That proof assumes a euclidean distance metric. With a non-Euclidean one, the "planes" could have kinks in them. I believe I have heard that the result still holds with, for instance, a city-block metric, but I cannot intuitively demonstrate it to myself by imagining volumes and planes as in this proof.
JQ 2011/7/13 <[email protected]> > Actually, any centrally symmetric distribution will do, no matter how many > dimensions. > > The property that we need about central symmetry is this: any plane (or > hyper-plane in higher > dimensions) that contains the center of symmetry C will have equal numbers > of voters on each side of > the plane.. > > To see how this guarantees a Condorcet winner, let A and B be candidates > such that A is nearer to the > center C than B is. Let pi be the plane (or hyper-plane in dimensions > greater than three) through C that > is perpendicular to the line segment AB. > > By the symmetry assumption there are just as many voters on one side of the > plane pi as on the other > side. > > Now move pi parallel to itself until it bisects the line segment AB. > > All of the voters that passed through the plane pi during this move went > from the B side to the A side of > the plane. So A beats B pairwise. > > Therefore, if there is a unique candidate that is closer to C than any of > the rest , that candidate will beat > each of the others pairwise. Otherwise, all of the candidates sharing the > minimum distance to C will be > perfectly tied for CW. > > > > > From: Bob Richard > > After looking up some old email threads, it now seems to me that > > I made > > a significant mistake in the post below. It is true that the > > model > > underlying Yee diagrams guarantees that there will always be a > > Condorcet > > winner. But apparently that has nothing to do with the two > > dimensions > > being orthogonal. It results from the fact that voters are > > normally > > distributed on both dimensions. > > > > --Bob Richard > ---- > Election-Methods mailing list - see http://electorama.com/em for list info >
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