Then on the intervals between the hour marks put the labels
A>>B>C (between 12:00 and 1:00), A>B>>C (between 1:00 and 2:00), B>A>>C (between 2:00 and 3:00), B>>A>C (between 3:00 and 4:00), B>>C>A (between 4:00 and 5:00), etc.
Ballots are ordinal rankings with approval cutoff. First and last place equality are allowed, but middle place equality doesn't make sense with only three candidates.
Thus there are 24 possible ways to fill in the ballot; one for each label on the clock face. For example A>B>>C means that the preference order is A > B > C, and that the approval cutoff is between B and C. The notation not(A) is an abbreviation for the ballot B=C>>A, which means that B and C are ranked equally first (and approved) while A is ranked last (and not approved).
A clock method for three candidates is a method that assigns winners (or winning probabilities) based on the distribution of ballots around the clock.
Approval can be thought of as a clock method:
The win goes to the candidate ranked above (i.e. ahead of) the approval cutoff on the greatest number of ballots. Geometrically, candidate A's approval is the number of ballots in the semicircle centered on 12:00 O'Clock (where we put the label A).
For another clock method, suppose that we center semicircles at A, B, C, not(A), not(B), and not(C), and count how many ballots in each of these six semicircles. The semicircle with the most ballots narrows the field to one or two candidates. If it is A, B, or C, then the winner is the approval winner. If it is not(A), not(B), or not(C), i.e. the opposite of the candidate with the greatest disapproval, then there has to be some way of deciding between the two remaining candidates.
Three possibilities are ...
(1) flip a coin
(2) random ballot between the two candidates
(3) the pairwise preference determined from the ballot orders
In the case of three candidates Kemeny can be considered as a restricted clock method, where all of the ballots must be located at odd hour labels (i.e. strict orderings without approval cutoffs). If candidates are ranked equally (corresponding to even hour labels) then Kemeny moves half to the previous hour and half to the following hour. This is called "symmetric completion."
Once there are only odd hour ballots, Kemeny finds the hour from which the sum of the arc lengths (i.e. hours of separation) to all of the ballots is minimal. This gives the Kemeny social order.
Any other ideas for clock methods?
One suggestion I have is a different form of "symmetric completion" for clock methods like my approval/disapproval methods described above:
If any ballots are located precisely at an hour label, nudge half into each of the adjacent intervals. Or better, if there are already ballots in those adjacent intervals, allocate the "nudged" ballots in the same ratio as the ballots that were actually voted into those intervals.
Note that this kind of symmetric completion changes neither the order nor the approval cutoff on any ballot that has already indicated it (or them).
With this kind of symmetric completion, the approval and disapproval for each candidate will add to one hundred percent.
Then my approval/disapproval methods can be formulated like this:
The candidate whose approval is furthest from fifty percent is either the winner or the first eliminated, depending on whether his approval is above fifty percent or below fifty percent, respectively.
Well, I hope that this stimulates some thought and experimentation.
Forest ---- Election-methods mailing list - see http://electorama.com/em for list info
