[EM] Condorcet's words
MIKE OSSIPOFF
nkklrp at hotmail.com
Mon Jan 5 06:35:17 PST 2004
Condorcet wrote ("Essai sur l'application de l'analyse
a la probabilite des decisions rendues a la pluralite
des voix," Imprimerie Royale, Paris, p. LXVIII of the
introduction, 1785):
>From the considerations we have just made we get the
>general rule that whenever we have to choose we have
>to take successively those propositions that have a
>plurality -- beginning with those that have the largest
>-- and to pronounce the result as soon as these first
>propositions create one.
Taking successively those propositions (pairwise defeats) that have the
largest plurality is Condorcet's "keep-strongest" proposal, of which
Ranked-Pairs is considered the best interpretation.
If Condorcet didn't specify the details, he certainly pointed the way to RP
& PC.
Condorcet wrote ("Essai sur l'application de l'analyse
a la probabilite des decisions rendues a la pluralite
des voix," Imprimerie Royale, Paris, p. 126, 1785):
>Create an opinion of those n*(n-1)/2 propositions that
>win most of the votes.
An "opinion" is a public collective opinion, consisting of the set of
"propositions" (pairwise defeats).
There are N*(N-1)/2 propositions because that's how many pairs of candidates
there are.
Half of those will be the winning propositions, where, say, A>B beats B>A.
>If this opinion is one of the
>n! possible ...
The possible opinions are the transitive combinations of propositions
(pairwise defeats), each of which corresponds to a transitive ordering of
the N candidates. N candidates can be ordered N! ways.
...then consider as elected that subject to
>which this opinion agrees with its preference. If this
>opinion is one of the (2^(n*(n-1)/2))-(n!) impossible
>opinions
There are n*(n-1)/2 pairs of candidates. Each pairwise comparison could go
either way. So the total number of possible configurations of pairwise
results is 2 to the power of the number of pairs of candidates. Subtract
from that the number of possible opinions, n!, and you get the number of
impossible opinnions.
So far that's just been an enumeration of the possible and impossible
propositions. Here's Condorcet's drop-weakest count proposal:
So, he says, assuming that what we've got from the balloting is one of the
impossible propositions, an election result with one or more cycles, he
suggests what to do.
One difference from PC is that PC only has to deal with a cycle if it
results in no one being unbeaten, while Condorcet seems to be saying that
the procedure below is used if there are _any_ cycles. I suggest that he
meant that it's to be used only if there's no unbeaten candidate.
>then eliminate of this impossible opinion
>successively those propositions that have a smaller
>plurality and accept the resulting opinion of the
>remaining propositions.
Since he says to do that if the opinion is impossible, if the election has
any cycles, that almost suggests that he's suggesting to drop weakest
defeats till you get a possible proposition, meaning that you get rid of all
the cycles. That could give you unnecessarily many winners. Surely he'd stop
as soon as someone is unbeaten. If so, then surely he wouldn't even use the
procedure if there's already someone unbeaten.
Hence, I suggest that PC is what he's proposing.
That's Condorcet's drop-weakest proposal, and apparently he's proposing PC.
Condorcet wrote ("Sur la Forme des Elections," 1789):
>To compare just 20 candidates two by two, we must examine
>the votes on 190 propositions, and for 40 candidates,
>on 780 propositons. Besides, this will often give us an
>unsatisfactory result; it may be that no candidate is
>considered by the plurality to be better than all the
>others, and then we would have to prefer the candidate
>who is just considered better than a larger number
Here Condorcet is proposing Copeland. Universally, when "Condorcet's method"
is used to refer to a particular circular tie solution, it's used to refer
to his keep-strongest or drop-weakest proposals. When someone wants to refer
to the circular tiebreaker known as Copeland, s/he calls it Copeland, not
Condorcet. (Of course "Condorcet" is sometimes used overbroadly to refer to
all pairwise count methods, but that's another issue).
So, by current usage, Copeland isn't spoken of as a Condorcet tiebreaker
version. Copeland is just called Copeland, while Condorcet, when it refers
to a particular circular tiebreaker, refers to Condorcet's drop-weakest or
Condorcet's keep-strongest.
Next, Condorcet suggests how to break the ties that Copeland is so prone to,
even in public elections with many voters.
>; and
>when several were considered better than the same number
>of candidates, we would have to choose the candidate who
>was either considered better by the greatest plurality,
>or worst by the smallest plurality.
Condorcet wrote ("Sur les Elections," Journal
d'Instruction Sociale, vol. 1, p. 25-32, 1793):
>A table of majority judgements between the candidates
>taken two by two would then be formed and the result
>-- the order of merit in which they are placed by the
>majority -- extracted from it. If these judgements
>could not all exist together, then those with the
>smallest majority would be rejected.
Again, Condorcet is making the drop-weakest proposal, more concisely this
time.
The majority judgements between candidates taken 2 by 2 are the pairwise
defeats.
Their order of merit is their order of strength.
If they cannot all exist together that's because there are cycles, and so
they cannot all exist together as a transitive ordering of the candidates.
Again it sounds as if Condorcet would get rid of all the cycles, since it's
being done because the defats aren't all consistent with a transitive
ordering. But he wouldn't do that if he's trying to find a single winner.
He'd stop when someone is unbeaten. If someone is initially unbeaten that
person would be chosen the winner without having to drop any defeats.
I suggest that when he says the defeats cannot all exist together, he really
means it's because they disagree on the matter of who should win. There's a
circular tie instead of someone unbeaten. If so then the smallest defeats
are successively dropped till someone is unbeaten.
If Condorcet had actually used these methods, he'd have written them down in
a more explicit and practical waly. But he's still pointed the way to the
best rank-counts.
Mike Ossipoff
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