[EM] Brief comments on FBC definitions

[Richard Moore]

MIKE OSSIPOFF wrote:

> 
> I noticed some time ago that Craig Carey was still saying that
> I never defined FBC. That topic was thoroughly discussed long before,
> at which time I finally had to stop taking the time to reply to
> Craig's 27K postings.
> 
> But, since it's brief, let me repeat that definition here:
> 
> Favorite Betrayal Criterion (FBC):
> 
> By voting another candidate over his/her favorite, a voter should
> never gain an outcome that s/he prefers to every outcome that s/he
> could get without doing so.


This may be nit-picking, but that wording is ambiguous.

Say there are five candidates: A, B, C, D, and E. Without voting another
candidate over her favorite, a voter might get any of the following results:
A, B, C, D, or E. When voting another candidate over her favorite, she
might also get any of the following results: A, B, C, D, or E. None of the
choices she could get by voting another candidate over her favorite is
preferred to all of the choices she could get by not doing so.

Therefore any method (provided it is capable of electing any one of the
candidates) could pass FBC under this (unintended) interpretation.

A better wording would be:

A method passes FBC if there is no scenario in which, by voting
another candidate over his or her favorite, a voter could gain an
outcome he or she prefers to any of the outcomes he or she could
gain in the same scenario without doing so.

It's a seemingly subtle change but it removes that ambiguity.

> What I mean by voting one candidate over another:
> 
> A voter votes A over B if s/he votes in such a way that one could
> contrive some configuration of other people's votes such that,
> if we delete from the ballots every candidate but A & B, A is
> the unique winner if & only if we count that voter's ballot.
> 
> [end of definition]
> 
> I suppose one could add to that "...and no one can contrive a
> configuration of other people's votes such that, if we delete from
> the ballots every candidate but A & B, the unique winner is B if
> & only if we count that voter's ballot."
> 
> I don't think that addition is necessary, but it could be added
> if someone devised a reasonable example where it seemed necessary.


Forest and I had an off-list discussion some time ago about defining
monotonicity, and the prerequisite definition of "changing a ballot
in a way that favors candidate X". Making such a definition generally
applicable (beyond fully ranked methods) is trickier than one would
think. For instance, in CR, if candidate X's rating is increased from
25 to 30, does this favor X? Yes, but what if candidate Y's rating
is increased by 10 points at the same time X's rating is increased?
We never came up with a completely satisfactory resolution. (As I recall,
one possible definition that looked promising turned out to be equivalent
to consistency, and the last definition I suggested was just too hard
to use to construct any proofs.)

I haven't studied it carefully enough, but I hope the above definition
of "voting one candidate over another" doesn't suffer from similar
problems. I think it might be OK, since it involves reducing the
ballots to only two candidates, but I just wanted to point out that
there are sometimes hidden "gotchas" in some of these definition
attempts.

> it's brief & simple enough to use in public
> discussion. Of course something more universally-applicable can be
> useful in mathematical discussion.


Yes, I agree there is value in both mathematically rigourous definitions
and in colloquial definitions. The general public won't be willing to
sit through lectures in set theory in order to understand why one method
is better than another.

Of course, for the general public, it might be sufficient to have the
colloquial FBC definition, without defining "voting one candidate over
another". Presumably anyone interested in voting knows what that phrase
means.

  -- Richard