I don't know if Juho is still cheering for MinMax as a public proposal. I used to be against it because of its clone dependence, but now that I realize that measuring defeat strength by AWP (Approval Weighted Pairwise) solves that problem, I'm starting to warm up more to the idea. MinMax elects the candidate that suffers no defeats if there is one, else it elects the one whose maximum strength defeat is minimal. There are various ways of measuring defeat strength. James Green Armytage has advocated one called AWP as making Condorcet methods less vulnerable to strategic manipulation. If all ranked candidates on a ballot are considered approved, then the AWP strength of a defeat of B by A is the number of ballots on which A is ranked but B is not. Then more recently I was reading a paper by Joaquin Pérez in which he shows that MinMax is the only commonly known Condorcet method that satisfies the following weak form of Participation: If A wins and then another ballot with A ranked unique first is added to the count, A still wins. Beatpath, River, Ranked Pairs, etc. fail this weak participation criterion, but they do satisfy this even weaker version: If A wins and then another ballot with only A ranked is added to the count, then A still wins. Proof: First add a ballot in which no candidate is ranked. The above mentioned methods allow this, and it doesn't affect their outcome since no mention of absolute majority is made in any of them. Then raise A while leaving the other candidates unranked. This cannot hurt A since all of the above mentioned methods are monotone. Knowing that Beatpath satisfies the weaker version but not the weak version may be an inducement for voters to bullet vote candidate A to make sure that they avoid the no show paradox. But MinMax is free of this temptation; they wouldn't have to truncate the other candidates.
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