[EM] FBC ambiguity & language for EM
MIKE OSSIPOFF
nkklrp at hotmail.com
Thu Jan 3 01:22:26 PST 2002
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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.
Richard wrote:
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.
I reply:
That's reasonable. My official definition of FBC is changed so as to specify
that
the outcomes that it refers to are outcomes of the same election. For
brevity
I might not include that specification when actually using the criterion in
public
debates on voting system proposals.
I'd said:
>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.
Richard replied:
Another "universal assumption"?
I Reply:
I don't know. I haven't asked anyone. But surely it's a safe assumption.
But let me say now that, again, I agree that my official definition of
voting Smith
over Jones should be changed to the more symmetrically-written version that
I'd
mentioned and that you suggest, so that even with the most bizarre
unproposable
methods, it will never say that you've voted Smith over Jones and Jones over
Smith.
But still, are you sure that it's inappropriate to say that, with such a
voting system
we can vote Smith over Jones and Jones over Smith? Doesn't it make sense to
say
that, when you have a voting system in which, when we've deleted everyone
from
the ballots except for Smith & Jones, your ballot could be what makes Smith
the unique
winner, and it could also be what makes Jones the uniquie winner? For a
voting system
like that, there seems, to me, to be nothing wrong with saying that you can
vote Smith
over Jones while voting Jones over Smith.
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.
Again, though that's my official version, I'll keep on using my briefer
version in
public debates about voting system proposals.
Richard continues:
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 reply:
Sure, but we both know that the public would never stand for that particular
change.
Old stupidities are hard to change, but new ones aren't well-accepted. IRV's
adoption failures are an example of that.
But, as I said, I change my official definitions of FBC & of voting Smith
over Jones
in the way that you suggested.
I'd said:
>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...
Richard replied:
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.
I reply:
Ok, then "informal" differs from "formal" not in meaning, but only in
language of
expression. So when you say that something is informal, you're just saying
that it
isn't in mathematical language.
But I don't know if you're right to say that only mathematical language is
"formal", because
, as many use the word "formal", many people who aren't mathematicians, but
who
are businessmen, government officials, butlers, etc. speak formally, without
using
mathematical language.
So it seems to me that to say that formal means mathematical would be an
incorrect appropriation of the meaning of "formal".
Why not instead speak of mathematical language as opposed to English (or
whatever
natural language is being spoken by participants in the discussion), since
that's
what you mean by "formal"?
I think it's good to have all-English definitions, since they're the ones
that are
useful when talking to most people. And one can't make much progress with
electoral reform without taliking to people.
Aside from that, I'm the 1st to admit that mathematical language can
sometimes be
more precise. But even definitions that use mathematical language as much as
possible
often have to rely on English in some parts, including the definition of
their
mathemcatical variable names.
That's why I said that using mathematical language can't always get rid of
all of
English's ambiguity, since English is still needed.
Richard continued:
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
I reply:
Of course that's true if "formal" means "mathematical". But you could speak
only
of elections, candidates, voters and ballots, in English that would be
suitable for
a formal meeting, and that would be formal even though it isn't
mathematical--according
to the most widely-used definition of "formal".
Richard continued:
, 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.
I reply:
Fine. That's one way of saying it, though certainly not the only way.
There's nothing
wrong with using different languages, as long as we don't mind translating
to our own
language.
I guess that, in the mathematical language above, it would also be specified
that
M is a function that relates each possible S' to one element of a certain
named set of
alternatives such as candidates or proposals, though I don't know how you'd
word that.
But doesn't a function have to have just one element of its range related to
each
element of its domain? And isn't it possible for some particular S' to
choose more
than one winner? And so wouldn't that be called a relation instead of a
function?
Or I suppose you could say that the function M relates each possible S' to
one
combination of the alternatives, so that the range of M is all the possible
combinations
that can be taken from the set of alternatives. In that way M is still a
function. Maybe
that's how voting system mathematicians mean it when they call a voting
system
a function. It's your language, and so I can be forgiven for not knowing the
answer to
that.
Mike Ossipoff
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