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<p>Just a preliminary remark. Divisor methods and quotas some times
distinguished. Thus there is the Droop quota and corresponding
D'Hont divisor method. Divisor methods regarded as belonging to
apportionment, as by Jefferson and by Webster, not to carve out
party seats, which is too restrictive of personal choice.<br>
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<div class="moz-cite-prefix">On 06/09/2023 05:05, Michael Ossipoff
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CAOKDY5A6GNwS4+CrXiYuYCr8gc2ORNrntnA7=PY1Vk4SMX_GKg@mail.gmail.com">
<div dir="ltr">
<p class="MsoNormal"><span>Greetings
list-members—<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>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).<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Instead
of defining “bias”, I’ll just let the derivation of BF tell
what it guarantees,
and anyone can decide whether that’s unbias.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Outline
of derivation of Bia-Free (BF):<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>First,
to define the terms in the explanation, I should say what a
divisor-method is:<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Divide
total votes by total seats. That’s the Hare Quota.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>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.)<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Allocate
seats according to those rounding-results.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>If
the number of seats thus allocated equals the
legally-ordained number of seats,
then that’s the final allocation.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>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.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>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.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>In
the explanation below, “quota” means “quota” as defined
above, or a number of
seats equal to the quota.<span> </span>The Hare Quote
too is a “quota” as the term is used below.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>The
object is for the average seats per quota to be unity,
averaged over an
interval between two integer numbers of quotas.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>q
= quotas.<span> </span>s = seats.<span> </span>R = the
round-up point between a & b.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Above
the round-up point, s/q = b/q.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Below
the round-up point, s/q = a/q.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>…because,
below the round-up point a party would have a seats, &
above the round-up
point a party would have b seats.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Integrate
b/q from R, to b.<span> </span><span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Integrate
a/q from a to R.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Add
the two integrals together.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>To
average over the interval, divide by b – a, the total amount
of quota in the
interval.<span></span></span></p>
<p class="MsoNormal"><span>…
<span></span></span></p>
<p class="MsoNormal"><span>i.e.
divide by 1.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Set
that average s/q in the interval equal to 1, because it’s
desired for it to be
1.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Solve
for R.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>R
= (1/e)((b^b)/(a^a)).<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>That
quantity is called (a special case of) the identric-mean of
a & b.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>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.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>The
identric-mean has been much discussed by mathematicians. <span> </span>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.<span> </span>There
were two academic journal-papers about that proposa, in
versions starting in
2008.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Here
are the two academic-journal references:<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>“The
Census and the Second Law: An Entropic Approach to Optimal
Apportionment for
the U.S. House of Representatives”.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>By
Andrew E. Charman<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>It
was in _Physics and Society__, or _Journal of Physics and
Society_, in 2017. <span></span></span></p>
<p class="MsoNormal"><span>(The
latest version of the article was in 2017)<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>The
citation said: <span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Cite
as arXiv.1712.09440v3 [<a href="http://physics.soc.ph"
moz-do-not-send="true">physics.soc.ph</a>]<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>I
don’t know the page or Journal-volume & the
issue-numberr, or if that
information is encoded in the numbers above.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>The
other paper was:<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>“Optimal
Congressional Apportionment”<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>By
Robert A. Agnew.<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>…in
The American Mathematical Monthly, for 2008, volume 115,
number 4 (April 2008).<span></span></span></p>
<p class="MsoNormal"><span>…<span></span></span></p>
<p class="MsoNormal"><span>Pp
297-303<span> </span>(7 pages)<span></span></span></p>
</div>
<br>
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