[EM] Saari's Basic Argument
Forest Simmons
fsimmons at pcc.edu
Thu Jan 16 17:06:21 PST 2003
On Thu, 16 Jan 2003, Steve Barney wrote:
> Forest:
>
> Isn't that just another way of saying Kemeny's Rule does not respect cyclic
> symmetry?
>
Or we could say that "cyclic symmetry" doesn't respect the minimal
distance criterion, since that is what Kemeny's rule is.
A more neutral statement is that minimal average distance and "cyclic
symmetry" are incompatible with each other.
The reason that I put "cyclic symmetry" in quotes is because, as I
mentioned before, it is only partial symmetry.
To see this put the orders ABC, ACB, CAB, CBA, BCA, and BAC clockwise
around the face of a clock at the even hour positions starting at the zero
position (which is the same as 12 mod 12).
Note that the pure candidate positions are A, C, and B, at the 1, 5, and 9
O'Clock positions, respectively.
Neither of the two cycles we've been talking about is symmetric with
respect to these candidate positions:
The ABC, BCA, CAB cycle is offset 30 degrees counter clockwise from the
pure candidate positions.
The CBA, BAC, ACB cycle is offset 30 degrees clockwise from the pure
candidate positions.
If you draw this on a piece of paper, you will see the rotational bias
that I mentioned.
The only (three member) cycle that is symmetric with respect to the
candidates (besides the pure candidate cycle) is the cycle A=B>C, B=C>A,
and C=A>B, which is both 60 degrees clockwise and 60 degrees
counterclockwise from the pure candidate cycle.
[Most rank based methods do not permit these reverse truncations.]
The reason I brought the Kemeny order into the discussion is because the
rotational bias is obvious when using this diagram to figure out which
order is the Kemeny order. In other words, when you focus on the order,
and not just the winner, the lack of symmetry becomes apparent.
Forest
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