[EM] Level of proportionality for two-seat PR methods

[Toby Pereira]

 Obviously I overcomplicated that. You're just going from q to 1-q in equal increments. Sorry for the multiple posts.
Toby
    On Saturday, 13 June 2026 at 18:01:23 BST, Toby Pereira <tdp201b at yahoo.co.uk> wrote:  
 
  Sorry - the formula for the lower extreme point would be 2q - 0.5. For the higher extreme point, it would be 1 minus that, so 1.5 - 2q.
Toby
    On Saturday, 13 June 2026 at 17:57:07 BST, Toby Pereira <tdp201b at yahoo.co.uk> wrote:  
 
  For n seats and a 0 to 1 scale for candidate positions, for your Spatial Droop 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.

So in the general case for Spatial Droop q is 1/(n+1) rather than specifically 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 extreme points for the far left and far right cell.
So in the 2-candidate case:
When q = 1/4, the extreme points are just 0 and 1When q = 1/3, the extreme points are 1/6 and 5/6When q = 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 majoritarian would always both be 1/2
q for the "wings" position is 1/(2n)q for Droop is 1/(n+1)q for majoritarian is 1/2
I think the formula for the extreme points would be 2q - 0.5 for a 0 to 1 scale.
Some of this might make sense.
Toby

    On Saturday, 13 June 2026 at 01:54:18 BST, Kristofer Munsterhjelm via Election-Methods <election-methods at lists.electorama.com> wrote:  
 
 As mentioned in my previous post, I extended my PR measuring code to 
consider different degrees of proportionality.

I haven't found a way to generalize proportionality degrees for any 
number of seats (I should read that post, I suppose...) but for two 
seats, I figured that it's not too hard. Since the voter opinion space 
distribution is a standard normal, it's symmetric around zero, so 
there's no reason for the method to prefer left-wing to right-wing 
candidates (or vice versa). Thus, the proportionality level can be 
parameterized by just how far from the median the two elected candidates 
lie.

That is, the error function is
    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 
position corresponding to the qth quantile of the voter opinion space 
distribution, and y_2 is the (1-q)th quantile; and x_1 and x_2 is the 
location of the leftmost and rightmost elected candidate in opinion space.

The "significant" values of q, or at least those that come most readily 
to mind as distinct, are, for two seats:
    q = 0
        as factional as possible, usually not a good idea, but perhaps useful 
for the unanimity setting I mentioned earlier.

    q = 1/4
        This is the "independent wings" position, where to elect a council, 
you split the voters into two halves (left-of-center and 
right-of-center) and elect the centrist from each (i.e. the 
left-wingers' internal median and the right-wingers' internal median). 
The median is at q = 1/2, so a median of the left half is 1/4.

    q = 1/3
        Spatial Droop proportionality.

    q = 1/2
        Bloc majoritarian voting (elect as many median voter candidates as you 
can).

The VSE is then a goodness-of-fit value (and is the maximum VSE that 
method can get at any q, grid search optimization inaccuracies 
notwithstanding). A low value means that even the best fit doesn't fit 
very well, and thus that the method has trouble being consistently 
proportional at any level. High values mean that the particular fit is a 
very good one.
-km
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