Three slot Bucklin is also known as MCA (Majority Choice Approval): if no alternative is "preferred" on more than fifty percent of the ballots, then the approval cutoff is lowered to include the middle slot.
Compare this with three slot WMA (Weighted Median Approval) based on the same ballots: (1) The random ballot probabilities are computed from the submitted ballots. (2) For each ballot b, if the total probability of the preferred candidates on ballot b is no more than fifty percent, then the approval cutoff is lowered to include the middle slot on ballot b. In the three candidate case WMA and MCA give identical approvals. But when there are four or more alternatives, WMA is less of a blunt instrument compared to MCA. When there is no majority preferred alternative, under WMA the approval cutoff is only lowered on the ballots where there is not a good chance that the winner will come from among the preferred alternatives on those ballots. As Chris Benham pointed out, this version of WMA satisfies the Participation Criterion, whereas MCA does not. On the other hand, MCA is efficiently summable by precinct, whereas WMA is not. Both methods satisfy Monotonicity, and both methods are as clone free as three slot Range is, based on the idea that the truer the clones, the more likely they will be rated the same, and when not rated the same they will be rated adjacently. Both methods satisfy Independence from Pareto Dominated Alternatives, provided that ties are broken by random ballot or by random ballot probabilities conditioned on the tied alternatives. MCA satisfies the FBC (Favorite Betrayal Criterion). I'm not sure if WMA satisfies the FBC. In other words, could raising one's true favorite from the middle slot to preferred status change the winner from someone else (compromise) with preferred status to a third alternative besides the recently raised favorite? Thanks, Forest ---- Election-Methods mailing list - see http://electorama.com/em for list info
