> haven't run the actual numbers, but looking at the mathematics behind
> KRACH shows that beating a team much weaker than you will raise your
> rating a little, losing to them will lower it a lot, beating a team
> much stronger than you will raise your rating a lot and losing to them
> will lower it a little.  As Gary Hatfield puts it, there is always a
> "non-positive response to losing" and a "non-negative response to
> winning".  And "strong" and "weak" opponents are defined based on
> something more robust than simple winning percentage.
 
I had a chance to run those numbers and confirm that this is the case.
Here's what you get if you use the KRACH (without fictitious games and
without UNO, so it doesn't quite agree with what Ken Butler posted
after the end of the conference tournaments) to calculate the
"Expected Round Robin Winning Percentages (RRWPs)"--the winning
percentages you'd predict if each team played wach other team
once--for Rensselaer and Quinnipiac, plus the head-to-head winning
probability (HHWP) for RPI against Q using the actual season's results
and those with an extra RPI-Q game added, depending on the outcome:
 
            Rens      Quin
          RRWP  Rk  RRWP  Rk   HHWP
no game   .643 #14  .146 #45   .972
RPI wins  .644 #14  .145 #45   .973
tie       .629 #15  .177 #45   .950
Q wins    .614 #16  .217 #43   .915
 
BTW, in the scenario where RPI loses to Quinnipiac, not only does Q
jump over Union and Air Force, but Air Force passes Union, because
Union's schedule, which included three games against RPI, now looks
weaker.  Various of RPI's and Q's opponents also move around a bit due
to a similar re-evaluation of their schedules.
 
So in the KRACH, a major conference team would have had very little to
gain and a lot to lose by scheduling a weak opponent, but that's
really to be expected, since the ratings tell us that a win is very
likely.  And the situation would be even worse in the ratings
percentage index if not for its blind spot about evaluating the
strength of opponents who play weak schedules: if Rensselaer had
played *and* *won* a game against Fairfield, their RPI would have
dropped from .533 to .530 (their Bradley-Terry RRWP would not have
change at all, since Fairfield, as a winless team, has a HHWP with
everyone else of exactly .000).
                                          John Whelan, Cornell '91
                                                  [log in to unmask]
                                     http://www.amurgsval.org/joe/
 
Play along at home at http://www.slack.net/~whelan/cgi-bin/tbrw.cgi?hypo
 
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