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

[Kristofer Munsterhjelm]

Replying to all of these in one post:

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

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.)

> What result to you get if you minimise the sum of squared distances of 
> voters to representatives?

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.

> Another question would be what to the Phragmen methods do under this model?

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