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At right are the
rankings of all NCAA Division 1-A college football teams thru the games
of January 8, 2009 (i.e., all bowl games) using MinV, a model developed by Jay Coleman of the University of North Florida. MinV generates a ranking that minimizes the number of
game score violations -- that is, the number of times a game's winner is
ranked behind the team it defeated. In other words, MinV guarantees the ranking with the best retrodictive accuracy.
The ranking at right
results in only 69 violations out of 717 Division 1-A
games, or a minimum violation percentage of 9.62%. Stated
otherwise, the ranking at right matches
the results of 90.38% of the games played this season, which is the
highest value possible. In
addition to minimizing the total number of violations, the ranking
this week minimizes the total weighted violations, where each
violation is weighted by the victory margin. (In other words, the ranking
shown violates the games in which the scores were as close as possible,
as opposed to violating games in which the victory margins were larger.)
The total weight (i.e., victory margins) of the violated games this week
is 573 points. Finally, the ranking shown at right at least
approximately matches a secondary target ranking – in this case the
current Sagarin ranking – as closely as
possible, while exceeding neither the minimum number of violations nor the
minimum weighted violations. Due to the size of the problem (the
number of games played thus far), MinV was not
able to guarantee that the ranking shown is the one that optimally
matches the secondary target ranking. However, the ranking shown is
likely a reasonably close approximation.
Note on the Texas-Oklahoma-Texas Tech “cyclic
triad”: These
three teams could obviously not be ranked in any order without incurring
a game violation. However, when enforcing the minimum weighted violations secondary objective,
the Texas Tech win over Texas
is the game result that should be violated. That game’s victory
margin was 6 points, whereas the Texas
win over Oklahoma was 10 points and the Oklahoma win over
Texas Tech was 44 points. With that established, Texas
should be ranked ahead of Oklahoma (and Oklahoma over
Texas Tech) in order to avoid incurring a second game score violation in
that grouping.
A minimum violations
ranking has never before been presented for college football (due in part
to the extreme computational difficulty involved for a problem with 120
teams). However, there are literally trillions of different
rankings at any given point in time that would yield the same minimum
number of violations; the ranking shown is only one of those.
"Minimizing Game
Score Violations in College Football Rankings," an article
describing MinV and its application to the 1994
through 2004 college football seasons, appears in the November-December
2005 issue of Interfaces,
a journal of the Institute for Operations Research and the Management
Sciences (INFORMS).
The final MinV ranking for the 2007 college football
season can be found here.
The final MinV ranking for the 2006 college football
season can be found here.
The final MinV ranking for the 2005 college football
season can be found here.
The final MinV ranking for the 2004 college football
season can be found here.
The final MinV pre-NCAA Tournament ranking for college
basketball in 2005 can be found here.
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