Condorect's successive deletions

Fri Jul 26 21:59:50 PDT 1996

Condorcet's Theory of Voting by H. P. Young, 82 American Political Science
Review 1231 (Dec. 1988), contains on page 1233----

In Condorcet's lexicon, an "opinion" is a series of pairwise comparisons on
the alternatives. Each pairwise combination is called a "proposition" and
written a>b, etc. An opinion is said to be "impossible", "contradictory", or
"absurd" of some of the propositions composing it form a cycle, such as a>b,
b>c, c>a. Normally, each individual voter is able to rank all of the
candidates in a consistent order. *** To break such cyclic majorities,
Condorcet proposed the following method.
[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 ]
1. All possible opinions that do not imply a contradiction reduce to an
indication of the order of merit that one judges among the candidates .....
therefore for n candidates one would have n(n-1).... 2 possibilities.
2. Each voter having thus given his or her opinion by indicating the
candidates' order of worth, if none compares them two by two, one will have
in each opinion n(n-1)/2 propositions to consider separately. Taking the
number of times that each is contained in the opinion of one the q voters,
one will have the number of voices who are for each proposition.
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.
Prof. Young noted that Condorcet unfortunately did not give any math example
with 4 or more candidates. 

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 ?

However, Prof. Young notes that if one or more pairings is dropped, then one
or more candidates may be undominated or tied for first.

Does Condorcet's own tie breaker have some "serious' defects ?

See also my--
Subj:  Total votes against tie breaker
Date:  Fri, Jul 19, 1996 11:40 PM EDT.

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