[EM] suggestion for MMPO/Approval hybrid

[Russ Paielli]

Folks,

Kevin has pointed out some interesting properties of MMPO. Although it 
fails CC, apparently it passes FBC and LNH, which Kevin argues are more 
important than CC. That may be debatable, but for the sake of this 
discussion, let's say he's right.

MMPO is an ordinal-only method, and I still think that cardinal 
information in the form of an Approval cutoff is indispensible. Why? 
Call it intuition at this point, or refer back to my earlier post on the 
topic at

http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2005-March/015216.html

I have been amusing myself trying to think of a way to combine MMPO with 
Approval. Here's what I've come up with.

Start with ranked ballots and Approval cutoffs as usual. Then arrange 
the pairwise matrix so the Approval scores are decreasing (or 
non-increasing) along the main diagonal, as in DMC. Now select two 
candidates as follows for a pairwise "runoff." The first candidate is 
the Approval winner. The second candidate is selected using the 
following variation of the MMPO procedure. In finding the candidate with 
the minimum number of maximum votes against, only consider the other 
candidates who are more approved than the candidate in question. In 
other words, consider only the upper-triangular portion of the pairwise 
matrix. That means the least-approved candidate has the most (n-1) other 
candidates over which to find the maximum votes against (hence his max 
votes against are more likely to be higher as a "penalty" for being 
least approved).

Anyway, I am just "brainstorming" at this point. I haven't analysed this 
method, but I think it may still pass FBC and LNH because it combines 
two methods that both pass those criteria if I am not mistaken. 
Admittedly, it *is* more complicated than MMPO but not a lot more, and 
the addition of cardinal information may add significant value. I am 
making no claims at this point, however.

--Russ