1. The mathematics of ranking sports teams Who’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!!!
What is so hard about ranking teams?
Strength of schedule matters.
Non-transitive property

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 Ranking http://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 Keep iterating and hope for convergence 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 1 2 3 4 5 6 7 8 9 10

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 Team Colley Rating Penn .792 Harvard .625
Columbia
.583
Princeton
Brown
.542
Dartmouth
.375
Cornell
.250
Yale
What is the best team?
Is Dartmouth better than Yale?

13. Massey Rating Method Ratings should predict score differential
Ratings should predict score differential
𝑟 𝑖 = rating of the 𝑖-th team
If team 𝑖 plays team 𝑗, want net point difference to be 𝑟 𝑖 − 𝑟 𝑗
𝑟 𝐵𝑟𝑜𝑤𝑛 − 𝑟 𝑌𝑎𝑙𝑒 =14
𝑟 𝐶𝑜𝑙𝑢𝑚𝑏𝑖𝑎 − 𝑟 𝐵𝑟𝑜𝑤𝑛 =14
𝑟 𝐶𝑜𝑙𝑢𝑚𝑏𝑖𝑎 − 𝑟 𝐶𝑜𝑟𝑛𝑒𝑙𝑙 =10
12 equations with 8 variables
- unique solution?

14. Massey – linear algebra formulation
# teams = n, # total games = m
m x n matrix 𝐵
Vector 𝑣 =( 𝑣 1 , 𝑣 2 ,…, 𝑣 𝑚 )
Rating vector 𝑟 =( 𝑟 1 , 𝑟 2 ,…, 𝑟 𝑛 )
In k-th game team team 𝑖 beats team 𝑗.
𝐵 𝑘𝑖 =1, 𝐵 𝑘𝑗 =−1, and 𝐵 𝑘𝑙 =0 if 𝑙≠𝑖,𝑗
𝑣 𝑘 = margin of victory
Massey equation:
𝐵 𝑟 = 𝑣
No unique solution – instead try to minimize
𝐵 𝑟 − 𝑣

15. Massey – Least squares Want to minimize 𝐵 𝑟 − 𝑣
Try 𝑟 = ( 𝐵 𝑡 𝐵) −1 𝐵 𝑡 𝑣 ???
𝐵 𝑡 𝐵 is not invertible
Add condition that 𝑟 ∙ 1 =0
New least squares problem

17. Ivy League Football - 2009 Team Massey Rating Penn 25.25 Harvard 10.75
Columbia
Princeton -3
Brown
-3.75
Yale -7
Cornell
-11
Dartmouth
-11.25
What is the best team?
Is Dartmouth better than Yale?

19. Colley – Massey comparison
Team
Colley Rating
Penn
.792
Harvard
.625
Columbia
.583
Princeton
Brown
.542
Dartmouth
.375
Cornell
.250
Yale
Team
Massey Rating
Penn
25.25
Harvard
10.75
Columbia
Princeton -3
Brown
-3.75
Yale -7
Cornell
-11
Dartmouth
-11.25

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 Team
3.5 B 4 C -5 D
-2.5
Team
Average Dominance A
2.33 B
2.67 C
-3.33 D
-1.67

22. Second Generation Dominance
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
Avg. 2nd Generation Dominance

23. Connection to Linear Algebra
Adjacency Matrix
Dominance Vector

24. Limiting Dominance

25. Limiting Dominance

26. 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?

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”?