[EM] FBC ambiguity?

[MIKE OSSIPOFF]

I haven't often had time to check the list, but I've just noticed Richard's 
reply
about FBC, and I'm replying to it now:

Richard quoted my definition of FBC:

>
>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.

Richard then said:

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.

I reply:

Richard, in order to pass a criterion, it isn't enough that you can find an 
example in
which the method doesn't fail. In order to pass, there must not be any 
example in which
the method fails to meet the criterion's requirement, when the criterion's 
premise
conditions are met.

So, if, for some particular method, you've found an example in which a 
certain voter
could elect any of the 5 candidates by voting someone over his favorite, and 
could
also elect any of the 5 candidates without voting someone over his favorite, 
that
isn't enough to allow you to say that the method meets FBC.

Richard continues:


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.

I reply:

Ok, so you're interpreting my wording to say that by voting another 
candidate over
his favorite, a voter should never gain an outcome that he prefers to every 
outcome that
he could get in that or some other election without doing so.

Actually, an outcome means an outcome of that election. Electing candidate A 
in
a different election at some later time isn't the same outcome as electing 
candidate A
in today's election. In that future election, the candidates who might win 
if A doesn't win
might be different, the social and other conditions in the country might be 
different,
and A's own policies might be different. It isn't reasonable to treat 
"outcome" as meaning
something other than "outcome of that election".

And although "any of the outcomes..." sounds good in everyday conversation,
"all of the outcomes..." has a more definite meaning. "I can beat _anyone_ 
at chess.
Can you beat anyone at chess?"

Don't think I don't appreciate your effort to find a problem with FBC. 
Without efforts like
that I wouldn't be able to say that the criterion has stood up to 
discussion.




>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.

Richard continues:

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.

I reply:

I too have noticed that wording Monotonicity precisely is trickier than one 
might
at first expect. But I think it's reasonable to assume that when we refer to 
changing
X's rating, that doesn't include changing someone else's rating too. I have 
a wordier
Monotonicity definition that is probably satisfactory, though I haven't 
finished work
on Monotonicity definitions.

Richard continues:

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.

I reply:

I don't know quite how to reply to that...He's saying that maybe it has a 
problem,
but he hasn't read it carefully enough to know. I suggest that it doesn't 
have a problem
until someone finds a problem with it.


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

Richard continues:

Yes, I agree there is value in both mathematically rigourous definitions
and in colloquial definitions.

I reply:

Even definitions that use mathematical language and substitute symbols for 
some
words, while defining the symbols in terms of those words, still usually 
need some
English, and so those definitions don't necessarily avoid the issues of what 
English
words mean. What you call "colloquial" is a definition that is entirely in 
English. I'd call
it colloquial if it contains words which the dictionary lists as colloquial.
We should resist the temptation to elevate mathematics to a priesthood, 
without whose
language no definition is valid.

One thing different about Forest's definition was that it covers more kinds
of examples, and, as I said, that's a good thing. But let me add that, when 
I said that
my definition isn't universally applicable, I _didn't_ mean that it isn't 
applicable to all
methods. I meant that it can't be tested in certain kinds of examples, such 
as examples
in which a voter has more than one equally favorite candidate.

Richard continues:

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.

I reply:

If I didn't define it, you or someone would point out that I hadn't defined 
it. No, I don't
tell the definition every time I use a criterion that speaks of voting one 
candidate over
another. Yes, it's necessary to have a definition of that in case someone 
asks.

Mike Ossipoff



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