Method Definition Considerations
Traditional math exposition, whether text book, lecture, journal article,
or monograph, tends to have a top down deductive logical structure that
belies all of the messy trial and error scratch work that accompanied the
creative process.
This was the style of Gauss ... unveil the finished work in all of its
polished, deductive logical glory, without any hint of the inductive
scaffolding or chisel marks that went into the finished work.
On the other hand, Euler led his reader through his discovery process ...
the chronological sequence of insights that led to his results without all
of the tidying up that would conceal the nitty gritty details.
Pure mathematicians tend to admire Gauss's style, while most engineers and
other applied mathematicians have a greater affinity for the style of Euler.
I have learned from over 40 years of teaching mathematics at all levels
from basic arithmetic to linear algebra, and differential equations, as
well as probability & statistics, that about 80 percent of students want to
learn the "how" before the "why" .... about half of the rest have zero
interest in the "why" ... and the others are completely content with a
cursory understanding of the "why" before getting down to the nitty gritty
...which is the healthiest attitude, in my humble opinion.
What does this have to do with election methods?
Consider this example:
Elect the candidate that outranks any other given candidate on more ballots
than not, if there is such a candidate. Otherwise, elect the candidate that
on the most ballots outranks at least one other candidate. Resolve any
further ambiguity in favor of the tied candidate with the greatest number
of first place votes.
To an election scientist this descripotion is a complete definition of what
has sometimes been called, "Implicit Approval Completed Condorcet."
It seems very concrete, down-to-earth, and nitty gritty. It specifies quite
precisely in plain English which candidate is to be elected on the basis of
a given set of voted ordinal ballots.
But we know from experience that the top FAQ will be, "How are we supposed
to find such a candidate?"
That question will never arise in connection with IRV.
Why not?
Because (1) the standard definition of IRV is entirely operational: step1,
step 2, etc. Furthermore (2) it is modeled on a well known run-off process.
My main point is that we need to keep this handicap in mind when broaching
any method for public consumption.
People expect an election method to consist of elimination steps described
in the language of vote transfers, in order to transparently preserve the
"one person, one vote" dictum.
Most people would not say that an election method "has" a counting
procedure ... rather they would consider the election method to "be" a
counting procedure.
To them, two different counting procedures are two different election
methods, never mind the possibility that they could be equivalent methods
derived from a common characterization.
We ignore these considerations at our own peril!
I'm talking about public consumption. Of course ... in technical discourse
(not intended for the lay public) it is a practical necessity to assume a
certain level of election science maturity.
The rule is to keep your intended readership in mind, and some kind of
"swim at your own risk" warning (explicit or implicit) for others.
Of course, some people want to read things for which they lack the
prerequisite knowledge. It might be good for them ... especially if it
contains words that excite the imagination!
-Forest
On 05.06.2022 03:14, Forest Simmons wrote:
Traditional math exposition, whether text book, lecture, journal
article, or monograph, tends to have a top down deductive logical
structure that belies all of the messy trial and error scratch work that
accompanied the creative process.
This was the style of Gauss ... unveil the finished work in all of its
polished, deductive logical glory, without any hint of the inductive
scaffolding or chisel marks that went into the finished work.
This is a little bit off topic, but one thing I wished someone would've
told me at university is just how difficult it is to come up with the
one trick that makes a proof work.
You tend to see (in mathematics/compsci papers at least) something like:
"We wish to reduce minimizing the spectral norm of this matrix function
of M to a convex second order cone program, which we can do by
augmenting the input matrix M like so
0 I
W = M M^T - I
and solving the associated optimization program, minimize f^Tx subject
to ||Wx||_2 < b as follows..."
There's then a brief proof of why just this choice of W works, but the
authors might as well have pulled the transformation out of a hat. The
problem is that this makes everything seem so easy: you just proceed
through the steps and the intended relation falls out at the end.
Then try to do it yourself in a novel setting and it's not so easy anymore.
Maybe students just want to know how, but I would say any aspiring
researcher would also want to know why... or at least in my case, know
that it's perfectly normal to be hitting one's head against a wall for a
long time before finding just what trick to use!
More generally, some problems are much easier to phrase than they are to
solve. (Particularly number theory, hence Barry Mazur's quote.) I think
providing an intuition of just what kind of problems are deceptive or
merely hard would be a good idea, although I'm not sure how to do it :-)
-km
On 05.06.2022 03:14, Forest Simmons wrote:
That question will never arise in connection with IRV.
Why not?
Because (1) the standard definition of IRV is entirely operational:
step1, step 2, etc. Furthermore (2) it is modeled on a well known
run-off process.
My main point is that we need to keep this handicap in mind when
broaching any method for public consumption.
People expect an election method to consist of elimination steps
described in the language of vote transfers, in order to transparently
preserve the "one person, one vote" dictum.
Most people would not say that an election method "has" a counting
procedure ... rather they would consider the election method to "be" a
counting procedure.
To them, two different counting procedures are two different election
methods, never mind the possibility that they could be equivalent
methods derived from a common characterization.
We ignore these considerations at our own peril!
Getting myself back on topic (ahem):
I agree. There's a related problem, that the operational procedure might
seem completely alien. Consider, for instance, my fpA-sum fpC, which can
be described operationally as:
"A candidate beats another pairwise if more voters prefer the first
candidate to the second, than the second to the first.
For each candidate, count the number of other candidates he beats
pairwise. If there exists one candidate who beats every other candidate
pairwise, elect him.
Otherwise, let each candidate's score be the number of first preferences
he has obtained, minus the sum of the number of first preferences
obtained by candidates who beat him pairwise. Elect the candidate with
the highest score."
(For simplicity, I've omitted descriptions of how you determine what
candidates are beaten pairwise by what other candidates. I also use sum
fpC instead of max fpC simply because it's easier to explain.)
Now suppose this is operational enough. The procedure itself looks
arbitrary if you're focused only on the counting. Why is it A's first
preferences minus the sum of candidates pairwise beating A? Why not some
other preference? Why not A's first preferences plus the sum of counts
of everybody A beats?
The reason it's like that is, of course, that this particular choice
makes fpA - sum fpC pass DMTCBR. But you can't easily explain that in an
operational manner.
The mathematically derived methods are procedurally ugly but
mathematically neat. IRV (and Plurality, Approval, Range, etc.) has it
the other way around.
So, are there any advanced Condorcet methods with understandable
procedural rules? I would perhaps say Ranked Pairs and Benham. The
former has the logic of "consider the landslides before you consider the
close races", and the latter could be understood as a patch on IRV to
not fail Burlington scenarios.
In the graded domain, perhaps MJ? It's the tiebreakers that get you, though.
(If you're exceptionally good at popularizing science, maybe Schulze.
The beatpath heuristic is more mathematical, the Schwartz set more
procedural. But I doubt it.)
-km
Reminds me of the professor who decided to mend his ways when a student
complained about not enough writing on the board for students taking notes.
At the end of a long explanation of quadratic reciprocity, remembering his
promise of more detailed notes, he says, "... which is as easy as one plus
one equals two," ... and then writes 1+1=2 on the black board.
For fpA-sumfpC ... I would have to say it came out of an attempt to
de-clone Copeland, where the fpA term was found empirically to restore
momotonicity to an educated guess that didn't quite work.... or something
like that. Not very inspiring to the students.
When we're lucky, once we get the right formula, we can see a shortcut to
it that would have saved us tons of time and agony if only we had seen it
earlier.
We cannot blame the students for not appreciating the shortcuts.
It reminds me of sci-fi stories about infrastructure that starts crumbling
centuries after all its engineers have died off.
Patience!
El dom., 5 de jun. de 2022 2:43 a. m., Kristofer Munsterhjelm <
km_elmet@t-online.de> escribió:
On 05.06.2022 03:14, Forest Simmons wrote:
Traditional math exposition, whether text book, lecture, journal
article, or monograph, tends to have a top down deductive logical
structure that belies all of the messy trial and error scratch work that
accompanied the creative process.
This was the style of Gauss ... unveil the finished work in all of its
polished, deductive logical glory, without any hint of the inductive
scaffolding or chisel marks that went into the finished work.
This is a little bit off topic, but one thing I wished someone would've
told me at university is just how difficult it is to come up with the
one trick that makes a proof work.
You tend to see (in mathematics/compsci papers at least) something like:
"We wish to reduce minimizing the spectral norm of this matrix function
of M to a convex second order cone program, which we can do by
augmenting the input matrix M like so
0 I
W = M M^T - I
and solving the associated optimization program, minimize f^Tx subject
to ||Wx||_2 < b as follows..."
There's then a brief proof of why just this choice of W works, but the
authors might as well have pulled the transformation out of a hat. The
problem is that this makes everything seem so easy: you just proceed
through the steps and the intended relation falls out at the end.
Then try to do it yourself in a novel setting and it's not so easy anymore.
Maybe students just want to know how, but I would say any aspiring
researcher would also want to know why... or at least in my case, know
that it's perfectly normal to be hitting one's head against a wall for a
long time before finding just what trick to use!
More generally, some problems are much easier to phrase than they are to
solve. (Particularly number theory, hence Barry Mazur's quote.) I think
providing an intuition of just what kind of problems are deceptive or
merely hard would be a good idea, although I'm not sure how to do it :-)
-km