Re: Level of proportionality for two-seat PR methods
Toby Pereira via Election-Methods <[email protected]> Sat, 13 Jun 2026 16:57:07 +0000 (UTC)
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For n seats and a 0 to 1 scale for candidate positions, for your Spatial D=
roop proportionality, the candidates would be at 1/(n+1), 2/(n+1), ... , n/=
(n+1). And for independent wings (if they're still called wings with multip=
le seats), it would be 0.5/n, 1.5/n, ... , (n-0.5)/n. Bloc majoritarian wou=
ld still be with them all in the middle.
So in the general case for Spatial Droop q is 1/(n+1) rather than specifica=
lly 1/3.
As for the overall formula, you can see it as looking for the mid-points in=
the cells given n equal-sized cells (for n seats), but with different extr=
eme points for the far left and far right cell.
So in the 2-candidate case:
When q =3D 1/4, the extreme points are just 0 and 1When q =3D 1/3, the extr=
eme points are 1/6 and 5/6When q =3D 1/2, the extreme points are 1/2 and 1/=
2 (the cells have no size, forcing everything into the middle)
In the general case:
The extreme points for wings would always be 0 and 1The extreme points for =
Droop would be 1/(2(n+1)) and 1-1/(2(n+1))The extreme points for majoritari=
an would always both be 1/2
q for the "wings" position is 1/(2n)q for Droop is 1/(n+1)q for majoritaria=
n is 1/2
I think the formula for the extreme points would be 2q - 0.5 for a 0 to 1 s=
cale.
Some of this might make sense.
Toby
On Saturday, 13 June 2026 at 01:54:18 BST, Kristofer Munsterhjelm via E=
lection-Methods <[email protected]> wrote: =20
=20
As mentioned in my previous post, I extended my PR measuring code to=20
consider different degrees of proportionality.
I haven't found a way to generalize proportionality degrees for any=20
number of seats (I should read that post, I suppose...) but for two=20
seats, I figured that it's not too hard. Since the voter opinion space=20
distribution is a standard normal, it's symmetric around zero, so=20
there's no reason for the method to prefer left-wing to right-wing=20
candidates (or vice versa). Thus, the proportionality level can be=20
parameterized by just how far from the median the two elected candidates=20
lie.
That is, the error function is
=C2=A0=C2=A0=C2=A0 sqrt((x_1 - y_1)^2 + (x_2 - y_2)^2)
and can be parameterized by a quantile level q, so that y_1 is the=20
position corresponding to the qth quantile of the voter opinion space=20
distribution, and y_2 is the (1-q)th quantile; and x_1 and x_2 is the=20
location of the leftmost and rightmost elected candidate in opinion space.
The "significant" values of q, or at least those that come most readily=20
to mind as distinct, are, for two seats:
=C2=A0=C2=A0=C2=A0 q =3D 0
=C2=A0=C2=A0=C2=A0 =C2=A0=C2=A0=C2=A0 as factional as possible, usually not=
a good idea, but perhaps useful=20
for the unanimity setting I mentioned earlier.
=C2=A0=C2=A0=C2=A0 q =3D 1/4
=C2=A0=C2=A0=C2=A0 =C2=A0=C2=A0=C2=A0 This is the "independent wings" posit=
ion, where to elect a council,=20
you split the voters into two halves (left-of-center and=20
right-of-center) and elect the centrist from each (i.e. the=20
left-wingers' internal median and the right-wingers' internal median).=20
The median is at q =3D 1/2, so a median of the left half is 1/4.
=C2=A0=C2=A0=C2=A0 q =3D 1/3
=C2=A0=C2=A0=C2=A0 =C2=A0=C2=A0=C2=A0 Spatial Droop proportionality.
=C2=A0=C2=A0=C2=A0 q =3D 1/2
=C2=A0=C2=A0=C2=A0 =C2=A0=C2=A0=C2=A0 Bloc majoritarian voting (elect as ma=
ny median voter candidates as you=20
can).
The VSE is then a goodness-of-fit value (and is the maximum VSE that=20
method can get at any q, grid search optimization inaccuracies=20
notwithstanding). A low value means that even the best fit doesn't fit=20
very well, and thus that the method has trouble being consistently=20
proportional at any level. High values mean that the particular fit is a=20
very good one.
-km
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Election-Methods mailing list - see https://electorama.com/em for list info
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<html><head></head><body><div class=3D"ydp56533b20yahoo-style-wrap" style=
=3D"font-family:Helvetica Neue, Helvetica, Arial, sans-serif;font-size:13px=
;"><div></div>
<div dir=3D"ltr" data-setdir=3D"false"><span><span style=3D"color: =
rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">F=
or n seats and a 0 to 1 scale for candidate positions, for your Spatial Dro=
op proportionality, the candidates would be at 1/(n+1), 2/(n+1), ... , n/(n=
+1). And for independent wings (if they're still called wings with multiple=
seats), it would be 0.5/n, 1.5/n, ... , (n-0.5)/n. Bloc majoritarian would=
still be with them all in the middle.</span></span><br></div><div dir=3D"l=
tr" data-setdir=3D"false"><span><span style=3D"color: rgb(0, 0, 0); font-fa=
mily: Helvetica Neue, Helvetica, Arial, sans-serif;"><br></span></span></di=
v><div dir=3D"ltr" data-setdir=3D"false"><span><span style=3D"color: rgb(0,=
0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">So in t=
he general case for Spatial Droop q is 1/(n+1) rather than specifically 1/3=
.</span></span></div><div dir=3D"ltr" data-setdir=3D"false"><span><span sty=
le=3D"color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, s=
ans-serif;"><br></span></span></div><div dir=3D"ltr" data-setdir=3D"false">=
<span><div><div>As for the overall formula, you can see it as looking for t=
he mid-points in the cells given n equal-sized cells (for n seats), but wit=
h different extreme points for the far left and far right cell.</div><div><=
br></div><div>So in the 2-candidate case:</div><div><br></div><div>When q =
=3D 1/4, the extreme points are just 0 and 1</div><div>When q =3D 1/3, the =
extreme points are 1/6 and 5/6</div><div>When q =3D 1/2, the extreme points=
are 1/2 and 1/2 (the cells have no size, forcing everything into the middl=
e)</div><div><br></div><div>In the general case:</div><div><br></div><div>T=
he extreme points for wings would always be 0 and 1</div><div dir=3D"ltr" d=
ata-setdir=3D"false">The extreme points for Droop would be 1/(2(n+1)) and 1=
-<span><span style=3D"color: rgb(0, 0, 0); font-family: Helvetica Neue, Hel=
vetica, Arial, sans-serif;">1/(2(n+1))</span></span></div><div>The extreme =
points for majoritarian would always both be 1/2</div><div><br></div><div>q=
for the "wings" position is 1/(2n)</div><div>q for Droop is 1/(n+1)</div><=
div>q for majoritarian is 1/2</div><div><br></div><div dir=3D"ltr" data-set=
dir=3D"false">I think the formula for the extreme points would be 2q - 0.5 =
for a 0 to 1 scale.</div></div><div dir=3D"ltr" data-setdir=3D"false"><br><=
/div><div dir=3D"ltr" data-setdir=3D"false">Some of this might make sense.<=
/div><div dir=3D"ltr" data-setdir=3D"false"><br></div><div dir=3D"ltr" data=
-setdir=3D"false">Toby</div></span></div><div dir=3D"ltr" data-setdir=3D"fa=
lse"><br></div><div><br></div>
=20
</div><div id=3D"ydpc47340f5yahoo_quoted_2234304791" class=3D"ydpc4=
7340f5yahoo_quoted">
<div style=3D"font-family:'Helvetica Neue', Helvetica, Arial, s=
ans-serif;font-size:13px;color:#26282a;">
=20
<div>
On Saturday, 13 June 2026 at 01:54:18 BST, Kristofe=
r Munsterhjelm via Election-Methods <[email protected]=
om> wrote:
</div>
<div><br></div>
<div><br></div>
=20
=20
<div><div dir=3D"ltr">As mentioned in my previous post, I e=
xtended my PR measuring code to <br></div><div dir=3D"ltr">consider differe=
nt degrees of proportionality.<br></div><div dir=3D"ltr"><br></div><div dir=
=3D"ltr">I haven't found a way to generalize proportionality degrees for an=
y <br></div><div dir=3D"ltr">number of seats (I should read that post, I su=
ppose...) but for two <br></div><div dir=3D"ltr">seats, I figured that it's=
not too hard. Since the voter opinion space <br></div><div dir=3D"ltr">dis=
tribution is a standard normal, it's symmetric around zero, so <br></div><d=
iv dir=3D"ltr">there's no reason for the method to prefer left-wing to righ=
t-wing <br></div><div dir=3D"ltr">candidates (or vice versa). Thus, the pro=
portionality level can be <br></div><div dir=3D"ltr">parameterized by just =
how far from the median the two elected candidates <br></div><div dir=3D"lt=
r">lie.<br></div><div dir=3D"ltr"><br></div><div dir=3D"ltr">That is, the e=
rror function is<br></div><div dir=3D"ltr"> sqrt((x_1 - y=
_1)^2 + (x_2 - y_2)^2)<br></div><div dir=3D"ltr"><br></div><div dir=3D"ltr"=
>and can be parameterized by a quantile level q, so that y_1 is the <br></d=
iv><div dir=3D"ltr">position corresponding to the qth quantile of the voter=
opinion space <br></div><div dir=3D"ltr">distribution, and y_2 is the (1-q=
)th quantile; and x_1 and x_2 is the <br></div><div dir=3D"ltr">location of=
the leftmost and rightmost elected candidate in opinion space.<br></div><d=
iv dir=3D"ltr"><br></div><div dir=3D"ltr">The "significant" values of q, or=
at least those that come most readily <br></div><div dir=3D"ltr">to mind a=
s distinct, are, for two seats:<br></div><div dir=3D"ltr">  =
; q =3D 0<br></div><div dir=3D"ltr"> a=
s factional as possible, usually not a good idea, but perhaps useful <br></=
div><div dir=3D"ltr">for the unanimity setting I mentioned earlier.<br></di=
v><div dir=3D"ltr"><br></div><div dir=3D"ltr"> q =3D 1/4<=
br></div><div dir=3D"ltr"> This is the=
"independent wings" position, where to elect a council, <br></div><div dir=
=3D"ltr">you split the voters into two halves (left-of-center and <br></div=
><div dir=3D"ltr">right-of-center) and elect the centrist from each (i.e. t=
he <br></div><div dir=3D"ltr">left-wingers' internal median and the right-w=
ingers' internal median). <br></div><div dir=3D"ltr">The median is at q =3D=
1/2, so a median of the left half is 1/4.<br></div><div dir=3D"ltr"><br></=
div><div dir=3D"ltr"> q =3D 1/3<br></div><div dir=3D"ltr"=
> Spatial Droop proportionality.<br></=
div><div dir=3D"ltr"><br></div><div dir=3D"ltr"> q =3D 1/=
2<br></div><div dir=3D"ltr"> Bloc majo=
ritarian voting (elect as many median voter candidates as you <br></div><di=
v dir=3D"ltr">can).<br></div><div dir=3D"ltr"><br></div><div dir=3D"ltr">Th=
e VSE is then a goodness-of-fit value (and is the maximum VSE that <br></di=
v><div dir=3D"ltr">method can get at any q, grid search optimization inaccu=
racies <br></div><div dir=3D"ltr">notwithstanding). A low value means that =
even the best fit doesn't fit <br></div><div dir=3D"ltr">very well, and thu=
s that the method has trouble being consistently <br></div><div dir=3D"ltr"=
>proportional at any level. High values mean that the particular fit is a <=
br></div><div dir=3D"ltr">very good one.</div><div dir=3D"ltr"><br></div><d=
iv dir=3D"ltr">-km<br></div><div dir=3D"ltr">----<br></div><div dir=3D"ltr"=
>Election-Methods mailing list - see <a href=3D"https://electorama.com/em" =
rel=3D"nofollow" target=3D"_blank">https://electorama.com/em</a> for list i=
nfo<br></div></div>
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