This is more of a mathematical question than a FileMaker one, but I
thought maybe someone in this group might have come across it in their
programming:

I have an FM11 database designed to distribute ranked items to four
participants using random assignment in rounds of four.  I want to be
able to grade how well each player did in getting to pick first,
second, third and fourth when all the rounds are completed.

Specifically, there are 100 items, each of which constitutes a
record.  There are four players interested in the items, so there are
four number fields in each record -- one for each of player.  Each
player uses his number field to rank his interest in an item from 1 to
100.  Taken as a whole, players will rank the items differently and
there may be some items for which no rank is given if the player isn't
interested at all.  Each round, the database randomly decides who gets
to pick first, second, third and fourth, then assigns the top ranked
item in that player's list to them, removing the item from contention.

I'd like to build some tolerances into the system so that if the
random distribution ends up favoring one player too much -- say with a
lot of first-choice picks -- or another player too little -- say with
a lot of last-choice picks, the players can decide ahead of time that
the distribution should be done over.  However, in order to grade how
well each player did in an overall distribution, I have to be able to
weight how much each position is worth relative to the others in the
round.  For example, here are two sample distributions:

Sample A
Pick Number:	1	2	3	4
Sam         	4	8	5	8
Raphael     	8	5	6	6
Peter        	5	5	9	6
Herb         	8	7	5	5

Sample B
Pick Number:	1	2	3	4
Sam         	4	7	7	7
Raphael     	9	5	9	2
Peter        	7	6	5	7
Herb         	5	7	4	9

In Sample A, Sam only gets 4 first-choice picks, which isn't that
great compared to Raphael's 8, but he gets 8 second-choice picks,
which is better than Raphael's 5.  Taking into account their third-
and fourth-place picks, what would the overall grade be for how well
they did in the lineup?  Is Sam's result in Sample B -- 4 first, 7
second, 7 third and 7 fourth -- better than Herb's, since Herb got one
more first-place pick but 3 fewer third-place picks?

How would I score this?  There must be a standard statistical model
that's been developed over the years so that I wouldn't have to make
something up that wasn't accurate.  And further complicating the issue
is that not everyone participates in every round, since not everyone
is interested in all 100 items.  So it wouldn't be a matter of just
saying how many first-place picks to each player get relative to each
other:  a player that participates in every round is much more likely
to get more first-place picks than a player that's only interested in
half the items.

Makes sense?  What do you think.

Thanks for your help,
-J.

On 2011-04-09 14:14:57 -0700, jahn <jahnbigbooty@yahoo.com> said:

> Each round, the database randomly decides who gets
> to pick first, second, third and fourth, then assigns the top ranked
> item in that player's list to them, removing the item from contention.
> 
> I'd like to build some tolerances into the system so that if the
> random distribution ends up favoring one player too much --

I have no idea how to score your game. Sounds like you want some kind 
of weighted total.  I'm only going to respond to the section quoted 
above.

This is where you're going to get into trouble, mathematically speaking.

If it's random, don't mess with it.

What people have a hard time understanding is that truly random results 
can include sequences that LOOK like they're non-random, or biased.  A 
truly random result is not going to be equitable, or reasonable or "not 
really favoring one player over another."  Sometimes random numbers 
come up all ones, or twos. Randomly. Low probability sequences do 
happen.

"Tolerances," to me, is just one more way of saying "the house always 
wins." Not that you're intending to win anything from these players, 
but you know what I'm talking about. Vegas casinos build in 
"tolerances" to make sure a certain percentage of bets fall their way, 
and that's enough to make them profitable.

If in the effort to make it seem that equitable choices are being made, 
you bias the results, then it's not random and you should acknowledge 
that to your players right up front, along with an explanation of the 
rules or algorithms you're using.

First, though, you have to make up some rules as to what's allowed.  
What's going to make it *seem* fair?  If one person has too many first 
picks, or too many last picks, they get a bump in the opposite 
direction?

The game would tend to even out all results toward the middle because 
of the rules. Is that what you want?

It sounds really complicated and likely to get more so as you develop 
your scoring system. Some people love that kind of complexity. Some, 
not so much. ;)
-- 
Lynn Allen
--
www.semiotics.com
Member FBA
FM 10 Certified Developer

Hi, Lynn -

Thanks as always for weighing for in.  I agree with your assessment of
the problems with randomness.  I think RadioLab did a show on this
recently -- great podcast.  The difference between casino tolerances
and the tolerances I'm looking for is that the casino is aiming to
bias the results in their favor, whereas I'm looking for a solution in
which results that are too biased (as a result of the problems of
randomness) are excluded.  If there's a way to weight numbers so that
an overall score can be assigned to a given distribution, then there's
a way to create a system in which only equal scores are accepted (or
relatively equal scores within a given tolerance).  So when you say
"first you have to make up some rules as to what's allowed" -- that
actually _is_ the question I'm asking; the rest is just programming.
What I'm looking for is to know whether there are established criteria
for grading these kinds of distributions.  But perhaps I need to find
a statisticians user group for that!

Thanks for your help.

Best,
-J.