[EM] "margins" Condorcet methods have a critical strategy problem

[James Green-Armytage]

	Here is my argument that "margins" methods have a critical strategy
problem. Some of this is a repeat from earlier posts, but there are subtle
and important differences in the argument, especially in my understanding
of how winning votes will work in this example.
	Let me introduce example 1, variations of which will be used throughout
this post.

	Ex. 1: Sincere preferences:
46: A>B>C
44: B>A>C
5: C>A>B
5: C>B>A
	Ex. 1: Pairwise comparisons:
A>B 51-49
A>C 90-10
B>C 90-10

	C is an extremely unpopular candidate who loses both pairwise comparisons
by 90 votes to 10. A is a Condorcet winner, but the A:B comparison is
quite close. Really, the only legitimate contest in the election is the
A:B comparison, and the fact that A wins it should rightfully seal the
result. However, if the B voters are very determined, they can vote B>C>A,
and stand a good chance of stealing the election by so doing.

	Ex. 2: Expressed preferences (some insincere): 
46: A>B>C
44: B>C>A
5: C>A>B
5: C>B>A
	Ex. 2: Pairwise comparisons:
A>B 51-49
C>A 54-46
B>C 90-10

	Now, A>B is the weakest defeat, and so B wins. This is a hideous result
that would shower Condorcet methods in shame for generations to come. 
	We should NOT assume that any other voters knew ahead of time whether the
B voters would execute this strategy or not. We should NOT assume that the
B voters' strategy relies on central coordination. These assumptions are
too optimistic, and thus by making them, we would fail to be appropriately
cautious. The question is: what could A>B voters have done to avoid this
bad result, without messing things up in case the strategic incursion did
not occur? 
	For one, the C>A>B voters could have voted C=A>B to begin with, but this
means abandoning their favorite to some extent, which is a lot to ask if
the B>A>C voters' strategy is not clearly known. Furthermore, the C voters
may have a lot to gain from leaving their votes as is and letting A and B
voters get involved in a strategy fight, as we will see later.
	Since the B voters are happy with the result as is, this leaves us with
the A>B>C voters. They want A to win, but they cannot get that result from
example 2. So, instead, they would have liked the B>A>C voters to know
ahead of time that their strategy could yield no benefit. We'll call this
a deterrent strategy. But how to do it? 
	Here's the punch line: Using margins, the deterrent strategy can't be
done without a very good chance of severely messing things up. Using
winning votes, it can be done without messing things up (at least in THIS
EXAMPLE... not necessarily in all examples!).
	How does a margins deterrent strategy mess things up? Well, first, we
need to figure out how the A>B>C voters can provide a genuine deterrent in
margins. Let's say that the A voters try to deter through mere truncation,
the B voters execute their burying strategy, and we get something like
this: 

	Ex. 2: Expressed preferences: 
46: A>B=C
44: B>C>A
5: C>A>B
5: C>B>A
	Ex. 2: Pairwise comparisons:
A>B 51-49
C>A 54-46
B>C 44-10

	A>B is still the weakest defeat in margins, but B>C is now the weakest
defeat in winning votes. Therein lies the key difference between the
methods. The consequence is that truncation tends to be an effective
deterrent in winning votes, but it tends not to be an effective deterrent
in margins. 
	So how can the A>B>C voters deter in margins? Only by voting A>C>B. Now
we have hit our problem. The only way that A voters can prevent their
hard-won A>B defeat from being overruled by a false C>A defeat is to rank
C in second place (and to vote this way in polls before the election,
announcing their intention), in the hopes of deterring the incursion
before it happens.
	However, what if the B voters weren't intending to bury A after all?
Since the A:B race is so close, we should not assume that the voters know
who will actually win it, before the election. Therefore we should not
assume that B voters will accept that A is the "rightful" winner and step
aside. Meanwhile, we have polls saying that most of the A voters are
listing this strange candidate C in second place, which means that a C>B
defeat could potentially overrule a genuine B>A defeat. The B voters will
not be happy about this; they will suspect it to be strategic, and they
will be sorely tempted to provide a deterrent of their own. If the B>A>C
voters respond by voting B>C>A, then candidate C becomes a Condorcet
winner! This brings us to a terribly complex game of chicken, played by
millions of voters simultaneously. At this point, any of the three
candidates could be elected with roughly equal probability, depending not
on voters' actual preferences, but rather on their predilections for
swerving rather than staying the course. This is a horrible result, which
would shower Condorcet methods in shame for generations to come.

	This is why margin methods are unusable. Now, let's go back to winning
votes. In this example, using winning votes, the A>B>C voters should vote
A>B=C, and the B>A>C voters should vote B>A=C. Then we get something like
this:

	Ex. 3: semi-sincere truncated votes:
46: A>B=C
44: B>A=C
5: C>A>B
5: C>B>A
	Ex. 3: Pairwise comparisons:
A>B 51-49
A>C 46-10
B>C 44-10

	This is a stable result, even though the voters don't know the result of
the A:B contest. If A wins it, as above, and the B voters bury A under C,
the B>C defeat will be the weakest of the A>B>C>A cycle. If B wins it (not
shown), and the A voters bury B under C, the result is similarly
counter-productive.
	**IMPORTANT: This deterrent strategy does NOT require the A voters to
DISCOVER that the B voters INTEND to commit a strategic incursion, and to
COORDINATE a response. Rather, this example is stable insofar as it is
intuitive from the beginning that the A and B voters ranking their primary
rivals is an unnecessary source of trouble. They do not truncate because
of any particular information that the 'enemy camp' is hatching a plan.
Rather they truncate because reporting a full ranking creates a liability
without creating a benefit.
	This particular example treats winning votes kindly, in that these two
primary rivals do not rely on each other's second preferences to be
viable. Winning votes may well exhibit a strategy problem when this is not
the case, but let me leave that issue for another time.
	So, I am not arguing that winning votes does not have significant
strategic vulnerability. Indeed, I have argued that it does, and I will
probably continue to argue this at a later time. However, I would first
like to work towards a broad understanding that margins Condorcet has a
very severe strategy problem, and that it should not be used for
contentious public elections.
>
Sincerely,
James Green-Armytage
http://fc.antioch.edu/~james_green-armytage/voting.htm