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

Michael Ossipoff email9648742 at gmail.com
Tue Jun 26 14:51:59 PDT 2012


Kristofer:
On Tue, Jun 26, 2012 at 4:06 PM, Kristofer Munsterhjelm <
km_elmet at lavabit.com> wrote:

> On 06/26/2012 04:34 PM, Michael Ossipoff wrote:
>
>> Kristofer:
>> You said:
>> Similarly, if you pick two states/parties j and k, Webster minimizes the
>> absolute value of S_j/P_j - S_k/P_k, i.e. the difference between "seats
>> per population" (share of influence per person) of state j and k.
>> [endquote]
>> Of course you can't mean that. You're referring to Webster's transfer
>> property:
>> If, starting with the Webster allocation, one state gives a seat to
>> another state, then that will increase the difference between their s/p.
>> It certainly will never decrease it.
>> That pair difference minimization only applies to two states during a
>> transfer of a seat.
>>
>
> The RangeVoting page says what you quoted me as saying. I'll quote more
> literally.
>
> Under "Pairwise optimality properties", http://rangevoting.org/**
> Apportion.html <http://rangevoting.org/Apportion.html> :
>
> "Let S_k be the number of seats for state k and P_k be its population.
>
> For all j,k, Adams minimizes |S_j - S_k P_j/P_k|.
> Dean minimizes |P_j/S_j - P_k/S_k|
> Huntington-Hill minimizes |S_j P_k/(S_k P_j) - 1|
> Webster minimizes |S_j/P_j - S_k/P_k|
> (...)".
>
> Seems pretty unambiguous to me!
>

[endquote]

Unambiguous, but not correct.

Say we're apportioning Congressional seats to the states in the U.S., by
Webster.

When the states populations have been divided by a common divisor, and the
resulting quotients rounded to the nearest whole number, and the sum of
those numbers equals the desired number of seats, of course, for each
state, that whole number is the number of seats that it gets.

I'll refer to each state's quotient, by that divisor, as its correct
proportional share, and abbreviate that "cps".

A state's cps divided by its population is the ideal s/p. If I say it a
lot, maybe I should abbreviate it "s/p_i".That s/p value is the one that,
ideally, all of the states should have.

With the smallest states, it often isn't possible to get states' s/p as
close to the s/p_i, as compared to large states.  For that reason, too,
they'll often be farther from eachother, too, in s/p, when they differ from
the s/p_i in opposite directions.

So, suppose that, in a particular apportionment, two large states, L1 and
L2, are on opposite sides of the s/p_i, but fairly close to it.

And two smaller states, S1 and S2 are also, compared to eachother, on
opposite sides of the s/p_i. They're farther from the s/p_i than L1 and L2
are.

Now, let's exchange a seat between L1 and L2. By Webster/SL's familiar
transfer property, that will put L1 and L2 farther apart than they were
before, and still on opposite sides of the s/p_i. And, by SL's most basic
and important optimization, it will put both L1 and L2 farther  from the
s/p_i than they were..

But that puts at least one of L1 and L2 closer to S1 or S2.

So, the Sainte-Lague/Webster allocation must not have minimized the
difference between the s/p of every pair of states.

Q.E.D.

You continued:

By the way, according to the same RangeVoting page, the method you wished
to see (that minimizes max s/p) is Jefferson (D'Hondt). See the global
properties list, #4. Unfortunately, Jefferson is biased in favor of large
parties.

[endquote]

I didn't ask to minimize the max s/p. I asked to minimize, over all of the
pairs of states, the amount by which one state's s/p differs from that of
the other.

I'm guessing that that can be only accomplished by a computation-intensive
trial-and-error process. But I don't claim to know that for sure.
I know that this subject is well-studied, and I should be looking it up
before saying anything. It was a long time ago when I read Balinski & Young.

I like that optimization because it minimizes the greatest injustice.

But SL's main optimization, putting each state's seats as close as possible
to that state's cps, and its s/p as close as possible to the s/p_i, does
and guarantees something for _every_ state. And SL is more easily
implemented, and has much precedent. So I'm quite satisfied with SL's
optimization.

Mike Ossipoff
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