# [EM] Two Condorcet Winners?

Adam Tarr atarr at purdue.edu
Mon Aug 19 00:02:49 PDT 2002

```At 04:49 PM 8/18/2002 -0400, Elisabeth Varin/Stephane Rouillon wrote:

>I believe there can be two different CW, one CW(wv) using winning votes
>and another CW(rm) using relative margins. "So with wv, truncation won't
>steal the election from a majority-supported CW(wv), as it will with
>margins and relative margins, but IT WILL STEAL THE ELECTION FROM A
>MAJORITY-SUPPORTED CW(rm) as it will with margins and relative margins.

It is a stretch of terminology to say you have two Condorcet winners
here.  It seems more accurate to say you have zero.

Craig Carey objections aside, I think everyone agrees what a Condorcet
winner is: a candidate who, based on submitted ranked ballots, beats every
other candidate in a pairwise comparison.  If such a candidate exists, he
or she will exist in Condorcet(m), Condorcet(wv), and Condorcet(rm), and it
will be the same candidate every time.  How you measure the strength of a
pairwise defeat is irrelevant if all the defeats point toward the same
candidate.

With that in mind, the only way I can interpret your statement ("two
different CW, one CW(wv) using winning votes and another CW(rm) using
relative margins") Is that you are referring not to a voted Condorcet
winner, but rather the winner of the election after some elimination
approach (say ranked pairs) has been used to eliminate cyclic
ambiguities.  These candidates are not Condorcet winners; they are the
declared winners of election methods that explicitly follow the Condorcet
criterion.

When Mike says "with wv, truncation won't steal the election from a
majority-supported CW", he's saying that truncation on some ballots
introduced a cyclic ambiguity into an election that otherwise had a
Condorcet winner (the one and only kind of Condorcet winner), but the
truncation does not change the outcome.  Your addition, "but IT WILL STEAL
THE ELECTION FROM A MAJORITY-SUPPORTED CW(rm)" implies that this candidate
was a Condorcet winner in some meaningful sense, which he or she simply was
not.  This candidate was the winner after the application of a certain
process for resolving cyclic ambiguities, and nothing else.  There is only
one Condorcet winner in this example, and that's the one that exists BEFORE
the truncation occurs.

This is not to say that your criterion have no merit, although I disagree
with them.  But your particular argument here essentially reduces to:
"RP(wv) is a bad method because it will not always reproduce the results of
RP(rm)."  Which is not really a useful criticism, i.e. it does nothing to
highlight the relative merits of the two approaches.