[EM] Saari's Basic Argument
Forest Simmons
fsimmons at pcc.edu
Fri Jan 17 12:42:31 PST 2003
I hate to beat a dead horse, but in order to see the fallacy of Saari's
symmetry arguments let's take this example a little further:
66 ABC
34 BCD
The 12 o'clock and 8 o'clock positions representing these two factions are
non adjacent on the clock face. Between them is the fully ranked order
BAC (at 10 o'clock).
To get them adjacent with Saari's symmetries, first add 34 copies of the
canceling pair {BAC, CAB} to get
66 ABC, 34 BAC, 34 BCD, 34 CAB.
Now remove 34 copies of the cycle {ABC,BCD,CAB} to get
32 ABC, 34 BAC.
According to Saari this ballot set is equivalent to the original.
How could these perfect symmetries bring about such a ridiculous
equivalence?
Well, symmetries are relative. The symmetry of the earth doesn't
necessarily concord with the symmetry of the front wheel of my bicycle.
The symmetry of the clock face distribution of possible orderings of the
candidates doesn't always jive with the symmetry of the distribution of
voter ballots in that space.
When there are only two factions, the ballot distribution has an axis of
symmetry through the two points representing those factions. In this
case, through the ABC (12 o'clock) and BCA (8 o'clock) positions.
The (supposedly) equivalent distribution has an axis of symmetry through
the BAC (10 o'clock) and ABC (12 o'clock) positions, a rotation of thirty
degrees clockwise.
How did symmetric transformations alter the axis of symmetry?
Well, neither the cycle nor the canceling pair respected the symmetry of
the original distribution. For example, neither had a center of gravity on
the axis of symmetry of the distribution. Consequently addition and
subtraction of these sets of ballots upset the original symmetry by
changing the center of gravity and the principal axes of rotation, not to
mention the radii of gyration.
On Fri, 17 Jan 2003, Alex Small wrote:
> Forest Simmons said:
> >
> >
> > On Thu, 16 Jan 2003, Steve Barney wrote:
> >
> >> Forest:
> >>
> >> In your example,
> >>
> >> >66 A>B>C
> >> >34 B>C>A
> > No need of giving weights to see all the mischief that could come from
> > giving the win to B.
>
> Moreover, if candidate C weren't there then we'd all agree that A trounced
> B conclusively. Then we throw in C, and because the A voters happen to
> agree that B is better than C, that point of agreement costs them what was
> a decisive victory. How can you justify that?
>
> Indeed, if C is a single-issue candidate then letting B wins is like
> saying that because the A faction agrees with the B faction on one
> particular issue, it's OK that B wins. In that case, I guess it's OK that
> Bush won, because even though he got fewer votes than Gore, he at least
> agrees with Gore on some things.
>
> Granted, all ranked methods are susceptible to violations of IIA
> (Independence from Irrelevant Alternatives) but some methods (e.g.
> Condorcet, IRV) at least avoid that problem when somebody has a majority.
>
> So, in the end, as nice as positional methods are for mathematical study,
> and as cool as Saari's pictures are, and as impressive and charismatic as
> Saari is (I've seen him talk in person), in the end Borda is a horrible
> method.
>
>
>
> Alex
>
>
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