[EM] (Fwd) Problem with Condorcet?
seppley at alumni.caltech.edu
Sun Mar 31 02:53:41 PST 1996
Somehow, another maillist I'm subscribed to has turned into another
election-methods list. (I wonder who's responsible for that! :-)
One the subscribers, Mike Saari, has a litmus test which Condorcet
fails. See below. Is the scenario farfetched?
------- Forwarded Message Follows -------
Date: Sat, 30 Mar 1996 21:28:18 -0500
From: Saari at aol.com
To: development at deliberate.com
Subject: Condorset counter-example
>Steve's definition of Condorset voting:
> "Each voter ranks the candidates from most preferred to least
> preferred. The info provided in the voters' rankings is used to
> determine the winners of each of the possible candidate pairings.
> The winner is the candidate whose worst pairing defeat is smallest."
OK, here's a situation where Condorset voting fails the "Twins"
Litmus Test. It's a variation on the Apple-Chocolate ambiguous
example in my paper. This example works because there "just happened"
to be a "voting cycle" between the 3 versions of Apple.
The example is a bit hairy, but I trust that other motivated people
will check my math/logic and let us know if I screwed up.
Basically, with this example Condorset voting yields winner=Apple
when the ballot contains (Apple, Chocolate), but yields
winner=Chocolate when the ballot contains (Apple-1, Apple-2, Apple-3,
Chocolate). Ready? Here goes!
Assume the following ratings (opinions):
Apple-1 Apple-2 Apple-3 Chocolate (Ranking)
20% Exc(99) Exc(98) Exc(97) Exc(95) A1>A2>A3>Ch
20% Exc(97) Exc(99) Exc(98) Exc(95) A2>A3>A1>Ch
15% Exc(98) Exc(97) Exc(99) Exc(95) A3>A1>A2>Ch
15% Bad(-30) Bad(-40) Bad(-50) Exc(95) Ch>A1>A2>A3
15% Bad(-50) Bad(-30) Bad(-40) Exc(95) Ch>A2>A3>A1
15% Bad(-40) Bad(-50) Bad(-30) Exc(95) Ch>A3>A1>A2
If the ballot contains only (Apple-x vs.Chocolate) then the
Condorset winner is Apple:
A > Ch 55%-45%
Apple wins easily.
But if the ballot contains (Apple-1, Apple-2, Apple-3, Chocolate)
then the Condorset winner is Chocolate:
A1 > A2 65-35
A2 > A3 70-30
A3 > A1 65-35
A1 > Ch 55-45
A2 > Ch 55-45
A3 > Ch 55-45
(Chocolate is the candidate whose worse pairing defeat is smallest.)
Because a situation can be created where adding "Twins" to the
ballot alters the outcome, I conclude that Condorset voting fails the
"Twins" Litmus Test and therefore is not worthy.
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