[EM] Proportionality: the small bias effect seems to be real for Harmonic

Toby Pereira tdp201b at yahoo.co.uk
Tue Sep 24 05:41:33 PDT 2024


 Just to add to this:
While the percentiles and "permeantiles"* would clearly match for a uniform distribution, I was assuming that they definitely wouldn't match for a non-uniform distribution. That might be true anyway but I'm not entirely sure. It might be that for a normal distribution they would also match, which would remove the tension I was discussing in my last post. In any case, it would be interesting to see results for a uniform distribution.
Also, in the case where delta approaches zero, resulting in only each voter's favourite winner counting towards the quality function, the results in general wouldn't look at all proportional, but I'm not sure it really matters because the 25/75 result that you wanted was likely an intuition that you could be persuaded out of!
*I'm not sure if anything like a "permeantile" is a recognised concept, but I do think it's useful for things like this. My working definition of the permeantile would involve finding the relevant weighted mean. If you wanted to find e.g. the 25th permeantile, you would find the point in the data that becomes the mean if you weight everything to the left of it by 75^2 and everything to right right of it by 25^2 (or just 3^2 to 1, or 9 to 1). To find the pth permeantile, you find the point that becomes the mean if you weight everything below it by (100-p)^2 and weight everything above it by p^2. The reason you square it is that you essentially have to weight twice - once for the weight of data and once for the distance.
Toby
    On Monday 23 September 2024 at 18:56:05 BST, Toby Pereira <tdp201b at yahoo.co.uk> wrote:  
 
  Interesting as always, Kristofer. A couple of things:
My intuition is that the most balanced result for 2 candidates is at 33 and 67 rather than 25 and 75. 25 and 75 seems to suggest you're just splitting the electorate into two and finding the best candidate for each half, rather than finding the best 2 candidates for the entire electorate. Given that harmonic voting works on the scores voters give to all elected candidates rather than simply their best candidate, I would suggest that the Sainte-Laguë delta giving the 33/67 result is what I would consider to be the correct behaviour. I would expect methods that assign voters to a single candidate to be 75/25.
Also, I think using a the normal distribution rather than a uniform one complicates matters, even if it is more realistic. The normal and uniform will have the mean and median the same, but the percentiles won't be the same as what I call the "permeantiles" (percentile equivalents when using the mean). So while I would expect the Sainte-Laguë delta to give 33/67 for a uniform distribution, I'm not sure I'd necessarily expect this result for the normal distribution, although your results suggest it does give this anyway. This is because harmonic voting works on scores rather than ranks, so I wouldn't expect it to particularly follow the percentile data. So your results are a bit of a surprise in that respect.
There's probably more to get from your post so I will go through it again and see if I have anything to add or change my mind about.
Toby


    On Monday 23 September 2024 at 13:07:31 BST, Kristofer Munsterhjelm <km-elmet at munsterhjelm.no> wrote:  
 
 I implemented some different measures of proportionality for my 
simulator, and they all favor small values of delta for the cardinal 
methods.

Since the result seemed so persistent, I decided to take a more 
mathematical approach with a 1D standard normal to see if I could 
reproduce it there. Infinite voters and candidates along the Gaussian, 
and treating it like an integration problem.

That Harmonic voting only cares about the ratings of the winners makes 
it easier, as I don't have to sum up infinite non-winning candidate 
terms per voter.

My simple proportionality idea for this model is: suppose the winners 
are x_1 and x_2, identified by their x coordinate on the standard normal 
and that WLOG x_1 is to the left of x_2. Then we want x_1's right wing 
to contain just as many voters (area under the curve) as x_1's left 
wing, and ditto for x_2's wings.

This means that x_1 should be at the 25th percentile and x_2 at the 
75th. Then x_1 covers/represents everybody from the minimum to the 
median, and x_2 covers everybody from the median to the maximum, with 
equal area on both sides.

We can then integrate over all the voters for some choices of x_1, x_2, 
and delta; and get the Harmonic's quality score for those choices. Since 
the normal is symmetric, we can also let x_1 = -x_2 and x_2 >= 0. We 
would then want to determine the delta where the maximum quality 
function value is attained at x ~= -0.6745. For that delta, Harmonic 
would pick winners who have equally strong left and right wings.

Doing the integral is pretty hairy but the general idea is that there 
are four types of voter:
    1. voters to the left of x_1
    2. voters between x_1 and x_2, but closer to x_1
    3. voters between x_1 and x_2, but closer to x_2
    4. voters to the right of x_2,

and they all rate x_1 and x_2 according to some constant (I set 20) 
minus the distance to the winner in question.

The first two voter types rate x_1 higher than x_2, and the second two 
rate x_2 higher than x_1, so we know whose rating will get divided by 
delta and whose will be divided by (1 + delta).

After a particularly long procedure (made possible by WolframAlpha), the 
integral is found to evaluate to:

2 * (20 - sqrt(2/pi) + x_1)/(2 + 2 * delta) + 2 * ((2 * x_1 * 
erfc(x_1/sqrt(2)) - 2 * sqrt(2/pi) * exp(-(x_1*x_1)/2) - 3 * x_1 + 
sqrt(2/pi) + 20) / (2 * delta)).

Some numerical testing later, and the optimum for delta=0.5 
(Sainte-Laguë) is x_1 ~= -0.43, which WolframAlpha states as x_1 = 
-sqrt(2) * erfc^-1(2/3) = -0.43073... x_1 is at the 33% percentile in 
this case.

For delta = 1, it is approximately -0.31864; -sqrt(2) erfc^-1(3/4): the 
37.5th percentile.

Further numerical testing suggests that the correct position, x ~= 
-0.6745, is only obtained in the limit of delta->0. E.g. delta=1e-6 
gives y ~= -0.67449.

Some fiddling and setting derivatives to zero appear to indicate that 
the optimum for a given delta is at sqrt(2) * erf^-1(-1/(2*delta+2)), 
and that this corresponds to the (2*delta+1)/(4*delta+4) quantile. Which 
gives the desired point exactly at delta=0 (and at-large Range at 
delta->infty).

So at least in this respect, the effect seems to be real. You can either 
have optimal proportionality for party list (at delta = 0.5) or for the 
1D gaussian (at delta -> 0), but not both at the same time.

One may argue that the "wings are balanced" definition of 
proportionality is kind of sketchy. I wouldn't entirely disagree; it 
would be better to have three candidates (one at zero, one at -x, and 
one at +x), and then set the requirement so that the number of voters 
closest to each is the same. But I wouldn't want to do *those* 
integrals; two winners was hard enough!

One could also argue that this kind of proportionality idea is too 
Monrovian in that it only takes into account the voter's favorite. 
Perhaps a better notion of proportionality would take the other winners 
into account. But how?

(In the limit of delta approaching zero, Harmonic reduces to simply: 
each voter contributes to the quality function the rating of the voter's 
favorite winner. Which shows the similarity to Monroe, although Monroe 
imposes an explicit limit on the fraction of voters assigned to each 
winner.)

-km
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