FBC Ambiguity?

[Richard Moore]

MIKE OSSIPOFF wrote:

> As a practical mattter, I haven't found a need to specify the same 
> scenario, but that
> doesn't mean that someone else shouldn't do so. But it seems to me that 
> it's universally
> understood in discussions of voting system criteria & examples that 
> "outcome" only has meaning  with reference to a particular election, and 
> that we aren't talking about outcomes
> that could be outcomes of other elections.


I'm very reluctant to assume that anything is "universally understood",
when the universe is such a big place. If some things are understood by
mutual consensus then we can use them as a basis to build upon; if that
consensus hasn't been established then it's best to state the assumptions
explicitly. If you want to present a definition to the masses then it
might be best to assume no consensus exists and go from there.

> Richard continues:
> 
> [regarding my definition of voting Smith over Jones]
> 
> 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).
> 
> I reply:
> 
> Maybe, but such a method could never be considered for actual use. I did 
> mention that
> the addition that you suggest  below could be helpful if someone comes 
> up with a voting
> system that requires it. For defining voting Smith over Jones for 
> mathematicians or
> logicians, I'd include that added wording that you describe.


Another "universal assumption"? Suppose some legislature decided there was
a risk of election fraud, and decided to put some non-linearity in the
counting algorithm (perhaps in the form of a higher-order polynomial
with sometimes negative slope) to discourage ballot-box stuffing. Now, you
and I agree that such a thing would be stupid, but have you never known a
legislature to do anything stupid? The electoral college doesn't make
any sense either, yet there are people who will defend it with all the
illogic they can muster.

I'll warrant that it is highly unlikely. But if it's in the realm of
possibility it should be taken into account.


> Maybe "colloquial" or "informal" should mean suitable for describing a 
> voting system
> or criterion to a citizen on a streetcorner, but not meeting the 
> necessary standards
> for a proposed electoral law. Sure, if I were writing an electoral law 
> about FBC, I
> probably _would_ specify that all the outcomes referred to are outcomes 
> of the
> same election. If I were writing an electoral law that referred to 
> voting Smith over Jones,
> I doubt that I'd bother writing it so that it doesn't act unexpectedly 
> when a voting system
> is nonmonotonic with 2 candidates, since it would never be tested with 
> such a voting
> system.


Well, "the necessary standards for a proposed electoral law" might
constitute a different sort of formal language, i.e., "legalese". It
isn't exactly the distinction I was going for. I was looking to distinguish
between what's "formal" in the analytical sense and what's "informal"
in the sense that it's an accurate mirror or tranlation of the "formal"
language, but expressed in a more philosophical or didactic manner.


> If the convention that all the outcomes referred to in a criterion are 
> outcomes of the
> same election is relied upon in some definitions in journal articles, is 
> it still
> informal or colloquial?


It would be hard for me to say, without seeing a specific example of the
usage. To me, formal/informal has more to do with whether we are using
pure mathematical concepts such as sets (which may be tied to 
nonmathematical
concepts, as in "sets of ballots"), or whether we simply stop at the level
of nonmathematical concepts such as elections, candidates, voters, and
ballots. There is a bridge between the two domains of discussion; the
formal side of that bridge could be very abstract but the abstractions
have definite analogies to the informal side. "Outcome" is thus analogous
to the result of a function whose input is a set (of ballots). I would think
that using the word "outcome" in a formal definition would require that it
be tied somewhere to the analogous function, and the inputs to the function
(including which are varied and which are held fixed) should be specified.
Along the lines of: "S is a set of ballots, M is a method (function), and
the outcome O is given by O = M(S)". There are ways we can define how
S might vary; S might be the union of sets R and T, and S' might be the
union of sets R and T' (so that the only ballots changed are those in T,
being changed to T'), and the new outcome O' is given by O' = M(S'). This
is similar to the format Forest and I were using in our discussion of
monotonicity.

  -- Richard