[EM] A distance based method

[fsimmons at pcc.edu]

Trying to build a metric from a set of ranked ballots is fraught with difficulties, and your outline of a 
procedure for doing it is interesting to me.

The simplest, least sophisticated idea I have so far that seems to have some use is to define the 
distance between two candidates X and Y to be the number of ballots on which at least one of the two is 
truncated.

The heuristic behind this is that if X and Y are polar opposites, then a typical voter will probably truncate 
at least one of them.

If we use this metric and simplify the method to always eliminate the pairwise loser of the current most 
polarized pair, then it gives negligible incentive to compromise.


----- Original Message -----
From: Kristofer Munsterhjelm 
> fsimmons at pcc.edu wrote:
> > First find a clone consistent way of defining distance between 
> candidates.
> This could be an interesting algorithm problem in itself. It is 
> possible 
> to triangulate points in space (assuming Euclidean distances) if 
> you 
> have the exact distances; but what if you have only the rank 
> order of 
> distances? You get a (usually) underconstrained problem of the sort:
> 
> Given v voters, n candidates, and a dimension integer k > 1,
> find v + n coordinates in k-dimensional space so that the 
> Euclidean 
> distance from V's coordinate to C's coordinate for some voter V 
> and 
> candidate C is less than the Euclidean distance from V's 
> coordinate to 
> D's coordinate, for the same voter V and another candidate D, 
> iff V 
> ranks C ahead of D. Break ties by assigning coordinates so that 
> the sum 
> of the Euclidean distances to the candidates from the origin is 
> minimized.If it is not possible to make such an assignment, make 
> one that 
> contradicts as few candidate rankings as possible.
> 
> I have no idea how you would actually do this, though, and it 
> would be 
> prone to overfitting. It might not be cloneproof, either, since 
> differences in clone rankings could eliminate some rotations 
> that would 
> otherwise be picked as the best choice by the tiebreaker. It 
> would 
> definitely not pass IIA, as the addition of "superbad" 
> candidates could 
> serve as anchors.
> 
> > Then while two or more candidates remain
> > of the two with the greatest distance from each other
> > eliminate the one with the greatest pairwise defeat
> > EndWhile.
> 
> Or, if you have the voters' coordinates too, you could use 
> histograms, 
> kernel density estimation, or some other estimation to try to 
> reconstruct opinion space, and then pick a subset of the 
> candidates that 
> "reproduces" that opinion space as best as is possible. E.g. if 
> each 
> candidate (and voter) is a Gaussian in opinion space and you 
> want p 
> seats, find the p Gaussians where the difference between the 
> space given 
> by the sum of the p Gaussians (of the prospective council) and 
> of the v 
> Gaussians (of every voter), normalized, is minimized.
> 
> Bandwidth selection would be a pain, as would finding the 
> "right" number 
> of dimensions.
> 
>