Hallo, > Suppose X is a monotone method. Then Smith,X and > Smith//X are monotone methods. For example, Smith//Borda violates monotonicity because it can happen that, by ranking candidate A higher, some other candidate B is kicked out of the Smith set so that, after re-calculating the Borda scores, candidate A is worse off. Markus Schulze

On 12/04/2017 11:48 AM, Markus Schulze wrote: > Hallo, > >> Suppose X is a monotone method. Then Smith,X and >> Smith//X are monotone methods. > > For example, Smith//Borda violates monotonicity > because it can happen that, by ranking candidate A higher, > some other candidate B is kicked out of the Smith set so > that, after re-calculating the Borda scores, candidate A > is worse off. Whoops, sorry about that, and good catch. Something about Borda's structure makes it hard to find a four candidate example of this (at least given the time I tried to do so), but here's one for Smith//Plurality: 1: A>B>C>D 1: A>B>D>C 2: B>D>C>A 3: C>A>B>D 2: D>A>B>C Plurality scores: A: 2, B: 2, C: 3, D: 2, Smith set: everybody Raise C on the BDCA ballots: 1: A>B>C>D 1: A>B>D>C 2: B>C>D>A 3: C>A>B>D 2: D>A>B>C The Smith set is now {A, B, C} and the Plurality scores within that set are A: 4, B: 2, C: 3. So C is raised and goes from winner to loser. Also, Smith//Antiplurality: 1: A>B>C>D 1: A>B>D>C 1: B>A>C>D 2: B>D>C>A 1: C>A>D>B 1: D>C>A>B Antiplurality scores: A: 5, B: 5, C: 6, D: 5, Smith set: everybody Raise C on the BDCA ballots: 1: A>B>C>D 1: A>B>D>C 1: B>A>C>D 2: B>C>D>A 1: C>A>D>B 1: D>C>A>B The Smith set is now {A, B, C} and Antiplurality scores within that set are A: 5, B: 5, C: 4. Thanks :-) Can we say anything about *when* Smith//X is monotone?