I knew I shouldn't shoot from the hip.  The first two reasons I gave are
true, but the third is badly mistated.  So here is the correct version:
 
3)  Given all linear models, the only one for which, in a round-robin
tournament, the number of wins by each team a complete and sufficient
set of statistics for the parameters (of the linear model) is the
Bradley-Terry model.  In short, if you believe that the only thing that
should matter in a round-robin tournament is how many wins each team
has,and you believe there is an underlying linear model, then the only
model available is Bradley-Terry.  I find this very compelling. The
reference here is 1963: Buhlmann and Huber, Pairwise Comparison and
Ranking in Tournaments. Annals of Mathematical Statistics Volume 34:
501-510. (Theorem 1)
 
4)  The parameter estimation method in KRACH is Maximum Liklihood, thus
it is maximally efficient. This means that of all consistent estimators,
it has the lowest variance. This *suggests* that KRACH is robust for
dealing with situations such as the MAAC (or as robust as one can
expect).
 
By the way, anyone who wants to see my article which addresses the
difficulties any rating scheme faces can look at:
 
http://www.gac.edu/~ghatfiel/sprt.ps  (Postscript) or
 
http://www.gac.edu/~ghatfiel/sprt.dvi  (DVI)
 
Gary Hatfield wrote:
>
> Greetings!
>
> This is an old argument.  Can a computer ranking be trusted and are its
> estimates valid.  I think that it's clear that RPI won't work if there
> is a subset of teams that have played almost no games against everyone
> else. I am not sure that KRACH is robust either, but I do think that
> KRACH should be considered ahead of RPI for several reasons.
>
> 1)  KRACH has an established mathematical/statistical history.  The
> ranking method is actually due to Zermelo(1929), who was trying rate
> chess players in an incomplete tournament.  It was later invented
> independently by Bradley and Terry(1952) and is still known in the
> biostatistics literature as the Bradly-Terry method.
>
> 2)  KRACH has a natural property that I think are desirable:   If the
> teams play a complete round robin schedule, then the KRACH ranking of
> the teams will be the same as ranking by wins.
>
> 3)  I also recall that the Bradley-Terry method satisifies some
> important optimality conditions (of the type statisticians like).
> Unfortunatley, it's been long enough since I looked at this stuff that I
> don't remember how to state it precisely.  I believe that if one assumes
> a linear model of pairwise comparisons that will have property 2), then
> the Bradley-Terry method is the maximally efficient consistent estimator
> of the probability that A will defeat B. NOTE: A linear model means that
> each team has a rating and that the probability that A will defeat B can
> be derived from the difference of their rating (For KRACH, you would
> take the LOG of the ratings before subtracting).  I will look up some of
> my references to see if I mistated this.
>
> In any event, I have long asserted that one cannot have the perfect
> computer ranking. See for example my article published in the UMAP
> Jounral (1998) (OK, so I plugged my own work, but I did make several
> references to College Hockey).
>
> Go GOPHERS!
>
> Gary Hatfield
> SKI-U-MAH!
>
> Citations:
>
> 1929: Zermelo,E. Die Berechnung der Turnier-Ergebnisse als ein
> Maximumproblem der
> Wahrscheinlichkeitsrechung.  Mathematische Zeitschrift Vol 29: 436-463.
>
> 1952:  Bradley, R.A. and Terry, M.E. Rank Analysis of Imcomplete Block
> Designs I: the method of paired comparisons.  Biometrika, Vol 39:
> 324-345.
>
> 1998:  Hatfield, G.A.  Sports Ranking Functions.  The UMAP Journal Vol
> 19 no. 2. :117-126.
>
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