[EM] IIAC. Juho: Census re-districting instead of PR for allocating seats to districts.

[Michael Ossipoff]

Juho--

You wrote:

> Let's say one quota is 1000 votes, and the district populations are A:50332, B: 1335, C:1333.
>
> If we allocate the seats 50-2-1, differences from accurate PR will be 332-665-333 persons.
> If we allocate the seats 51-1-1, differences from accurate PR will be 668-335-333 persons.
>
> I don't think the idea of allocating the seats so that the sum of overrepresentations and underrepresentations is as small as possible is an arbitrary way to allocate seats. Quite natural, don't you think.

Certainly. LR isn't without its own optimization. But name one person
who benefits from it. Yes, it looks good on paper.

But remember that, to get one thing, you give up something else. What
are you giving up to get LR's optimization? You know the answer to
that. When you give up SL's  smaller differences in S/Q, the number of
representatives for each Hare quota in the particular districts, you'd
better be getting something for it. So what are you getting? Anything
that counts for equal representation the way that SL's properties do?
And wasn't equal representation or individuals the original goal of
proportional allocation?

So sure, LR minimizes the overall difference, summed over all the
districts, of the difference between the districs' seats and their
numbers of Hare quotas. Sounds good, but be sure to ask, to make sure
that it's ok with the people to whom you're giving drastically and
unnecessarily less representation per person.

> Divisor methods focus on ratios of people and representatives. Why should that be the only approach that people should use?

Because equal representation for all people is the goal.

Because if you give someone drastically and unnecessarily less
representation, you'd better have a good justification for him/her.

It's a question of how many seats a Hare quota of people in your
district gets in comparison to how many seats a Hare quota of people
in my district gets. In my example, where you live in district C, and
I live in district B, my district is getting about twice as much
representation per hare quota of people, compared to your district. Of
course, even if you aren't in the shorted district, it's necessary to
consider the people who are. Ask them how they like getting only half
as much representation per person as the neighboring district.



>
> If we measure the biggest differences in persons (i.e. not in ratios like you did), then the differences are 997 (min -332, max +665) and 1003 (min -335, max +668)

Yes, LR minimizes the difference, summed over all of the districts, of
the difference between the districts' seats and their numbers of Hare
quotas.
See above.


> One more way to read your example is to assume that district A first had first exactly 50 quotas and 50 seats, and B had 1 quota and 1 seat. Then we > annex some new areas to those districts (332 and 335 persons). The question is why the 335 extra (over 1 quota) people in district B are "less > valuable" than the extra 332 (over 50 quota) of district A? Isn't this a valid concern, at least from one point of view?

Certainly. I'm glad you brought that up, because I was going to. If
those districts each had whole numbers of Hare quotas, qualifying
thereby for whole numbers of seats, and we give to them those whole
numbers of seats, that's quite right and fair. That, of course is what
the first part of LR does.

Now, new story: We find ourselves with a situation in which each of
several districts has a certain number of people, and there is one
seat. What should we do. Why not give the seat to the district with
the most people. That makes perfect sense.

In both of the two paragraphs before this, we've done the right thing,
and there's no doubt about it.

In other words, each half of LR makes perfect sense.

But it's a fiction, because we don't really have just that 1st
paragraph, or that 2nd paragraph. We don't just have one of those
stories. Both stories are false, because we have something quite
different. We have a situation where several districts have various
non-integer numbers of Hare quotas, and we want to give to everyone
the same representation per person, as nearly as possible. That means
that we want the S/Q to differ as little as possible. That means that
we want what SL does.

And yes, it is about S/Q, because the whole original purpose was so
that people would have equal representation.

>> Surely no one would deny that the number of representatives that a Hare quota of people has is its "representation".
>
> I note that although you wrote these words to support Saint-Laguë, they work also against it. Let's say we have proportions 61-13-13-13. SL allocates the seats 2-1-1-1. The number of quotas of each district/party has is 3.05 - 0.65 - 0.65 - 0.65. The third full quota of the largest district/party does not get its seat. Shouldn't all quotas get their representation?

Yes. Every Hare quota in my district should have as much
representation as do the Hare quotas in your district. But look at
what you're doing: Again, you're fragmenting the situation. ...the
Hare quotas this time. Looking at a particular piece of a Hare quota
and saying "This fraction of a Hare quota has no representation." But
that's a fiction. Everyone in that district has representation. Their
district has a certain number, S, of representatives. They aren't all
seized by one fraction of the district's people, leaving the other
fraction without representation. That's where the fiction comes in.

In reality, the district's representatives are shared by _everyone_ in
the district. If there are S representatives, and Q Hare quotas of
people in the district, then there are S/Q representatives per Hare
quota of people.

All sorts of fallacies involve some sleight-of-hand. Don't deceive
yourself in that way.

Let's look at your example, above. I'll re-quote what you said:

> Let's say we have proportions 61-13-13-13. SL allocates the seats 2-1-1-1. The number of quotas of each district/party has is 3.05 - 0.65 - 0.65 - 0.65. > The third full quota of the largest district/party does not get its seat. Shouldn't all quotas get their representation?

Ok, let's compare how many seats a Hare quota of voters has, in the
different districts. Let's look at the most that that differs among
the districts, with SL and with LR.

In terms of subtractive difference, SL and LR are pretty similar in
this example. In SL, the greatest difference is 5/6 of a seat. In LR,
the greatest difference is 1 seat.

But what about the greatest _factor_ by which they differ? In SL, the
greatest factor is 9/4. In LR, you give 0 seats to one of the small
parties (let's say that in any real election, there won't be 3
districts with an exact tie). That district has a finite number of
Hare quotas, and you give to it 0 seats.

Each of the other parties has infinitely many times as many seats per
Hare quota as that district has.

Explain LR's on-paper optimization to the people in that district.

> Is this in line with "SL's optimal proportionality"? SL is one good allocation method (for certain needs)

...like the wish to give people equal representation, as nearly as possible.

You continued:

> but I have hard time defining it as optimal.

Optimal is a strong word, in seat allocation. For one thing, there are
various different things to optimize. Bias? LR and SL are not optimal
in regard to bias, because, though they're both unbiased, given a
uniform probability distribution for districts, with respect to
district population, they're only unbiased under that condition. That
isn't optimal, because there are other seat allocation methods that
are unconditionally unbiased. So, as I said, SL and LR are not optimal
with regard to bias.

So, is SL optimal with regard to closeness of the districts' S/Q?  Again, no.

Yes, SL has a transfer property with regard to subtractive difference
between two districts' S/Q. In an SL allocation, if a district gives a
seat to another, that will always put their S/Q farther apart, in
terms of subtractive difference.

That sounds pretty good, but SL doesn't guarantee that it will
minimize the greatest difference in S/Q among the districts. Nor does
it guarantee that it will minimize the sum of the differences in S/Q,
summed over all of the district-pairs.

Those optimizations can, of course be achieved, using a trial and
error process that is quite feasible with a computer (but which wasn't
feasible in previous centuries).

Therefore, SL/Webster can't be called optimal in terms of S/Q difference.

The same can be said for Hill's method (I mistakenly called it
"Hall"), with regard to the ratio of districts' S/Q. It isn't optimal
for the factor by which S/Q can differ among districts.

Then what can we say for SL?

We can say that it does a good job of keeping difference in S/Q low.
After all, when it rounds off, it puts each district as close as
possible to its correct equal-representation share of the seats, for
the particular number of seats in the Parliament. And, when it puts
all the districts' S/Q as close as possible to the same value, you
know that it's also putting them close to eachother too. So then, a
lot can be said for what SL does.

As for bias, it counts for something that SL and LR are unbiased,
given the uniform probability distribution that I spoke of above.

Mike Ossipoff