[EM] five monotonic methods that choose from the uncovered set

[Forest W Simmons]

I would like to mention that I proposed the first of these five methods 
back in December of 2004:

http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-December/014293.html

"Ballots are ordinal with approval cutoff, equal rankings allowed.

Let U(A) be the set of uncovered candidates that cover the approval 
winner A.  The member of U(A) with the highest approval is the method 
winner W."

Shortly after that I was discouraged by an assertion that methods that 
always picked from the uncovered set could not be monotonic.

Recently, I realized that, nevertheless, this and similar methods are 
monotonic.

The key lemma is this:  If an uncovered candidate X covers Y, and this 
X moves up in rank on one or more ballots while all of the other 
candidates retain their original relative ranks to each other, then X 
remains uncovered and the set of candidates that cover Y does not 
change.

Proof:  Suppose that X is uncovered and covers Y.  Then clearly X is 
still uncovered, because it still has a beatpath of length two or less 
to each of the other candidates.  Also, X's upward mobility doesn't 
change the set of candidates that Y beats, nor does it change the set 
of candidates that beat Y.  Therefore it doesn't change the set of 
candidates that cover Y. Q.E.D.

A simple application of this lemma shows that in the above method, if W 
improves its position relative to the other candidates (either in rank 
or in approval or both) then W is still the highest approval uncovered 
candidate that covers A.

Similar proofs work for similar methods.

Forest