[EM] Clock Methods (for Three Candidates)

[Forest Simmons]

Take a clock face and put labels A, B, and C at 12:00, 4:00, and 8:00, 
respectively. At 2:00, 6:00, and 10:00 put the labels not(C), not(A), and 
not(B), respectively.

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