[EM] DMC/RAV and FBC

[Russ Paielli]

Folks,

Kevin Venzke recently posted a message in which he showed that "WV 
("winning-votes" Condorcet) methods fail FBC (Favorite Betrayal 
Criterion) with 3 candidates."

I am interested in how DMC/RAV (Definite Majority Choice/Ranked Approval 
Voting) fares in this regard. I think I can easily show that it won't 
pass, but I'd like to estimate the likelihood that a voter will "shoot 
himself in the foot" by ranking his favorite candidate in the first 
slot. I'd also like to know all the conditions under which it could happen.

If I am not mistaken, the algorithms for all ordinal-only Condorcet 
methods are third order or higher, whereas the algorithms for DMC/RAV 
are only second order, which greatly simplifies the analysis.

Let me pose a specific question and propose a method of solution.

Let's say we're having our first DMC/RAV election (after plurality) and 
we have three candidates:

F = "Favorite"
L = "Lesser of 2 evils"
G = "Greater of 2 evils"

Let's say you've already decided to approve F and L and disapprove G, so 
your possible votes are

F > L
F = L
L > F

The first point to note is that, given all the votes of the other 
voters, these 3 possible votes will all produce the same approval 
ordering (because they all approve F and L and disapprove G).

The second point to note is that these votes all have the same effect on 
the two pairwise races involving G, hence they all have the same effect 
on the G row and G column of the pairwise matrix (because they all rank 
F and L above G).

The solution I propose starts with an enumeration of all the possible 
configurations of the other votes such that your vote can possibly 
change the winner. This is much easier for DMC/RAV than it would be for 
WV or any other ordinal-only Condorcet method.

The total number of approval orderings is six:

F L G
F G L
L F G
L G F
G L F
G F L

As mentioned above, your vote will have no effect on the pairwise races 
involving G. In DMC, unlike ordinal-only Condorcet, all that matters in 
each pairwise race is who wins or loses; the actual vote counts do not 
matter beyond that. That greatly simplifies the problem. So the total 
number of possible outcomes of the races involving G are four: G can 
beat or be beaten by each of F and L. (We neglect ties here.)

Multiplying the six possible approval orderings by the four possible 
outcomes involving G gives us 24 permutations so far.

Now we consider the number of possible outcomes of the F and L pairwise 
race for which your vote can possibly affect the outcome. Your vote can 
change the winner if F and L are tied or within one vote of each other 
in their pairwise race. That is three possibilities.

Multiplying those 3 possibilities by the 24 permutations gives us 72 
total permutations of vote configurations in which your vote can change 
the winner.

The next step is to determine the effect of each of your three possible 
votes for each of those 72 permutations of the other votes. The possible 
effects are

F -> G  -2
F -> L  -1
G -> F  +2
G -> L  +1
L -> F  +1
L -> G  -1

where the signed number in the third column gives the "desirability" of 
each effect. Note your vote will not change the winner in all cases. For 
example, if F was leading G by one vote and you voted F over G, your 
vote will not change the winner.

That gives the complete mapping of the possible effects of your vote. We 
could then count the number of cases in which each of your three 
possible votes produces each of the six possible effects.

That's not quite the entire story, however, because not all of the 72 
permutations of the other votes are equally likely. For example, F is 
certainly less likely to win the approval vote than L or G. The 
probabilities of each of the six approval permuatations would have to be 
estimated to give the results practical significance.

This problem could probably be done with pencil and paper in a few very 
tedious hours, but a nice little script would sure come in handy. I may 
give it a try one of these days, but I am very busy at work. Any other 
DMC/RAV advocate willing to give it a try?

I think the results will say a lot about the quality and effectiveness 
of DMC/RAV.

--Russ