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The mathematics of ranking sports teams Whos #1? Jonathon Peterson Purdue University

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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 –

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Ivy League Football What is the best team? Is Dartmouth better than Yale?

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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

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Ranking Methods Mathematical methods Ranking based on a mathematical model Minimize ad-hoc choices Based on simple principles Examples – Colley matrix – Masseys method – Generalized point-difference ranking

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Colley Matrix Ranking Team i Data: Schedule Data: Only simple statistics needed (wins, losses, & schedule) Doesnt depend on margin of victory Does include strength of schedule

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Colley Matrix Method RankingSOS Adjustment Keep iterating and hope for convergence

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

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Iteration – Simple Example Iterationr1r

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Colley Matrix - Solution Two equations:

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

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Ivy League Football TeamColley 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?

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Massey Rating Method

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Massey – linear algebra formulation

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Massey – Least squares

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Ivy League Football TeamMassey Rating Penn25.25 Harvard10.75 Columbia0 Princeton-3 Brown-3.75 Yale-7 Cornell-11 Dartmouth What is the best team? Is Dartmouth better than Yale?

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Colley – Massey comparison TeamMassey Rating Penn25.25 Harvard10.75 Columbia0 Princeton-3 Brown-3.75 Yale-7 Cornell-11 Dartmouth TeamColley Rating Penn.792 Harvard.625 Columbia.583 Princeton.583 Brown.542 Dartmouth.375 Cornell.250 Yale.250

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

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Average Dominance TeamAverage Dominance A2.33 B2.67 C-3.33 D-1.67 TeamAverage Dominance A3.5 B4 C-5 D-2.5 Average margin of victoryAdd self-comparisons

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Second Generation Dominance Avg. 2 nd Generation Dominance TeamDominance2 nd Gen. Dominance A B C D

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Connection to Linear Algebra Adjacency MatrixDominance Vector

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Limiting Dominance

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Ivy League Football TeamDominance Rating Penn24.34 Harvard10.06 Columbia-0.09 Brown-2.84 Princeton-2.91 Yale-7.13 Dartmouth Cornell What is the best team? Is Dartmouth better than Yale?

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Conclusion Linear Algebra can be useful! – Matrices can make things easier. Complex Rankings, with simple methods. Methods arent perfect. – What ranking is best?

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