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
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