Besides this disagreement, there is the problem that if the win is given to B for this ballot set, then if the 24 faction voters' true preferences were B>A>C, they would be sorely tempted to truncate so that B would win instead of A.
I think that the best way out of this dilemma is appropriate randomization in "situations like this".
To randomize this example I would use "symmetric depletion" as opposed to symmetric completion, to split the last faction in two:
48 C, 24 B, 14 A, 14 not(C).
Then I would cancel the 14 not(C) against 14 of the 48 C to get
34 C, 24 B, 14 A.
Then I would subtract 14 from each of these factions to get
20 C, 10 B.
Finally, I would roll a die and pick B if a one or a two came up, or C in the case of 3, 4, 5, or 6.
An equivalent way of finding these 2:1 odds is to subtract the least Borda Count from the other two, and take the ratio of the two differences.
In this example, the A>B, B>C, and C>A margins are 4, 4, and 20, respectively, so the relative Borda Counts for A, B, and C are
(4-20)=-16, (4-4)=0, and (20-4)=16, which are proportional to
-1, 0, and 1, respectively. When we subtract the negative one (by adding its opposite) from the other two scores we get 0, 1, and 2, respectively.
So the required odds are 2:1.
Suppose that the members of the second faction secretly rate A, B, and C at x, 1, and 0, respectively. Then with this randomization each member of this faction has expectation of 1/3. If they don't truncate, then there is no randomization, and A (with rating x) wins.
So it is to their advantage to truncate only if x is less than 1/3. In that case, candidate A SHOULD BE the loser!
The full payoff matrix for A and B with sincere preferences
48 C, 24 B > A, 28 A > B is
\ B A \____truncate_____|__show preference__ | | truncate| (0,0) | ( 1, x ) | | ------------------------------------ show | | pref. | (y/3, 1/3) | ( 1, x )
where y is the third faction's rating of B.
Note that for A, the strategy "show pref." dominates, no matter the value of x or y. Since B knows this, he crosses row one out. Then he chooses "truncate" iff 1/3 is greater than x.
I think that this is a good way out of the dilemma. For those that don't like randomization, let them contemplate that different methods give different winners, so both B and C have a legitimate claim. Giving the win to C cheats B in the case when B sincerely doesn't like A at all. Giving the win to B cheats A in the case when A is the sincere CW.
I would rather have a clean, up front randomization in a case like this, then leave the voters with this dilemma.
Exactly which ballot sets should be considered appropriate for randomization, I don't know... perhaps whenever there is a beat cycle, perhaps only when margins and wv disagree, or perhaps whenever Black and wv disagree.
Forest ---- Election-methods mailing list - see http://electorama.com/em for list info
