Empathy List Archives

KM

Kristofer Munsterhjelm

Fri, May 15, 2026 6:32 PM

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

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

RR

Ryan Regier

Fri, May 15, 2026 9:19 PM

Hi Kristofer,

I know of a way to generalize the Droop quota based on how it is derived.
Generally, in a plurality election where every voter gets to cast their
vote for one candidate and the top s candidates win, any candidate with
more than a Droop quota of voters is guaranteed to win regardless of the
other votes, while any candidate with less can lose in some election.
Likewise, any faction with a Droop quota of voters is guaranteed
representation by a candidate of their choice provided they coordinate
their vote perfectly. We can generalize this to the case where voters may
cast votes for m distinct candidates. The threshold for a candidate to be
guaranteed election is now mk/(s+1) votes. But the size a faction needs to
be is lower: mk/(s+m) voters. Proof of the formula is below. When m=s you
have k/2 and you recover the majority threshold, m=1 is the Droop quota.
The proof assumes an integer m, but you might be able to extend this
formula beyond its original definition and still get a reasonable
proportionality threshold.

Proof: Suppose v voters in a faction all vote for m candidates and none of
their candidates win. Importantly, none of the voters in the faction vote
for any other candidates. Then the other k - v voters must have
collectively put at least v votes on each of the s winners. Thus, (k-v)m >=
sv. Isolating v yields v <= mk/(s+m). Therefore, any faction larger than
this must have one of their chosen candidates win. If they exactly match
this threshold one of their candidates must at least tie for the win.

Since the faction threshold is below the candidate threshold, you get an
interesting middle ground where factions of a middle size can ensure one of
the candidates they vote for will win, but not which one. If the candidates
receive no other votes they will tie and a tiebreaker will determine who
will be elected, but you could also have other people outside the faction
break the tie by voting for one of the candidates in their m candidate set.
Proportionality really is a combination of two meanings: being able to have
a representative, and not letting others decide who your representative is.
These diverging formulas show that there can be different thresholds to
qualify for each right.

We can extend this idea to generalize proportionality in other ways. For
instance, for an STV-like definition, we can define the quota for election
and the quota of ballots exhausted as separate numbers. So in an election
for s=3, we could keep the election threshold at 25% but lower the
exhaustion threshold to 5% to get a majoritarian-leaning proportionality.
If the exhaustion threshold was set to 40%, you would get more of what I
call an egalitarian election where different factions each get one
representative even in spite of large differences in vote count. If you set
the election threshold to 40% but the exhaustion threshold to 25%, you get
a proportional representation but force the majority to pick more consensus
candidates with the approval of the minority rather than pick whomever the
majority would prefer. Of course, there are tactical considerations in all
of these setups.

For the Harmonic case, that leads to another parameterization is of course
dividing party votes by n^p where n is the number of candidates they have
already elected. When p=1 we achieve proportionality along the lines of
d'Hondt / Jefferson divisors. For p>1 you have more egalitarian systems
where minorities are over-represented, and for p < 1 you have more
majoritarian systems where minorities are underrepresented. At p=0 the
faction with the most votes gets all the seats, and in the limiting case as
p->infinity the s biggest factions each get one seat regardless of vote
count.

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.

I disagree. With an egalitarian system (the opposite side of majoritarian)
all factions get equal representation regardless of vote count. So as long
as the center party got any votes at all, then left, right, and center
should all get a third of the seats at the extreme end. As I discussed in
the p-series parameterization that is exactly what happens.

Best,
Ryan Regier

On Fri, May 15, 2026 at 12:27 PM Kristofer Munsterhjelm via
Election-Methods election-methods@lists.electorama.com wrote: