[EM] Approval-Completed Condorcet vs. Criterion
atarr at ecn.purdue.edu
Wed Mar 20 10:00:36 PST 2002
I'm trying to figure out if Approval-Completed Condorcet does better than
Beatpath or Ranked Pairs or the other popular Condorcet implementations around
here. So I decided to look at it against Mike's criteria. Well, it fails
Favorite Betrayal, like everything this side of Approval does. And depending on
how we define it, it also fails the Generalized Condorcet and Generalized
Strategy-Free criterion. Still, the idea of Approval-Completed Condorcet is
quite appealing. I'm not convinced it's better, but it may well be. Let me
define what I mean by Approval-Completed Condorcet first.
For the purposes of measuring ACC against various criteria, lets propose a
system where the voter ranks the candidates as he/she wishes, and then
designates a cutoff where candidates above a certain point on the ballot are
approved, and those below are not. Forest has suggested that this could be
easily implemented by inserting "none of the below" as a candidate that can be
ranked. In the case where there is no Condorcet winner, we elect the candidate
with the highest approval count. We can restrict our choice to the Smith set,
but I'm not sure this is a good idea; I'll explain why later.
Forest has further suggested the use of five-slot ballots (A,B,C,D,F) for use in
such elections. I don't think 5 grades is quite enough, but mostly I don't like
the fact that there are not an even number of grades. Forest has suggested that
plusses and minuses could be used for more resolution if desired, but 13 or 15
slots seems like overkill, and it still does not allow for an even split between
approved and unapproved grades. I'd suggest a six-slot ballot: A,B,C,D,E, and
F. A,B, and C are "passing" (approved) grades, while D, E, and F are "failing"
(disapproved) grades. I think this would be intuitive to the average voter
(some schools actually grade this way) and I think 6 slots gives enough room to
It's entirely possible that the 6-slot ballot implementation of ACC will fail to
pass certain criterion that the smoother "none of the below" implementation
would pass. These would only happen when there were a lot of (i.e. more than
four) candidates running very close to one another, though, so I doubt it would
be a serious problem in real elections. If the number of competitive candidates
in elections began to ramp up into these ranges, the ballot could be expanded to
have more slots to accommodate that. But 6 slots is both intuitive (due to an
even number of slots) and fine-grained enough for current electoral conditions.
But anyway, on to my subject... ACC does fail Favorite Betrayal. Strong FBC
could be roughly stated as, "you never have any incentive to insincerely rank
another candidate equal to or higher than your favorite." Remove the "equal to
or" part to get regular FBC. Approval passes FBC, but IRV/Borda/Condorcet/etc
do not. Nothing passes SFBC, as far as we know.
If there is a cyclic tie in ACC, then we go to the approval counts, where I can
rank Compromise below Favorite in ACC and still end up casting a vote for
Compromise over Worst (by approving the former and not the latter). So ACC
either passes both FBC and SFBC, or it fails both. As far as I can see, there
are two types of cases that could cause ACC to fail.
One case would be if Worst wins the approval count, and a cyclic tie exists
where Favorite beats Compromise beats Worst beats Favorite. By reversing
Favorite and Compromise on your ballot, you can make Compromise the Condorcet
winner, and avoid the approval runoff altogether. This shows that ACC fails
The other case would be where Compromise wins the approval runoff, but does not
reach the Smith set. I don't think this is a possible situation with three
candidates. In order for Compromise to not get into the runoff, Compromise must
already lose pairwise to both Favorite and Worst. This means that, with only
three candidates, the pairwise winner of Favorite and Worst is the Condorcet
winner. If Worst is the Condorcet winner, then you can't force a three way tie
by putting Compromise in front, and if Favorite is the Condorcet winner, then
you don't want to change things.
This case does become possible if we can introduce a fourth candidate, who
reaches the Smith set. Say that the approval rankings are Compromise, Worst,
Favorite, and finally New. In the pairwise elections, New beats Worst beats
Favorite beats New (or the other way around; it doesn't matter) but all three
beat Compromise. Compromise is not in the Smith set, so Worst wins if we
restrict the approval choice to the Smith set. In this case, by insincerely
ranking Compromise over Favorite, we might get Compromise into the Smith set,
where Compromise wins the Approval runoff.
If we remove the Smith set restriction, then the second case of ACC failing FBC
and SFBC is no longer possible. But doing so causes ACC to fail GSFC and GCC.
I tend to think it is worth dumping these two criterion for better FBC
compliance. My reasoning is:
1) Removing the mention of the Smith Set simplifies the method.
2) The second case where FBC fails seems like the more significant one to me.
That is the case where favorite betrayal was needed to protect the approval
winner. That seems like a significant problem. The first (unavoidable) case
involves using favorite betrayal to BEAT the approval winner. The fact that
this can be difficult to do doesn't bother me as much.
3) The only time the difference comes up is when there is no Condorcet winner,
but the Approval winner is not in the Smith set. With no Condorcet winner, I'm
inclined to go with the Approval winner.
At any rate, I'm pretty sure ACC satisfies all the rest of Mike's criteria. ACC
seems like a good method. Unfortunately, it's not as great as I hoped I'd find.
On the borderline cases where the various Condorcet methods (ACC/RP/Beatpath)
disagree, ACC seems to be just as vulnerable to strategic manipulation as the
others. Still, it seems like a good idea. If anyone has any other strong
arguments for or against ACC, compared to RP or Beatpath, I'd like to hear them.
One last argument for ACC is that it would render irrelevant the winning votes
vs. margins argument. Much rejoicing (yay).
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