[EM] A distance based method

[fsimmons at pcc.edu]

> I doubt it's monotonic, though it's probably not a practical 
> problem. That
> is, it would probably be totally impractical to try to use the
> nonmonotonicity for anything strategic, and it wouldn't even 
> lead to Yee
> diagram ugliness.

You would have to have at lest four candidates to get a non-monotonic example.  The reason it is so 
hard to find an example is that moving the winner up delays its entry into the fray, so it is competeing 
with only a proper subset of the candidates that it came out on top of before.


> fsimmons at pcc.edu wrote:
> > If we abandon the Euclidean metric, then we also abandon Voronoi
> > Polygons; the corresponding idea for more general metrics is 
> that of
> > a Dirichlet region.
> 
> That's strange. The Wikipedia article on Voronoi diagrams 
> mention 
> diagrams based on L_1 and Mahalanobis distance. Is the article 
> being 
> incorrect when it uses the term "Voronoi diagram" for these?

I'm sure tht would still get polygons for some metrics, but in general you wouldn't.

> 
> > It would be amusing to see Yee diagrams based on L_1 and L_infinity
> > metrics

> To my knowledge, L_inf is just L_1 rotated.

In two dimensions that is the case for the basic balls because a diamond is a rotated square.  But in 
three dimensions that is not the case, since an octagon is not just a rotated cube.
> 
> > Think of the Huntington Hill method of apportionment that is 
> used in
> > this country after every census. How many voters understand its
> > details? Less than one in a thousand, but that doesn't 
> matter; they
> > understand the proportionality goal of apportionment, and 
> they are
> > willing to let the experts take care of the details.
> 
> In that case, I think letting the experts decide produced the 
> wrong 
> outcome, and that Webster should have been used instead. But so 
> it goes.
> 

> From: Kevin Venzke 
> Hi,
> 
> --- En date de?: Jeu 14.7.11, Kristofer Munsterhjelm 
> a ?crit?:
> > Nonmonotonicity could be considered an error even with
> > honest voters. The argument would go something like: "Okay,
> > if we raise X, then X goes from winner to loser. That means
> > that the method is either wrong about who should have won in
> > the ballot set before we raised X (it shouldn't have been
> > X), or after we raised X (it should have been X). We have no
> > way of knowing which is the 'right' result, and so other
> > results could also be suspect".
> 
> I'm inclined to lump this in with IIA problems more generally. 
> It seems
> to tend to be the case that raising the winner adjusted the method's
> perceived relative strengths of other candidates, with unpredictable
> results.
> 
> I think I will try implementing the "eliminate the pairwise 
> loser of
> the most distant pair" method. I am curious how well it would 
> discourageburial. I wonder also how often it would fail Plurality...

Kevin,  the method fails Plurality, and is vulnerable to burying.  Its strong point should be with regards to 
compromising.

Compare it with the version that starts with closest candidates, and you will notice a differnce.  In 
particular, the "closest" version will be much more non-monotonic.