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<font face="Helvetica, Arial, sans-serif">Perhaps some people will
be interested in another conclusion I came to from reading
Condorcet's Essai. He proposed a method of breaking cycles which
generated a lot of confusion until Peyton Young glossed it as a
garbled account of the Kemeny-Young method. His reading has been
widely accepted; Tideman (in his 2006 book) declared that
"Condorcet's intent is decoded to my satisfaction" by Young.<br>
<br>
Condorcet described his method twice: forwards in the Preliminary
Discourse and backwards in the body of the work. Young only
discusses the backwards version. In both cases Condorcet starts
from a list of pairwise comparisons, sorted by margin. In the
backwards version he writes: "We will successively discard from
the contradictory set the preferences which have the smallest
majority, and elect the candidate preferred by those which
remain". Presumably he stops discarding when the residue is
consistent; the flaw is that by this point there may not be enough
comparisons left to determine a unique winner. Young noticed this
and remarked that "It seems more likely that Condorcet meant to
*reverse*, rather than to *delete* the weakest proposition". This
is nonsense: no one writes "delete" when they mean "swap", and
Young's reading doesn't fit the forwards version. <br>
<br>
The forwards statement is clearer: "We thus obtain the following
general rule, that whenever we are required to elect a candidate,
we must take in turn all the pairwise preferences which have
majority support, starting with the largest majorities, and make a
decision according to these initial preferences as soon as they
imply one, without worrying about the less probable later
preferences." In other words, given a list sorted in decreasing
order of margin, take an initial part which is small enough to be
consistent but large enough to determine a unique winner. This has
a corresponding flaw, which is that as you work through the list,
you may be forced to include a comparison which contradicts those
already present before reaching the point at which you have a
winner. But there's nothing here which you can interpret as
meaning "swap" rather than something else. <br>
<br>
It seems to me as clear as daylight that Condorcet had an
incomplete grasp of Tideman's Ranked Pairs. Tideman recognised the
risk that a new pair may contradict the ones already in the list,
and he saw what to do about it, namely throw it away.<br>
<br>
CJC<br>
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