[EM] Richard: PR, for the last time

[Michael Ossipoff]

On Wed, Oct 18, 2023 at 04:08 Toby Pereira <tdp201b at yahoo.co.uk> wrote:

> I think Webster/Sainte-Laguë is generally accepted as the most
> mathematically accurate method of apportionment / party list PR,
>

Absolutely, Webster/Sainte-Lague is by far the most unbiased
allocation-rule to have ever been used for PR on apportionment. It’s bias
is very slight.

It’s bias is so slight that, with a 150-seat at-large, no districts
Parliament, if there are 17 small parties, each with 3% of the vote,
summing to 51% or the vote, & one big party with 49% of the vote, the small
parties together will get a majority of the seats.

Sainte-Lague is what I propose for allocating Parliamentary seats.

The entirety absolutely unbiased Bias-Free would be a little better, but
the difference is tiny, insignificant.

If it were up to me, the allocation would be by Bias-Free. But SL’s use of
the arithmetical mean of the two consecutive integers as its round-off
point, rounding to the nearest integer is so obvious, natural &
plainly-motivated, & also traditionally-established, that plainly SL is the
proposal more likely to be accepted.


and any large party bias comes from assumptions about the distribution,
>

No. SL really is slightly biased in favor of large parties…just not enough
to matter.

But the allocation-rule currently used for allocating seats in the
U.S.House of representatives, called “Equal-Proportions” is about twice as
biased as Webster/Sainte-Lague.  “Equal Proportion” is biased in favor of
small states, which of course is why it was adopted & is still in use.

Not that the House apportionment method makes any difference when each
state, regardless of how small it is, get the same number of senators
(two), & at least one House representative.

No need to take my word for SL’s bias. Check out a number of sources. Some
will mention SL’s slight large-favoring bias. …& yes some, especially the
earlier writings, might mistakenly say that SL is unbiased, because its
bias is so small as to easily not be noticed.



and this also calls into question what features we need to reduce the bias
> of. The following example has four parties (A, B, C, D) and the percentage
> voting for them:
>
> A: 38
> B: 38
> C: 12
> D: 12
>
> Under Sainte-Laguë, with four seats A and B will get two each. So if this
> is the kind of distribution you get in general, you will see a large party
> bias. Whereas if the distribution was:
>
> A: 37
> B: 37
> C: 13
> D: 13
>
> Then with four seats each party will win one. So this gives a small party
> bias. These are very simplistic examples just for illustrative purposes,
> and it might well be that under realistic distributions, you end up with
> more of a large party bias on average
>


The fewer the seats, the less often that small bias will show up & make a
difference between SL & Bias-Free (BF).

I once tried SL, BF, & “Equal Proportions” on  the votes in an actual
many-party at-large European List-PR election.

All three methods gave the same seat-allocation.

But in our 435-seat House of Representatives, it isn’t unusual for them to
give different results.

> . However, this doesn't change the fact that for a given election, Sainte-Laguë
> gives the most proportionally accurate, and therefore least biased by a
> reasonable measure of bias, result, even if large parties fair better on
> average.
>

Yes, & the bias changes the result from the BF allocation fairly rarely
unless the at-large Parliament is huge.  …& then the difference witl mostly
be for the 2nd seat.  (SL specifies the rounding-point for the 1st seat as
.7 instead of.5.  That .7 rounding-point should be used with BF.  …for the
same reason, to disincentivize party-splitting strategy.

>
> Changing the divisor to shift the bias towards smaller parties may seem
> like a good solution,
>

Divisor-methods, such as SL, BF, d’Hondt, & “Equal Proportions”, choose the
divisor (sometimes called a “quota”) so that when you divide each parties
quotas (result of dividing the party’s votes by the quota), & then
round-off by the particular method’s rounding-rule, the total number of
seats allocated  sums to the desired house-size.

Start with the Hare-quota (total-votes. divided by desired house-size).

Divide each party’s votes by that quota.

If that gives the desired total number of seats, then you’re done.

But if there are too many seats allocated, then try a slightly larger quota.

If there are slightly too few seats allocated, then try a slightly smaller
quota.

Find the right quota, the one that gives the right house-size, by trial &
error.

>From what I read, that’s how the divisor-methods were defined in the early
days of House-apportionment, & for some time after.

Of course, for PR, a systematic procedure is used, to avoid speaking of
trial & error in the definition.

The procedure starts with all parties having zero seats (as the would with
a sufficiently large quota), & with the seats assigned to the parties as
they round up one at a time as the quota is gradually lowered.

That’s the basis of the systematic-procedure by which the divisor methods
are defined.

For each next seat, they divide each party’s total votes by the round-up
point for that party’s next seat.

…which shows which party will round up next as the quota is lowered.

Note that it isn’t necessary to refer to the constantly decreasing quota
when doing or defining the method.

The methods differ from eachother by how the rounding-point is determined
from the pair of consecutive integers consisting of the party’s current
number of seats (“a”), & the next integer (“b”).

i.e. For SL, the rounding-point, R, is the arithmetical-mean of a & b.

(a+b)/2

For BF:

R = (1/e)((b^b)/(a^a)).

i.e. divide b^b by a^a. Then divide the result by e.

e is the base of the natural logarithms, equal to about 2.718…

d’Hondt’s R = b.

i.e. you don’t get your 3rd seat unless you have at least 3 quotas. In
other words, everyone rounds down.

So SL’s R is the arithmetical mean of a & b.

“Equal-Proportions” R is the geometric mean of a & b.

Some people aesthetically like that, but it results in about twice as much
bias as SL.

Incidentally, BF’s R is called the identric-mean of a & b.


> but as well as coming at the cost of proportional accuracy, it works under
> the assumption that large parties form an entity or themselves so that if
> one large party gets more than its proportional share, it's better for a
> small party rather than another large party to also get more than its
> proportional share.
>
> So in my first example, to reduce bias, you might decide to give A 2
> seats, B 1 seat, C 1 seat and D 0 seats. So if you consider A and C as a
> pair, your bias is toward the large party and with B and D, your bias is
> towards the small party. However, what we actually have is four separate
> parties. A and B are not in league with each other and neither are C and D.
> So trying to reduce bias on party size in this way is based on false
> assumptions and you end up biasing the result in other ways - it's not just
> about large v small. The most proportionally accurate result is 2, 2, 0, 0,
> not 2, 1, 1, 0.
>
> As for the other point, STV (and other candidate-based PR methods) do not
> exclude parties; they just give voters the choice whether to vote for party
> candidates or any independents that might be standing. People are also just
> as likely to vote for parties based on the face, hairdo or personality of
> their leader, and party promises are also not enforceable. Party-list PR is
> more democratically limited than candidate-based PR.
>
> And to finish on a more hypothetical note, one solution that would remove
> large/small party bias and also make determining the result of the election
> much simpler would be to use a non-deterministic method. E.g. party A in my
> first example could win more or less than their proportional share, but
> would win 38% of the seats on average.
>
> Toby
>
>
> On Tuesday, 17 October 2023 at 22:09:41 BST, Michael Ossipoff <
> email9648742 at gmail.com> wrote:
>
>
> Though you’re certainly certainly welcome to your theories, sorry but your
> source is mistaken.
>
> If you’d do a little reading, you’d find that there’s a consensus that
> Webster/Sainte-Lague, while very nearly unbiased, & while the most unbiased
> of the allocation rules that are or have been used, is slightly biased in
> favor of larger parties (or states).
>
> I’ve told the unbiased allocation rule, & have supplied journal-paper
> references, & have, last month here, outlined its derivation.
>
> Thank you for reminding us that you prefer voting only for faces, hairdos,
> & personalities, with their vague, unreliable & unenforceable promises,
> instead of for policy platforms.
>
> Though about 2/3 of the world’s countries use PR, only a tiny fraction of
> them use STV. They nearly all use Party-List PR, a referendum on policy.
>
> Open-list PR includes voting for the people who will be seated by the
> platform-lists.
>
> Open-List PR is incomparably more easily, transparently &
> easily-verifiably  counted  than the cumbersome days-long STV-count.
>
> …which, even with computers, & even when used to elect only one winner,
> has a way of taking days.
>
> I don’t have time to keep replying to these posts. I’ll make us if the
> settings to re-route them from the inbox.
>
>
>
>
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>
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