Colignatus, Thomas (2018): Measures of policy distance and inequality / disproportionality of votes and seats.

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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 let w = v / V and z = s / S be the perunages (or per ten or percentages). Let d[w, z] be the inequality / disproportionality of votes and seats. This can be the angle between the vectors (AID) and the sinediagonal (SDID) measure based upon this. Parties can also be scored with policy vector p, using a "lefttoright" policy scale [0, 10]. A common voterlegislative distance is the weighted average a = p' (z  w). With AID d[w, z] the present paper looks into the properties of d[p w, p z]. The latter term for variable w and z given p works as a disproportionality measure, and for variable p given w and z works as policy congruence. We can define an angular policy distance (APD) pd[w, z, p] that employs this d[p w, p z] properly. The APD is much more sensitive than the weighted average, but Sqrt[Abs[a]] has remarkably similar behaviour.
Item Type:  MPRA Paper 

Institution:  Thomas Cool Consultancy & Econometrics 
Original Title:  Measures of policy distance and inequality / disproportionality of votes and seats 
Language:  English 
Keywords:  Votes, Seats, Electoral System, Policy Distance, Disproportionality, Angular Distance, SineDiagonal Inequality / Disproportionality, LoosemoreHanby, Gallagher, Descriptive Statistics, Education 
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:  84324 
Depositing User:  Thomas Colignatus 
Date Deposited:  03 Feb 2018 14:21 
Last Modified:  28 Sep 2019 02:07 
References:  Colignatus is the name in science of Thomas Cool, econometrician and teacher of mathematics, Scheveningen, Holland, http://econpapers.repec.org/RAS/pco170.htm This text is also available at https://www.wolframcloud.com/objects/thomascool/Voting/20180202PolicyDistance.nb Colignatus, Th. (2017a), "Comparing votes and seats with cosine, sine and sign, with attention for the slope and enhanced sensitivity to disproportionality", https://mpra.ub.unimuenchen.de/81389/ Colignatus, Th. (2017b), "One woman, one vote. Though not in the USA, UK and France", https://mpra.ub.unimuenchen.de/82513/ Colignatus, Th. (2017c), "Statistics, slope, cosine, sine, sign, significance and Rsquared", https://boycottholland.wordpress.com/2017/10/21/statisticsslopecosinesinesignsignificanceandrsquared/ Colignatus, Th. (2018), "Comparing the Aitchison distance and the angular distance for use as inequality or disproportionality measures for votes and seats", https://www.wolframcloud.com/objects/thomascool/Voting/20180118Aitchison.nb Golder, M. & Stramski, J. (2010), "Ideological congruence and electoral institutions", American Journal of Political Science, 54(1), 90106, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.606.8475&rep=rep1&type=pdf Stokman, F., Knoop, J. van der & Oosten, R. C. H. van (2013), "Modeling collective decision making", in Nee, V., Snijders, T. A. B. & Wittek, R. (eds.) (2013), "Handbook of Rational Choice Social Research", Stanford University Press, p. 151182. See also https://www.rug.nl/staff/f.n.stokman, while Dutch readers may look at http://decide.nl 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/84324 