[EM] Another PR method based on ranked ballots

[Forest Simmons]

Olli,

Here's another idea for PR that makes use of Borda style ballots:

Suppose that we have a family F of subsets of candidates, and that we want
to see which member of the family would better represent the voters
according to the information contained on their ranked preference ballots.

First convert all of the ballots into rank vectors, so that if the fifth
candidate is ranked eleventh on a ballot, then the fifth component of the
vector representing that ballot is the number eleven.

Next, for each candidate X in the union of (the members of) F, let A(X) be
the average of all these ballot vectors that rank X in first place..

Next, for each ballot vector V and each candidate X in the union of F, let
d(V,A(X)) be the distance from V to A(X) in the Euclidean metric (i.e. the
root of the sum of squares of the respective component differences).

For each member H of the family F, and each candidate X in H, let
Voronoi(H,X) be the set of ballots whose distance to A(X) is smaller than
the distance to A(Y) for all Y in H-{X}.

In other words, Voronoi(H,X)  represents the voters that are more similar
(in preferences) to the average supporter of X than to the average
supporter of any other candidate of the set H.


Now, for each H in F ...

let D(H) be the
   Max over X in H, of the
      Sum over V in Voronoi(H,X), of the
         distance d(V,A(X))
      Next V
   Next X .

The smaller D(H), the better proportional representation H affords the
voters (according to their ballot information).

This method can be used to compare the outputs of other more
computationally efficient methods when the number of candidate subsets
prohibits an exhaustive search for the best subset by this method (as a
stand alone method).


Forest


On Sat, 15 Mar 2003, Olli Salmi wrote:

> Forest,
>
> Thanks. I'll play with it for a while now.
>
> >Borda PR inherits this problem from Borda, so Borda PR should be limited
> >to applications where Borda's defects don't matter, i.e. in sports,
> >robotics, and as a benchmark of the limits of social utility for more
> >viable election methods.
>
> Or it can be used for elections of officers in a department of mathematics.
> http://math.asu.edu/~math/dept_overview/bylaws.html
>
> Best regards,
> Olli
>