Left or right loser

[DEMOREP1 at aol.com]

Yet another Condorcet circular tie with 15 voters--

3 AC
4 AB

5 BC

2 CA
1 C

A v. B-- 9 to 5
B v. C-- 9 to 6
C v. A-- 8 to 7

According to Mr. Ossipoff, A should win by having the fewest votes against A
in his/her worst (and only) defeat (8).  Note that all of the B and C voters
combined thus could not defeat A (even if, horror of horrors, B and C were
"progressives" against a "reactionary" A)

However, B gets the fewest number of votes for him/her in his/her worst (and
only) defeat (5). If B loses, then C wins by defeating A.

If each vote is an approval vote, then there is-- 
A 9, B 9, C 11. 

The general case again for 3 candidates is
N1 A1 + N8 CA > N2 B1 + N9 CB
N2 B1 + N4 AB > N3 C1 + N5 AC
N3 C1 + N6 BC > N1 A1 + N7 BA

The "1" after A, B and C means first choice votes.  The  N amounts (N1, N2,
..., N9) are numbers of votes.  Assume there are minimums on both sides of
the greater than symbol (>) and assumes no ties.
The left side or right side minimum tiebreaker amount also applies when there
are 4 or more candidates in a circular tie.  If the right side minimum is
used, then after a loser is dropped the data is looked at to see if there is
a Condorcet winner among the remaining candidates.  If there is no Condorcet
winner, then the next right side minimum loser is dropped. 
-----
I repeat part of a July 27, 1996 posting--
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 ?   Note Condorcet's use of *successively deletes* and
*the least plurality*.

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