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


  • Colignatus, Thomas


When v is a vector of votes for parties and s is 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 - and when V = 1'v is 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] is a measure of disproportionality along the diagonal. 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 slopes of the regressions through the origin of z given w and 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 element is that voting theorists speak about "proportionality" only for the diagonal while in mathematics and statistics any line through the origin is proportional.

Suggested Citation

  • Colignatus, Thomas, 2017. "Comparing votes and seats with a diagonal (dis-) proportionality measure, using the slope-diagonal deviation (SDD) with cosine, sine and sign," MPRA Paper 80833, University Library of Munich, Germany, revised 17 Aug 2017.
  • Handle: RePEc:pra:mprapa:80833

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    References listed on IDEAS

    1. Draper, Norman R. & Yang, Yonghong (Fred), 1997. "Generalization of the geometric mean functional relationship," Computational Statistics & Data Analysis, Elsevier, vol. 23(3), pages 355-372, January.
    2. Colignatus, Thomas, 2007. "Correlation and regression in contingency tables. A measure of association or correlation in nominal data (contingency tables), using determinants," MPRA Paper 3394, University Library of Munich, Germany, revised 07 Jun 2007.
    3. Colignatus, Thomas, 2010. "Single vote multiple seats elections. Didactics of district versus proportional representation, using the examples of the United Kingdom and The Netherlands," MPRA Paper 22671, University Library of Munich, Germany, revised 06 Jul 2007.
    4. Colignatus, Thomas, 2017. "Two conditions for the application of Lorenz curve and Gini coefficient to voting and allocated seats," MPRA Paper 80297, University Library of Munich, Germany, revised 21 Jul 2017.
    5. Colignatus, Thomas, 2009. "Elegance with substance," MPRA Paper 15173, University Library of Munich, Germany.
    6. Moshe Koppel & Abraham Diskin, 2009. "Measuring disproportionality, volatility and malapportionment: axiomatization and solutions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 33(2), pages 281-286, August.
    7. Colignatus, Thomas, 2017. "The performance of four possible rules for selecting the Prime Minister after the Dutch Parliamentary elections of March 2017," MPRA Paper 77616, University Library of Munich, Germany, revised 17 Mar 2017.
    8. Tofallis, C., 2000. "Multiple Neutral Regression," Papers 2000:13, University of Hertfordshire - Business Schoool.
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    More about this item


    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;

    JEL classification:

    • A10 - General Economics and Teaching - - General Economics - - - General
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

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