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 0.500000

1 0.666667 0.333333

2 0.611111 0.388889

3 0.629630 0.370370

4 0.623457 0.376543

5 0.625514 0.374486

6 0.624829 0.375171

7 0.625057 0.374943

8 0.624981 0.375019

9 0.625006 0.374994

10 0.624998 0.375002

Colley Matrix - Solution

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Solution – Simple Example

Two teams and one game (team 1 wins)

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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

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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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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”?