Jobst, You wrote (Thur.Mar.24): "First, I'd like to emphasize that DMC, AWP, and AM can be thought of as being essentially the same method with only different definition of defeat strength, so it seems quite natural to compare them in detail as you started.
Recall that the DMC winner is the unique immune candidate when defeat strength is defined as the approval of the defeating candidate, so with that definition, Beatpath, RP, and River become equivalent to DMC. Perhaps it is helpful to look at the defeat strength like this: When A defeats B, then the defeat strength is composed as a linear combination of the following three components:" AM AWP DMC no. of voters approving A but not B + + + no. of voters approving A and B 0 0 + no. of voters approving B but not A - 0 0 CB: I like this table. Doesn't AM look like the most "natural" and "balanced"? I was wondering if it is possible in AM for a candidate who is both the sincere CW and sincere AW to successfully Buried, and I've come up with an example that shows that unfortunately it is, but AWP and DMC likewise fail in the same example. Sincere preferences: 48: A>B>>C 01: A>>C>B 03: B>>A>C 48: C>>B>A B is the CW and AW. Then 45 of the 48 A>B voters Bury B "strongly", i.e. with both rankings and approval, while the other 3 of the 48 only Bury with their rankings (not approving C). This gives: 45: A>C>>B 04: A>>C>B 03: B>>A>C 48: C>>B>A C>B>A>C. Approvals: A49, B3, C93. AM and AWP both elect A, while DMC elects C Approval Margins AWP DMC C>B 93-3 (m+90) 93 B is eliminated B>A 3-49 (m-46) 3 A>C 49-93 (m-44) 4 A's defeat is the weakest in both AM(-46) and AWP(3), while DMC eliminates B. All three methods elect the Burier's candidate, A. When there are three candidates in the top cycle, AM has the property that the candidate with the lowest voted approval score can't win. Suppose there are three candidates in the top cycle. >From highest to lowest, in order of approval scores they are A,B,C. If A>C, that defeat will be positive number. Then C>B, a negative number; and B>A, another negative number. The weakest defeat will of course be one of the negative numbers, so C can't win. If on the other hand B>C, that defeat will be a positive number. Then C>A, a negative number; and A>B, a positive number. Obviously the weakest defeat will be the negative number (A's) and so again C can't win. This 3-candidate analysis also shows that AM always elects a member of P (i.e. it never elects a candidate who is pairwise beaten by a more approved candidate). If, in descending order of approval score, candidates A,B,C are in a cycle; then the following is true in AM: (1) If B>A, and B's approval score is closer to A's than to C's; then B wins (2) Otherwise A wins. To sum up the case for AM versus Approval-Weighted Pairwise (AWP): AWP is only somewhat better than AM at electing the sincere CW (both being good but fallible) and that seems to be balanced by it being worse at electing the sincere AW. Therefore AWP's ability to give a majority an unanswerable complaint, by electing a candidate from outside P, cannot be justified. Jobst wrote: "Here you state the obvious problem when looking at both approval and defeat information. Forest's ingenious argument was that we should at least not elect a candidate where both kinds of information agree that the candidate is defeated, leaving us with his set P of candidates which are not strongly defeated. But when we take both kinds of information serious, it does not seem appropriate to me to always elect a candidate from the two extremes of P like Approval and DMC do. Still, DMC has the obvious advantage of extreme simplicity. I would find it much more natural if the winner was somewhere in the "middle" of P!" Doesn't Approval Margins fill the bill? Welcome to the AM fan club! Chris Benham Find local movie times and trailers on Yahoo! Movies. http://au.movies.yahoo.com ---- Election-methods mailing list - see http://electorama.com/em for list info