[EM] A question in classroom creation

[James Green-Armytage]

Proposal 2: Same as proposal 1, but with the following teacher-assignment
method built in:
	In addition to rating other children from -3 to 3, children also rate
each teacher from -3 to 3. (The teachers do not rate the children, and
hence utility scores for teachers are not calculated.)
	The utility for each child in each example is calculated as a combination
of their "teacher utility" (tu) and their "classmate utility" (cu). It is
possible to give different weights to teacher and classmate utilities.
E.g., a student's overall utility might be calculated via .5tu*.5cu, or
.4tu*.6cu. You could call these fractions the "teacher fraction" (tf) and
the "classmate fraction" cf. tf and cf should add up to 1 for all
children. 
	It might be possible to allow children to choose their cf and tf,
provided that they sum to 1. Thus, if children were not very concerned
about what teacher they got, but they were very concerned about being in
class with their friends, they might choose e.g. cf=.9, tf=.1.

Commentary:
	I think that Paul is right about the number of possible arrangements
given 100 children and 4 classes, which means that I am most likely wrong.
My second guess at the magic number is (100!)/((25!)^4)... is that
correct, Paul? I obviously need to brush up on my probability and
statistics. Any idea what is the order of magnitude for (100!)/((25!)^4)?
	I wonder whether it makes sense to maximize the children's' ratings in
scale, e.g. to automatically turn a rating set of (-1, 0, 2, 3) into (-3,
-1.5, 1.5, 3).

- James Green-Armytage

I wrote:
>
>Proposal 1: Use a range ballot, e.g. integers from -3 to 3 inclusive. 
>	Consider every possible arrangement of children into the 4 classes. 
>	Give each possible arrangement a score as follows: Measure each child's
>utility as the sum of his rating of all other children in the class. Sum
>the utility scores for each child to find the total utility of the
>arrangement.
>	Choose the arrangement with the highest total utility. (If multiple
>arrangements are tied, choose randomly between the arrangements with the
>highest score.)
>
>Commentary:
>	This method, while perhaps optimal from a results point of view, seems
>like it would take a lot of computing power. Given 100 children and 4
>classrooms, how many possible arrangements are there? Is it somewhere
>close to (4!)^25? So, 24^25? More than that? Yikes. I'm not much of a
>computer expert, so someone else will have to tell me whether that's a
>prohibitive computational cost.
>	Is there a computationally cheaper method with a similar effect?
>
>
Michael Rouse wrote (previous to text immediately above):	
>
>>Here's a rather different (and more complicated) voting problem than
>>usual:
>>
>>In the interest of classroom harmony, a school decides to let the 
>>children vote for which classmates they want in their home room. 
>>Assuming each class is the same size, what kind of ballot and what 
>>method of grouping students should be used? Also, should top-ranked 
>>(most liked) or bottom-ranked (most disliked) preference take precedence?
>>
>>Some possibilities and problems that come to mind:
>>Ranked ballots  -- difficult to make it a "secret ballot," but it gives 
>>a fine-grained preference listing.
>>Approval/Anti-Approval -- rating classmates as approved, disapproved, 
>>and unknown. Also difficult to use with secret ballot. Probably the 
>>easiest to use.
>>Classroom grouping -- let students make their own classroom groupings 
>>(kind of like the districting problem), possibility of secret ballot but 
>>a *lot* of work.
>>
>>If an example is needed -- and just to give some numbers -- let's say 
>>the school has 4 teachers and 100 students in the same grade, which 
>>would give 25 students per home room. For extra credit (heh), if they 
>>can also vote for which teacher they want, what would be the fairest way 
>>of resolving ties if more than one class prefers the same teacher?
>>
>>This would also have an interesting application in voting district 
>>creation -- if voters can choose which precincts go into a voting 
>>district, what would be the fairest way of doing so?
>