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<div dir="ltr" data-setdir="false">Using PAV to force the result you want might seem arbitrary, but PAV is also just the fixed-winner version of the Nash Product Rule (which would allow for any number of candidates to be elected in varying proportions). (As you increase the number of candidates to be elected while allowing unlimited clones, PAV converges on the Nash result.) The point being that I wouldn't consider it to simply be some hack used to produce a desired result. And it certainly wasn't designed for this election scenario in any case. So I would say its result in this example at least counts for something.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">What result to you get if you minimise the sum of squared distances of voters to representatives?</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 Monday 17 November 2025 at 00:58:12 GMT, Kristofer Munsterhjelm <km-elmet@munsterhjelm.no> wrote:
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<div><div dir="ltr">On 2025-11-06 23:38, Toby Pereira wrote:<br clear="none">> My thinking is that if voters all fit on a neat line (e.g. left to <br clear="none">> right) then electing at 25 and 75 makes sense only if you consider each <br clear="none">> voter to have a specific representative assigned to them. In that case <br clear="none">> you just split the electorate neatly in half and take the mid-point of <br clear="none">> each half. Whereas I think it makes more sense to consider that every <br clear="none">> voter is affected by each elected candidate so their opinions on all of <br clear="none">> them should be taken into account to some extent. If you're not <br clear="none">> pre-splitting the electorate into two, 33 and 67 seems the most balanced <br clear="none">> rather than 25 and 75.<br clear="none"><br clear="none">I'm inclined to agree, but I just haven't found any natural measure, <br clear="none">based on proximity or some feature in the underlying opinion space, <br clear="none">where the optimum naturally falls out as 33/67. Most either produce a <br clear="none">bloc result (a bunch of centrists at the median voter position) or a <br clear="none">Monrovian result (each vote having "their one" rep, and thus producing<br clear="none">25/75).<br clear="none"><br clear="none">For instance, if we let the voter's satisfaction be the sum of<br clear="none">distances to all representatives, then everybody being located as close<br clear="none">to the median as possible is optimal. Trying to use a reduction from <br clear="none">k-median gives a Monroe optimum (because the objective only considers <br clear="none">each voter's distance to their closest representative).<br clear="none"><br clear="none">Even something as seemingly unrelated as minimizing a simple model of <br clear="none">gerrymandering susceptibility produces a Monroe optimum. (E.g. taking a <br clear="none">standard normal and letting the leftmost third be one single-winner <br clear="none">district and the rightmost 2/3 a two-seat district; the median rep, i.e. <br clear="none">the leftmost of the right-district winners, is the median voter for both <br clear="none">districts as a whole if the 2/3 district uses a 25-75 outcome to elect <br clear="none">its two winners.)<br clear="none"><br clear="none">We could, of course, assume the PAV metric itself to be a standard of <br clear="none">desirability, i.e. a voter should care 1 unit about his closest winner,<br clear="none">1/3 about his next closest, etc.; but that feels too much like begging<br clear="none">the question. The Sainte-Laguë numbers make sense in a party list <br clear="none">setting, but it's hard to make something that generalizes smoothly to <br clear="none">less disjoint settings.<br clear="none"><br clear="none">I have found some possibly useful properties, though. Like this one,<br clear="none">which we could tentatively call "median representative". If the <br clear="none">multiwinner method elects an odd number of candidates, the underlying <br clear="none">space has a concept of a median, and the voters have single-peaked <br clear="none">preferences, then the median representative of the outcome hould be the <br clear="none">one closest to the median voter. STV fails this for similar reasons that <br clear="none">IRV fails Condorcet. I would imagine consistently Condorcet multiwinner <br clear="none">methods like Schulze STV to pass, but I haven't checked this.<br clear="none"><br clear="none">> Slightly tangentially, rigidly assigning voters to candidates can<br clear="none">> lead to what I would consider undesirable results.<br clear="none">> Take these approval ballots (each letter is a candidate):<br clear="none">> <br clear="none">> 1000 voters: ABC<br clear="none">> 1000 voters: ABD<br clear="none">> 1 voter: C<br clear="none">> 1 voter: D<br clear="none">> <br clear="none">> I would prefer AB to CD, whereas assigning voters just one candidate who <br clear="none">> is "their" candidate is likely to lead to CD.<br clear="none"><br clear="none">Perhaps there's some way to formalize the intuition... I'm wondering if<br clear="none">it could be done in a spatial model. I think you'd need at least two<br clear="none">dimensions, and the setup would be that most voters care about dimension <br clear="none">1 and break ties by dimension 2 (hence ABC and ABD), with a few voters <br clear="none">being very focused on the second dimension.<br clear="none"><br clear="none">Then the desideratum is that we don't want the method to be pulled <br clear="none">"off-axis" by a very small minority. (With one dimension per party, we <br clear="none">would probably also get a reduction to party list.)<br clear="none"><br clear="none">> I also hope all is well, Kristofer.<br clear="none"><br clear="none">Thank you. It's been kind of rough, worse and then better and then<br clear="none">not sure at the moment. Maybe I'll give you more information in private;<br clear="none">I'm not the kind of person to put these kind of things on list,<br clear="none">particularly not given certain people who may or may not be lurking<div class="ydp57306546yqt4004197651" id="ydp57306546yqtfd86492"><br clear="none">here.<br clear="none"><br clear="none">-km<br clear="none"></div></div></div>
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