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<div dir="ltr" data-setdir="false">The other thing with this is that looking solely at proportionality is not necessarily the best thing to do. It needs to be combined with some reasonable degree of monotonicity as well. Take the following approval ballots:</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">99: ABC</div><div dir="ltr" data-setdir="false">99: ABD</div><div dir="ltr" data-setdir="false">1: C</div><div dir="ltr" data-setdir="false">1: D</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">With two to elect, arguably CD is the most proportional result (and this is what Phragmen philosophy prefers). However, C and D each have just 50% of the support, whereas A and B each have 99%. The 99s in the ballots could be increased arbitrarily and you'd get the same thing. This is why Phragmen-based methods tend to be only weakly monotonic.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">I've written about optimal proportionality before, but if you are allowed to elect any number of candidates with any amount of weight (rather than having a fixed number with equal weight), then optimal PAV and COWPEA both have some very nice properties, including a strong degree of monotonicity and passing independence of irrelevant ballots. They differ slightly in their "philosophy" so can produce different results, but I see them as possibly the only two contenders for the "ultimate" in optimal PR.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Unfortunately, they do not carry all of their nice properties over to elections to elect a fixed number of candidates with equal weight. Well, there isn't a definitive version of COWPEA for deterministic elections of this sort. PAV (and by extension Harmonic Voting) can give disproportional results, although this goes away if candidates all have infinite clones. While this is unrealistic, it means that PAV is proportional in the case where it is an approval-based party-list election where all parties field enough candidates.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">PAV and COWPEA do retain their nice properties if you allow non-deterministic elections. For this you would use the candidate weights instead as probabilities in a lottery and elect sequentially. The PAV version of this is likely to be computationally intractable, but the COWPEA version is very simple.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">But the bottom line is that there is no known method that keeps all the nice properties you'd want for all deterministic elections where a fixed number of candidates are to be elected with equal weight. Some compromise is likely to always be needed. It's not that surprising since proportionality and monotonicity are essentially orthogonal things, and it's not a priori obvious that they could be perfectly combined in a non-arbitrary and clean way. It seems they can be anyway for optimal PR (with two nice methods - an embarrassment of riches) but not for fixed candidates with equal weight.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">So I do largely trust Harmonic Voting with the <span><span style="color: rgb(38, 40, 42); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Sainte-Laguë delta to give good results in cases where a lack of clones is not an issue.</span></span></div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Toby</div><div><br></div>
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On Wednesday 25 September 2024 at 18:28:16 BST, Toby Pereira <tdp201b@yahoo.co.uk> wrote:
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<div dir="ltr">Thanks for the response and added information. On proportionality, it can be difficult to pin down a definition precisely, but obviously I would see it in terms of balance in some sense. But given that my existing intuition was that 33/67 was the better result and <span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">the Sainte-Laguë delta gave this result anyway, rather than me having to rationalise the result, I'm not too concerned at the moment.</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;">In any case, I don't see the more polarised version of proportionality where it's all about your best candidate as the best definition. E.g.</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;">50: A=10, B=0, C=9, D=9</span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">50: A=0, B=10, C=9, D=9</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"><font color="#26282a">I see CD as a "better" result than AB, and so a definition that prefers this would be more appealing to me. This is likely to lead to 33/67 rather than 25/75 in the other example.</font></div><div dir="ltr"><font color="#26282a"><br clear="none"></font></div><div dir="ltr"><font color="#26282a">For approval voting, my working definition of proportionality is achieving perfect representation in the limit as the number of candidates increases, as long as the ballots make this possible (voters approve enough candidates). Converting this definition to score voting might involve making some arbitrary decisions about what scores should mean. E.g. if one simply assumes that the KP-transformation is what you are supposed to do, then just using an approval method that passes perfect representation in the limit with the KP-transformation will do the job.</font></div><div dir="ltr"><font color="#26282a"><br clear="none"></font></div><div dir="ltr"><a shape="rect" href="https://electowiki.org/wiki/Perfect_representation" class="ydpea2b3083yiv3009596738" rel="nofollow" target="_blank">https://electowiki.org/wiki/Perfect_representation</a><font color="#26282a"><br clear="none"></font></div><div><br clear="none"></div><div>On permeantiles, I was really just thinking of something that corresponds to the mean the same way as percentiles correspond to the median. Given what you say about the uniform and normal distributions behaving in the same way under Harmonic voting, it probably doesn't matter at this point anyway.<br clear="none"></div><div><br clear="none"></div><div dir="ltr">Toby</div><div><br clear="none"></div>
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On Wednesday 25 September 2024 at 14:51:27 BST, Kristofer Munsterhjelm <km-elmet@munsterhjelm.no> wrote:
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<div><div dir="ltr">On 2024-09-23 19:56, Toby Pereira wrote:<br clear="none">> Interesting as always, Kristofer. A couple of things:<br clear="none">> <br clear="none">> My intuition is that the most balanced result for 2 candidates is at 33 <br clear="none">> and 67 rather than 25 and 75. 25 and 75 seems to suggest you're just <br clear="none">> splitting the electorate into two and finding the best candidate for <br clear="none">> each half, rather than finding the best 2 candidates for the entire <br clear="none">> electorate. Given that harmonic voting works on the scores voters give <br clear="none">> to all elected candidates rather than simply their best candidate, I <br clear="none">> would suggest that the Sainte-Laguë delta giving the 33/67 result is <br clear="none">> what I would consider to be the correct behaviour. I would expect <br clear="none">> methods that assign voters to a single candidate to be 75/25.<br clear="none"><br clear="none">I can see that argument. The question is really what proportionality means.<br clear="none"><br clear="none">We have to determine if 33/67 being more desirable than 25/75 is because <br clear="none">33/67 is more proportional *as such*, or if we're deliberately giving up <br clear="none">some raw proportionality to get an assembly that's better adjusted as a <br clear="none">whole (more able to work as a unified body, related to the "utility" <br clear="none">measure).<br clear="none"><br clear="none">If it's the former, then we'd need to find a better definition of what <br clear="none">we mean by proportional. Whatever it is should use hidden information <br clear="none">(something depending on opinion space) so that the 1D normal reaches <br clear="none">optimal proportionality by 33/67 instead of 25/75. Just saying "what <br clear="none">harmonic/psi/Phragmen says is good is good" blinds us to the possibility <br clear="none">that there may exist (yet undiscovered) methods that do what the <br clear="none">yardstick does, but better.<br clear="none"><br clear="none">If it's the latter, then that means that what we're balancing is the <br clear="none">ability of the method to evenly cover opinion space with its winners, <br clear="none">and the ability of the winning assembly to work in a way similar to a <br clear="none">high-quality single winner.<br clear="none"><br clear="none">That this balance isn't "all proportionality at all costs" makes sense. <br clear="none">Party list PR methods with Plurality ballots can't do that balance well, <br clear="none">even when they're proportionally unbiased, and thus exhibits other <br clear="none">problems: an assembly elected by Sainte-Laguë may still have kingmaker <br clear="none">problems, something that would be improved by having a "centrist center <br clear="none">of gravity" to break the ties instead. But D'Hondt can't do that because <br clear="none">it doesn't know who the centrists are, since all it's got are first <br clear="none">preference votes.<br clear="none"><br clear="none">As you point out, it's not that strange for Harmonic to deviate from the <br clear="none">Monrovian proportionality concept, since it does, after all, take <br clear="none">ratings of all winners by all voters into account. What I'm saying is <br clear="none">that we should try to find out what the ideal of proportionality ought <br clear="none">to be. Or find some of the different aspects of the elephant, if we <br clear="none">can't characterize it exactly.<br clear="none"><br clear="none">> Also, I think using a the normal distribution rather than a uniform one <br clear="none">> complicates matters, even if it is more realistic. The normal and <br clear="none">> uniform will have the mean and median the same, but the percentiles <br clear="none">> won't be the same as what I call the "permeantiles" (percentile <br clear="none">> equivalents when using the mean). So while I would expect the <br clear="none">> Sainte-Laguë delta to give 33/67 for a uniform distribution, I'm not <br clear="none">> sure I'd necessarily expect this result for the normal distribution, <br clear="none">> although your results suggest it does give this anyway. This is because <br clear="none">> harmonic voting works on scores rather than ranks, so I wouldn't expect <br clear="none">> it to particularly follow the percentile data. So your results are a bit <br clear="none">> of a surprise in that respect.<br clear="none"><br clear="none">I'm not quite sure I get what you're saying. By permeantile, do you mean <br clear="none">something related to the expected value of a truncated distribution, the <br clear="none">way the mean is the expected value of the full distribution?<br clear="none"><br clear="none">I agree that Harmonic doesn't try to place itself at a given percentile <br clear="none">for the normal, since it hasn't been designed to do so. It is <br clear="none">surprising, though, that my first proportionality measure (opinion space <br clear="none">proportionality) seems to prefer deltas that lead to 25-75 to ones that <br clear="none">lead to 33-67.<br clear="none"><br clear="none">If there is a Monroe component to this measure, then I would expect <br clear="none">Monroe itself to show up as being very proportional by that measure. I <br clear="none">should check that - only problem is that implementing the assignment <br clear="none">algorithm is kind of a hassle.<br clear="none"><br clear="none">> While the percentiles and "permeantiles"* would clearly match for a<br clear="none">> uniform distribution, I was assuming that they definitely wouldn't match<br clear="none">> for a non-uniform distribution. That might be true anyway but I'm not<br clear="none">> entirely sure. It might be that for a normal distribution they would<br clear="none">> also match, which would remove the tension I was discussing in my last<br clear="none">> post. In any case, it would be interesting to see results for a uniform<br clear="none">> distribution.<br clear="none"><br clear="none">I did some quick integral evaluations, so take this with a bit of salt, <br clear="none">but it seems like the uniform distribution percentiles are the same as <br clear="none">the normal.[1]<br clear="none"><br clear="none">Suppose we have a uniform between -1 and 1. the pdf is 1/2. Each voter <br clear="none">rates a candidate equal to the negative of their distance (I'm skipping <br clear="none">the constant of 20 this time as it shouldn't matter.) As before, the <br clear="none">winners' positions are at x_1 and -x_1. Then, if my calculations are <br clear="none">correct, the Harmonic score for given delta and x_1 is:<br clear="none"><br clear="none">(-2 * x_1^2 * delta - 2 * x_1^2 - 2 * x_1 - 2 * delta - 1)/(2 * delta * <br clear="none">(delta + 1))<br clear="none"><br clear="none">The optimum is achieved when x_1 = -1/(2(delta+1)), which might look a <br clear="none">bit familiar.<br clear="none"><br clear="none">The quantile is (2 * delta + 1) / (4 * delta + 4), which *definitely* <br clear="none">looks familiar.<br clear="none"><br clear="none">For different values of delta:<br clear="none"> delta = 0 x_1 = -1/2 25th percentile<br clear="none"> delta = 0.5 x_1 = -1/3 33rd percentile<br clear="none"> delta = 1 x_1 = -1/4 37.5th percentile.<br clear="none"><br clear="none">> Also, in the case where delta approaches zero, resulting in only each<br clear="none">> voter's favourite winner counting towards the quality function, the <br clear="none">> results in general wouldn't look at all proportional, but I'm not<br clear="none">> sure it really matters because the 25/75 result that you wanted was<br clear="none">> likely an intuition that you could be persuaded out of!<br clear="none"><br clear="none">Oh, delta=0 definitely isn't proportional in every case. Monroe proper <br clear="none">has a constraint that each candidate can only be assigned to a certain <br clear="none">number of voters, so that each candidate represents the same number of <br clear="none">voters. On symmetric distributions like the normal and uniform with <br clear="none">symmetrically positioned candidates, you get that anyway, but I would <br clear="none">imagine that Harmonic with delta=0 would be disproportional if the <br clear="none">distribution is skewed: some winners would be backed by more voters than <br clear="none">others. Meanwhile, Monroe's constraint would keep that from happening, <br clear="none">even for a Monroe-type idea of proportionality.<br clear="none"><br clear="none">To put it differently, Harmonic delta=0's results would be more like <br clear="none">Chamberlin-Courant but with the weights discarded after election. E.g. <br clear="none">for something like<br clear="none"><br clear="none">100: A (10) > B (9) > C (0)<br clear="none"> 1: C (10) > B (1) > A (0)<br clear="none"><br clear="none">the intuitively proportional result for two winners is {A, B}, but <br clear="none">Harmonic with delta=0 would choose {A, C}. Monroe would pick {A, B} but <br clear="none">Chamberlin-Courant would do something like: A (weight 100/101), C <br clear="none">(weight 1/101).<br clear="none"><br clear="none">In this sense delta=0 is akin to at-large IRV and LPV0+.<br clear="none"><br clear="none">This also shows that methods that are on-average proportional (for some <br clear="none">model) may still fail quite badly outside that model, or even some of <br clear="none">the time in-model. That's a weakness of VSE-type analysis. We could <br clear="none">possibly get some information about how often that happens (still <br clear="none">in-model) by reporting something that is to the standard deviation what <br clear="none">the VSE is to the mean.<div id="ydpea2b3083yiv3009596738ydp3421b0e6yqtfd08078" class="ydpea2b3083yiv3009596738ydp3421b0e6yqt6878415498"><br clear="none"><br clear="none">-km</div><br clear="none"><br clear="none">[1] A better mathematician might be able to determine if this is true of <br clear="none">every symmetric distribution. Such a strategy would probably use that <br clear="none">the pdfs of distributions that are symmetric about 0 are odd, and then <br clear="none">something with voters on the far right canceling out voters on the far <br clear="none">left. But turning that into an actual proof would still be quite a lot <br clear="none">of work.<div id="ydpea2b3083yiv3009596738ydp3421b0e6yqtfd22127" class="ydpea2b3083yiv3009596738ydp3421b0e6yqt6878415498"><br clear="none"></div></div></div>
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