Sorry to hear Kristofer, I hope it's all resolved as best it can.

I'd say a qualified yes to both: at the extreme where random ballot is used
in the assembly (as opposed to electing the assembly itself), it very much
would matter how proportional the body is. Most would be quite happy for a
very majoritarian assembly in such conditions, I'd think, to avoid the risk
of being subjected to the vote of the 5% 5% of the time.

At the other extreme of some complicated iteration of quadratic
negotiations or what have you, so some procedure that somehow irons out all
issues to produce an ultimately unanimous vote, proportionality would also
matter less.

In practice neither of these two extremes apply, and so the proportionality
of the assembly matters to most, but in principle it needs not.

These are all ways to achieve utility, the intermediate steps are open to
improvement.

Regards,

On Mon, 27 Oct 2025, 10:39 pm Kristofer Munsterhjelm via Election-Methods, <
election-methods@lists.electorama.com> wrote:

On 2025-10-27 13:11, Etjon Basha wrote:

I hope so too. Things are looking a bit better now, but I'm not sure
yet. I'll try to reply to this, though, and hope I'm not pulled away again.

What I found to be a problem with utility when I explored
proportionality measures is that it's relatively weak.

Here's an example: Suppose that the voter opinion distribution is a
standard normal (i.e. one-dimensional). Suppose that the assembly
reviews successive measures and that the representatives vote to accept
or reject the measure by a majority vote.

The people, could we ask all of them, will accept the measure if "yes"
is closer to the mean/median opinion of zero than "no" is. But any
assembly with an odd number of seats and a symmetric distribution of
representatives around zero will also behave this way.

So if a voter's utility for passing (failing) a measure is the negative
of the distance between his position in opinion space and the measure's
"yes" ("no") position in opinion space, then the balanced council
accepts the measure iff the people does.

In particular, the degenerate centrist case where every rep is at most
an epsilon away from zero also does this, as long as the choice is
symmetric around zero. So the model doesn't answer what level of PR is
enough.

As a simplification of what I'm (or was) trying to figure out, consider
a similarly simple case: the people's opinion distribution is a standard
normal, and just about every voter stands as a representative. The
assembly has two seats, so by the argument above, the elected reps would
be at quantiles 0.5-x and 0.5+x.

But what is the correct level of PR? If x = 0, you get a pure
centrist/bloc situation. x->0.5 gives a very polarized assembly. The
k-median optimum is at 25% and 75%, and that's what Monroe does; but
Toby Pereira argued that 33% and 67% is better (and he has a point).

If I were asked what the benefits of PR are, I'd say that it keeps the
representatives accountable, it keeps the factions from becoming too
complacent, and allows shifting coalitions if the people's opinion space
distribution is not simple.

The first point is that the voters can see what the reps are accepting
or rejecting, so they have a record of how each faction puts their money
where their mouth is, so to speak. This can be hard to determine if all
the negotiation happens inside the big-tent party.

The second being that if we have a bunch of centrists, they may start to
get sloppy because they're so ideologically similar; that they'll take
each other's support for granted and thus may start to drift from the
population's center. There's a lack of competition, so to speak, and
it's worse if all the centrists come from the same party.

And the third is pretty clear, and does happen in PR countries. But from
a pure utility metric, one could imagine that sufficiently virtuous
centrists would do this "inside their own heads", and track the popular
center because that's what they were elected to do.

But at least the first two of these points are hard to formalize and do
simulations about. We could add parameters and say, suppose that similar
reps lose contact with reality or get corrupted at some given rate, but
different parameter choices would give different results about how much
PR is too much: where the drawbacks of increased polarization start to
outweigh the benefits of increased diversity of the representatives'
positions.

-km

I'm unaware of Pereira's argument but it seems intuitively obvious that P33
and P67 are the optimal when picking two representatives.

I suppose there'd be no material difference between proportionality and
centrism in these simple models but in reality issues don't get decided one
by one in isolation, but in bundles, and in those cases, a proportional
assembly would (if at all it could) decide differently to a centrist one
and the practical outcomes would be quite different between the two.

Not to mention the nitty gritty of committee work, some of which might be
populated by skewed representations of the assembly by sheer chance in a
proportional system, a null issue under a proper centrist system, and so on.

If we ignore these issues, I see no difference between the Dutch and
Australian Assmeblies, and indeed no reason to have an assembly at all
beyond a single president (or perhaps a triumvirate of them).

Regards,

On Tue, 4 Nov 2025, 12:48 am Kristofer Munsterhjelm, <
km-elmet@munsterhjelm.no> wrote:

My thinking is that if voters all fit on a neat line (e.g. left to right) then electing at 25 and 75 makes sense only if you consider each voter to have a specific representative assigned to them. In that case you just split the electorate neatly in half and take the mid-point of each half. Whereas I think it makes more sense to consider that every voter is affected by each elected candidate so their opinions on all of them should be taken into account to some extent. If you're not pre-splitting the electorate into two, 33 and 67 seems the most balanced rather than 25 and 75. Slightly tangentially, rigidly assigning voters to candidates can lead to what I would consider undesirable results. Take these approval ballots (each letter is a candidate):
1000 voters: ABC1000 voters: ABD1 voter: C1 voter: D
I would prefer AB to CD, whereas assigning voters just one candidate who is "their" candidate is likely to lead to CD.
I also hope all is well, Kristofer.
Toby
On Tuesday 4 November 2025 at 08:44:20 GMT, Etjon Basha via Election-Methods election-methods@lists.electorama.com wrote:

I'm unaware of Pereira's argument but it seems intuitively obvious that P33 and P67 are the optimal when picking two representatives.
I suppose there'd be no material difference between proportionality and centrism in these simple models but in reality issues don't get decided one by one in isolation, but in bundles, and in those cases, a proportional assembly would (if at all it could) decide differently to a centrist one and the practical outcomes would be quite different between the two.
Not to mention the nitty gritty of committee work, some of which might be populated by skewed representations of the assembly by sheer chance in a proportional system, a null issue under a proper centrist system, and so on.
If we ignore these issues, I see no difference between the Dutch and Australian Assmeblies, and indeed no reason to have an assembly at all beyond a single president (or perhaps a triumvirate of them).
Regards,

On Tue, 4 Nov 2025, 12:48 am Kristofer Munsterhjelm, km-elmet@munsterhjelm.no wrote:

On 2025-10-27 13:11, Etjon Basha wrote:

I hope so too. Things are looking a bit better now, but I'm not sure
yet. I'll try to reply to this, though, and hope I'm not pulled away again.

What I found to be a problem with utility when I explored
proportionality measures is that it's relatively weak.

Here's an example: Suppose that the voter opinion distribution is a
standard normal (i.e. one-dimensional). Suppose that the assembly
reviews successive measures and that the representatives vote to accept
or reject the measure by a majority vote.

The people, could we ask all of them, will accept the measure if "yes"
is closer to the mean/median opinion of zero than "no" is. But any
assembly with an odd number of seats and a symmetric distribution of
representatives around zero will also behave this way.

So if a voter's utility for passing (failing) a measure is the negative
of the distance between his position in opinion space and the measure's
"yes" ("no") position in opinion space, then the balanced council
accepts the measure iff the people does.

In particular, the degenerate centrist case where every rep is at most
an epsilon away from zero also does this, as long as the choice is
symmetric around zero. So the model doesn't answer what level of PR is
enough.

As a simplification of what I'm (or was) trying to figure out, consider
a similarly simple case: the people's opinion distribution is a standard
normal, and just about every voter stands as a representative. The
assembly has two seats, so by the argument above, the elected reps would
be at quantiles 0.5-x and 0.5+x.

But what is the correct level of PR? If x = 0, you get a pure
centrist/bloc situation. x->0.5 gives a very polarized assembly. The
k-median optimum is at 25% and 75%, and that's what Monroe does; but
Toby Pereira argued that 33% and 67% is better (and he has a point).

If I were asked what the benefits of PR are, I'd say that it keeps the
representatives accountable, it keeps the factions from becoming too
complacent, and allows shifting coalitions if the people's opinion space
distribution is not simple.

The first point is that the voters can see what the reps are accepting
or rejecting, so they have a record of how each faction puts their money
where their mouth is, so to speak. This can be hard to determine if all
the negotiation happens inside the big-tent party.

The second being that if we have a bunch of centrists, they may start to
get sloppy because they're so ideologically similar; that they'll take
each other's support for granted and thus may start to drift from the
population's center. There's a lack of competition, so to speak, and
it's worse if all the centrists come from the same party.

And the third is pretty clear, and does happen in PR countries. But from
a pure utility metric, one could imagine that sufficiently virtuous
centrists would do this "inside their own heads", and track the popular
center because that's what they were elected to do.

But at least the first two of these points are hard to formalize and do
simulations about. We could add parameters and say, suppose that similar
reps lose contact with reality or get corrupted at some given rate, but
different parameter choices would give different results about how much
PR is too much: where the drawbacks of increased polarization start to
outweigh the benefits of increased diversity of the representatives'
positions.

-km

Election-Methods mailing list - see https://electorama.com/em for list info

Thanks Toby,

Makes sense. the voters would mind how far away the "away" representative
is from them, and would be unlikely to care about the home team only.
Minimising total distance as opposed to distance from the closest
representative makes sense.

Furthermore, and tangentially related but still related, the P33 and P67
are going to have a much easier time understanding one another than the P25
and P75 are.

Regards,

Etjon

On Fri, 7 Nov 2025, 9:39 am Toby Pereira, tdp201b@yahoo.co.uk wrote:

On 2025-11-06 23:38, Toby Pereira wrote:

I'm inclined to agree, but I just haven't found any natural measure,
based on proximity or some feature in the underlying opinion space,
where the optimum naturally falls out as 33/67. Most either produce a
bloc result (a bunch of centrists at the median voter position) or a
Monrovian result (each vote having "their one" rep, and thus producing
25/75).

For instance, if we let the voter's satisfaction be the sum of
distances to all representatives, then everybody being located as close
to the median as possible is optimal. Trying to use a reduction from
k-median gives a Monroe optimum (because the objective only considers
each voter's distance to their closest representative).

Even something as seemingly unrelated as minimizing a simple model of
gerrymandering susceptibility produces a Monroe optimum. (E.g. taking a
standard normal and letting the leftmost third be one single-winner
district and the rightmost 2/3 a two-seat district; the median rep, i.e.
the leftmost of the right-district winners, is the median voter for both
districts as a whole if the 2/3 district uses a 25-75 outcome to elect
its two winners.)

We could, of course, assume the PAV metric itself to be a standard of
desirability, i.e. a voter should care 1 unit about his closest winner,
1/3 about his next closest, etc.; but that feels too much like begging
the question. The Sainte-Laguë numbers make sense in a party list
setting, but it's hard to make something that generalizes smoothly to
less disjoint settings.

I have found some possibly useful properties, though. Like this one,
which we could tentatively call "median representative". If the
multiwinner method elects an odd number of candidates, the underlying
space has a concept of a median, and the voters have single-peaked
preferences, then the median representative of the outcome hould be the
one closest to the median voter. STV fails this for similar reasons that
IRV fails Condorcet. I would imagine consistently Condorcet multiwinner
methods like Schulze STV to pass, but I haven't checked this.

Perhaps there's some way to formalize the intuition... I'm wondering if
it could be done in a spatial model. I think you'd need at least two
dimensions, and the setup would be that most voters care about dimension
1 and break ties by dimension 2 (hence ABC and ABD), with a few voters
being very focused on the second dimension.

Then the desideratum is that we don't want the method to be pulled
"off-axis" by a very small minority. (With one dimension per party, we
would probably also get a reduction to party list.)

Thank you. It's been kind of rough, worse and then better and then
not sure at the moment. Maybe I'll give you more information in private;
I'm not the kind of person to put these kind of things on list,
particularly not given certain people who may or may not be lurking
here.

-km

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.
What result to you get if you minimise the sum of squared distances of voters to representatives?
Toby
On Monday 17 November 2025 at 00:58:12 GMT, Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:

On 2025-11-06 23:38, Toby Pereira wrote:

I'm inclined to agree, but I just haven't found any natural measure,
based on proximity or some feature in the underlying opinion space,
where the optimum naturally falls out as 33/67. Most either produce a
bloc result (a bunch of centrists at the median voter position) or a
Monrovian result (each vote having "their one" rep, and thus producing
25/75).

For instance, if we let the voter's satisfaction be the sum of
distances to all representatives, then everybody being located as close
to the median as possible is optimal. Trying to use a reduction from
k-median gives a Monroe optimum (because the objective only considers
each voter's distance to their closest representative).

Even something as seemingly unrelated as minimizing a simple model of
gerrymandering susceptibility produces a Monroe optimum. (E.g. taking a
standard normal and letting the leftmost third be one single-winner
district and the rightmost 2/3 a two-seat district; the median rep, i.e.
the leftmost of the right-district winners, is the median voter for both
districts as a whole if the 2/3 district uses a 25-75 outcome to elect
its two winners.)

We could, of course, assume the PAV metric itself to be a standard of
desirability, i.e. a voter should care 1 unit about his closest winner,
1/3 about his next closest, etc.; but that feels too much like begging
the question. The Sainte-Laguë numbers make sense in a party list
setting, but it's hard to make something that generalizes smoothly to
less disjoint settings.

I have found some possibly useful properties, though. Like this one,
which we could tentatively call "median representative". If the
multiwinner method elects an odd number of candidates, the underlying
space has a concept of a median, and the voters have single-peaked
preferences, then the median representative of the outcome hould be the
one closest to the median voter. STV fails this for similar reasons that
IRV fails Condorcet. I would imagine consistently Condorcet multiwinner
methods like Schulze STV to pass, but I haven't checked this.

Perhaps there's some way to formalize the intuition... I'm wondering if
it could be done in a spatial model. I think you'd need at least two
dimensions, and the setup would be that most voters care about dimension
1 and break ties by dimension 2 (hence ABC and ABD), with a few voters
being very focused on the second dimension.

Then the desideratum is that we don't want the method to be pulled
"off-axis" by a very small minority. (With one dimension per party, we
would probably also get a reduction to party list.)

Thank you. It's been kind of rough, worse and then better and then
not sure at the moment. Maybe I'll give you more information in private;
I'm not the kind of person to put these kind of things on list,
particularly not given certain people who may or may not be lurking
here.

-km

Another question would be what to the Phragmen methods do under this model?
Toby
On Monday 17 November 2025 at 13:34:05 GMT, Toby Pereira tdp201b@yahoo.co.uk wrote:

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.
What result to you get if you minimise the sum of squared distances of voters to representatives?
Toby
On Monday 17 November 2025 at 00:58:12 GMT, Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:

On 2025-11-06 23:38, Toby Pereira wrote:

I'm inclined to agree, but I just haven't found any natural measure,
based on proximity or some feature in the underlying opinion space,
where the optimum naturally falls out as 33/67. Most either produce a
bloc result (a bunch of centrists at the median voter position) or a
Monrovian result (each vote having "their one" rep, and thus producing
25/75).

For instance, if we let the voter's satisfaction be the sum of
distances to all representatives, then everybody being located as close
to the median as possible is optimal. Trying to use a reduction from
k-median gives a Monroe optimum (because the objective only considers
each voter's distance to their closest representative).

Even something as seemingly unrelated as minimizing a simple model of
gerrymandering susceptibility produces a Monroe optimum. (E.g. taking a
standard normal and letting the leftmost third be one single-winner
district and the rightmost 2/3 a two-seat district; the median rep, i.e.
the leftmost of the right-district winners, is the median voter for both
districts as a whole if the 2/3 district uses a 25-75 outcome to elect
its two winners.)

We could, of course, assume the PAV metric itself to be a standard of
desirability, i.e. a voter should care 1 unit about his closest winner,
1/3 about his next closest, etc.; but that feels too much like begging
the question. The Sainte-Laguë numbers make sense in a party list
setting, but it's hard to make something that generalizes smoothly to
less disjoint settings.

I have found some possibly useful properties, though. Like this one,
which we could tentatively call "median representative". If the
multiwinner method elects an odd number of candidates, the underlying
space has a concept of a median, and the voters have single-peaked
preferences, then the median representative of the outcome hould be the
one closest to the median voter. STV fails this for similar reasons that
IRV fails Condorcet. I would imagine consistently Condorcet multiwinner
methods like Schulze STV to pass, but I haven't checked this.

Perhaps there's some way to formalize the intuition... I'm wondering if
it could be done in a spatial model. I think you'd need at least two
dimensions, and the setup would be that most voters care about dimension
1 and break ties by dimension 2 (hence ABC and ABD), with a few voters
being very focused on the second dimension.

Then the desideratum is that we don't want the method to be pulled
"off-axis" by a very small minority. (With one dimension per party, we
would probably also get a reduction to party list.)

Thank you. It's been kind of rough, worse and then better and then
not sure at the moment. Maybe I'll give you more information in private;
I'm not the kind of person to put these kind of things on list,
particularly not given certain people who may or may not be lurking
here.

-km

Replying to all of these in one post:

On 2025-11-17 17:42, Toby Pereira wrote:

Yes, that's a point, but I think the asymptotics are more like
constraints than they set a particular method.

I ran into a comparable case some years ago when experimenting with
party list methods. There's a similarity between Sainte-Laguë and the
statistical chi square test, which is an approximation of more
discerning tests like the G-test. (And that's also related to the
entropy/Shannon and Nash result, if I recall correctly.)

So I thought, why not just make a party list method derived from the
G-test the way Sainte-Lague is from the chi-square? It should converge
to the same thing, and be "better" if the G-test is better than the
chi-square.

It turned out it had the same convergence, but for a limited number of
seats was much too favoring of/biased in the direction of small
states/parties.

So I think asymptotics can narrow down the field, but they don't
necessarily determine the method outright; in particular, they don't say
what's the right balance in situations with a small number of seats.

(I remember Forest suggested using lotteries to create PR methods. The
problem is similar - you need to "trade off" properly with small
assemblies, so a static lottery is of limited use as the number of seats
vary.)

It still rewards an outcome where everybody is in one location. The
intuition is this: suppose you have enough clones that you could fill
the whole assembly with people from the same position in opinion space.
Then if your method is given a ballot set v and works by finding the
winner set W that minimizes a function

obj(W,v) = sum over winners w in W: f(w,v)

and f is IIA-like in that it doesn't care about the other members of W,

then if you have a proportional winner set W, and some winner w* gives
the minimum f(w, v) of the chosen winners, then replacing everybody else
with a clone of w* can't make the objective value worse and may make it
better.

This works for all functions decomposable in that way - median distance,
squared distance, etc.

This took a lot of time to figure out because, as far as I understand,
nobody's done a rated-ballot version of Phragmen, so I had to use
approval ballots.

But what I found was very surprising, so much so that I'd appreciate if
anybody could try to reproduce it. Suppose that we're using max-Phragmen
with mean-utility threshold approval voting, on a standard normal
spatial model (i.e. a fraction p(x) of the voters hold the opinion of
coordinate x), and their utility for a candidate is some constant minus
the distance from themselves to that candidate.
Then suppose two candidates are located at -x and +x, and a new entrant
at some y between 0 and x. Let these candidates be A (at -x), B (at y),
and C (at +x).

Then the optimal max-Phragmen two-seat outcome is {A,B}... no matter
what x is.

So in this simple model, there's always an incentive to position oneself
closer to the center - at least if the current candidate pool is
balanced. There's no "equilibrium point" where the inward and outward
pressure balances out.

Maybe this is an artifact of mean-utility thresholding and KPT can
circumvent it. But if so, that's all the more damaging for mean-utility
threshold approval, and it would show a particularly severe IIA failure.

I haven't done var-Phragmen because turning the quadratic optimization
into something that can be parameterized (over all values of y for any
particular x) is much harder.

I also found something initially surprising, but understandable in
hindsight, about sequential methods. Suppose we have a sequential method
like sequential Harmonic voting, and we're electing a two-seat group.
One of the winners will be the Range winner, and this winner will have
an incentive to migrate to the center (since that's how Range works). If
the method has no proportional equilibrium (like Phragmen above), the
second winner is also incentivized to migrate to the center. On the
other hand, if the method has a proportional equilibrium, the second
candidate might migrate away from the center.

E.g. suppose that the first winner was originally left-of-center and the
second was right-of-center. With the first winner moving to the center,
this puts pressure on the second to distance itself further to the right
(since more of the voters right of center are now closer to the first
winner). One could then end up with a series of unbalanced councils with
the first winner being at center and the second winner being either
quite far to the left or to the right, and potentially the winner set
flipping back and forth between {center, left} and {center, right} based
on noise or minor differences in voter opinion.

More seats will help. But it does feel like something of a flaw of
sequential methods as such.

-km