[EM] FBC ambiguity & changes
nkklrp at hotmail.com
Sat Jan 5 16:55:58 PST 2002
>>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.
Quite so. That is an error. I meant to say "...is the unique winner
if & only if we count that ballot."
But maybe, for more definite meaning, it should say: "...is the unique
winner, meaning the unique winner by the count rule, with no tie,
and no tiebreaker applied..."
That would be to clarify that "unique winner" means unique winner
by the count itself, rather than the one winner who is declared
after he wins a tiebreaker. To avoid the messiness & complication
of including the random tiebreaker as part of the voting system.
That wording could probably be improved.
[stating a possible definition of voting A over B]
"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*."
But when the word "unique" is included, as it should be,
isn't it only necessary to count that vote once? To change a tie
between A & B into a result with A as the unique winner?
Somone could propose a method that says no one wins unless they
win by a margin of 2 (or 100), and that's another problem, one
that I should check out some more before I speak. The method could
say that if you don't win by a 100 vote margin, then it's considered
a tie, and the random tiebreaker is used.
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.
True, if, by John voting a certain way, there's a small possibility
that he could make X into the unique winner, and a greater possibility
that he could make Y into the unique winner, he isn't voting someone
over X, by the definition, but he's voting in a way more likely to
favor Y over X.
To avoid having to compare the probabilities of making X or Y the
unique winner, it probably _is_ better to speak of only counting
John's ballot, if that would avoid the problem. So that's my
official definition now. But there's still that other problem mentioned
above, which I don't yet know what to do about.
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