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:  16 Sep 2017 14:09 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/81389 
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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)
 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)