[EM] n% of seats for n% of votes is unattainable. LR can't do it either.

[Michael Ossipoff]

Juho:

You said that you want n% of the votes to get n% of the seats. You
implied that, in Raph's example, LR achieves that and SL doesn't. What
you didn't notice is that LR doesn't achieve it either.

In the LR allocation, some of the small parties have 1.5% of the vote,
and 0%  of the seats. Give them a seat and they have about 2% of the
49 seats. If you give them a seat, their seat% is off by 1/2. If you
don't give them a seat, their seat percent is off by 1.5%  LR, when it
doesn't give them a seat, puts their seat% off by 3 times what it
would be off by if they'd gotten a seat. LR is not putting their seat%
as close as possible to their vote%.
And yes, I admit that SL doesn't put each party's seat% as close as
possible to its vote%. Neither does LR, as I've shown above.

What would give n% of the seats for n% of the votes?

Giving to each party a number of seats exactly equal to its number of
Hare quotas, that's what would achieve that.

Again, of course that's unattainable.

You said that you want to make a party's seat% as close as possible to
its vote%.  That means that you want to put its number of seats as
close as possible to its number of Hare quotas.

That you want to do that was the premise of one of my arguments. It
was an argument in one of my postings about 12 hours ago. I also gave
that argument on the day before that.

I refer you to that argument. In that argument, I showed that, if
that's what you want, then SL is what you want.

I'll repeat that argument again here:

Say you want to put everyone's seats as close as possible to their
number of Hare quotas.

If so, then, instead of just guaranteeing that their seat # is one of
the two integers immediately above and below their number of Hare
quotas, you'd especially want to give them the integer that is
_closest_ to
their number of Hare quotas.

You say we can't do that because it doesn't give the preferred
house-size? Forget that. We're talking about what we'd _like_. So
round their number of Hare quotas off to the nearest integer, and give
them that number of seats.

But we can do that, and _still_ give them that preferred house-size. Here's how:

Let's pretend that we started with a different "preferred" house-size.
Choose one so that with it, and its resulting "Hare quota", we get the
desired house-size.

Surely you're not going to say that the allocation is proportionally
fair with one "preferred" house-size, but not with another.

Therefore, the allocation is just as fair with the new pretended
"preferred" house-size. And Voila! The initially-desired house-size,
even though we rounded the number of "Hare quotas" to the nearest
integer, as you said (or implied) that you wanted to do.

By the way, another way to say this would be to point out that the
Hare quota is a divisor. You divide each party's votes by the same
divisor, which, in this case is the Hare quota. And then you round off
to the nearest integer. (Because you want to put seats as near as
possible to Hare quotas). You're doing  a divisor method. But
obviously, when it comes to proportional fairness, there is nothing
privileged about one divisor, as opposed to another.

The Hare quota is just one of infinitely many possible divisors.

If you use the Hare quota as the divisor, and round off to the nearest
integer, and that's your final allocation, then you're doing Webster
the way it was initially done. If you use a different divisor, in
order to achieve a certain pre-chosen house-size,  then you're using
Webster as it was later used--and the way that it's currently used, in
Sainte-Lague PR.

As I said, what you really want is Sainte-Lague.

Mike Ossipoff