John Haeussler posted a message where he stated that multiplying the
three factors that are used to calculated the RPI is equivalent to a
1/3, 1/3, 1/3 weighting.  This is not true.  I assume that John
reasoned that because one is multiplying three factors, they are all
treated evenly, and thus they each count a third.  The mistake is that
the relationship is not linear when one multiplies factors.
 
As an example, assume that the three factors (record, opponents'
record, and opponents' opponents' record) for two teams are .200, .900,
.500 and .500, .500, .500, respectively.  If one uses the 1/3, 1/3, 1/3
weighting the first team is at .5333 and the second at .5000, i.e., the
first team has a better score.  However, if one multiplies the three
factors (the normalization is unimportant, it just gives neater
numbers) one gets .0900 and the second .1250, i.e. the second has a
better score.
 
I agree with John that 1/3, 1/3, 1/3 seems better than the .25, .5, .25
weighting that is used in the RPI calculation.  Last spring, I posted a
message where I derived a formulation that gave .5, .25, .25 (actually
it gave .5, .25, .125, .0625, .03125 ... with the other terms for
opponents' opponents' opponents' record, etc.).  Over the summer I
derived a revised version of this which I never posted.  For those who
remember my original posting, please note that in my calculation of
opponents' records, I did include the games played with the team in
question while in the explanation of RPI and RPICH by Erik Biever
recently, he stated that these games are omitted in this calculation.
(This probably will be a minor difference.)
 
Ralph Baer
RPI '68, '70, '74
Maybe we want to become Rensselaer so no one thinks we are responsible
for the Ratings Percentage Index.  :-)