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

[Jobst Heitzig]

Dear Scott,

what you are searching for is, I think, the so-called "spacial median".
The spacial median of a sample of points (in our case the voters' positions) is defined as a point where the sum of all the distances from this point to a sample point is minimal. This kind of median depends on the choice of a distance function and it is not always unique. 

Also, it is not something which you can get with linear algebra. With linear algebra, you can only find the mean (which is the point where the sum of all the *squared* distances from this point to a sample point is minimal).

Yours, Jobst
 
> -----Ursprüngliche Nachricht-----
> Von: Scott Ritchie <scott at open-vote.org>
> Gesendet: 21.02.07 12:03:06
> An: Warren Smith <wds at math.temple.edu>
> CC: election-methods at electorama.com
> Betreff: Re: [EM] L1 distance, Condorcet, etc [Ossipoff]


> 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
> 
> ----
> election-methods mailing list - see http://electorama.com/em for list info
> 


_____________________________________________________________________
Der WEB.DE SmartSurfer hilft bis zu 70% Ihrer Onlinekosten zu sparen!
http://smartsurfer.web.de/?mc=100071&distributionid=000000000066