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<p>It's gratifying to be praised for helping something you don't
yourself comprehend. Forest has that good nature.</p>
<p>It may be worth mentioning that the binomial stv use of an
election count and an exclusion count may both benefit a
candidate. If the candidate is popular enough to gain an election
quota, but not too unpopular to gain an exclusion quota, then a
poor exclusion count can consolidate a good election count, for
the same candidate..</p>
<p>Unlike classical Borda count, but in iine with the Gregory count,
later preferences do not count against former preferences. When
there is a conflict with a candidate both popular (in the election
count) and unpopular (in the exclusion count) it is a reflection
of real differences in opinion between different factions of the
electorate.<br>
</p>
<p>Unlike traditional stv, the Gregory count, via an extension of
Meek method keep-value counting to all candidates, not just
elected candidates, is repeated on the voters preferences, in
reverse, as an unpreference vote. This necessitates the counting
of abstentions, to ascertain the balance of liking and disliking
of the candidates, by the voters.</p>
<p>This in turn means all the preferences are counted, which amounts
to a complete dimension of choice counted -- not merely an
uncertain fraction of a dimension counted.</p>
<p>Real elections with binomial stv should show how the entropy of
the voters usual exponential decline in preferences -- or
unpreferences -- works or fails to work for decisive results, in
practise.<br>
</p>
<p>Sorry, if this explanation was too cryptic.</p>
<p>Regards,</p>
<p>Richard Lung.</p>
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<div dir="auto">To give credit where credit is due, I want to thank
Richard Lung for getting us thinking along the inclusion/exclusion
axis corresponding to near-first/near-last ballot ranks ... That
contributed to my confidence in the lottery/anti-lottery idea as a
means of de-cloning Borda, Copeland. and Kemeny-Young, which has
panned out so beautifully.
<div dir="auto"><br>
</div>
<div dir="auto">Thanks, Richard!!!</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">El vie., 14 de ene. de 2022 8:18
a. m., Forest Simmons <<a
href="mailto:forest.simmons21@gmail.com"
class="moz-txt-link-freetext">forest.simmons21@gmail.com</a>>
escribió:<br>
</div>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="auto">
<div>Two equivalent ways of decloning Borda ... both relying
on a combination of the benchmark and anti-benchmark
lotteries BML and ABML.
<div dir="auto"><br>
</div>
<div dir="auto">First method:</div>
<div dir="auto"><br>
</div>
<div dir="auto">Ballot B contributes to candidate X's score
the sum of BML(j) [summed over candidates j ranked below
X] minus the sum of ABML(k) [summed over candidates k
ranked above X].</div>
<div dir="auto"><br>
</div>
<div dir="auto">Think of it this way ... the importance of
candidate X to the voter of ballot B is the estimated
strength of the candidates she likes less than X, minus
the estimated weakness of the candidates she likes better:</div>
<div dir="auto"><br>
</div>
<div dir="auto">[If the ones you like better are weak and
the ones you dislike in comparison are strong, you should
support X.]</div>
<div dir="auto"><br>
</div>
<div dir="auto">[Also this is a natural DSV kind of way to
specify an approval cutoff]<br>
<div dir="auto"><br>
</div>
<div dir="auto">Method 2.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The score for X is the difference in the
BML expectation of X's pairwise matrix row and the ABML
expectation of X's pairwise matrix column.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The former is a source of pride for X
...the weighted average pairwise scores of X over the
candidates, where the weights are estimates of the
strengths of those candidates.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The latter expectation is a source of
chagrin for X ... the weighted average of the scores
against X by the other candidates, where the weights are
estimates of the weakness of those candidates... if you
are soundly trounced by a weak candidate, you should
feel chagrin, if not shame!</div>
<div dir="auto"><br>
</div>
<div dir="auto">This gives another idea for a kind of
MinMax/MaxMin method:</div>
<div dir="auto"><br>
</div>
<div dir="auto">Let M(X) be the Max value of the product
of ABML(k)*P(k,X), where P(k, X) is the percentage of
ballots on which k is ranked ahead of X.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Similarly, let m(X) be the min value of
P(X, j)*BML(j), where P(X,j) is the percentage of
ballots ranking X above j.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Elect argmin(M(X)-m(X)).</div>
<div dir="auto"><br>
</div>
<div dir="auto">[Maybe this is another key to fpC-fpA]</div>
<div dir="auto"><br>
</div>
<div dir="auto">Note the strong symmetry of this method,
also.</div>
<div dir="auto"><br>
</div>
<div dir="auto">All in all, I'm pretty excited about the
usefulness of the anti-benchmark lottery in conjunction
with the benchmark lottery.</div>
<div dir="auto"><br>
</div>
<div dir="auto">If BML(X) is an estimate of X's viability,
then ABML(X) is an estimate of X's weakness.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Of course, the BML is not the only
clone-free lottery ...Jobst's MaxParC, the Martin Harper
Lottery, the Random Approval Lottery, the Nash Lottery
and its "Ultimate Lottery" generalization ...all of
these and others have their corresponding anti-lottery
counterparts ... just reverse the ballots to get them.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Why didn't we think of this before? It
was staring us right in the face! </div>
<div dir="auto"><br>
</div>
<div dir="auto">What other obvious ideas are still
invisible to us?</div>
<div dir="auto"><br>
</div>
<div dir="auto">-Forest</div>
</div>
<br>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">El vie., 14 de ene. de
2022 6:24 a. m., Forest Simmons <<a
href="mailto:forest.simmons21@gmail.com"
target="_blank" rel="noreferrer"
class="moz-txt-link-freetext">forest.simmons21@gmail.com</a>>
escribió:<br>
</div>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="auto">
<div>Kristofer,
<div dir="auto"><br>
</div>
<div dir="auto">I have a feeling you might like this
method:</div>
<div dir="auto"><br>
</div>
<div dir="auto">For simplicity assume complete
rankings.</div>
<div dir="auto"><br>
</div>
<div dir="auto">For each candidate X, let fp(X) and
lp(X), respectively, be the total number of first
place votes and last place votes, respectively, of
the candidates that are pairwise defeated by X,
and pairwise defeat X, respectively. </div>
<div dir="auto"><br>
</div>
<div dir="auto">Elect argmax(fp(X)-lp(X))</div>
<div dir="auto"><br>
</div>
<div dir="auto">What do you think?</div>
<div dir="auto"><br>
</div>
<div dir="auto">-Forest</div>
<div dir="auto"><br>
</div>
<div dir="auto"><br>
</div>
<br>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">El vie., 14 de
ene. de 2022 2:11 a. m., Kristofer Munsterhjelm
<<a href="mailto:km_elmet@t-online.de"
rel="noreferrer noreferrer" target="_blank"
class="moz-txt-link-freetext">km_elmet@t-online.de</a>>
escribió:<br>
</div>
<blockquote class="gmail_quote" style="margin:0 0
0 .8ex;border-left:1px #ccc
solid;padding-left:1ex">On 14.01.2022 11:01,
Kristofer Munsterhjelm wrote:<br>
> On 14.01.2022 04:05, Forest Simmons wrote:<br>
>> I've always admired the basic idea of
Kemeny-Young, namely a cost metric<br>
>> for conversion of one ranking into
another ... simply the number of<br>
>> transpositions (elementary swaps)
required.<br>
>><br>
>> The deal-breaker draw-back of the
Kemeny cost metric is that it is clone<br>
>> dependent ... if it were independent of
clones the " cost" of<br>
>> transposing AB to BA would be the same
as the net cost of transposing<br>
>> the rank order of respective clones of
A and B:<br>
>><br>
>> the order a1a2a3b1b2b3<br>
>> to the order b3b2b1a3a2a1<br>
> <br>
> Of course, I have to restate here that
Ranked Pairs is such a method:<br>
> the metric is just leximax. (ordering a is
better than ordering b if the<br>
> greatest pairwise victory consistent with a
is higher than the greatest<br>
> pairwise victory consistent with b, with
tiebreaks for second greatest,<br>
> third greatest, etc.) And it's cloneproof.<br>
<br>
Here's a thought: let parameterized p-Kemeny be:
take the lp norm of the<br>
vector of all pairwise victories consistent with
the output ordering.<br>
Break ties by gradually lowering p, i.e.
l(p-epsilon), further ties by<br>
l(p-2 epsilon), etc.<br>
<br>
Then Kemeny is l_1 and RP is the limit as p
approaches infinity, l_inf.<br>
I think? Are any other values of p interesting?<br>
<br>
-km<br>
</blockquote>
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