[EM] Tideman, Hoffman, outcome-space

[MIKE OSSIPOFF]

Hoffman's method is what Tideman's method is an approximation of.

They use outcome-space. Outcome-space has as many dimensions as there
are candidates. The outcome of the election is represented by an
outcome-point in outcome-space. That point's co-ordinate in a particular
dimension is the vote-total of the candidate to whom that dimension
corresponds.

With 3 candidates, the outcome space is in the form of a cube.
The length of the cube's size could be the number of voters. In that
case the locus of the outcome point is the whole cube. Or the length'
of the cube could be the total number of votes cast, and, in that
case the locus of the outcome point is different. Say X's vote total
is equal to the number of votes cast. That puts the outcome point's
x co-ordinate at the extreme value, and the y & z co-ordinates at zero.

Likewise if candidates y or z get all the votes. These 3 outcome
points are at the 3 corners of a triangle positioned diagonally in
the cube, and that triangle is the surface that's the locus of the
outcome point.

The advantage of letting the side of the cube be the number of voters
is that it's easier to explain. I intend to change the barnsdle
website explanation to say it that way. The advantage of the side of
the cube being the total number of votes cast is that it reduces the
number of dimensions of the locus of the outcome-point. The number
of dimensions is one less than the number of candidates. When there
are 4 candidates, that's right about the point at which I passionately
prefer the the approach that uses fewer spatial dimensions.

Either way, though, of course all the win-zones have the same volume
and shape, and all the tie-zones have the same volume & shape.

Tideman merely pointed out that, since the AB tie zone is between the
A win-zone and the B win-zone, we can typically expect the probability
density in the AB tie zone to be the mean of what it is in the
2 win-zones. The geometric mean is more usable, and is also what
you get by the other approach that I suggested at the barnsdle website.

Of course if the most likely position for the outcome point is in
the AB tie zone, then the probability density there can be expected tob
be greater than its value in the A & B win zones, but maybe it won't
differ too much from them. Anyway, typically it could be expected to
be reasonably estimated by the average of the densities in the 2
win zones.

The difference with Hoffman is that he actually integrates the
probability density in a particular tie-zone.

That means that, unlike with Tideman's estimate, you really have to
deal with the many-dimensional geometry, of course, and the problem
gets more complicated and calclulation-labor-intensive when candidates
are added.

Another approach, which I believe a few people here have already
mentioned, would be to consider each candidate or party separately,
and, based on previous elections, to write a probability density
distribution for that candidate's vote total, maybe a percentage of
the total. The, from that, one could estimate the probabilities of
the ties & near ties between various pairs of candidates.

Tideman's method is intended for 2-way ties, but Hoffman's method
and the individual candidate vote total distribution method could
estimate probabilities of the n-member ties too, for small committee
voting strategy.

Crannor's (Cranor's?) method resembled the individual candidate
probability distribution method that I outlined, and maybe it was
the same thing, but, as I said, I didn't understand the descripion
at her website. Crannor's & Hoffman's methods are described at
her DSV website, at the Pivotal Probabilities page.

Mike Ossipoff



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