The mathematics of ranking sports teams W hos 1 Jonathon Peterson Purdue University The Ranking Problem Why is ranking of sports teams important College football BCS College basketball NCAA tournament ID: 765574
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The mathematics of ranking sports teamsWho’s #1? Jonathon PetersonPurdue University
The Ranking ProblemWhy is ranking of sports teams important?College football – BCSCollege 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 - 2009What is the best team? Is Dartmouth better than Yale?
Ranking MethodsStatistical MethodsGather as much data as possibleCook up a good predicting function ExamplesJeff SagarinRPIProblemsad-hoc techniques Dependent on parameters
Ranking MethodsMathematical methodsRanking based on a mathematical modelMinimize ad-hoc choicesBased on simple principles ExamplesColley matrixMassey’s methodGeneralized point-difference ranking
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
Colley Matrix MethodRanking SOS Adjustment Keep iterating and hope for convergence
Iteration – Simple ExampleTwo teams and one game (team 1 wins)
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:
Solution – Simple ExampleTwo teams and one game (team 1 wins) Matrix Form Solution
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 n et point difference to be 12 equations with 8 variables - unique solution?
Massey – linear algebra formulation # teams = n, # total games = mm 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 squaresWant 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.250Yale.250
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 TeamAverage DominanceA 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 DominanceAvg. 2nd Generation Dominance Team Dominance 2 nd 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 Adjacency Matrix Dominance Vector
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?
ConclusionLinear Algebra can be useful!Matrices can make things easier.Complex Rankings, with simple methods. Methods aren’t perfect.What ranking is “best”?