[EM] Voting, Grading, etc.

[Forest Simmons]

Greetings to all election methods aficionados.

I'm a new guy on the list, so I hope it's OK to jump right in and join the
fun.

Here's a line of thought concerning voting methods in general, as well as
Approval Voting in particular, that some of you might find interesting:

The first time I heard of Approval Voting was about twenty years ago when
I was sent a ballot from the Mathematical Association of America for the
election of various officers. While trying to guess the rationale behind
the method, it occurred to me that voting in general was somewhat
analogous to grading papers (though with a different purpose), and that
Approval Voting in particular was like eschewing partial credit. 

As a young math instructor I was aware that in the long run it didn't
matter if partial credit were allotted, since the "roundoff errors" would
tend to cancel out in the course of the academic term (for the student)
and in the course of the election (for the candidates) given sufficiently
many problems or voters in the respective cases.

In the analogy I have in mind the students are like the candidates and
each problem on a departmental exam is like a voter whose vote is
expedited by its grader. (Traditionally each problem has its own grader on
departmental exams. The problem/grader combination is like the voter with
whatever criteria he uses to judge the candidates.)

In this analogy the traditional "one man one vote" method would force each
problem to "choose" (as facilitated by its grader) exactly one student to
get the credit for the problem. 

If the purpose of the exam was to choose one student to represent the
department in a state contest, this
"one-problem-one-student-gets-it-right" rule would be slightly less
ridiculous, but still inferior to the approval voting method, where full
credit for the problem could be awarded to as many students as
appropriate.

Thoughts along these lines started to convince me that Approval Voting was
a good idea.

The analogy gets more interesting if the exam is both short and important
so that it is deemed advisable to give partial credit in an effort to
squeeze out every bit of information possible from this one exam. 

For awhile this led me to believe that a rating system (on a scale of zero
to 100%) would be an improvement in the case of small elections. 

How could the partial credit be assigned as impartially as possible? In
the paper grading setting the first thing that came to mind was (for each
problem) sorting the responses from worst to best answer, and assigning
partial credit in proportion to the papers' positions in the list,
analogous to the Borda Count.

I actually tried that as a young instructor, but gave it up because it was
too tedious, and more importantly , it was almost always the case that
even spacing of credit was obviously wrong; I could do better by surveying
a random sample of the answers, and then quickly assign partial credit on
the basis of the seriousness of the mistakes relative to the average. 

Taking the analogy a step further, suppose that some or all of the graders
have hidden agendas so that they are not impartial expediters of what the
problem might reveal about the students' abilities. Some graders might
sense that certain favorite students need a little extra boost.  Since
there is a certain honor associated with having your student represent the
department, an instructor might maximize his chances of having this honor
by giving all of his own students full credit and none for the other
students.

In the grading setting, this difficulty can be taken care of by not
revealing the identity of the students until their papers are fully
graded.  In the voting setting, the voters have to know for whom they are
voting. 

On the other hand, an informed citizen has a responsibility to exert as
much influence as possible in the direction that she believes would serve
the best interest of her country.  She has an obligation to optimize some
strategy or another. 

If the range of choices is an hypercube whose dimension is given by the
number of candidates, then most simple strategies (linear, for example)
will be optimized at some corner of the hypercube ... no partial credit
for the candidates. 

I tried taking a survey in our department about various options for
incorporating graphing calculators into our curriculum.  I allowed them to
rate the various options on a scale from -1 to 1.  Almost all respondents
stuck to the two extremes in all of their ratings. 

As you can see, this journey has brought me back to approval voting again,
at least for typical single winner elections. 

If anybody is interested, I have some thoughts about multiwinner elections
also.  In the analogy, this would be like using the departmental exam to
pick a team to represent the department in the state competition. 

Just taking the top scorers would not be a good idea if they all missed
the same problems on the exam, and there were other students that had
expertise on those kinds of problems.  In other words, the usual method of
adapting approval voting to multiwinner elections is not necessarily a
good idea. 

I have a way of adapting approval voting to proportional representation
that I would like to share in my next letter.  It uses the "front end" of
approval voting, but the voted ballots are used in a different way to
determine the winners. 

I know that Steve Brams has done something similar with the aim of better
representation, but not necessarily proportional, by using linear
programming to minimize something he calls "misrepresentation".

If I don't get shot down too harshly for this letter, I'll write more
about "Proportional Representation thru Approval Voted Ballots."
                                                         
Peace,

In remembrance of MLK, Jr.

Forest