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

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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.

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