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

[Michael Ossipoff]

Greetings list-members—

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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).

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Instead of defining “bias”, I’ll just let the derivation of BF tell what it
guarantees, and anyone can decide whether that’s unbias.

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Outline of derivation of Bia-Free (BF):

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First, to define the terms in the explanation, I should say what a
divisor-method is:

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Divide total votes by total seats. That’s the Hare Quota.

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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.)

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Allocate seats according to those rounding-results.

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If the number of seats thus allocated equals the legally-ordained number of
seats, then that’s the final allocation.

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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.

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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.

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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.

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The object is for the average seats per quota to be unity, averaged over an
interval between two integer numbers of quotas.

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q = quotas.   s = seats.  R = the round-up point between a & b.

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Above the round-up point, s/q = b/q.

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Below the round-up point, s/q = a/q.

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…because, below the round-up point a party would have a seats, & above the
round-up point a party would have b seats.

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Integrate b/q from R, to b.

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Integrate a/q from a to R.

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Add the two integrals together.

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To average over the interval, divide by b – a, the total amount of quota in
the interval.

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i.e. divide by 1.

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Set that average s/q in the interval equal to 1, because it’s desired for
it to be 1.

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Solve for R.

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

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That quantity is called (a special case of) the identric-mean of a & b.

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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.

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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.

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Here are the two academic-journal references:

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“The Census and the Second Law: An Entropic Approach to Optimal
Apportionment for the U.S. House of Representatives”.

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By Andrew E. Charman

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It was in _Physics and Society__, or _Journal of Physics and Society_, in
2017.

(The latest version of the article was in 2017)

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The citation said:

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Cite as arXiv.1712.09440v3 [physics.soc.ph]

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I don’t know the page or Journal-volume & the issue-numberr, or if that
information is encoded in the numbers above.

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The other paper was:

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“Optimal Congressional Apportionment”

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By Robert A. Agnew.

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…in The American Mathematical Monthly, for 2008, volume 115, number 4
(April 2008).

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Pp 297-303  (7 pages)
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