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 _________________________________________________________________ Join the world�s largest e-mail service with MSN Hotmail. http://www.hotmail.com ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
