I want to thank Markus for keeping me from going too far off track. And the link he gave below to a great message of Chris Benham was valuable for more than showing us that Bucklin violates mono-add-top: Chris also pointed out that WMA (weighted median approval) does satisfy Participation.
I never fully appreciated before what a good method WMA was. We assume that equal rankings are possible and that we have the probabilities of some monotonic, clone independent lottery L, like random ballot, at our disposal. The weighted median approval cutoff is the highest "level" R above which the ranked alternatives have more than half the total probability of the lottery L. All alternatives ranked above R are approved. Approval ties are broken by the lottery L. It may be that the "level" R is below the truncation level, i.e. on some ballots the untruncated candidates have barely half the probability or less. If that is the case on all of the ballots, all of the candidates are tied, so the election is decided by the lottery L. Note that by this method any alternative with maximum approval will have above average approval, and the average approval will represent more than half of the lottery probability, since on each ballot more than half the lottery probability is located above the cutoff R. This method satisfies monotonicity and a marginal version of clone independence: If clone family members are ranked equal, then when a winner is cloned in this way, the new winner will be a member of the old winner's clone family, and if a loser is cloned in this way, no member of the loser family will advance to winner. In other words, this WMA version satisfies clone independence about the same way Range does. In fact, the easiest way to implement it is probably through the use of Range style ballots. Any clone families that are not adjacent to the approval cutoff R can relax and spread out a little. This method makes use of a lottery to break ties (which should be rare) and makes use of lottery probabilities to determine approval cutoffs. Since the lottery probabilities are not random variables once the ballots are ready to be counted, the method is as deterministic as any method that uses randomness to break ties. ----- Original Message ----- > > Hallo, > > Bucklin violates mono-add-top. See: > > http://lists.electorama.com/pipermail/election-methods- > electorama.com/2004-April/012752.html > > Markus Schulze > ---- Election-Methods mailing list - see http://electorama.com/em for list info
