Colignatus, Thomas (2017): Comparing votes and seats with a diagonal (dis) proportionality measure, using the slopediagonal deviation (SDD) with cosine, sine and sign.
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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  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 WeberFechner 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 slopediagonal 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 nonnormalised space) any line through the origin is proportional.
Item Type:  MPRA Paper 

Institution:  Thomas Cool Consultancy & Econometrics 
Original Title:  Comparing votes and seats with a diagonal (dis) proportionality measure, using the slopediagonal deviation (SDD) with cosine, sine and sign 
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, 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 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:  80965 
Depositing User:  Thomas Colignatus 
Date Deposited:  25 Aug 2017 16:23 
Last Modified:  07 Oct 2019 12:59 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/80965 
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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) [Currently Displayed]