[EM] FBC ambiguity?

[Richard Moore]

MIKE OSSIPOFF wrote:

> 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.


Exactly. In this (mis)interpretation of the definition, the criterion
is never failed under any conditions.


> 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.


Since the potential misinterpretation could be applied under all 
circumstances,
we are not dealing with just an example.

Granted, it's not logical to stick with that interpretation once you realize
it's a wrong interpretation. But if a reader is confused because he/she
initially only sees the wrong interpretation, without seeing the correct
interpretation, that reader may conclude the criterion isn't useful or
important. He/she may then miss the correct interpretation, and miss the
point of FBC altogether.

That is why I am suggesting a simple way to remove the ambiguity: for better
communication of the idea of 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.


Well, I'm not *personally* interpreting that way, but that is a good
characterization of the misinterpretation I would like to prevent.


> 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?"


OK, "all of the outcomes" works in my definition if you substitute it
for "any of the outcomes", and I agree that it's better. The key change
I was after is adding the part about "in the same scenario". And in fact,
that is the additional information you provide when you say "an outcome
means an outcome of that election". But that additional information is
not in the original definition you gave.


> 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.


I don't think there's a problem with FBC itself -- just with a particular
wording of it. I know Forest worked hard to come up with a formal 
definition.
I just want the colloquial definition to come across clearly. Hence my
"devil's advocate" role.


> 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.


The problem I was referring to is one of uniformity: You can restrict the
definitions for ratings methods so that only one candidate's rating is
changed (because the ratings are independent), but then the definition
isn't good for ranking methods, because if you swap two candidates'
rankings, you've changed both rankings. We were looking for a uniform
definition that covers both cases (as well as lone-mark plurality
and approval, which are special cases of ranking and rating methods).
Separate definitions are possible, but if you have two separate definitions
how will you convince people that they are really the same property? This
was only one of the problems Forest and I wrestled with. I haven't returned
to this issue since two or three months ago.

 > I have a wordier
 > Monotonicity definition that is probably satisfactory, though I haven't
 > finished work
 > on Monotonicity definitions.

I hope when you get through with your monotonicity definition you will
post it to this list. Comparing notes would be helpful then.


> 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.


OK, at first I thought it was unlikely (but possible) that the definition
of "voting one candidate over another" would cause trouble. I at least
thought it would be less problematic than a definition of "a change that
favors a certain candidate", that is needed for monotonicity. But you've
forced me to look again, and I think I may have spotted a hole. In fact,
it's a hole very reminiscent of one of the problems Forest and I struggled
with for defining monotonicity.

Where one might get into trouble is in nonmonotone methods, where
it isn't clear when a voter is "voting one candidate over another".
In such a method, I might contrive a case where my ballot helps my
candidate (when all but the two candidates are eliminated) and another
case where my ballot hurts my candidate (when all but the two candidates
are eliminated). Now, this wouldn't occur in ordinary nonmonotone methods
(such as IRV), because with only two candidates the nonmonotonicity does
not show up. But imagine a severely nonmonotone method, such as one based
on modular arithmetic or some other non-linear function of the number of
votes. While such methods are not practical, they do exist (in the
mathematical sense), and need to be considered (at least if we want our
definition to be complete).

By your first definition ("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."), my ballot
under this unusual method votes A over B, but it also votes B over A. So
let's tack on your suggested addition: "...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." By the modified definition, my ballot neither
votes A over B nor B over A.

Come to think of it, I wonder if Forest's FBC definition addresses this
issue? It's getting late here so I don't feel up to checking the archives
just now. Tomorrow, perhaps.

One thing that might be said is that a method that plays so much havoc with
the FBC definition, has problems far worse than failing FBC.


> 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.


My distinction wasn't at all based on whether a natural language (such
as English) is used. In one case, the definition is given in a form that
is suitable for proving theorems and other analytical purposes, and in
the other case it is given in a form suitable for enlightening others
without relying extensively on pure mathematical concepts.

Many people would call the colloquial definition the "layman's" version,
but that expression suggests the existence of a "priesthood" far more
than "colloquial" does. Furthermore, I think many -- perhaps most --
mathematicians would be comfortable with discussing abstract concepts in
a "colloquial" style. So, I am not using "colloquial" pejoratively.

My dictionary gives two definitions of colloquial: (1) informal, and
(2) conversational. I intended the word in the former sense. Instead of
colloquial, we could call this the "informal" definition and the other
one the "formal" definition.

 -- Richard