[EM] (Fwd) Problem with Condorcet?

Steve Eppley 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.

Mike Saari

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