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<p>On 24/06/2023 5:55 am, Forest Simmons wrote:<br>
<br>
<blockquote type="cite">
<div dir="auto">Example 3</div>
<div dir="auto">49 C</div>
<div dir="auto">26 A>B</div>
<div dir="auto">25 B</div>
</blockquote>
<blockquote type="cite">In this example A wins the sincere
comparison 51 to 49.</blockquote>
<br>
Forest,<br>
<br>
You don't specify in the example that any of the votes are
insincere. Does A win because all the B voters are defecting
against A and will all show up for the runoff?<br>
<br>
</p>
<blockquote type="cite">
<div dir="auto">Example 1.</div>
<div dir="auto"><br>
</div>
<div dir="auto">45 A>B (sincere A>C)</div>
<div dir="auto">40 B>C</div>
<div dir="auto">15 C>A</div>
</blockquote>
<blockquote type="cite">The sincere winner between A and C is C. </blockquote>
<br>
<div dir="auto">Quite an impressive effort to save such a relatively
forlorn sincere CW. I wouldn't think too many people from FPP or
IRV land would be overly disturbed by just electing<br>
A in a one-round process.<br>
<br>
Chris<br>
<br>
<br>
</div>
<div class="moz-cite-prefix">On 24/06/2023 5:55 am, Forest Simmons
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CANUDvfqVmtATcNEVaK2MvbO+dQXT6PnC=H0nAciK+LhcwzRa=g@mail.gmail.com">
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
<div dir="auto">Before the examples, note that the Condorcet Loser
elimination step is redundant because the removal of covered
candidates will automatically take care of it.
<div dir="auto"><br>
</div>
<div dir="auto">The method thus simplified can be cast as ...</div>
<div dir="auto"><br>
</div>
<div dir="auto">0. Elect the Condorcet Winner if there is one.
Else ...</div>
<div dir="auto"><br>
</div>
<div dir="auto">1. Let K be the candidate with the smallest
MaxPO among the remaining candidates.<br>
</div>
<div dir="auto">2. Let X be the candidate with the smallest PO
against K.</div>
<div dir="auto">3. Let S be the set of remaining candidates that
are currently covered by X.</div>
<div dir="auto">4. Eliminate every member of S.</div>
<div dir="auto">5. Eliminate X.</div>
<div dir="auto">6 Repeat steps one thru five until only two
candidate remains.</div>
<div dir="auto">7. Elect the winner of a sincere runoff between
the two remaining candidates.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Note that once any Smith candidate is
eliminated, only Smith candidates will remain ... since each
Smith candidate covers every candidate outside of Smith.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Example 1.</div>
<div dir="auto"><br>
</div>
<div dir="auto">45 A>B (sincere A>C)</div>
<div dir="auto">40 B>C</div>
<div dir="auto">15 C>A</div>
<div dir="auto"><br>
</div>
<div dir="auto">The MMPO candidate is A with a maxPO of only 55,
frim C.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The candidate with the least PO against A is B,
with a PO of only 40 against A.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Candidate B is the only one eliminated because
it covers nobody else.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The sincere winner between A and C is C. </div>
<div dir="auto"><br>
</div>
<div dir="auto">Example 2.</div>
<div dir="auto"><br>
</div>
<div dir="auto">46 A>B</div>
<div dir="auto">44 B>C</div>
<div dir="auto">5 C>A</div>
<div dir="auto">5 C>B</div>
<div dir="auto"><br>
</div>
<div dir="auto">The MMPO candidate is B whose max opposition is
only 51, from A.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The min opposition to B is only ten, from C.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Eliminating C leaves A and B for the runoff.</div>
<div dir="auto"><br>
</div>
<div dir="auto">If the given ballots are sincere, then A will
win.</div>
<div dir="auto"><br>
</div>
<div dir="auto">But the cycle brings into question the sincerity
of the ballots; false cycles are more likely than sincere
cycles.</div>
<div dir="auto"><br>
</div>
<div dir="auto"> If the cycle was caused by B's unilateral
burial of A under C... by changing 44 B>A to 44 B>C
... then A wins the sincere comparison, removing the doubt.</div>
<div dir="auto"><br>
</div>
<div dir="auto">If the cycle was caused by burial of B under A
... by changing 10 C>A to 5 each of C>A and C>B ...
then B wins the sincere comparison, and is restored to its
rightful place as sincere CW.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Example 3</div>
<div dir="auto">49 C</div>
<div dir="auto">26 A>B</div>
<div dir="auto">25 B</div>
<div dir="auto"><br>
</div>
<div dir="auto">The MMPO candidate is A based on its second
highest PO score of 25 (from B), because A and B were tied for
the smallest MaxPO (of 49 from C).</div>
<div dir="auto"><br>
</div>
<div dir="auto">The candidate with min PO against A is B with
25, (compared to 49 from C).</div>
<div dir="auto"><br>
</div>
<div dir="auto">So B is eliminated, leaving the sincere runoff
between A and C. </div>
<div dir="auto"><br>
</div>
<div dir="auto">In this example A wins the sincere comparison 51
to 49.</div>
<div dir="auto"><br>
</div>
<div dir="auto">I hope these examples are sufficient to show how
the method works.</div>
<div dir="auto"><br>
</div>
<div dir="auto">fws</div>
<div dir="auto"><br>
</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Fri, Jun 23, 2023, 12:56 AM
Forest Simmons <<a href="mailto:forest.simmons21@gmail.com"
moz-do-not-send="true" class="moz-txt-link-freetext">forest.simmons21@gmail.com</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="auto">Chris has been keeping me humble ... shooting
down my worst nonsense ... while Kristofer has been busy
cleaning up code ... so watch out for an explosion of
progress!
<div dir="auto"><br>
</div>
<div dir="auto">Think of the MMPO candidate C as being
located in the midst of the other candidates in such a way
that the smallest neighborhood centered on C that is large
enough to contain the other candidates is as small as
possible.</div>
<div dir="auto"><br>
</div>
<div dir="auto">If you extend a line segment from C to
another candidate X on the outskirts of that neighborhood,
and then cut that segment with a perpendicular bisecting
plane ... that plane will separate the voters closer to X
from the voters closer to C. </div>
<div dir="auto"><br>
</div>
<div dir="auto">The number of voters closer to X is known as
X's Pairwise Opposition to C in that pairwise contest.</div>
<div dir="auto"><br>
</div>
<div dir="auto">This imaginary schematic suggests that the
candidate X with the least PO to C, would be a good one to
eliminate in a single winner method.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Accordingly ... when every candidate suffers
at least one pairwise defeat, we suggest the following
elimination procedure:</div>
<div dir="auto"><br>
</div>
<div dir="auto">While more than two candidates remain ...
Eliminate the candidate that is defeated pairwise by all
of the other candidates, if there is such a candidate ...
Otherwise ...</div>
<div dir="auto">1. let X be the candidate with the least PO
to the candidate C whose Max PO is smallest among the
remaining candidates. Then</div>
<div dir="auto">2. Eliminate X, as well as any candidate(s)
covered by X relative to the remaining candidates.</div>
<div dir="auto">EndWhile </div>
<div dir="auto"><br>
</div>
<div dir="auto">Finally, elect the sincere runoff winner
between the two remaining candidates.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Examples to follow soon ....</div>
<div dir="auto"><br>
</div>
<div dir="auto">For now, we only mention that the
eliminations are designed to make it highly likely that
when a sincere undefeated candidate exists; it will be one
of the finalists ... and also to point out that the
elimination of candidates covered by other candidates
being eliminated ... this precaution ensures that no
covered candidate will be elected ... thus avoiding a
potential liability that plauges nearly all other extant
methods.</div>
<div dir="auto"><br>
</div>
<div dir="auto">fws</div>
</div>
</blockquote>
</div>
</blockquote>
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