Re: Ideas for another proportionality measure
Etjon Basha via Election-Methods <[email protected]> Tue, 19 May 2026 08:34:33 +1000
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--===============0870263852706602604== Content-Type: multipart/alternative; boundary="000000000000600b6706521f2b64" --000000000000600b6706521f2b64 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Hi Kristofer and all, Perhaps not super practical, but the Monte Carlo random vote appeals to me: in your scenario above, run it 100 times with two random voters voting each time in sequence. So, 81% of the outcomes would be A and B, 9% E and D, and the rest some mix of A and E. Sum over all outcomes and see the top two? Would this be the platonic proportional benchmark, that doesn't require us to splice the voters into n dimensions of proportionality? It does a pretty shonk job at acquiescing the minority in this case, but there are far worse outcomes. And in you example specifically, it doesn't feel right for E to be elected to me, though I could see a case for D (the benchmark will never elect them though, too deep down) Regards, On Sat, 16 May 2026, 5:27=E2=80=AFam Kristofer Munsterhjelm via Election-Me= thods, < [email protected]> wrote: > I've been trying, on and off, to quantify proportionality for > multiwinner methods. (My first post on this list was about that, even.) > But usually, the metrics I tried to use, though seemingly reasonable, > ended up closer to measuring the degree to which the method gives each > group "their own" representative. > > For very large elections or party-list ones, that's not much of a > problem, but it seems intuitive that multi-winner methods electing fewer > fewer seats have to balance broad support and factional support. A > Condorcet-type bloc vote would be all broad support and would elect a > number of clones at the median position, while something that's entirely > based on factional support would divide the voters into sections, each > of which get a candidate elected based on their own center regardless of > what the distribution of opinion outside their chunk happens to be. > > (Multiwinner methods that are not proportional might elect candidates > that are further still from the center. For instance, suppose for the > sake of the argument that we want to hold an assembly vote with a very > high supermajority threshold; but first, we want to elect > representatives to that assembly from a greater number of candidates. > Then with preferences something like > 90: A>B>C>D>E > 10: E>D>C>B>A > it might make more sense to elect {A,E} than {A,B} even though the > latter is more proportional than the former; the point being that if the > threshold is above 90%, then electing {A,B} could lead to a proposition > being passed which would not pass the 90% threshold among the voters.) > > So, because I've had little luck in finding a good proportionality > measure from first principles, here's an idea that's a lot more > pragmatic, but should work. > > Let opinion space be the real line and the voters' distribution of > opinions (i.e. fractions holding each opinion value x) be some > statistical distribution, e.g. a standard normal. Then a possibly > reasonable (?) extension of majority rule is Droop: that the candidates > closest to quantile k/(s+1) should be elected, where s is the number of > seats and 1 <=3D k <=3D s. So for one winner, that's the closest to 50% (= the > median); for two winners it's 33% and 67%; for three winners it's 25%, > 50%, and 75%; and so on. > > So pick some random quantiles for the set of candidates and generate an > election consistent with the voters' preferences over these candidates > based on how close the voters are to the candidates. (This can be done > by sampling, or with very high precision for something like a normal > distribution.) Let the set of candidate quantiles be Q_C, the number of > seats be s, and Q_W some winner subset of s members. > > Let Q_W_1, ..., Q_W_s be the quantiles (members) of Q_W in sorted > increasing order. > > Then a quality measure relative to the Droop heuristic could be > something like > f(Q_W, s) =3D sum k=3D1...s: ( k/(n+1) - Q_W_k )^2 > > which we'd want to minimize. If the winners are exactly at the Droop > points, then f =3D 0. Then we could use usual approaches like VSE to take > into account that a randomly selected number of candidates might not > have such a perfect subset. > > -- > > Other ideas and observations: > > - The variance in f over multiple rounds (each of "pick a Q_C, generate > ballots, run a method, see what winner set it outputs, construct Q_W > based on it") could be used to determine if the method is consistently > proportional or all over the place. > > - If we had a way of generalizing the "optimal" quota points beyond the > k/(n+1) that Droop suggests, then for any method, we could find the > quantile distribution that fits the method best (i.e. produces the > minimal values of the penalty function f). This would then return what > behavior the method has to winner selection, from "always elect > centrists" to "always elect candidates with factional support". > > - Combining the two would give an indication of what kind of > proportionality a method (in effect) seeks to obtain, and how consistent > it is at doing so. Then we could try to make a method that takes the > proportionality level as an input and gives good performance (at that > level) no matter what level it's set to. > > - I don't know how to generalize this to multiple dimensions. That's a > problem with using a "pragmatic" measure like this. > > - A possible way to generalize the quota would be like this: let the > voting opinion distribution be a standard normal. Let delta be the > tunable parameter for Harmonic voting as in > https://rangevoting.org/QualityMulti.html. Then let the ideal candidate > locations for delta and s seats be the positions whose candidates are > always elected in an s of (s+1) election with Harmonic voting no matter > where the last candidate is located. This is pragmatic and would make > Harmonic's level of proportionality equal to its delta variable. But > it's also kind of arbitrary and finding the quantile values in practice > would be a real pain. > > - Whatever parameterization is used for proportionality, it should > probably range from "entirely bloc" (all seats at the median voter) at > one end, through Droop, to a step-like function that prefers half the > council (minus one if odd) to be far left, the other half (minus one) to > be far right, and the last, if any, to be center. > > -km > ---- > Election-Methods mailing list - see https://electorama.com/em for list > info > --000000000000600b6706521f2b64 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable <div dir=3D"auto"><div>Hi Kristofer and all,<div dir=3D"auto"><br></div><di= v dir=3D"auto">Perhaps not super practical, but the Monte Carlo random vote= appeals to me: in your scenario above, run it 100 times with two random vo= ters voting each time in sequence. So, 81% of the outcomes would be A and B= , 9% E and D, and the rest some mix of A and E. Sum over all outcomes and s= ee the top two? Would this be the platonic proportional benchmark, that doe= sn't require us to splice the voters into n dimensions of proportionali= ty?</div><div dir=3D"auto"><br></div><div dir=3D"auto">It does a pretty sho= nk job at acquiescing the minority in this case, but there are far worse ou= tcomes.=C2=A0 And in you example specifically, it doesn't feel right fo= r E to be elected to me, though I could see a case for D (the benchmark wil= l never elect them though, too deep down)</div><div dir=3D"auto"><br></div>= <div dir=3D"auto">Regards,</div><br><br><div class=3D"gmail_quote gmail_quo= te_container"><div dir=3D"ltr" class=3D"gmail_attr">On Sat, 16 May 2026, 5:= 27=E2=80=AFam Kristofer Munsterhjelm via Election-Methods, <<a href=3D"m= ailto:[email protected]">[email protected]= ama.com</a>> wrote:<br></div><blockquote class=3D"gmail_quote" style=3D"= margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">I've bee= n trying, on and off, to quantify proportionality for <br> multiwinner methods. (My first post on this list was about that, even.) <br= > But usually, the metrics I tried to use, though seemingly reasonable, <br> ended up closer to measuring the degree to which the method gives each <br> group "their own" representative.<br> <br> For very large elections or party-list ones, that's not much of a <br> problem, but it seems intuitive that multi-winner methods electing fewer <b= r> fewer seats have to balance broad support and factional support. A <br> Condorcet-type bloc vote would be all broad support and would elect a <br> number of clones at the median position, while something that's entirel= y <br> based on factional support would divide the voters into sections, each <br> of which get a candidate elected based on their own center regardless of <b= r> what the distribution of opinion outside their chunk happens to be.<br> <br> (Multiwinner methods that are not proportional might elect candidates <br> that are further still from the center. For instance, suppose for the <br> sake of the argument that we want to hold an assembly vote with a very <br> high supermajority threshold; but first, we want to elect <br> representatives to that assembly from a greater number of candidates. <br> Then with preferences something like<br> =C2=A0 =C2=A0 =C2=A0 =C2=A0 90: A>B>C>D>E<br> =C2=A0 =C2=A0 =C2=A0 =C2=A0 10: E>D>C>B>A<br> it might make more sense to elect {A,E} than {A,B} even though the <br> latter is more proportional than the former; the point being that if the <b= r> threshold is above 90%, then electing {A,B} could lead to a proposition <br= > being passed which would not pass the 90% threshold among the voters.)<br> <br> So, because I've had little luck in finding a good proportionality <br> measure from first principles, here's an idea that's a lot more <br= > pragmatic, but should work.<br> <br> Let opinion space be the real line and the voters' distribution of <br> opinions (i.e. fractions holding each opinion value x) be some <br> statistical distribution, e.g. a standard normal. Then a possibly <br> reasonable (?) extension of majority rule is Droop: that the candidates <br= > closest to quantile k/(s+1) should be elected, where s is the number of <br= > seats and 1 <=3D k <=3D s. So for one winner, that's the closest = to 50% (the <br> median); for two winners it's 33% and 67%; for three winners it's 2= 5%, <br> 50%, and 75%; and so on.<br> <br> So pick some random quantiles for the set of candidates and generate an <br= > election consistent with the voters' preferences over these candidates = <br> based on how close the voters are to the candidates. (This can be done <br> by sampling, or with very high precision for something like a normal <br> distribution.) Let the set of candidate quantiles be Q_C, the number of <br= > seats be s, and Q_W some winner subset of s members.<br> <br> Let Q_W_1, ..., Q_W_s be the quantiles (members) of Q_W in sorted <br> increasing order.<br> <br> Then a quality measure relative to the Droop heuristic could be <br> something like<br> =C2=A0 =C2=A0 =C2=A0 =C2=A0 f(Q_W, s) =3D sum k=3D1...s: ( k/(n+1) - Q_W_k = )^2<br> <br> which we'd want to minimize. If the winners are exactly at the Droop <b= r> points, then f =3D 0. Then we could use usual approaches like VSE to take <= br> into account that a randomly selected number of candidates might not <br> have such a perfect subset.<br> <br> --<br> <br> Other ideas and observations:<br> <br> - The variance in f over multiple rounds (each of "pick a Q_C, generat= e <br> ballots, run a method, see what winner set it outputs, construct Q_W <br> based on it") could be used to determine if the method is consistently= <br> proportional or all over the place.<br> <br> - If we had a way of generalizing the "optimal" quota points beyo= nd the <br> k/(n+1) that Droop suggests, then for any method, we could find the <br> quantile distribution that fits the method best (i.e. produces the <br> minimal values of the penalty function f). This would then return what <br> behavior the method has to winner selection, from "always elect <br> centrists" to "always elect candidates with factional support&quo= t;.<br> <br> - Combining the two would give an indication of what kind of <br> proportionality a method (in effect) seeks to obtain, and how consistent <b= r> it is at doing so. Then we could try to make a method that takes the <br> proportionality level as an input and gives good performance (at that <br> level) no matter what level it's set to.<br> <br> - I don't know how to generalize this to multiple dimensions. That'= s a <br> problem with using a "pragmatic" measure like this.<br> <br> - A possible way to generalize the quota would be like this: let the <br> voting opinion distribution be a standard normal. Let delta be the <br> tunable parameter for Harmonic voting as in <br> <a href=3D"https://rangevoting.org/QualityMulti.html" rel=3D"noreferrer nor= eferrer" target=3D"_blank">https://rangevoting.org/QualityMulti.html</a>. T= hen let the ideal candidate <br> locations for delta and s seats be the positions whose candidates are <br> always elected in an s of (s+1) election with Harmonic voting no matter <br= > where the last candidate is located. This is pragmatic and would make <br> Harmonic's level of proportionality equal to its delta variable. But <b= r> it's also kind of arbitrary and finding the quantile values in practice= <br> would be a real pain.<br> <br> - Whatever parameterization is used for proportionality, it should <br> probably range from "entirely bloc" (all seats at the median vote= r) at <br> one end, through Droop, to a step-like function that prefers half the <br> council (minus one if odd) to be far left, the other half (minus one) to <b= r> be far right, and the last, if any, to be center.<br> <br> -km<br> ----<br> Election-Methods mailing list - see <a href=3D"https://electorama.com/em" r= el=3D"noreferrer noreferrer" target=3D"_blank">https://electorama.com/em</a= > for list info<br> </blockquote></div></div></div> --000000000000600b6706521f2b64-- --===============0870263852706602604== Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Disposition: inline ---- Election-Methods mailing list - see https://electorama.com/em for list info --===============0870263852706602604==--