# Condorect's successive deletions

Mike Ossipoff dfb at bbs.cruzio.com
Sat Jul 27 13:18:49 PDT 1996

```DEMOREP1 at aol.com writes:

[Mike adds: If you're interested in Condorcet's own wording, then also
check out the translation in Duncan Black's _Theory of Committees &
Elections_. I'm not sure of the exact wording of the title, but I
believe the way I said it is probably correct. Black's book, it seems
to me, was written in 1958]

>
> 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.

So, if 1 alternative beats each of the others, then it wins.

> 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.

That's the same as saying that the winner is the candidate who is
least beaten in his worst pairwise comparison. Though Condorcet didn't
specify whether to judge that by votes-against, or margin, or ratio,
or even votes-for, the use of the word "plurality" of a proposition
seems to imply votes-against. In any case, if Condorcet didn't
with Condorcet's words.  Votes-against gives the method the important
properties that I've been talking about.

> -----
> 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 ?

That's what the man said. Except that you've assumed that least plurality]
means least margin--you're judging pairwise defeats by margin. Condorcet
didn't say to do that. As I said, though Condorcet may not have specifically
said how to judge defeats, the reference to a proposition's 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.

The object of dropping propositions is to _get rid of_ ties for winner,
and that's the result. A direct quotation of what Young said about
that would be better, because he couldn't have said what you said
he did. Of course in a small election there could be _pairwise_ ties,
where neither of two alternatives beats the other, because equal numbers
of people rank A over B & B over A. That can't be called a fault of
Condorcet's choice rule, though. When there are pairwise ties, there
can, of course be several unbeaten alternatives, and that's the
kind of thing that could be worthwhile to deal with in small committee
elections (if the participants are into the additional rules needed),
but it's something of no relevance for public elections. But if we
wanted to get fancy, for elections on EM, we could use the Schwartz
set, and have a rule to deal with several unbeaten alternatives.
I emphasize that such refinements are optional, even in small
committees, and of no relevance in public elections.

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

That depends on how defeats are judged. If we interpret a proposition's
plurality to mean votes-against, for that propopsition's pairing, then
the answer is no: Condorcet's proposal has no serious defect.

True, the academics don't like plain Condorcet, because it doesn't
meet the academic criteria, which I don't consider nearly as
important as the LO2E criteria & GMC, which plain Condorcet
meets. But, as you know, Smith//Condorcet meets those criteria.

And it seems to me that Condorcet was proposing Smith//Condorcet,
because he proposed his choice rule as a way of solving circular
ties. If the object is to break a circular tie, isn't it reasonable
to suppose that Condorcet meant for his choice rule to choose from
that circular tie?

So then, if we (reasonably, if Young's translation is literal)
he intended his choice rule to choose from among the circular
tie, then Condorcet's original proposal is free of defects,
even in terms of the academic candidate-counting criteria.

I'm not claiming that it's for sure that Condorcet meant votes-against,
only that that is what's implied by "a proposition's plurality".

>