[EM] Strong FBC Yet Again--This time it's personal! ;)

[Alex Small]

(I tried sending this several hours ago but it didn't post, so I'm trying
again.  I apologize to those who get multiple messages.)

Strong FBC:  Revisited.

I won’t actually prove that strong FBC is impossible, but I will prove
that Approval Voting is superior to any ranked method if your goal is to
let voters defend their interests without betraying their favorite.  I
apologize for the length, but the methods I’m using require some
explanation.  I know that a year ago I posted a retraction of a similar
argument.  I wasn’t sufficiently careful with my use of symmetry
conditions, so my conclusions were nullified.  Here I’m more careful,
and instead of trying to tackle the most ambitious aspect of the problem
(showing that strong FBC is impossible) I just show that any method
satisfying strong FBC would be inferior to Approval Voting.



PRELIMINARIES:

I’ll consider the case of 3 candidates.  I’ll consider a method that
uses ranked ballots.  For simplicity we won’t consider equal rankings.
Since all voters are treated equally, and since there are 6 possible
preference orders, the space of all possible electorates is
6-dimensional.  We can think of a ranked voting method as one that
partitions the 6-dimensional space of all possible electorates into 3
regions.  Each region corresponds to victory for one of the 3
candidates.

I will be considering the properties of the surfaces that separate these
3 regions.  I will usually represent these surfaces with the linear
equation Nxy*E = 0, where Nxy is a vector normal to the boundary between
the region where x wins and the region where y wins, E is a vector
representing a point in the space of all possible electorates, and “*”
denotes a scalar product, AKA dot product, AKA inner product, for those
who know linear algebra.

Say that we’re looking at the boundary between the region where A wins
and the region where B wins.  We’ll call the normal to this boundary
Nab.  If Nab*E >0, A wins.  If Nab*E < 0 B wins.  If Nab*E = 0 then
there’s a tie.  (We’re assuming that in this region of electoral space C
cannot win.)  We won’t require that Nab be the same everywhere.  For
instance, consider Condorcet methods.  In some regions of electoral
space the condition determining whether A or B wins will be that A
defeats B pairwise.  In other regions of electoral space the condition
might be that A’s margin of victory over B exceeds B’s margin of victory
over C.  The form of Nab will depend on which region of electoral space
we’re in.

When we actually write down the form of Nab, we’ll use a simple
notation:  The notation “ABC” will denote unit vector pointing in the
direction of adding more voters with the preference A>B>C.  Remember,
this is a six-dimensional space of all possible electorates.  Say we
want to represent an electorate where 30% of the voters have the
preference A>B>C, 45% have the preference B>A>C, and 25% have the
preference C>A>B.  We could denote this electorate as

E = 30 ABC + 45 BAC + 25 CAB

Now say that some of the C>A>B voters want to change their ballots to
insincerely say A>B>C, because B is the plurality winner and they would
prefer A over B.  We could denote this change by saying that the change
in the electorate vector is

delta E = 16(ABC – CAB).




THE STRONG FAVORITE BETRAYAL CRITERION:

We know that we can’t stipulate that a method never give a voter an
incentive to vote insincerely.  The Gibbard-Satterthwaite Theorem says
that any non-dictatorial and Pareto efficient ranked method will be
manipulable.  We examine a weaker condition:  The voting method should
never give us an incentive to rank our favorite candidate below first
place, or rank another candidate equal to our sincere favorite.  Or, if
there is an incentive to rank your favorite below the first place, it
should be possible to achieve the same outcome through some other means,
namely by reversing the rankings of your second and third choices, while
leaving your favorite in first place.

We call this criterion “strong FBC” to distinguish it from “weak FBC”,
which stipulates that you should never have an incentive to rank another
candidate ahead of your favorite, but you _might_ have an incentive to
rate another candidate _equal to_ your favorite.  Approval voting
satisfies weak FBC but flunks strong FBC.

(Some might find the condition less than compelling, but other people
have a strong desire to sincerely support their favorite, so the goal is
to find out whether it can be satisfied.)

An example of a method that satisfies strong FBC (the only strictly
ranked method that does for 3 candidates, to the best of my knowledge)
is “Negative Voting”:  You indicate your preference order, and the
method assigns one vote apiece to your favorite and middle candidates,
and no vote to your least favorite.  This still violates the “spirit” of
strong FBC, however, because it doesn’t give any special status to your
favorite.
 This method and its various properties will come up throughout this
posting.



SYMMETRY

We’ll assume and require that our election method treats all candidates
equally.  (Sorry, Florida... ;)  Say that we’re near the A-B boundary in
voter space.  It’s possible that that one of the factions (e.g. A>C>B)
might have no incentive to vote insincerely for the purpose of helping A
win instead of B.  It’s possible that another faction (e.g. A>B>C) might
have an incentive to vote insincerely for the purpose of helping A
defeat B.  It all depends on the details of our ranked method.  However,
symmetry makes the following demands on us:

1)  If a faction with a given preference order (e.g. A>C>B) has no
incentive to vote insincerely for the purpose of helping A defeat B,
then a faction that reverses A and B in its preference order (e.g.
B>C>A) should likewise have no incentive to vote insincerely.  The
method should not treat A’s supporters any differently from B’s
supporters.

2)  Conversely, if a given faction (e.g. A>B>C) has an incentive to
insincerely swap its second and third choices to help A defeat B (e.g.
insincerely report A>C>B) then a faction that has the reverse opinion of
A and B (e.g. B>A>C) should also have an incentive to insincerely swap
its second and third choices (e.g. report B>C>A) to help B defeat A.

3)  If a faction with a given preference order (e.g. A>C>B) has no
incentive to vote insincerely to help A defeat B, then when we permute
the candidates (e.g. A->B, B->C, C->A) the statement should still be
true. (e.g. voters with the preference B>A>C should have no incentive to
vote insincerely to help B defeat C.)

4)  If a faction with a given preference order (e.g. A>B>C) has an
incentive to insincerely swap its second and third candidates to help A
defeat B, then when we permute the candidates (e.g. A->B, B->C, C->A)
the statement should still be true.  (e.g. voters with the preference
B>C>A should have an incentive to swap C and A to help B defeat C.)

These symmetry conditions greatly simplify our analysis.  Instead of
considering the strategic incentives faced by all 6 factions at the 3
different boundaries, we only need to consider the strategic incentives
faced by 3 factions at a single boundary, and everything else followed
from symmetry.



MORE ON THE GEOMETRY OF ELECTORATE SPACE:

Let’s consider the A-B boundary.  Say that a XYZ faction prefers A to B.
 The condition for A to win is Nab*E > 0.  If that faction has no
incentive to vote insincerely for the purpose of helping A defeat B,
then XYZ*Nab must be greater than or equal to RST*Nab, where RST is any
other faction.  Otherwise, voters from the faction XYZ could increase
E*Nab by reporting their preference as RST.  Likewise, if XYZ preferred
B to A, and we stipulated that XYZ never have an incentive to vote
insincerely to help B defeat A, then we’d require that XYZ*Nab be less
than or equal to RST*Nab, where RST is any other preference order.

On the other hand, suppose that XYZ has an incentive to help A defeat B
by reporting the (insincere) preference X>Z>Y.  Then Nab*XZY must be
less than or equal to Nab*RST, where RST is any other preference order.
Or, suppose that XYZ prefers B to A, and that it has incentive to help B
defeat A by reporting the (insincere) preference X>Z>Y.  Then Nab*XZY
must be less than or equal to Nab*RST, where RST is any other
preference.

In both cases where a faction XYZ has an incentive to vote insincerely,
the strong FBC criterion tells us absolutely nothing about Nab*XYZ, but
it does tell us something about Nab*XZY.  It is in the cases where a
faction has an incentive to vote insincerely that we have some freedom
in
specifying the form of Nab.  The form of Nab can then vary from place to
place in electoral space, as it does in many methods.  (e.g. in IRV the
form of Nab depends on whether a candidate won with an outright majority
of first-place votes, or if instead an elimination had to be held.  In
Condorcet it depends on whether or not there’s a cycle to be resolved.)

So, strong FBC tells us something about which components of Nab must be
a maximum or minimum.  We’ll assume that the maximum and minimum
components of Nab are +1 and -1.  One of those components can be chosen
completely arbitrarily.  The other one requires careful consideration,
because it is NOT necessarily true that Nab will change sign if we swap
A and B.  (To persuade yourself of this, consider any method that
selects the first choice of a majority when such a candidate exists.  If
we’re testing to see whether A is the first choice of the majority, then
Nab will be ABC + ACB – BAC – BCA – CAB – CBA.)  However, we will impose
the reasonable requirement that if one person’s vote helps A defeat B,
there should always be a way for another person to vote so that he or
she cancels out the first person’s vote, at least with regard to the
choice between A and B.  In that case, if the maximum component of Nab
is +1, then the minimum component must be -1.



ANALYSIS  Here’s where the good stuff begins.

Now, let’s consider the boundary between the A region of electorate
space and the B region.  We’ll start with the faction C>A>B.  If this
faction never has an incentive to vote insincerely (for the purposes of
helping A defeat B) then Nab*CAB = 1 and Nab*CBA = -1.  On the other
hand, if the faction C>A>B sometimes has an incentive to vote
insincerely to help A defeat B then CBA*Nab = +1, and CAB*Nab = -1.
Either way, those two components of Nab will be constant rather than
varying.  We have no freedom to pick them except for choosing their
signs.  For convenience, we will confine our attention to the case where
Nab*CAB = +1 and Nab*CBA = -1.  This also implies Nbc*ABC = +1, Nbc*ACB
= -1, Nca*BCA = +1, and Nca*BAC = -1.  We will see below that this
convention does not undermine our main argument.

Next consider the factions A>B>C and A>C>B (as well as the other two
factions with opposite opinions of A and B).  If neither faction can
help A defeat B by voting insincerely, then Nab*ABC and Nab*ACB will
both be +1, and Nab*BAC and Nab*BCA will both be -1.  The same form of
Nab would obtain if ABC and ACB both had an incentive to help A defeat B
by voting insincerely.  ABC’s incentive would imply that Nab*ACB = +1,
and ACB’s incentive would imply that Nab*ABC = +1.  Leave aside the
logical
inconsistency there, the point is that the same form would obtain.

It is straightforward to obtain the forms of Nbc and Nca as well, and
see that Nbc, Nca, and Nab will be linearly independent because they are
constant rather than variable.  Linear independence leads to paradoxes,
because we can have an electorate that’s on the A side of the A-B
boundary, the B side of the B-C boundary, and the C side of the C-A
boundary.  This paradox occurs regardless of which convention we pick
with regard to Nab*CAB.  So it is clear that either ABC or ACB (but not
both) must, at least sometimes, have an incentive to help A defeat B by
insincerely swapping the second and third candidates.

We will limit ourselves to the case where the A>B>C faction has an
incentive to vote insincerely, but not the A>C>B faction.  By following
this convention, as well as our convention for Nab*CAB, we have a
monotonic method.  Monotonicity may or may not be a crucial feature for
a voting rule, but we will see later that it does not undermine our
basic argument.  Since it does not undermine our results, and since many
people consider it desirable, we will for now stick to monotonic
methods.

The form of Nab is therefore:

Nab = ACB + CAB – BCA – CBA + g1(E)*ABC – g2(E)*BAC

g1(E) and g2(E) are functions of E, our vector in electorate space.  The
signs in front of g1 and g2 are arbitrary and chosen for convenience.
To find Nbc and Nca, one would permute A, B, and C in the components of
Nab, and also permute A, B, and C in E (the argument of g1 and g2).  We
note that if g1 and g2 are identically zero then we have Negative
Voting.

Instead of exploring the properties of g1(E) and g2(E) to determine
whether these functions can ever be non-zero without producing a
paradox, we will explore how voters will act when faced with an election
using a method from the family proposed here.

Suppose that the electorate is in a region of electorate space close to
the A-B boundary.  This amounts to a race with 2 chief contenders.
Anybody who has the preference A>B>C will insincerely vote A>C>B when
g1<1, and anybody who has the preference B>A>C will insincerely vote
B>C>A when g2<1.  The result is de facto Negative Voting.

Of course, the problem with Negative Voting is that everybody voting
against either A or B is obliged to vote for C.  This is fine for those
who consider C their favorite, but not for those who consider A or B
their favorites, and even worse for those who consider C their least
favorite.  It is obvious that Approval Voting and other rated methods,
which give the option of equal rankings for first place, are superior to
any ranked method that attempts to satisfy Strong FBC.  Personally, my
favorite such method is Majority Choice Approval, since it gives voters
a way to make support for their second choice contingent on how other
people vote.




Final Thoughts:

1)  Can we go beyond my analysis and actually prove that paradoxes are
inevitable if g1 and g2 are non-zero in a region of electorate space
with non-zero measure?

2)  Would anybody here be interested in helping me clean up these
results and make it more readable for publication?  I’m looking for a
co-author.  Surely there must be a few people here who are accustomed to
publishing scholarly papers and interested in adding to the literature a
convincing case that Approval Voting is superior to any ranked method
when one considers strategy and sincerity.

3)  Does anybody have ideas for how to extend these results to 4 or more
candidates?  It’s difficult because right off the top of my head I can
think of numerous methods that satisfy strong FBC (albeit in the
disappointing way that Negative Voting does):  Vote for the top 2, or
vote for the top 3, or give 1 vote apiece to the top 2 and a half vote
to the third-choice candidate on your ballot, etc.  With a little
thought I could probably come up with variants on other methods as well.