# [EM] FBC ambiguity

Richard Moore rmoore4 at home.com
Sat Jan 5 00:52:10 PST 2002

```MIKE OSSIPOFF wrote:

> I'd said:
>
>> Still, I'll say that my official version of my definition of voting
>> Smith over Jones is the
>> one that adds that it must _not_ be possible to contrive a
>> configuration of the
>> other people's ballots such that, if we delete everyone but Smith &
>> Jones from the
>> ballots, then Jones wins if & only if we count that voter's ballot.

First, I think there's an unrelated error, since "Jones wins if & only
if we count that voter's ballot" would require that a single ballot is
capable of converting Jones from a definite loser to a definite winner.
If a random tie-breaker is used, then, in many methods, a single ballot
added (not changed) at best can cause a tie-breaker to be invoked, or
prevent a tie-breaker from being invoked, so one of the two outcomes is
not definite.

I'll sidestep that issue for now by assuming that the ballot in question
will be counted twice, for puposes of determining if it votes A over B.
My modification of your definition (which I got from combining the two
conditions in your earlier message) is:

"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 *twice*,
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 *twice*."

This have the same effect as your original definition, without the
tie-breaker issue. There may be a better way to fix the wording to get
away from the issue.

> Richard replied:
>
> If you add this second condition, and it *is* possible, for each possible
> ballot, to contrive a configuration of the other ballots that converts
> the winner to Jones, then there is no such thing as a "Smith over Jones"
> ballot...
>
>
> Are we talking about a voting system in which it isn't possible to write
> an example
> in which someone can vote one candidate over another, by my definition
> that I

Yes.

> Then no one could gain by voting someone over their favorite, since no
> one could
> vote anyone over their favorite (or over anyone else either).
> I guess, then, that I'd have to say that that, by the above-quoted
> definition, that
> method passes FBC. Maybe that would be a fault of FBC, to the extent
> that such
> a method is important, but maybe it's ok. After all, my goal with FBC is
> that no one
> should ever have strategic need to vote someone over their favorite. If
> we're using
> a method in which it's impossible to vote anyone over anyone, then my
> goal is
> satisfied. A person can only vote in a way that could, with just 2
> candidates left on
> the ballots, make Smith beat Jones, or could make Jones beat Smith. I
> wouldn't call
> that dumping Smith or Jones, and so that method doesn't sound bad in
> that regard.
> Of course it would have considerable other problems, as you said.
>
> It seems to pass FBC fair & square, but does so at the cost of taking
> away the
> possibility of voting anyone over anyone.
>
> But if you're just talking about a special example that can be written
> in which no
> voter can vote anyone over anyone, then that isn't a problem, since all
> we have to
> do is write a different example by which we _can_ test the method by FBC.
>
> Richard continued:
>
> Then it is impossible to test this sort of method for FBC, using
> this definition.
>
>
> But FBC's wording, mostly for the purpose of brevity, says "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 in that election without doing so."
>
> No one can gain a better outcome by voting someone over his favorite,
> because no
> one can vote anyone over anyone.

Then the method passes FBC, as you concluded above. In my mind, a
method like this fails FBC miserably. I could invent an obviously
defective method that would have this problem. Suppose the candidate
with the most approval wins, unless that candidate has exactly 10
approval votes, in which case that candidate loses and the candidate
with the next most approval votes (also not equal to 10) wins. So
in almost every conceivable case, my ballot will help my candidate
if I approve my candidate. But in the one case where my candidate
has 9 votes (not counting my ballot), my ballot can hurt that candidate,
and in the case where my candidate and candidate Z both have 9 votes
(not counting my ballot), and everyone else gets less than 9 votes,
then if I vote for my candidate (and not for Z) I give the victory to
Z; if I betray my candidate by voting for Z and not my candidate I
give the victory to my candidate. I would call this an FBC violation,
but if the definition says my ballot in the second case doesn't vote
Z over my candidate then this doesn't qualify as an FBC violation.

What I am saying is that your definition works, provided any monotonicity
violations in the method in question is confined to cases of 3 or more
candidates. As long as the method is monotonic for 2 candidates, you
can always create a ballot that votes X over Y (using your definition).
In the case above, the monotonicity is violated with only two candidates,
so you can't have any ballot that tests the method, so the method passes
FBC by default.

> Richard continued:
>
> You could say that voting Smith over Jones means that,
> if every candidate but Smith and Jones is removed from the election,
> and if this ballot is the only ballot that is counted, then Smith
will win.
>
>
> I'd considered that, but resisted it. Is it absolutely necessary? Is it
> necessary to
> avoid the possibility, with a method like you describe, of voting Smith
> over Jones &
> Jones over Smith? Or the possibility of FBC not being violated by a
> method in which
> it's impossible to vote anyone over anyone?
>
> But that definition that speaks of only counting that one ballot--that
> should be listed
> as one definition of FBC. It might be interesting to take a poll, on
which
> Condorcet Criterion definition, and which Monotonicity Criterion
> definition, and which
> FBC definition, and which definition of voting Smith over Jones, should
> be official on EM. I wouldn't argue with the results. I suggest
> separate categories for English & Mathematical definitions, so as not to
> pit them
> against eachother, since neither could replace the other.
>
> I can't say for sure that the 1-ballot definition isn't better than the
> ballot-configuration
> definition, of FBC. But it's more likely to make people balk when we
> speak of only
> counting one ballot.

My justification is that it creates a stronger version of FBC. I don't know
why people would balk at a definition of voting X over Y on a ballot that
examines only that ballot and no other ballots.

-- Richard

```