# Condorcet Intended "Margin of Defeat" in Tiebreak

Hugh Tobin htobin at redstone.net
Sun Feb 16 23:04:09 PST 1997

```[Having perused Condorcet's Essai, I now attempt to correct an
ill-informed comment that I made previously on DEMOREP's analysis of
Condorcet's tiebreak method, and to suggest that DEMOREP's reading is
correct.]

Hugh Tobin wrote:
>
> DEMOREP1 at aol.com wrote:

> [snip]
> > [Condorcet's comments translated (by Prof. Young ?) from his Essai sur
> > l'application de l'analyse a la probabilite des decisions rendues a la
> > probabilite des voix (1785), pp. 125-126 ]
> > [snip]
> > 3. One forms an opinion from those n(n-1)/2 propositions that agree with the
> > most voices. If this opinion is among the n(n-1)... 2 possible opinions, one
> > regards as elected the subject to whom this opinion accords the preference.
> > If this opinion is among the 2 (to the exponent n(n-1)/2) minus n(n-1) .....
> > 2 impossible opinions, then one successively deletes from that impossible
> > opinion the propositions that have the least plurality, and one adopts the
> > opinion from those that remain.
> > -----
> > [snip]
> >
> > My comments----In view of Condorcet's definition of a "proposition" (i.e. a x
> > versus y pairing), if there is a cycle (i.e. tie- such as the above a>b, b>c,
> > c>a), then did Condorcet mean to drop the pairing(s) in the cycle with the
> > least pluralities (the one or more lowest x minus y, if x>y) to produce a
> > result without a tie ?   Note Condorcet's use of *successively deletes* and
> > *the least plurality*.
> >
> [snip]
>
> Why do you read "plurality" as "margin" or "difference"?  May we assume
> it is "pluralité" in the original French?  [snip] if Condorcet
> said "le moindre pluralité," the above sources [dictionaries] suggest this bears
> Ossipoff's reading better than yours. [Actually, I should not have said his "reading" --  Mike's position is that Condorcet is ambiguous but that the better rule is "least votes against in one's worst defeat" --H.T.]
>
> -- Hugh Tobin

I have now examined the original; Condorcet wrote that one discards
successively from the impossible opinion "les propositions qui ont une
moindre pluralité."  In the immediate context, given the usage of
"pluralité" in surrounding text (for example, contrasted with
"minorité"), the most natural reading is that it refers to the total
number of votes, not the difference.  However, in the larger context of
Condorcet's Essai, I now think it very likely that he intended what
Demorep said -- the margin of victory.

In at least one place in the Essai, p. 242, Condorcet does use
"pluralité" to mean the margin of victory.  At another point, attacking
a run-off system, he refers to both the top two candidates in the
primary as having "the plurality," p. 118.  At other points he uses the
term to refer to a required vote for success, such as a supermajority.
Given his use of "pluralité" with different meanings, any attempt to
decipher his
intent should focus on his underlying theoretical construct and the
wording of the example immediately following his general statement of
his tiebreak rule.  These suggest that had he considered the effect of
incomplete rankings, he would have used margins of defeat as the measure
of the tiebreaker.

Condorcet's whole framework was based upon the notion that the election
rules should maximize the likelihood of the "true" result, on these
assumptions: that each voter sought to vote for that result (rather than
voting out of corrupt or selfish motives); that each voter had a better
than even (but less than 100%) chance of choosing the true result in a
pairwise race; and that each voted independently of the others.  (In the
section of the Essai in question, Condorcet expressly assumes away "bad
faith").
Condorcet would define the worst defeat as the one which makes it least
probable, on the above assumptions, that the selection of the loser
would be the "true" result.  Indeed, in his example of a circular tie
immediately following his general statements, Condorcet states that A
will win a circular tie if his loss to C is "le moins probable" of A>B,
A<C, and B>C (p. 127).  Under his assumption, the "probability" of a
proposition (what we might call posterior probability) is a function of
both the votes for and the votes against that proposition (and of each
voter's probability of getting the vote right, "v").  Thus, Condorcet
would consider a loss by 1-10 with ten abstentions worse than a loss by
10-11, because in the latter case the probability that the loss was
"true," though greater than 50%, would be lower than in the former
case.
Indeed, if I understand Condorcet's analysis and Bayes' rule (which may
well be doubted), on Condorcet's assumptions the posterior probability
that a given proposition is "true" for any value of "v" depends only on
the margin of victory, i.e., for any "v" a 10-5 result has the same
probability of "truth" as a 100-95 result.  That this seems implausible
seems to be because the observed results are likely to affect our
estimate of "v". I invite correction here from the mathematically
minded.

The main point is that on Condorcet's assumptions another abstention
does not have the same effect in reducing the "probability" that one's
loss is the "true" result as does another vote in one's favor.

Thus, Condorcet's goal of accepting the least improbable result in order
to form a possible opinion and break a circular tie is more compatible
with Demorep's reading -- using margin of defeat as the measure of loss
-- than with the use of total votes against.

-- Hugh Tobin

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