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

[Toby Pereira]

 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 at 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 at munsterhjelm.no> wrote:

On 2025-10-27 13:11, Etjon Basha wrote:
> Sorry to hear Kristofer, I hope it's all resolved as best it can.

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.

> 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.

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

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