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

[Juho Laatu]

On 15.7.2012, at 11.34, Michael Ossipoff wrote:

> 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.

The target was to approximate this ideal. No method can offer ideal proportionality because of rounding errors if we use whole seats. (My mail (one before the last) was btw not about Largest Remainder but about Sainte-Laguë, although I mentioned quotas in association with SL.)

> 
> 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?

No.

> 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

In in the discussion about the properties of Saint-Laguë I mentioned that in one of the examples SL could deviate 8 seats from the ideal allocation (possible but very unlikely). My interpretation of your answers is that you say that s/q values should still be respected in the way that SL does, and seat allocation (to whole seats) should be done as in SL to get best possible allocation results. Right? We might agree on that although I might not agree that the SL approach to rounding the seats to whole numbers is the only correct one.

(Btw, one problem I have is that your justification chain seems to apply pretty much to all methods that try to approximate ideal proportionality in some way. I.e. the proofs are not LS specific since they don't address the LS rounding approach but talk generally about "as close as possible".)

Juho