Comparing votes and seats with cosine, sine and sign, with attention for the slope and enhanced sensitivity to disproportionality

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 Weber-Fechner 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 / Sainte-Laguë (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 ...