The mathematics of ranking sports teams
Who’s #1?
Jonathon PetersonPurdue University
The Ranking Problem
Why is ranking of sports teams important?• College football – BCS• College basketball – NCAA tournament• Win $1 billion!!!
– http://www.quickenloansbracket.com/
What is so hard about ranking teams?• Strength of schedule matters.• Non-transitive property
– http://www.myteamisbetterthanyourteam.com
Ivy League Football - 2009
What is the best team?Is Dartmouth better than Yale?
Ranking Methods
Statistical Methods• Gather as much data as possible• Cook up a good predicting function• Examples– Jeff Sagarin– RPI
• Problems– ad-hoc techniques– Dependent on parameters
Ranking Methods
Mathematical methods• Ranking based on a mathematical model• Minimize ad-hoc choices• Based on simple principles • Examples– Colley matrix– Massey’s method– Generalized point-difference ranking
Colley Matrix Rankinghttp://www.colleyrankings.com
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•Only simple statistics needed (wins, losses, & schedule)•Doesn’t depend on margin of victory•Does include strength of schedule
Colley Matrix Method
Ranking SOS Adjustment
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Iteration – Simple Example
Two teams and one game (team 1 wins)
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Iteration – Simple ExampleIteration r1 r2
0 0.500000 0.5000001 0.666667 0.3333332 0.611111 0.3888893 0.629630 0.3703704 0.623457 0.3765435 0.625514 0.3744866 0.624829 0.3751717 0.625057 0.3749438 0.624981 0.3750199 0.625006 0.374994
10 0.624998 0.375002
Colley Matrix - Solution
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Solution – Simple Example
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Ivy League Football - 2009
Team Colley Rating
Penn .792
Harvard .625
Columbia .583
Princeton .583
Brown .542
Dartmouth .375
Cornell .250
Yale .250
What is the best team?Is Dartmouth better than Yale?
Massey Rating Methodhttp://www.masseyratings.com
Ratings should predict score differential
rating of the -th team
If team plays team , want net point difference to be
12 equations with 8 variables - unique solution?
Massey – linear algebra formulation
# teams = n, # total games = m• m x n matrix • Vector • Rating vector
In k-th game team team beats team . • , , and if • margin of victory
Massey equation:
No unique solution – instead try to minimize
Massey – Least squares
Want to minimize • Try ???– is not invertible– Add condition that
New least squares problem
Ivy League Football - 2009
Team Massey Rating
Penn 25.25
Harvard 10.75
Columbia 0
Princeton -3
Brown -3.75
Yale -7
Cornell -11
Dartmouth -11.25
What is the best team?Is Dartmouth better than Yale?
Colley – Massey comparison
Team Massey Rating
Penn 25.25
Harvard 10.75
Columbia 0
Princeton -3
Brown -3.75
Yale -7
Cornell -11
Dartmouth -11.25
Team Colley Rating
Penn .792
Harvard .625
Columbia .583
Princeton .583
Brown .542
Dartmouth .375
Cornell .250
Yale .250
Another Ranking Method“A Natural Generalization of the Win-Loss Rating System.”
Charles Redmond, Mercyhurst CollegeMathematics Magazine, April 2003.
Compare teams through strings of comparisons
Yale vs. Columbia
•Columbia is 14 better than Brown•Brown is 14 better than Yale•So… Columbia is 28 better than Yale
•Columbia is 20 worse than Harvard•Harvard is 4 better than Yale•So… Columbia is 16 worse than Yale
Average of two comparisons: Columbia is 6 better than Yale
Average Dominance
Team Average Dominance
A 2.33
B 2.67
C -3.33
D -1.67
Team Average Dominance
A 3.5
B 4
C -5
D -2.5
Average margin of victory Add self-comparisons
Second Generation Dominance
Avg. 2nd Generation Dominance
44.3933
912190251250
Team Dominance 2nd Gen. Dominance
A 2.33 3.44
B 2.67 3.22
C -3.33 -4.11
D -1.67 -2.56
Connection to Linear Algebra
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Limiting Dominance
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Ivy League Football - 2009
Team Dominance Rating
Penn 24.34
Harvard 10.06
Columbia -0.09
Brown -2.84
Princeton -2.91
Yale -7.13
Dartmouth -10.56
Cornell -10.88
What is the best team?Is Dartmouth better than Yale?
Conclusion
• Linear Algebra can be useful!– Matrices can make things easier.
• Complex Rankings, with simple methods.
• Methods aren’t perfect.– What ranking is “best”?