Dear EM aficionados, Here's a method that elects the candidate with the best ratio of offensive strength to defensive weakness. Until a better name comes up, call it Offense/Defense or O/D. For each pair of candidates X and Y, let F(X,Y) be the number of ballots on which X is ranked equal first with Y, plus the number of ballots on which X is ranked ahead of Y. Similarly, let L(X,Y) be the number of ballots on which X is equal last with Y (i.e. on which neither is ranked ahead of any other candidate), plus the number of ballots on which X is ranked behind Y. Then let m(X) = min (over Y) of F(X,Y), and let M(X) = max (over Y) of L(X,Y). Elect the candidate for whom the ratio m(X)/M(X) is the largest. Note that the size of m(X) is an indication of the offensive strength of X, while the size of M(X) is an indication of the defensive weakness of X. The m(X)/M(X) ratios yield a social ordering of the candidates. The method satisfies both FBC and what we might call RFBC for "reverse FBC." A method satisfies RFBC if you never regret truncating the candidate that you like least (even if she isn't one of the front runners). If all of the voters completely rank all of the candidates, then this method gives the same result as MMPO, which in turn gives the same result as MinMax(wv) and MinMax(margins) in that fully ranked case. If all of the voters rank their approved candidates equal first and truncate all of the rest, this method gives the same result as Approval. The method works equally well with cardinal and ordinal style ballots. However, for practical purposes I would suggest a three slot ballot, with the "+" option for Favorite. With this style ballot, the voters would use the highest of the three slots for "equal first," the bottom slot (truncation) for "equal last," and the middle slot for all other candidates. A check box can be used if the voter wants the candidates in the middle slot to be ranked by Favorite. If a voter marks this check box and only one candidate is ranked, then that candidate's entire ballot will be replicated for this voter. In this O/D+ version of O/D, if two or more candidates tie for the highest O/D ratio, then the candidate marked "+" on a random ballot picks the winner from among the tied candidates. In my opinion, no method more complicated than O/D+ is a better public proposal than it. And no simpler method is likely to perform as well in the public arena. In particular, DMC which does pretty well at the FBC, does not satisfy it absolutely. And absolute compliance with the FBC (Mike has argued convincingly) is an extremely desireable, if not indispensable, feature of public proposals. I believe that O/D+ is better than MDDA, except perhaps for MDDA's simplicity. There are two things that bother me about MDDA: (1) The two cases that fall back on Approval are diametrically opposed: the first case is when every candidate is majority defeated, and the second case is when several are not majority defeated. Why not just use Approval no matter how many are majority defeated? (2) I'm coming around to the view that strongly defeated candidates (e.g. majority defeated candidates) are not necessarily irrelevant in determining the winner from among the remaining candidates. Otherwise Raynaud would be the best Ranked Ballot method. [This is another weakness of DMC. In the case of doubly defeated candidates, it seems clear that none of them should be the winner. But perhaps they still have relevance in determining which of the remaining candidates should win. The best argument in favor of DMC is that in order to have immunity from second place complaints, you have to elect the beats all candidate from among the non-doubly defeated.] In my opinion, the methods that are both simpler than O/D+ and that might be better public proposals than O/D+ make up a pretty short list , including Approval Plus (A+), MCA+, and Candidate Transfer (the simplest version of Asset Voting), and perhaps Grade Voting (A thru F) and Olympic Ratings (one thru 10). No offense if I left out your favorite method, but I'm only talking about proposals simple enough to be practical for general purpose political elections, and among those, only methods that satisfy the FBC. Forest
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