[EM] L1 distance, Condorcet, etc [Ossipoff]

[Scott Ritchie]

On Tue, 2007-02-20 at 16:21 -0500, Warren Smith wrote:
> >Ossipoff:
> Say there’s a candidate at the multidimensional median point. As Warren 
> said, s/he is the CW.
> 
> --WDS: false.  Ossipoff was wrong. I, in agreeing with him, was also wrong.
> 
> >Ossipoff:
> Starting from the CW, say we move some distance from the CW, in a direction 
> parallel to one of the axes by which we measure city-block distance. By 
> moving in that direction from the median candidate, we’re moving away, by 
> city-block distance, from the CW, and from everyone on the other side of the 
> CW--that entire half of the candidates who are on the other side of the CW, 
> in the dimension in which we’re moving. Of course we’re also moving away 
> from everyone whom we’ve passed while moving. Now, say, from there, we move 
> in direction perpendicular to the one in which we initially moved. Again, by 
> city-block distance, we’re moving away from the CW, and from the half of the 
> candidates who are on the other side of the CW, and from everyone whom we’ve 
> passed, in the dimension that we’re now moving. In both of those moves, 
> we’re moving away from more voters than we’re moving toward, by the same 
> distance.
> 
> Maybe that isn’t rigorously-stated, and maybe, taking more time, I could 
> word it better. But it seems convincing
> 
> -WDS:
> your "proof" is wrong.
> 
> A counterexample to both the claim and the proof, is collected (with other coutnerexamples)
> here
>    http://rangevoting.org/BlackSingle.html#condmyth
> 
> Also, Ossipoff in his original post did not have the assumption
> "say there's a candidate at the multidimensional median point".
> He had claimed it was simply true without that asumption.  But either way, he was wrong.
> 
> Bottom line:
> L1 distance has nothing good to do with Condorcet winners.
> L2 distance is in fact more related than L1 distance, far as I can see.
> I agree with Ossipoff L1 distance is better, and therefore I conclude that Condorcet 
> methods are worse than they had appeared based on L2 distance.  IEVS will implment L1
> distance soon and so we'll see.  Ossipoff in trying to make an argment
> for COndorcet methods exactly failed.
> 
> Warren D Smith
> http://rangevoting.org


Warren, I don't like your definition of "median" here.  The axes and
origin are completely arbitrary!

Here, I've duplicated your example in image form, with your original
axes:  The cyan circle is your candidate O, and the purple X is your
candidate X.

http://tuzakey.com/~scott/2dimensional-median-counter.png

Now, I'll leave every voter in the same place, but look at it
differently by moving the origin and rotating the axes:

Look at what's happened when we moved the origin to X's position and
rotated the axes: instead of O, now X is the "median", with the exact
same voters!


How did this happen?  Well, one thing I noticed about your example was
that voters seemed to be a little bit more "dense" along the diagonal
line - naturally that seemed like a better basis for a dimensional axis.

I'd like to propose an alternative definition for median here, but for
the life of me I can't remember the linear algebra term that I'm looking
for.  There is, however, a unique way to choose an axes such that we
minimize the amount of data loss if we drop a subsequent dimension - in
this case the first axes is the diagonal line I drew that's closest (by
some norm) to the points.  Note that if we choose the standard euclidean
norm then this first axis will be exactly the best fit least squares
regression line.

In other words, I don't think you've shown a counter-example, because I
think X is a better median here.

Thanks,
Scott Ritchie