[EM] A question about proportionality and... something else.

[Toby Pereira]

 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 at munsterhjelm.no> wrote:  
 
 On 2025-11-06 23:38, Toby Pereira 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.

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.

> 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: ABC
> 1000 voters: ABD
> 1 voter: C
> 1 voter: D
> 
> I would prefer AB to CD, whereas assigning voters just one candidate who 
> is "their" candidate is likely to lead to CD.

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.)

> I also hope all is well, Kristofer.

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
  
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