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<p><br>
<blockquote type="cite">
<div dir="auto">This suggests another method ...</div>
<div dir="auto">elect the candidate with the highest ratio given
by the expression</div>
<div dir="auto"><br>
</div>
<div dir="auto">((MaxPS+MinPS)/2)/MaxPO</div>
<div dir="auto"><br>
</div>
<div dir="auto">which is an estimate of the ratio of the
approval the candidate would get if it were the approval
cutoff candidate to the max approval any other candidate would
get with the same cutoff.</div>
<div dir="auto"><br>
</div>
<div dir="auto">In other words, it is candidate expected to
bear up the best under Approval voting if it were the
projected winner ... therefore (adjacent to) the approval
cutoff in the next round of repeated voting, say.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Restricting this to Smith should be good.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Example:</div>
<div dir="auto"><br>
</div>
<div dir="auto">48 C</div>
<div dir="auto">28 A>B</div>
<div dir="auto">24 B</div>
<div dir="auto"><br>
</div>
<div dir="auto">The respective ratios for A, B, and C are</div>
<div dir="auto"><br>
</div>
<div dir="auto">26/48, [(52+24)/2]/48, 48/52</div>
<div dir="auto"><br>
</div>
<div dir="auto">So C wins.</div>
</blockquote>
<br>
Is there a typo here? Where does the 26 in "26/48" come from?
Should it be 28?<br>
</p>
<p>I'm not really switched on to the positive point of this
relatively complicated method.<br>
<br>
In your example it fails Minimal Defense. Does it meet Chicken
Dilemma?<br>
<br>
<blockquote type="cite">*Eliminate all the candidates not in the
Smith set. Give each remaining <br>
candidate a score equal to the number of ballots<br>
on which it is ranked (among remaining candidates) below no
other <br>
candidate minus the number of ballots on which it<br>
is ranked (among remaining candidates) above no other candidate.<br>
<div dir="auto"><br>
</div>
<div dir="auto">This score can be approximated as the average of
the Max and Min Pairwise Supports (restricted to Smith) of the
candidate.</div>
</blockquote>
<br>
I can see that that would nearly always (or always?) be the same
thing, and that it could be just read off the pairwise matrix <br>
(which might streamline the counting process a lot).<br>
<br>
Chris B.<br>
<br>
<br>
</p>
<div class="moz-cite-prefix">On 11/08/2023 1:37 am, Forest Simmons
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CANUDvfr0VhdLRs3-zb1dksnJEVM186=B4U47RSV94cgRXSBTjQ@mail.gmail.com">
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
<div dir="auto">
<div><br>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Sun, Aug 6, 2023, 2:59
PM C.Benham <<a href="mailto:cbenham@adam.com.au"
moz-do-not-send="true" class="moz-txt-link-freetext">cbenham@adam.com.au</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex"><br>
I think Condorcet methods that don't allow voters to enter
an approval <br>
threshold have to choose between trying to<br>
minimise Compromise incentive or trying to reduce
Defection incentive.<br>
<br>
The methods I like in this category allow voters to rank
however many <br>
candidates they like and also approve all but<br>
one or only one or any number in between of the candidates
(consistent <br>
with their rankings). Equal-ranking is allowed.<br>
<br>
Default approval goes only to candidates ranked below no
other candidate.<br>
<br>
I suggest that voters can just mark one of the candidates
as the lowest <br>
ranked one they approve (i.e. only that candidate<br>
and those ranked higher or equal to it are approved).<br>
<br>
But other ways of doing it could be fine.<br>
<br>
Regarding which algorithm, I very much like Forest's
Sorted Approval <br>
Margins.<br>
<br>
I also like another method of his, the exact name of which
I've <br>
forgotten (something about "Chain" building or climbing):<br>
<br>
*Begin the chain with the most approved candidate. Then
add the most <br>
approved candidate that covers that candidate.<br>
Then add the most approved candidate that covers all the
candidates <br>
already in the chain.<br>
<br>
Keep doing that as many times as possible, and then elect
the last added <br>
candidate*.<br>
<br>
I think nearly always this will elect the same candidate
as <br>
Smith//Approval, but is more elegant and ensures that the<br>
winner is Uncovered.<br>
<br>
For a practicable Condorcet method that uses plain ranked
ballots <br>
(equal-ranking and truncation allowed), I like<br>
Smith//Ranked below none minus ranked above none.<br>
<br>
*Eliminate all the candidates not in the Smith set. Give
each remaining <br>
candidate a score equal to the number of ballots<br>
on which it is ranked (among remaining candidates) below
no other <br>
candidate minus the number of ballots on which it<br>
is ranked (among remaining candidates) above no other
candidate.<br>
</blockquote>
</div>
</div>
<div dir="auto"><br>
</div>
<div dir="auto">This score can be approximated as the average of
the Max and Min Pairwise Supports (restricted to Smith) of the
candidate.</div>
<div dir="auto"><br>
</div>
<div dir="auto">This suggests another method ...</div>
<div dir="auto">elect the candidate with the highest ratio given
by the expression</div>
<div dir="auto"><br>
</div>
<div dir="auto">((MaxPS+MinPS)/2)/MaxPO</div>
<div dir="auto"><br>
</div>
<div dir="auto">which is an estimate of the ratio of the
approval the candidate would get if it were the approval
cutoff candidate to the max approval any other candidate would
get with the same cutoff.</div>
<div dir="auto"><br>
</div>
<div dir="auto">In other words, it is candidate expected to
bear up the best under Approval voting if it were the
projected winner ... therefore (adjacent to) the approval
cutoff in the next round of repeated voting, say.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Restricting this to Smith should be good.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Example:</div>
<div dir="auto"><br>
</div>
<div dir="auto">48 C</div>
<div dir="auto">28 A>B</div>
<div dir="auto">24 B</div>
<div dir="auto"><br>
</div>
<div dir="auto">The respective ratios for A, B, and C are</div>
<div dir="auto"><br>
</div>
<div dir="auto">26/48, [(52+24)/2]/48, 48/52</div>
<div dir="auto"><br>
</div>
<div dir="auto">So C wins.</div>
<div dir="auto"><br>
</div>
<div dir="auto"><br>
</div>
<div dir="auto"><br>
</div>
<div dir="auto"><br>
</div>
<div dir="auto">
<div class="gmail_quote">
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
Elect the candidate with the highest score."<br>
<br>
Given how rare top cycles will likely be, I think this is
probably good <br>
enough.<br>
<br>
Obviously it meets Plurality. It fails both Minimal
Defense and Chicken <br>
Dilemma, but never both at once :)<br>
<br>
It looks fair and gives a pretty-enough winner. I'll be
back later with <br>
some examples.<br>
<br>
Chris Benham<br>
<br>
<br>
</blockquote>
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