On Wednesday, I wrote the following:
>Seeing how important the RPI calculation has become to deciding who
>should make the NC$$ tournament (and in filling my mailbox the last few
>days), does anyone know if there is any scientific basis for the way
>that the multipliers for three factors (record, opponents' record, and
>opponents' opponents' record) were chosen? I realize that these
>multipliers were changed this year, but I am curious as to why they
>were chosen. Although I can easily see that some combination of these
>factors are an indication of the relative strength of teams, I don't
>recall having ever seen any reason for why the multipliers were chosen
>as they are. I also wonder why further terms as opponents' opponents'
>opponents' record were not also somehow included although the terms
>will have a tendency to approach 1/2.
A note from Wayne where he stated that he had not seen a reason for
these factors made me think if I could come up with some logical
formulation which would give the RPI(CH) formulation of
.25R0+.5R1+.25R2 or the old formulation of .2R0+.4R1+.4R2 where R0
stands for the team's record R1 the opponents record and R2 the
opponents opponents record. R3, R4, etc which will appear below are
defined similarly.
What I need is to somehow give more credit to a team for beating a good
team than for beating a bad team and similarly take away less credit
when losing to a good team than losing to a bad team. Furthermore, in
the spirit of RPI(CH), the scores of the games are not to be taken into
account. Thus, there is no way of rewarding a team for having a close
lose against a very good team or making them lose a lot of credit for
being blown out by a very poor team.
The first thing that I came up with was simply to give a team the
percentage of the opponent in a win, and subtract "one minus the
percentage of the opponent" for a game lost. These factors then are
added up for each game and then divided by the number of games played.
Because this gives something between -1 and +1 instead of the usual 0
and 1, we can normalize by adding 1 and then dividing by 2. So, say a
team plays 3 games, beating a team with a .400 percentage and another
with a .750 pecentage, and then losing to a team with a .500
percentage, this would result in
( (.400 + .750 - (1-.500)) / 3 + 1 ) / 2 = .608.
This can be simplified rather easily, and turns out to be .5 * (R0 +
R1). [I am omitting the Math because I find it rather difficult to
write summation signs in ASCII.] Note that for the team I described in
the paragraph above, the record is .667 (two wins in three games) and
the opponents' record is the average of .400, .750, and .500 which is
.550. Finally, .5 * (.667 +.550) = .608.
If you have followed this so far, the next thing to note is that when I
use an opponent's record as a measure of a quality of a win, I have
ignored the quality of the teams that the opponent itself has faced.
What can be done is to substitute for the opponents' record what would
be obtained by the formula itself. That is, in the formula .5 * (R0 +
R1) one can substitute for R1 the term .5 * (R1 + R2) giving .5R0 +
.25R1 + .25R2. Actually, one should do this on an individual game
basis and then go through the entire derivation, but the result is the
same.
One can continue like this because in the formula .5R0 + .25R1 + .25R2
I have ignored the quality of the opponents' opponents' opponents.
This next results in .5R0 + .25 R1 + .125R2 + .125R3. One more time,
one obtains .5R0 + .25R1 + .125R2 + .0625R3 + .0625R4. You can see
where this is going. At some point the terms Rn become too close to .5
to make any difference (look at any listing of RPICH, except from near
the beginning of the season, to see that R2 is already usually very
near .500).
Anyway, I obtain the formula .5R0 + .25R1 + .25R2 if I stop at the same
point that RPICH stops. This formula is quite different from either
that used currently (.25R0+.5R1+.25R2) or the old formula
(.2R0+.4R1+.4R1). It gives significantly more weight to the record of
the team. As I didn't have the data at home to do the calculation, I
do not know which teams would benefit by this. However, it has seemed
to me, ever since I first saw it, that the weight on R0 in RPICH is too
small -- whether you win or lose is more important than how the team
you play has performed in other games, which does not say that such
factors should not be included also.
Another formula that goes out to the term R2 as RPICH does can be
obtained by assuming that all terms starting with R3 are equal to .500.
This gives .5R0 + .25R1 + .125R2 + .0625 or if you do not want to just
add a constant, you can leave it off and renormalize to get .571R0 +
.286R1 + .143R2. I prefer the .5, .25, .25 formula which I suspect
would give similar results to this.
Clearly, no matter what method is used, team #13 will grumble.
Comments?
Ralph Baer RPI '68, '70, '74 Beat UNH!
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