[EM] Approval-Completed Condorcet vs. Criterion

[Adam Tarr]

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 
differentiate candidates.

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 
FBC.

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).

-Adam