<div dir="ltr">
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Greetings
list-members—<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Outline
of derivation of Bia-Free (BF):<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">First,
to define the terms in the explanation, I should say what a divisor-method is:<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Divide
total votes by total seats. That’s the Hare Quota.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Allocate
seats according to those rounding-results.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">q
= quotas.<span> </span>s = seats.<span> </span>R = the round-up point between a & b.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Above
the round-up point, s/q = b/q.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Below
the round-up point, s/q = a/q.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Integrate
b/q from R, to b.<span> </span><span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Integrate
a/q from a to R.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Add
the two integrals together.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">To
average over the interval, divide by b – a, the total amount of quota in the
interval.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…
<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">i.e.
divide by 1.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Solve
for R.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">R
= (1/e)((b^b)/(a^a)).<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">That
quantity is called (a special case of) the identric-mean of a & b.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Here
are the two academic-journal references:<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">“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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">By
Andrew E. Charman<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">It
was in _Physics and Society__, or _Journal of Physics and Society_, in 2017. <span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">(The
latest version of the article was in 2017)<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">The
citation said: <span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Cite
as arXiv.1712.09440v3 [<a href="http://physics.soc.ph">physics.soc.ph</a>]<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">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" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">The
other paper was:<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">“Optimal
Congressional Apportionment”<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">By
Robert A. Agnew.<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…in
The American Mathematical Monthly, for 2008, volume 115, number 4 (April 2008).<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">…<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:normal;font-size:11pt;font-family:"Calibri","sans-serif""><span style="font-size:10pt;font-family:"Courier New"">Pp
297-303<span> </span>(7 pages)<span></span></span></p>
</div>