<div dir="auto">BTW, Bias-Free is my favorite allocation-rule for list-PR.</div><div dir="auto"><br></div><div dir="auto">But I propose Sainte-Lague, because…</div><div dir="auto"><br></div><div dir="auto">…though SL has a tiny amount of bias in favor of larger parties, it’s so slight that it’s completely insignificant.</div><div dir="auto"><br></div><div dir="auto">e.g. </div><div dir="auto"><br></div><div dir="auto">150 seats</div><div dir="auto"><br></div><div dir="auto">17 small parties, each with 3% of the vote </div><div dir="auto"><br></div><div dir="auto">… together amounting to 51% of the vote </div><div dir="auto"><br></div><div dir="auto">One big party with 49% of the vote </div><div dir="auto"><br></div><div dir="auto">The 17 small parties each with 3% of the vote together get a majority of the seats, as they should, with Sainte-Lague.</div><div dir="auto"><br></div><div dir="auto">They can form a majority coalition & a government.</div><div dir="auto"><br></div><div dir="auto">With d’Hondt, the big party gets a big majority.</div><div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Wed, Sep 6, 2023 at 22:04 Michael Ossipoff <<a href="mailto:email9648742@gmail.com">email9648742@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)"><div><br></div><div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Tue, Sep 5, 2023 at 23:59 Richard Lung <<a href="mailto:voting@ukscientists.com" target="_blank">voting@ukscientists.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)">
<div>
<p><br>
</p>
<p>Just a preliminary remark. Divisor methods and quotas some times
distinguished. </p></div></blockquote><div dir="auto"><br></div><div dir="auto">I don’t know what you mean by that.</div><div dir="auto"><br></div><div dir="auto">I didn’t say that the divisor methods are quotas. I didn’t say that quotas are divisor methods.</div><div dir="auto"><br></div><div dir="auto">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.</div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)"><div><p dir="auto">Thus there is the Droop quota and corresponding
D'Hont divisor method. </p></div></blockquote><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">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.</div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)"><div><p dir="auto">Divisor methods regarded as belonging to
apportionment</p></div></blockquote><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">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.</div><div dir="auto"><br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)"><div><p dir="auto">as by Jefferson and by Webster, not to carve out
party seats, which is too restrictive of personal choice.</p></div></blockquote><div dir="auto"><br></div><div dir="auto">Carve out?</div><div dir="auto"><br></div><div dir="auto">I’m not sure, but you seem to be saying that party-list PR restricts personal choice.</div><div dir="auto"><br></div><div dir="auto">But you didn’t say why you think so.</div></div></div><div><div class="gmail_quote"><div dir="auto"><br></div><div dir="auto"><br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)"><div><p dir="auto"><br>
</p></div><div>
<p></p></div></blockquote><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto"><br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-color:rgb(204,204,204)"><div><p><br>
</p>
<div>On 06/09/2023 05:05, Michael Ossipoff
wrote:<br>
</div>
</div><div><blockquote type="cite"></blockquote></div><div><blockquote type="cite">
<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" target="_blank">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>
<fieldset></fieldset>
</blockquote></div><div><blockquote type="cite"><pre style="font-family:monospace">----
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