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<div dir="ltr" data-setdir="false">Just to add to this:</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">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.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">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!</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">*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.</div><div dir="ltr" data-setdir="false"><span><br></span></div><div dir="ltr" data-setdir="false">Toby</div><div><br></div>
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On Monday 23 September 2024 at 18:56:05 BST, Toby Pereira <tdp201b@yahoo.co.uk> wrote:
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<div dir="ltr">Interesting as always, Kristofer. A couple of things:</div><div dir="ltr"><br clear="none"></div><div dir="ltr">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 <span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">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.</span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><br clear="none"></span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">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 <span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">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.</span></span></span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><br clear="none"></span></span></span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">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.</span></span></span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><br clear="none"></span></span></span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">Toby</span></span></span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><br clear="none"></span></span></span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><br clear="none"></span></span></span></span></div><div><br clear="none"></div>
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On Monday 23 September 2024 at 13:07:31 BST, Kristofer Munsterhjelm <km-elmet@munsterhjelm.no> wrote:
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<div><div dir="ltr">I implemented some different measures of proportionality for my <br clear="none"></div><div dir="ltr">simulator, and they all favor small values of delta for the cardinal <br clear="none"></div><div dir="ltr">methods.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">Since the result seemed so persistent, I decided to take a more <br clear="none"></div><div dir="ltr">mathematical approach with a 1D standard normal to see if I could <br clear="none"></div><div dir="ltr">reproduce it there. Infinite voters and candidates along the Gaussian, <br clear="none"></div><div dir="ltr">and treating it like an integration problem.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">That Harmonic voting only cares about the ratings of the winners makes <br clear="none"></div><div dir="ltr">it easier, as I don't have to sum up infinite non-winning candidate <br clear="none"></div><div dir="ltr">terms per voter.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">My simple proportionality idea for this model is: suppose the winners <br clear="none"></div><div dir="ltr">are x_1 and x_2, identified by their x coordinate on the standard normal <br clear="none"></div><div dir="ltr">and that WLOG x_1 is to the left of x_2. Then we want x_1's right wing <br clear="none"></div><div dir="ltr">to contain just as many voters (area under the curve) as x_1's left <br clear="none"></div><div dir="ltr">wing, and ditto for x_2's wings.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">This means that x_1 should be at the 25th percentile and x_2 at the <br clear="none"></div><div dir="ltr">75th. Then x_1 covers/represents everybody from the minimum to the <br clear="none"></div><div dir="ltr">median, and x_2 covers everybody from the median to the maximum, with <br clear="none"></div><div dir="ltr">equal area on both sides.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">We can then integrate over all the voters for some choices of x_1, x_2, <br clear="none"></div><div dir="ltr">and delta; and get the Harmonic's quality score for those choices. Since <br clear="none"></div><div dir="ltr">the normal is symmetric, we can also let x_1 = -x_2 and x_2 >= 0. We <br clear="none"></div><div dir="ltr">would then want to determine the delta where the maximum quality <br clear="none"></div><div dir="ltr">function value is attained at x ~= -0.6745. For that delta, Harmonic <br clear="none"></div><div dir="ltr">would pick winners who have equally strong left and right wings.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">Doing the integral is pretty hairy but the general idea is that there <br clear="none"></div><div dir="ltr">are four types of voter:<br clear="none"></div><div dir="ltr"> 1. voters to the left of x_1<br clear="none"></div><div dir="ltr"> 2. voters between x_1 and x_2, but closer to x_1<br clear="none"></div><div dir="ltr"> 3. voters between x_1 and x_2, but closer to x_2<br clear="none"></div><div dir="ltr"> 4. voters to the right of x_2,<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">and they all rate x_1 and x_2 according to some constant (I set 20) <br clear="none"></div><div dir="ltr">minus the distance to the winner in question.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">The first two voter types rate x_1 higher than x_2, and the second two <br clear="none"></div><div dir="ltr">rate x_2 higher than x_1, so we know whose rating will get divided by <br clear="none"></div><div dir="ltr">delta and whose will be divided by (1 + delta).<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">After a particularly long procedure (made possible by WolframAlpha), the <br clear="none"></div><div dir="ltr">integral is found to evaluate to:<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">2 * (20 - sqrt(2/pi) + x_1)/(2 + 2 * delta) + 2 * ((2 * x_1 * <br clear="none"></div><div dir="ltr">erfc(x_1/sqrt(2)) - 2 * sqrt(2/pi) * exp(-(x_1*x_1)/2) - 3 * x_1 + <br clear="none"></div><div dir="ltr">sqrt(2/pi) + 20) / (2 * delta)).<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">Some numerical testing later, and the optimum for delta=0.5 <br clear="none"></div><div dir="ltr">(Sainte-Laguë) is x_1 ~= -0.43, which WolframAlpha states as x_1 = <br clear="none"></div><div dir="ltr">-sqrt(2) * erfc^-1(2/3) = -0.43073... x_1 is at the 33% percentile in <br clear="none"></div><div dir="ltr">this case.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">For delta = 1, it is approximately -0.31864; -sqrt(2) erfc^-1(3/4): the <br clear="none"></div><div dir="ltr">37.5th percentile.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">Further numerical testing suggests that the correct position, x ~= <br clear="none"></div><div dir="ltr">-0.6745, is only obtained in the limit of delta->0. E.g. delta=1e-6 <br clear="none"></div><div dir="ltr">gives y ~= -0.67449.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">Some fiddling and setting derivatives to zero appear to indicate that <br clear="none"></div><div dir="ltr">the optimum for a given delta is at sqrt(2) * erf^-1(-1/(2*delta+2)), <br clear="none"></div><div dir="ltr">and that this corresponds to the (2*delta+1)/(4*delta+4) quantile. Which <br clear="none"></div><div dir="ltr">gives the desired point exactly at delta=0 (and at-large Range at <br clear="none"></div><div dir="ltr">delta->infty).<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">So at least in this respect, the effect seems to be real. You can either <br clear="none"></div><div dir="ltr">have optimal proportionality for party list (at delta = 0.5) or for the <br clear="none"></div><div dir="ltr">1D gaussian (at delta -> 0), but not both at the same time.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">One may argue that the "wings are balanced" definition of <br clear="none"></div><div dir="ltr">proportionality is kind of sketchy. I wouldn't entirely disagree; it <br clear="none"></div><div dir="ltr">would be better to have three candidates (one at zero, one at -x, and <br clear="none"></div><div dir="ltr">one at +x), and then set the requirement so that the number of voters <br clear="none"></div><div dir="ltr">closest to each is the same. But I wouldn't want to do *those* <br clear="none"></div><div dir="ltr">integrals; two winners was hard enough!<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">One could also argue that this kind of proportionality idea is too <br clear="none"></div><div dir="ltr">Monrovian in that it only takes into account the voter's favorite. <br clear="none"></div><div dir="ltr">Perhaps a better notion of proportionality would take the other winners <br clear="none"></div><div dir="ltr">into account. But how?<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">(In the limit of delta approaching zero, Harmonic reduces to simply: <br clear="none"></div><div dir="ltr">each voter contributes to the quality function the rating of the voter's <br clear="none"></div><div dir="ltr">favorite winner. Which shows the similarity to Monroe, although Monroe <br clear="none"></div><div dir="ltr">imposes an explicit limit on the fraction of voters assigned to each <br clear="none"></div><div dir="ltr">winner.)<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">-km<br clear="none"></div><div dir="ltr">----<br clear="none"></div><div dir="ltr">Election-Methods mailing list - see <a shape="rect" href="https://electorama.com/em" rel="nofollow" target="_blank">https://electorama.com/em</a> for list info<br clear="none"></div></div>
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