Ok, sure, its obvious (when my claim is challenged) that, though a candidate at the voter median is the CW, its plausible that the CW neednt be at the voter median: Suppose no candidate is at the voter median. Might not a candidate who is much closer (in some sense) to the voter median than any other candidate is be the CW?
And if the CW isnt at the voter median point, its possible to suggest a scenario in which s/he doesnt maximize SU: Maybe, though most voters are closer to X than to Y, those who are closer to X are far away, but those who are closer to Y precisely share Ys position in issue-space. So, though X is preferred to Y by a majority, Y has more SU than X does.
But, with just one issue-dimension, when X is preferred to Y by more voters than prefer Y to X, I couldnt make Y have more SU than X. Even if all those preferring Y to X share Ys position, while those preferring X to Y are far away.
I could be worthwhile to find out if the CW can fail to maximize SU when there are more dimensions and candidates, and what that would take--what kind of voter-distribution it would take. Could it happen if the voters are normally distributed in each dimension, for instance?
Maybe Warrens authors (you know, the ones he says have their heads up their ass) have answered that question. If Warren has their books or papers, and if they answer that question, and if their answer isnt unfavorable to Warrens position, maybe hell tell us their h.u.t.a. answer. <smiley>
In any case, though, if the a candidate at the voter median point is CW, and if, with city block distance, a candidate at the voter median point maximizes SU (as Ive demonstrated here a few days ago), then at least that means that theres some connection between CW and SU maximization. And if the candidates tend to thickly populate the region of the voter median, as will probably be the case, that could make it more difficult for a candidate away from the voter median point to be CW.
The fact that the CW might not be at the voter median isnt quite so bad when you consider that there might not even be a CW. At least one can say that, in spatial voting, the wv Condorcet methods will tend to do well by SU, even if one cant make the stronger guarantee that I wanted to make.
Mike Ossipoff
---- election-methods mailing list - see http://electorama.com/em for list info
