[EM] Derivation of Identric-mean as round-up point for Bias-Free method

Wed Sep 6 22:16:06 PDT 2023

BTW, Bias-Free is my favorite allocation-rule for list-PR.

But I propose Sainte-Lague, because…

…though SL has a tiny amount of bias in favor of larger parties, it’s so
slight that it’s completely insignificant.

e.g.

150 seats

17 small parties, each with 3% of the vote

… together amounting to 51% of the vote

One big party with 49% of the vote

The 17 small parties each with 3% of the vote together get a majority of
the seats, as they should, with Sainte-Lague.

They can form a majority coalition & a government.

With d’Hondt, the big party gets a big majority.

On Wed, Sep 6, 2023 at 22:04 Michael Ossipoff <email9648742 at gmail.com>
wrote:

>
>
> On Tue, Sep 5, 2023 at 23:59 Richard Lung <voting at ukscientists.com> wrote:
>
>>
>> Just a preliminary remark. Divisor methods and quotas some times
>> distinguished.
>>
>
> I don’t know what you mean by that.
>
> I didn’t say that the divisor methods are quotas. I didn’t say that quotas
> are divisor methods.
>
> I defined “quota” for its use in what I was saying. That use of that word
> isn’t new or unusual. It’s found in divisor method discussion.
>
>> Thus there is the Droop quota and corresponding D'Hont divisor method.
>>
>
>
> I don’t know what you’ve heard of, but I’ve never heard of Droop quota in
> a definition or discussion of d’Hondt.
>
> But maybe someone has proposed a method that he calls “d’Hondt”, & maybe
> his method uses the Droop quota, which I’ve heard of being sometimes
> proposed, & sometimes used, in STV.
>
>> Divisor methods regarded as belonging to apportionment
>>
> That term was probably first applied to apportionment proposals, but
> nonetheless d’Hondt is the Jefferson divisor method, & Saints-Lague is the
> Webster divisor method.
>
> Those two list-PR methods are usually defined, & at least partly
> implemented by a systematic procedure, rather than the by the
> implementation often or usually specified by the definitions of the divisor
> methods proposed & used for apportionment.  ..at least in earlier
> apportionment discussion.
>
> as by Jefferson and by Webster, not to carve out party seats, which is too
>> restrictive of personal choice.
>>
>
> Carve out?
>
> I’m not sure, but you seem to be saying that party-list PR restricts
> personal choice.
>
> But you didn’t say why you think so.
>
>
>
>>
>
>
>
>
>> On 06/09/2023 05:05, Michael Ossipoff wrote:
>>
>> Greetings list-members—
>>
>> …
>>
>> In 2006, I proposed an allocation divisor-method that I called Bias-Free,
>> which eliminates bias. I’d like, in this message, to better explain my
>> derivation of Bias-Free (BF).
>>
>> …
>>
>> Instead of defining “bias”, I’ll just let the derivation of BF tell what
>> it guarantees, and anyone can decide whether that’s unbias.
>>
>> …
>>
>> Outline of derivation of Bia-Free (BF):
>>
>> …
>>
>> First, to define the terms in the explanation, I should say what a
>> divisor-method is:
>>
>> …
>>
>> Divide total votes by total seats. That’s the Hare Quota.
>>
>> …
>>
>> Divide each party’s votes by the Hare Quota, & round off to one of the
>> two closest integers. (Each divisor-method uses a different round-up point.)
>>
>> …
>>
>> Allocate seats according to those rounding-results.
>>
>> …
>>
>> If the number of seats thus allocated equals the legally-ordained number
>> of seats, then that’s the final allocation.
>>
>> …
>>
>> Otherwise, try the procedure using another number to replace the Hare
>> Quota, & call that new number the quota. Repeat the above procedure, using
>> that new quota instead of the Hare Quota.
>>
>> …
>>
>> Find (by trial-&-error, or by some systematic-procedure) a quota such
>> that the resulting number of seats allocated equals the legally-ordained
>> number of seats.
>>
>> …
>>
>> In the explanation below, “quota” means “quota” as defined above, or a
>> number of seats equal to the quota.  The Hare Quote too is a “quota” as
>> the term is used below.
>>
>> …
>>
>> The object is for the average seats per quota to be unity, averaged over
>> an interval between two integer numbers of quotas.
>>
>> …
>>
>> q = quotas.   s = seats.  R = the round-up point between a & b.
>>
>> …
>>
>> Above the round-up point, s/q = b/q.
>>
>> …
>>
>> Below the round-up point, s/q = a/q.
>>
>> …
>>
>> …because, below the round-up point a party would have a seats, & above
>> the round-up point a party would have b seats.
>>
>> …
>>
>> Integrate b/q from R, to b.
>>
>> …
>>
>> Integrate a/q from a to R.
>>
>> …
>>
>> Add the two integrals together.
>>
>> …
>>
>> To average over the interval, divide by b – a, the total amount of quota
>> in the interval.
>>
>> …
>>
>> i.e. divide by 1.
>>
>> …
>>
>> Set that average s/q in the interval equal to 1, because it’s desired for
>> it to be 1.
>>
>> …
>>
>> Solve for R.
>>
>> …
>>
>> R = (1/e)((b^b)/(a^a)).
>>
>> …
>>
>> That quantity is called (a special case of) the identric-mean of a & b.
>>
>> …
>>
>> Someone expressed concern that the unbias would be spoiled because the
>> size of parties has a nonuniform probability-distribution. But he didn’t
>> say why he thinks so, I don’t know what that probability-distribution has
>> to do with anything said in the derivation.
>>
>> …
>>
>> The identric-mean has been much discussed by mathematicians.  But,from
>> what was said in an academic paper (I’ll cite it below), it wasn’t proposed
>> as the round-up point for an unbiased divisor-method before I proposed it
>> here in 2006.  There were two academic journal-papers about that
>> proposa, in versions starting in 2008.
>>
>> …
>>
>> Here are the two academic-journal references:
>>
>> …
>>
>> “The Census and the Second Law: An Entropic Approach to Optimal
>> Apportionment for the U.S. House of Representatives”.
>>
>> …
>>
>> By Andrew E. Charman
>>
>> …
>>
>> It was in _Physics and Society__, or _Journal of Physics and Society_, in
>> 2017.
>>
>> (The latest version of the article was in 2017)
>>
>> …
>>
>> The citation said:
>>
>> …
>>
>> Cite as arXiv.1712.09440v3 [physics.soc.ph]
>>
>> …
>>
>> I don’t know the page or Journal-volume & the issue-numberr, or if that
>> information is encoded in the numbers above.
>>
>> …
>>
>> The other paper was:
>>
>> …
>>
>> “Optimal Congressional Apportionment”
>>
>> …
>>
>> By Robert A. Agnew.
>>
>> …
>>
>> …in The American Mathematical Monthly, for 2008, volume 115, number 4
>> (April 2008).
>>
>> …
>>
>> Pp 297-303  (7 pages)
>>
>> ----
>> Election-Methods mailing list - see https://electorama.com/em for list info
>>
>>
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