[EM] Ideas for another proportionality measure

[Etjon Basha]

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 am Kristofer Munsterhjelm via Election-Methods, <
election-methods at lists.electorama.com> 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 <= k <= 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) = sum k=1...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 = 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
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> info
>
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