The mathematics of ranking sports teams W ho’s #1?

1. The mathematics of ranking sports teamsWho’s #1? Jonathon Peterson Purdue University

2. 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

3. Ivy League Football - 2009 What is the best team? Is Dartmouth better than Yale?

4. 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

5. 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

6. Colley Matrix Rankinghttp://www.colleyrankings.com Team i Data: Schedule Data: • Only simple statistics needed (wins, losses, & schedule) • Doesn’t depend on margin of victory • Does include strength of schedule

7. Colley Matrix Method Ranking SOS Adjustment Keep iterating and hope for convergence

8. Iteration – Simple Example Two teams and one game (team 1 wins)

9. Iteration – Simple Example

10. Colley Matrix - Solution Two equations:

11. Solution – Simple Example Two teams and one game (team 1 wins) Matrix Form Solution

12. Ivy League Football - 2009 What is the best team? Is Dartmouth better than Yale?

13. Massey Rating Method http://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?

14. 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

15. Massey – Least squares Want to minimize • Try ??? • is not invertible • Add condition that New least squares problem

17. Ivy League Football - 2009 What is the best team? Is Dartmouth better than Yale?

19. Colley – Massey comparison

20. Another Ranking Method “A Natural Generalization of the Win-Loss Rating System.” Charles Redmond, Mercyhurst College Mathematics 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

21. Average Dominance Average margin of victory Add self-comparisons

22. Second Generation Dominance Avg. 2nd Generation Dominance

23. Connection to Linear Algebra Adjacency Matrix Dominance Vector

24. Limiting Dominance

25. Limiting Dominance

26. Ivy League Football - 2009 What is the best team? Is Dartmouth better than Yale?

27. Conclusion • Linear Algebra can be useful! • Matrices can make things easier. • Complex Rankings, with simple methods. • Methods aren’t perfect. • What ranking is “best”?