[EM] Richard: PR, for the last time

[Michael Ossipoff]

BTW, the definition of SL doubles those rounding-points by which parties’
votes are divided in the systematic procedure.

That’s to get rid of the fractions, & get integers with fewer digits. Maybe
to simplify the calculation in those pre-computer days.

On Wed, Oct 18, 2023 at 05:45 Michael Ossipoff <email9648742 at gmail.com>
wrote:

>
>
> 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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