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<p class="MsoNormal"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">Quota
averages</span></p>
<p class="MsoNormal"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> </span></p>
<p class="MsoNormal"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">Representative
Government, by John Stuart Mill, describes what we now call
the Hare quota, as
an average. Add the district electorates and divide by the
number of districts.
That is the arithmetic mean voters per seats.</span></p>
<p class="MsoNormal"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">The
Droop quota allows an election, to a single vacancy, with only
half the votes
to the candidate. Whereas, with the Hare quota, only the total
vote is
elective. The latter is undemocratic, because it requires
total deference to
one candidate. The Droop quota is also undemocratic, in that
it allows the
election of a candidate, with a negligibly greater vote than
another.</span></p>
<p class="MsoNormal"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">To
avoid the undemocratic extremes of maximum and minimum
proportional
representation, I introduced average proportional
representation. The average
of the Hare and Droop quotas, that is to say, a quota
representative of their
range, is their harmonic mean, which I therefore called the
Harmonic Mean
quota, V/(S + 1/2).</span></p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">The working is:
invert the Hare and Droop</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> quotas; add, and divide by two; then re-invert.
Or: </span></p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">[S/V + (S+1)/V]/2
= (2S+1)/2V. Hence: 2V/(2S+1).</span></p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">The Monty Hall
(multiple doors) problem is</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> a successive
choice and exclusion procedure that
probability sums to Droop</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> quota proportional
representation, the ratio: S/(S+1). <br>
</span></p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">The Harmonic Mean quota</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> series is alternate terms to the Droop quota.
Therefore,
the Harmonic Mean </span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">quota follows
Monty Hall procedure. As must an alternate
quota for</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> alternate terms of
the Droop quota.</span>
</p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">The alternate HM
quota PR is: (2S-1)/2S.</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> It is an average
of the Hare quota PR ratio and alternate
Droop quota ratio: S/S and</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> (S-1)/S. The latter
multiplied by the inverse Droop</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> quota ratio gives
a squared geometric mean quota ratio:
(S-1)(S+1)/S^2 = (S^2 –</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">1)/S^2. </span>
</p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">{The GM ratio is:
[(S^2 -1)^1/2)]/S. Multiply</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> by the Hare quota,
V/S, for the geometric mean quota: V[(S^2
-1)^1/2)]/S^2.}</span></p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> </span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">The effect of
multiplying alternate and inverse Droop
quota ratios is to give an</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> almost complete or
unitary proportional representation, while retaining the
essential Droop quota
ratio, at a much higher level. The Monty Hall
procedure, per number of door choices, produces Droop quota
proportional
representation. </span></p>
<span style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">Suppose the Monty
Hall problem starts with three doors, then the above GM formula
squares the
number of doors to 9, and the PR is in the ratio 8/9.</span>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">The contestant is
asked to make a choice, from nine doors, with a prize behind
one of them, and a
probability of success of one ninth.</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> The compere opens
one of the other doors, <span style="mso-spacerun:yes"> </span>lacking
the
prize, and offers the contestant another choice. The
probability of success is
now one eighth times a smaller statistical “universe” eight
ninths the size of
the original: 1/8 x 8/9 = 1/9. This<span
style="mso-spacerun:yes">
</span>choice may be repeated eight times, giving an eight
ninths probability
of success to finding the prize behind one of the nine doors.</span>
</p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">If the prize has
not been found behind one of the chosen eight doors, it is
certain to be behind
the ninth door. But equally, the compere is certain to remove
the one remaining
door, leaving zero probability of success. This is comparable
to the situation,
coming to the residual unrepresented quota, caused by the
Droop quota
denominator, in terms of seats plus one.</span></p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> </span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">A quota is an
average, and can be different kinds of average, the arithmetic</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> mean, harmonic
mean and geometricmean. The
geometric mean is a name</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> for a power
arithmetic mean. There is also a power harmonic mean, in which
one</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> of the binomial theorem terms is a fraction. This
was the
fourth average, in FAB</span><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> STV: Four Averages
Binomial Single Transferable Vote.
There, I used the power harmonic means, of individual
candidates keep values,
to</span><span style="font-size:16.0pt;font-family:"Arial
Rounded MT Bold""> average higher
orders of count, in accord with the binomial theorem.</span>
</p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""> Regards,</span></p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold"">Richaed Lung.</span></p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""><br>
</span></p>
<p class="MsoNormal"
style="margin-left:36.0pt;text-indent:-36.0pt"><span
style="font-size:16.0pt;font-family:"Arial Rounded MT
Bold""><br>
</span></p>
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