[EM] Distance measure--Are issue-position differences additive?

[MIKE OSSIPOFF]

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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