I've seen examples in which Bucklin (with equal ratings) fails the Participation criterion, AKA Woodall's mono-add-top criterion for deterministic methods:
"the participation criterion says that the addition of a ballot, where candidate A is strictly preferred to candidate B, to an existing tally of votes should not change the winner from candidate A to candidate B." (from Wikipedia) In a Bucklin single-winner election with 3 or more levels, it is possible that in an election in which the quota is not met at the first or second level threshold, candidate A may be selected after the threshold has dropped to the third level, but after adding some number of A > B ballots, B then has enough votes to exceed the quota at the second threshold, thus failing Participation. So the extra A > B voters might as well have not shown up. However, if there are only two approval levels in the Bucklin election, it appears that this problem could not occur, and the no-show paradox would be avoided. The failure above hinges on the fact that lower-ranked B fails to make quota at the 2nd level before the new ballots are cast, but exceeds the quota afterward. With levels compressed to two instead of three, B would exceed the quota at the second level threshold initially. [Chris Benham has made me aware that ER-Bucklin 2-level still fails the Independence of Irrelevant Alternatives, but that is a different situation.] Does anyone know of any 2-level ER-Bucklin Participation failures? Ted -- araucaria dot araucana at gmail dot com ---- Election-Methods mailing list - see http://electorama.com/em for list info