# [EM] Saari's Basic Argument

Forest Simmons fsimmons at pcc.edu
Wed Jan 15 16:00:15 PST 2003

```On Wed, 15 Jan 2003, Alex Small wrote:

....

> I'm not convinced
> that symmetry is a particularly compelling reason to pick an election
> method,

....

especially not the symmetry of {ABC,BCA,CAB}, which has a rotational bias.

True, it favors no candidate, but it does favor its three orders over the
other three orders.

One way to see this is that the average Kemeny distance from any order of
the cycle to the three orders in the cycle is (0+2+2)/3, while the average
Kemeny distance from a fully ranked order outside the cycle to orders of
the cycle is (1+1+3)/3.

On the other hand reverse symmetry has no effect on the Kemeny distance:

If xyz is any order of the three candidates, and {rst, tsr} is any
opposite pair, then the average Kemeny distance (d(xyz,rst)+d(xyz,tsr))/2
is 3/2.

So removal of a reverse pair cannot affect the Kemeny order.

In the fully ranked three candidate case the Kemeny order is the same as
the Ranked Pairs order, etc.

Forest

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