James Green-Armytage
Severe strategic vulnerability in "margins"-based pairwise methods
Here is my argument that "margins" methods have a critical strategy problem. First, I should define "margins"-based pairwise methods as opposed to "winning votes"-based pairwise methods.
In "margins" methods, the strength of a defeat is defined by the number of votes in strict agreement with the defeat, minus the number of votes in strict disagreement with the defeat. (Winning votes minus losing votes.)
In "winning votes" methods, the strength of a defeat is defined simply by the number of votes in strict agreement with the defeat. The two approaches produce identical results when none of the voters assign equal rankings to any candidate. (I also define margins and winning votes here and here.)
Now
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.
I argue that we should not assume that any other
voters knew ahead of time whether the B voters would execute this strategy or
not, and that we should not assume that the B voters' strategy relies on central
coordination. These assumptions are optimistic, and thus by making them, we
would fail to be appropriately cautious. My 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 (in example 2), this leaves us with the A>B>C voters. They want A to win,
but they cannot get that result directly from example 2. That is, if the B and C
voters vote as in example 2, the A voters can do nothing to elect A. 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. This reflects a key difference between the
methods: 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 note: 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.
My conclusion is that margins-based pairwise methods have a strategy problem that is severe enough to make them unusable in contentious public elections.