# [EM] New thoughts on strong FBC

Alex Small asmall at physics.ucsb.edu
Sun Dec 8 17:30:02 PST 2002

```I already said that Negative Voting (a person's first and second choices
are each given a single vote) passes strong FBC (at least when voters have
sufficient information).  Granted, it passes in a very dubious way,
because the method makes no distinctions between the first and second
choices.  Nonetheless, if we use ranked ballots it technically does pass
strong FBC.

As far as I can tell, negative voting is the only possible system that
satisfies the following requirements when there are 3 candidates:

(1) strong FBC
(2) swapping first and second choices doesn't change outcome
(3) symmetry:  If every voter swaps his rankings of A and B, and A was the
winner, then B now wins, and if C won then C still wins.

I haven't proven this, but let's take it as a conjecture and examine its
consequences.

Now, as always, we have 6 voter types.  Constraining the number of voters
in each category to add up to some constant gives us 5 essential a
5-dimensional space of possible electorates.  Normally, a tie corresponds
to a 4-D subset of the 5-D space of possible electorates.

Let's resolve at least some 2-way ties with a pairwise comparison of the
tied candidates.  Our augmented method now makes a meaningful distinction
between first and second rankings in at least some circumstances (a 4D
subset of the space of possible electorates, to be precise).

As far as I can tell, the method satisfies strong FBC.  The only time the
method will distinguish between your first and second choices is when they
tie, and in those cases you never have an incentive to list #2 ahead of
#1.  You might, of course, have an incentive to demote #2 to third place
if your favorite would lose the pairwise contest.  But that's consistent
with strong FBC.

Now, suppose we increase the margin needed to trigger a pairwise contest
between the top two.  How large the margin is really doesn't matter, as
long as it's greater than zero.  When the margin is small enough to
trigger a pairwise comparison, you're essentially voting on who the 2
contestants will be.  We now have a 5 dimensional section of "voter space"
where the method makes a meaningful distinction between first and second
choices.

It's easy to show that although a single person cannot change the outcome
to something that he prefers, two voters acting in concert can.  Consider
this example, where the critical margin for a top-2 runoff is greater than
zero:

Society's pairwise preferences:  C>A, B>C, A vs. B irrelevant here.

There's a runoff between A and C, and C (your least favorite) wins.  If
you were to insincerely give the ranking B>C>A, A and B would be tied and
the result would be indeterminate (who goes into the runoff?).  However,
if you and a like-minded friend both give the insincere ranking B>C>A, the
runoff is between B and C.

Now, technically, since a single voter acting alone cannot obtain a better
result by favorite betrayal, one could say that the method passes strong
FBC.  However, that is torturing the concept of strategic voting.  If one
person notices from polls that the race is really close, the odds are good
that a second person will as well, and both will realize "Hey, I have an
incentive to do such-and-such."

THE POINT OF THIS LONG MESSAGE:

We took a method (perhaps the only method?) that (sort of) passes strong
FBC but doesn't make a meaningful distinction between first and second
choices.  We modified the method to make a meaningful distinction between
the top two choices on a 4D subset of the 5D "voter space" and saw that it
still satisfied strong FBC.

When we modified the method further, to make that meaningful distinction
in a small 5D subset of voter space we saw that strong FBC is violated,
provided that at least 2 people have the same incentives.  Although we
didn't consider every possible modification of the method, we considered
the only obvious modification of the method and saw that, no matter how
small of a modification we make, strong FBC is violated.

If only I could generalize, and show that ANY modification of negative
voting causes a violation of strong FBC (assuming that the modification
doesn't violate the symmetry condition).....

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