Measures of policy distance and
inequality / disproportionality of votes
and seats
Colignatus, Thomas
Thomas Cool Consultancy Econometrics
2 February 2018
Online at
https://mpra.ub.uni-muenchen.de/84324/
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Discussion of electoral systems: https://mpra.ub.uni-muenchen.de/82513/
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https://en.wikipedia.org/wiki/United_States_House_of_Representatives_elections,_2016#Results_summary
vts UnitÐ votes , sts UnitÐ seats N
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Show ListLinePlot 0, 0 , 6 1, 1 ,
ListPlot MapThread Labeled, Transpose vts, sts , Dem, Rep, Other , PlotStyle PointSize Large , AxesLabel "Votes Ð ", "Seats Ð " , AspectRatio 1, BaseStyle FontSize 13
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https://mpra.ub.uni-muenchen.de/81389/
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votes dem 61 776 554, rep 63 173 815, 128 627 010 rep dem ; seats 194, 241, 0 ;
w UnitÐ votes , z UnitÐ seats N
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policy 4, 6, 5 ;
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pdorg AngularPolicyDistance w, z, policy N
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0.877463, 0.259108
eq AngularPolicyDistance votes, seats, 4, 6 dp, 5 AngularPolicyDistance votes, newseats, 4, 6, 5 ;
sol FindRoot eq, dp, 0
dp 0.832026
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AxesLabel "t", "d p w, p z " , PlotRange 0, Ð
GraphicsGrid plaid 0, .1, 4
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AngularPolicyDistance w, z, p
, t, 0, Ð , r, 0, Ð , AxesLabel "t", "r", " d p w, p z \n & APD "
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r, 0, Ð , AxesLabel "t", "r", " d w, z \nd p w, p z "
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r, 0, Ð , AxesLabel "t", "r", " Sqrt d w, z \nd p w, p z "
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, r, 0, Ð ,
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t, 0, Ð , AxesLabel "t", " d w, z and d w^2, z^2 "
2 4 6 8 10 t
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2 4 6 8 10
Seats Democrats
Ð
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p . z w Ð , t, 0, Ð , r, 0, Ð , AxesLabel "t", "r", " W. Av."
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4, 6 . seats3 ds 1, 1 votes3 , dp Simplify
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p1 Plot wavlin1 t , wavlin2 t, dif , t, 0, Ð ,
AxesLabel "Seats\n Democrats\n Ð ", "Policy distance to the right Ð " , PlotRange 4.8, 5.2 , .03, .06 ;
p2 Graphics Line new, 0 , new, wavlin1 new , 4.9, wavlin2 4.9, dif , 4.9, 0 ; Show
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t, 0, Ð , r, 0, Ð , AxesLabel "t", "r", "Disp or\nW. Av."
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Plot Evaluate p r, Ð r ;
AngularPolicyDistance votes3, seats3, p , Sqrt Abs p . seats3 votes3 , r, 0, Ð ,
AxesLabel "Policy stance\nfirst party Ð \n& 2nd party opposite", "Policy distance to the right Ð "
2 4 6 8 10
Policy stance first party Ð & 2nd party opposite 0.5
1.0 1.5
Policy distance to the right Ð
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Plot Evaluate p r, 7 ;
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, r, 0, Ð , AxesLabel "Policy stance\nfirst party Ð \n& 2nd party at 7", "Policy distance to the right Ð "
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Policy stance first party Ð & 2nd party at 7 0.5
1.0 1.5
Policy distance to the right Ð
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Plot Evaluate p r, Ð r ; wa p . seats3 votes3 Ð; AngularPolicyDistance votes3, seats3, p , Ð Sqrt Abs wa Ð , WAvSqrtPolicyDistance votes3, seats3, p , wa
, r, 0, Ð ,
AxesLabel "Policy stance\nfirst party Ð \n& 2nd party opposite", "Policy distance to the right Ð "
2 4 6 8 10
Policy stance first party Ð & 2nd party opposite
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Policy distance to the right Ð
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z t, Ð t ; w Ð t, t ; p r, Ð r ;
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2 4 6 8 10 ALHID Ð
0.5 1.0 1.5 2.0 Abs a Ð
ParametricPlot Evaluate
z t, Ð t ; w Ð t, t ; p r, Ð r ;
AbsLoosemoreHanbyID w, z , Abs p . z w Ð . t 4 , r, 0, Ð , AxesLabel "ALHID Ð ", "Abs a Ð "
1 2 3 4 ALHID Ð 0.5
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Plot Evaluate p r, 10 r ; wa p . seats3 votes3 10; AngularPolicyDistance votes3, seats3, p ,
Sqrt Abs wa 10 10, AngularID votes3, seats3 , wa , r, 0, 10 , AxesLabel
"Policy stance\nfirst party 10 \n& 2nd party opposite", "APD & WA scale 10 "
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Plot3D Evaluate z t, Ð t ; w Ð t, t ; p r, Ð r ;
AngularID w, p , t, 0, Ð , r, 0, Ð , AxesLabel "t", "r", " d w, p "
Plot3D Evaluate z t, Ð t ; w Ð t, t ; p r, Ð r ; Max AngularID z, p , AngularID w, p ,
t, 0, Ð , r, 0, Ð , AxesLabel "t", "r", " Max "
Max:Invalid comparison with 0. 1.34158 107 attempted. Max:Invalid comparison with 0. 1.34158 107 attempted. Max:Invalid comparison with 0. 1.34158 107 attempted.
General:Further output ofMax::nordwill be suppressed during this calculation.
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PrecisionGoal but was unable to find a sufficient decrease in the function. You may need more than MachinePrecision digits of working precision to meet these tolerances.
0.691983, polpos 7.17136
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polpos, 0, Ð ,
AxesLabel "Policy stance\n first party Ð ", "d v, p , d s, p Ð " , Plot MaxAngularPolicies 4.9, 5.1 , 6, 4 , polpos, 6 ,
polpos, 0, Ð , AxesOrigin 0, 0 ,
AxesLabel "Policy stance\n first party Ð ", "Max"
2 4 6 8 10
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2 4 6 8 10
Policy stance first party Ð 1 2 3 4 5 6 Max
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r, 0, Ð , AxesLabel "t", "r", " Combined "
Max:Invalid comparison with 0. 2.10734 108 attempted. Max:Invalid comparison with 0. 2.10734 108 attempted. Max:Invalid comparison with 0. 2.10734 108 attempted.
General:Further output ofMax::nordwill be suppressed during this calculation.
CombinedPolicyDistance a, b , a, b , r, Ð r
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FindMinimum:The line search decreased the step size to within the tolerance specified by AccuracyGoal and
PrecisionGoal but was unable to find a sufficient decrease in the function. You may need more than MachinePrecision digits of working precision to meet these tolerances.
0.0149279, polpos 6.
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Plot IndepAngPolicyDistance 4.9, 5.1 , 6, 4 , polpos, 6 ,
CombinedPolicyDistance 4.9, 5.1 , 6, 4 , polpos, 6 , polpos, 0, Ð
2 4 6 8 10
1 2 3 4
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differenceAndAngle p : AngularPolicyDistance votes3, seats3, p , APD p . seats3 votes3 Ð, WA or a
WAvAngPolicyMeasure votes3, seats3, p , wavapm
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policies 7, 2 , 6, 4 , 5, 5 , 7, 7 , 7, 8 , 4, 6 , 2, 7 , 0, 10 ; TableForm differenceAndAngle policies, TableHeadings
policies, "APD", "WA", "wavapm v, s, p ", "apm v, s, p ", "d p v, p s "
APD WA wavapm v, s, p apm v, s, p d p v, p s
7, 2 1.23481 0.1 0.131335 0.120049 0.134565
6, 4 0.731773 0.04 0.0689112 0.0195672 0.235047
5, 5 0 0 0 0 0.254614
7, 7 0 0 0 0 0.254614
7, 8 0.50673 0.02 0.0381195 0.00225085 0.252363
4, 6 0.731773 0.04 0.0689112 0.0195672 0.235047
2, 7 1.23481 0.1 0.131335 0.120049 0.134565
0, 10 1.80965 0.2 0.2 0.254614 0
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eq WAvAngPolicyMeasure votes, seats, 4, 6, 5 , Exponent f 0.12
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