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

[Michael Ossipoff]

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)
>
> ----
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>
>
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