The Mathematics of Ranking Sports Teams - Insights and Methods
Exploring the importance and challenges of ranking sports teams, this content delves into various ranking methods, including statistical and mathematical approaches like the Colley Matrix, highlighting the complexities and iterative processes involved in determining team rankings.
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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!!! 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 Ranking http://www.colleyrankings.com = # # wins n , w i = losses n Team i Data: , l i = = + # games n n n , , , tot i w i l i games # = G ( ) i j Schedule Data: ij 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 = 2 n n + 1 n j , , + w i l i eff w G n r , = w i r , i ij j i + 2 n , tot i + eff w 1 n n n j , = i ' r , , = + w i l i eff w G ' ' n r i + 2 n , i ij j 2 , tot i Keep iterating and hope for convergence
Iteration Simple Example Two teams and one game (team 1 wins) 1 , + i tot n + eff w n 2 n n = i r , , = + w i l i eff w n r i 2 , i j ,
Iteration Simple Example Iteration 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 Two equations: 2 + eff w n n 1 n j , , = + w i l i , = eff w i G n r r , i ij j i + 2 n , tot i G n n 1 j ij , , = + + w n i l ) 2 i Combine both : r r i j + + + 2 ( 2 2 n n , , , tot i tot i tot i = + A Matrix Form : r b r
Solution Simple Example + + i tot i tot n n ( 2 2 , , G n n 1 j ij , , = + w i l ) 2 i r r i j + + 2 n , tot i Two teams and one game (team 1 wins) / 1 2 0 / 1 3 / 1 = = A b / 1 6 3 0 Matrix Form Solution / 1 1 / 1 3 2 r / 1 1 = = I A ( ) r b / 1 3 1 6 r 2 / 5 8 625 . r 1 = = / 3 8 375 . r 2
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 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 ?? ?? ?????? ?????= 14 ????????? ??????= 14 ????????? ????????= 10 12 equations with 8 variables - unique solution?
Massey linear algebra formulation # teams = n, # total games = m m x n matrix ? Vector ? = (?1,?2, ,??) Rating vector ? = (?1,?2, ,??) In k-th game team team ? beats team ?. ???= 1, ???= 1, and ???= 0 if ? ?,? ??= margin of victory Massey equation: ? ? = ? No unique solution instead try to minimize ? ? ?
Massey Least squares Want to minimize ? ? ? Try ? = (???) 1?? ? ??? ??? is not invertible Add condition that ? 1 = 0 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 Colley Rating Team Massey Rating Penn .792 Penn 25.25 Harvard .625 Harvard 10.75 Columbia .583 Columbia 0 Princeton .583 Princeton -3 Brown .542 Brown -3.75 Dartmouth .375 Yale -7 Cornell .250 Cornell -11 Yale .250 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 Average Dominance Team Average Dominance A 3.5 A 2.33 B 4 B 2.67 C -5 C -3.33 D -2.5 D -1.67
Second Generation Dominance 2nd Gen. Dominance Team Dominance A 2.33 3.44 Avg. 2nd Generation Dominance B 2.67 3.22 + + + + 0 5 12 5 2 0 19 12 33 = = . 3 44 C -3.33 -4.11 9 9 D -1.67 -2.56
Connection to Linear Algebra Adjacency Matrix Dominance Vector 1 1 0 1 7 1 1 1 0 8 = = M S 0 1 1 1 10 1 0 1 1 5 1 = 1st dominance avg. gen. S 3 M 1 1 1 = + = + 2nd M dominance avg. gen. 3 ( 9 ) S S S S 3 3 3 k M 1 - n 1 = th n dominance avg. gen. S 3 3 = k 0
Limiting Dominance k M 1 - n 1 = k Does the limit exist? lim S 3 3 n 0 / 1 / 1 2 / 1 2 0 2 / 1 2 0 / 1 2 / 1 2 = = = = , , , v v v v 1 2 1 1 / 1 / 1 / 1 2 2 0 2 / 1 / 1 2 0 2 / 1 2 , 1 = , 3 / 1 = , 3 / 1 = = / 1 3 1 2 3 4 = + + + 0 17 13 3 S v v v v 1 2 3 4 k k k k M 1 1 1 = + + 17 13 3 S v v v 2 3 4 3 3 3 3
Limiting Dominance = 3 3 0 k k k k k M 1 17 1 13 1 3 1 = k = k = k = + + S v v v 2 3 4 3 3 3 3 3 3 0 0 0 17 1 13 1 1 = + + v v v 2 3 4 + 3 1 / 1 3 3 1 / 1 3 1 / 1 3 17 13 3 = + + v v v 2 3 4 2 2 4 31 / 8 . 3 875 29 / 8 . 3 625 = = 37 / 8 . 4 625 23 / 8 . 2 875
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 ?
