[EM] Saari's Basic Argument

[Steve Barney]

Forest:

In your example,

>66 A>B>C
>34 B>C>A

If you give second preferences any more than 16/33 of the weight which you 
give to the first prefs, the winner is B; since:

  34+66x=66
  66x=32
  x=32/66
  x=16/33


Let's look at another example:

  1,000,001 A>B>C
  1,000,000 B>C>A

  1,000,000+1,000,001x=1,000,001
  1,000,001x=1
  x=1/1,000,001


In this case, a second pref must be given less than 1/1,000,001 of the weight 
of a first pref, if A should win. That's a stretch.



SB

>===== Original Message From Forest Simmons <fsimmons at pcc.edu> =====
>Here's an example with the symmetry already subtracted out. [Add it back
>in and then subtract it out again if you want to.]
>
>66 A>B>C
>34 B>C>A
>
>A should be the winner, but Borda picks B.
>
>Forest
>
>On Thu, 16 Jan 2003, Steve Barney wrote:
>
>> Forest:
>>
>> That's quite a strong claim you made there. I must have missed it. Please 
give
>> me the number of the EM message in which you did what you describe below.
>>
>> SB
>>
>> >From: Forest Simmons <fsimmons at pcc.edu>
>> >To:  <election-methods-list at eskimo.com>
>> >Subject: RE: [EM] Saari's Basic Argument
>> >
>> >I've already given an example in which Borda gives the wrong answer after
>> >the symmetry is removed.  Now you have given an example in which symmetry
>> >removal shows the CW to be wrong. So that evens the score :-)
>> >
>> >In other words, neither Borda nor Condorcet can claim to be superior on
>> >the basis of symmetry removal.
>> >
>> >So here's a new method (for three candidate races only): first remove all
>> >of the symmetry, and then the candidate with a majority of first place
>> >votes (on the remaining ballots) is the winner.
>> >
>> >This method beats both Borda and Condorcet by the "symmetry removal
>> >criterion."
>> >
>> >Forest

Steve Barney

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