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<p>I missed the earlier discussion on this. <br>
<br>
So what if an elimination round <i>does </i>have a
"pairwise-losing candidate", what then? <br>
<br>
And what exactly is a "pairwise-losing candidate"? <br>
<br>
Just curious.<br>
<br>
Chris Benham<br>
</p>
<div class="moz-cite-prefix">On 13/01/2020 12:14 pm, VoteFair wrote:<br>
</div>
<blockquote type="cite"
cite="mid:b495b3c8-a12d-9bb9-1ce5-85da72371800@votefair.org">Based
on a suggestion from a user on Reddit, I have revised the
definition of the Instant Pairwise Elimination method that
previously I published at Democracy Chronicles and then discussed
here.
<br>
<br>
The method still successively eliminates pairwise (Condorcet)
losers.
<br>
<br>
Now, instead of resolving Condorcet (rock-paper-scissors) cycles
using an "upside-down" version of instant-runoff voting (IRV), it
uses pairwise counts as described here:
<br>
<br>
"If an elimination round has no pairwise-losing candidate, then
the method eliminates the candidate with the largest pairwise
opposition count, which is determined by counting on each ballot
the number of not-yet-eliminated candidates who are ranked above
that candidate, and adding those numbers across all the ballots.
If there is a tie for the largest pairwise opposition count, the
method eliminates the candidate with the smallest pairwise support
count, which similarly counts support rather than opposition. If
there is also a tie for the smallest pairwise support count, then
those candidates are tied and all those tied candidates are
eliminated in the same elimination round."
<br>
<br>
Below are my guesses for which fairness criteria it fails and
passes. Please tell me which guesses are not correct.
<br>
<br>
Condorcet: fail
<br>
Condorcet loser: pass
<br>
Ranks equal: pass
<br>
Ranks greater than 2: pass
<br>
Polytime: pass
<br>
Resolvable: pass
<br>
Majority: fail
<br>
Majority loser: fail
<br>
Mutual majority: fail
<br>
Smith/ISDA: fail
<br>
LIIA: fail
<br>
IIA: fail
<br>
Cloneproof: fail
<br>
Monotone: fail
<br>
Consistency: fail
<br>
Reversal symmetry: fail
<br>
Later no harm: fail
<br>
Later no help: fail
<br>
Burying: fail
<br>
Participation: fail ?
<br>
No favorite betrayal: fail ?
<br>
Summable: O(N!) ?
<br>
<br>
As I've said many times, it's the frequency with which the
failures occur that is much, much more important than simply
counting how many criteria it fails. I suspect that its
frequencies of failure will be quite low compared to most other
single-winner methods, and may approach the low frequencies that I
believe characterize the Condorcet-Kemeny method.
<br>
<br>
I've created a page for this method on Electowiki. You are welcome
to edit that page with any corrections.
<br>
<br>
BTW, I realize that it's possible that the alternate elimination
method always identifies the pairwise/Condorcet loser (if there is
one). If so, this would mean that the description could be
"simplified" to a single step (actually two steps in case there is
a tie). However, for the benefit of most voters who are not
comfortable with mathematics it's important to explicitly state
that the first priority is to eliminate the pairwise loser.
<br>
<br>
Of course software that implements the method would do the
calculations using a much faster method than the counting method
described above. The description above is written to be
understandable to people who are not already familiar with
pairwise counting.
<br>
<br>
In advance, thank you for any feedback.
<br>
<br>
Richard Fobes
<br>
----
<br>
Election-Methods mailing list - see <a class="moz-txt-link-freetext" href="https://electorama.com/em">https://electorama.com/em</a> for
list info
<br>
</blockquote>
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