Bart--
You wrote:
This doesn't seem possible for more than one dimension-- don't Merrill's models show sincere Borda yeilding slightly higher SU than the CW in two dimensions, and Approval higher than both when there are only three candidates?
I reply:
I don't know; I'd have to check.
But it can be demonstrated that if distance is city-block distance, then the CW always maximizes SU, and that if distance is Euclidean distance, then the CW maximizes SU under the conditions I described, including when the population density is a normal function about some center, in each dimension.
Why does the CW maximize SU with city-block distance?
Say we start at the median point, the point that's at the voter-median in each dimension.
(By "going away from" or "going toward", I mean increasing or decreasing distance to).
Say we depart from that point in one of the issue-dimensions. Immediately after departure, we're going away from more voters than we're going toward, in that dimension. With city-block distance, if half of the voters are on the +X side of the central point, and half on the -X side, and if we're on the +X side, and going farther in the +X direction, we're going away from every voter whose X co-ordinate is less than ours, at the same rate at which we're going toward all the voters whose X co-ordinate is more than ours. So, as soon as we've gone any distance in the +X direction, then, continuing in that direction, we're going away from more voters than we're going toward, because we've added some voters who are in the -X direction from us, because we've passed the X co-ordinate of those voters.
That's true for any issue-dimension, and it's true for any position away from the voter median point.
Excuse that hasty argument. But you see that it's true, that going away from the voter-median point increases the summed distance to the voters.
On another day I"ll demonstrate the correctness of my claim about Euclidean distance.
By the way, though, I've told why city-block distance is more meaningful in spatial models.
I could add that von Neuman & Morgenstern spoke of using hypothetical lotteries to put completely different things on a single common utility scale.
That further strengthens the argument for city-block distance.
Also, even when Euclidean distance is used, doesn't the relative scale of issue-distances in the various dimensions matter? If so, and if there's an effort to make that right, then doesn't that mean relating the importance of distances in the various issue-dimensions? And if that can be done, why not just add them up? If the various issue-space distances are all just amounts of the same quantity, disutility.
Again, these arguments are hasty, and probably not well-worded.
Mike Ossipoff
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