# The mathematics of ranking sports teams Who’s #1?

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The mathematics of ranking sports teams Who’s #1?
Jonathon Peterson Purdue University

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

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 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

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

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

Iteration – Simple Example
1 2 3 4 5 6 7 8 9 10

Colley Matrix - Solution
Two equations:

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

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?

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?

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

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

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?

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

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

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

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

Connection to Linear Algebra

Limiting Dominance

Limiting Dominance

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”?