[EM] visualizing the procedure for RAV (or whatever you call it)

[Russ Paielli]

Forest Simmons simmonfo-at-up.edu |EMlist| wrote:
> Here's the original recursive procedure that I gave for Approval Seeded 
> Bubble Sort:
> 
> 1. List the candidates in order of approval, from top to bottom.
> 
> 2. Percolate the bottom candidate as far as possible up the recursively 
> sorted list of the other candidates.
> 
> How's that for concise?
> 
> Jobst is right: the winner is the lowest approval score candidate that 
> beats all of the candidates with greater approval, which turns out to be 
> the same as the first CW that eventually appears as low approval 
> candidates are eliminated successively.

I agree with Forest here, but I think the way he phrases it may be 
slightly misleading because it seems to imply that a lower Approval 
score is advantageous.

Let me explain how I visualize this procedure.

Let's start first with a way to visualize a standard Condorcet pairwise 
matrix. Imagine "blacking out" the non-diagonal elements corresponding 
to winning scores and "whiting out" the elements corresponding to losing 
scores. Let's say the diagonal elements are blacked out too. A CW then 
must have a completely blacked out row.

In no CW exists, put the Approval scores on the diagonal and reorder the 
matrix so that the diagonal is strictly non-increasing starting from the 
upper left and going down.

The least-approved candidate can only win by being the CW, meaning that 
his entire row must be blacked out. But I just said that no CW exists, 
so that didn't happen. So eliminate the last row and column. Now the 
next-to-last-approved candidate can only win by having a completely 
blacked out row, which would be blacked out all the way to the diagonal 
of the original matrix. If that isn't the case, eliminate the last row 
and column again.

The winner is the first candidate, starting at the bottom and going up, 
to have his row blacked out all the way to the diagonal.

Note, however, that as you proceed up toward the Approval winner, the 
number of blacked-out elements needed to win decreases. The AW himself 
needs none. In other words, the Approval winner needs no pairwise wins 
to win the election. That's the mirror image of the fact that the 
least-approved candidate can be the CW, as Forest pointed out the other day.

--Russ