_09.pdf Carey, J. M., & Hix, S. (2011), "The Electoral Sweet Spot: LowMagnitude Proportional Electoral Systems", American Journal of Political Science, 55(2), 383397, Open Source http://onlinelibrary.wiley.com/doi/10.1111/j.15405907.2010.00495.x/full and http://personal.lse.ac.uk/hix/Working_Papers/CareyHixAJPS2011.pdf Caulfield, M.J. (2010), "Apportioning Representatives in the United States Congress", MAA Convergence, https://www.maa.org/press/periodicals/convergence/apportioningrepresentativesintheunitedstatescongressintroduction 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.unimuenchen.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.unimuenchen.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.unimuenchen.de/77616/ Colignatus, Th. (2017b), "Two conditions for the application of Lorenz curve and Gini coefficient to voting and allocated seats", https://mpra.ub.unimuenchen.de/80297/ Colignatus, Th. (2017c), "Comparing votes and seats with a diagonal (dis) proportionality measure, using the slopediagonal deviation (SDD) with cosine, sine and sign", first version of this paper, at MPRA, August 17, https://mpra.ub.unimuenchen.de/80833/, 1st revision August 24, https://mpra.ub.unimuenchen.de/80965/ 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 LaaksoTaagepera'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 355372; 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, p7680 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%2Fs1206401502208.pdf or https://link.springer.com/article/10.1007%2Fs1206401502208 Gallagher, M. (1991), "Proportionality, disproportionality and electoral systems", Electoral Studies, 10:1, 3351, https://www.tcd.ie/Political_Science/draft/staff/michael_gallagher/ElectoralStudies1991.pdf Gallagher, M. (1992), "Comparing proportional representation electoral systems: Quota's, thresholds, paradoxes and majorities", B.J. Pol.S. 22, 469496 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 20170731, http://www.tcd.ie/Political_Science/staff/michael_gallagher/ElSystems/Docts/ElectionIndices.pdf Goldenberg, J. & S.D. Fisher (2017), "The SainteLaguë index of disproportionality and Dalton’s principle of transfers", Accepted for publication in Party, March 2017, Hill, I.D. (1997), "Measuring proportionality", Voting matters, 8, p78, http://www.mcdougall.org.uk/VM/VOL1/ISSUE0123.pdf Johnston, J. (1972), "Econometric methods", 2nd edition, McGrawHill Karpov, A. (2008), "Measurement of disproportionality in proportional representation systems", Mathematical and Computer Modelling, Volume 48, Issues 9–10, November 2008, Pages 14211438, http://www.sciencedirect.com/science/article/pii/S0895717708001933 Politics Kestelman, P. (2005), "Apportionment and Proportionality: A Measured View", Voting Matters, 20, p 1222, 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 Kozak, A. and R. A. Kozak (1995), "Notes on regression through the origin", The Forestry Chronicle, May/June, Vol 71 no 3, p326330 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/estimatinginvariantprincipalcomponentsusingdiagonalregression(d3379080bda24d76ab8db557c7fabea1).html Lovell, D. V. PawlowskyGlahn, 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/featurecolumn/fcarcapportion1 Malkevitch, J. (2017), "Pairwise Equity in Apportionment (2017)", https://www.york.cuny.edu/~malk/gametheory/tc2017pairwiseapportion.html NCSS Statistical Software (undated), "Lin's concordance correlation coefficient", Chapter 301 of the documentation, https://ncsswpengine.netdnassl.com/wpcontent/themes/ncss/pdf/Procedures/NCSS/Lins_Concordance_Correlation_Coefficient.pdf PawlowskyGlahn, V., J.J. Egozcue, R. Meziat (2007), "The statistical analysis of compositional data: The Aitchison geometry", https://laboratoriomatematicas.uniandes.edu.co/cursocoda/04Verageometry.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/electoraldisproportionalitywhatisitandhowshouldwemeasureit/ Samuelson, P. (1942), " A Note on Alternative Regressions", Econometrica, Vol. 10, No. 1 (Jan., 1942), pp. 8083, : 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 seatsvotes disproportionality and interelection volatility", Party Politics, 9(6), p659677, http://escholarship.org/uc/item/0m9912ff#page1 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 Tullock, G. (2008), "Public Choice", The New Palgrave Dictionary of Economics, Second Edition, http://www.dictionaryofeconomics.com/article?id=pde2008_P000240&q=rational%20choice&topicid=&result_number=10 Quinn, T. (2017), "An introduction to proportionality", https://cran.rproject.org/web/packages/propr/vignettes/a_introduction.html Young, H.P. (2004), “Fairness in Apportionment”, Mimeo, Prepared for the U. S. Census Bureau Symposium, http://www.census.gov/history/pdf/Fairness_in_Apportionment_Young.pdf 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.mathematicajournal.com/2015/09/graphicalrepresentationofproximitymeasuresformultidimensionaldata/ 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/81389 
Available Versions of this Item

Comparing votes and seats with a diagonal (dis) proportionality measure, using the slopediagonal deviation (SDD) with cosine, sine and sign. (deposited 18 Aug 2017 22:21)

Comparing votes and seats with a diagonal (dis) proportionality measure, using the slopediagonal deviation (SDD) with cosine, sine and sign. (deposited 25 Aug 2017 16:23)
 Comparing votes and seats with cosine, sine and sign, with attention for the slope and enhanced sensitivity to disproportionality. (deposited 16 Sep 2017 14:08) [Currently Displayed]

Comparing votes and seats with a diagonal (dis) proportionality measure, using the slopediagonal deviation (SDD) with cosine, sine and sign. (deposited 25 Aug 2017 16:23)