Presentation on theme: "The mathematics of ranking sports teams Who’s #1?"— Presentation transcript:
1 The mathematics of ranking sports teams Who’s #1? Jonathon PetersonPurdue University
2 The Ranking Problem Why is ranking of sports teams important? College football – BCSCollege basketball – NCAA tournamentWin $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 functionExamplesJeff SagarinRPIProblemsad-hoc techniquesDependent on parameters
5 Ranking Methods Mathematical methods Ranking based on a mathematical modelMinimize ad-hoc choicesBased on simple principlesExamplesColley matrixMassey’s methodGeneralized 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 victoryDoes include strength of schedule
7 Colley Matrix Method Keep iterating and hope for convergence Ranking SOS AdjustmentKeep iterating and hope for convergence
8 Iteration – Simple Example Two teams and one game (team 1 wins)
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.583PrincetonBrown.542Dartmouth.375Cornell.250YaleWhat is the best team?Is Dartmouth better than Yale?
13 Massey Rating Method Ratings should predict score differential Ratings should predict score differential𝑟 𝑖 = rating of the 𝑖-th teamIf team 𝑖 plays team 𝑗, want net point difference to be 𝑟 𝑖 − 𝑟 𝑗𝑟 𝐵𝑟𝑜𝑤𝑛 − 𝑟 𝑌𝑎𝑙𝑒 =14𝑟 𝐶𝑜𝑙𝑢𝑚𝑏𝑖𝑎 − 𝑟 𝐵𝑟𝑜𝑤𝑛 =14𝑟 𝐶𝑜𝑙𝑢𝑚𝑏𝑖𝑎 − 𝑟 𝐶𝑜𝑟𝑛𝑒𝑙𝑙 =1012 equations with 8 variables- unique solution?
14 Massey – linear algebra formulation # teams = n, # total games = mm x n matrix 𝐵Vector 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,𝑗𝑣 𝑘 = margin of victoryMassey equation:𝐵 𝑟 = 𝑣No unique solution – instead try to minimize𝐵 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 invertibleAdd condition that 𝑟 ∙ 1 =0New least squares problem
20 Another Ranking Method “A Natural Generalization of the Win-Loss Rating System.”Charles Redmond, Mercyhurst CollegeMathematics Magazine, April 2003.Compare teams through strings of comparisonsYale vs. ColumbiaColumbia is 14 better than BrownBrown is 14 better than YaleSo… Columbia is 28 better than YaleColumbia is 20 worse than HarvardHarvard is 4 better than YaleSo… Columbia is 16 worse than YaleAverage of two comparisons: Columbia is 6 better than Yale
21 Average Dominance Average margin of victory Add self-comparisons Team 3.5B4C-5D-2.5TeamAverage DominanceA2.33B2.67C-3.33D-1.67
22 Second Generation Dominance TeamDominance2nd Gen. DominanceA2.333.44B2.673.22C-3.33-4.11D-1.67-2.56Avg. 2nd Generation Dominance
23 Connection to Linear Algebra Adjacency MatrixDominance Vector
26 Ivy League Football - 2009 Team Dominance Rating Penn 24.34 Harvard 10.06Columbia-0.09Brown-2.84Princeton-2.91Yale-7.13Dartmouth-10.56Cornell-10.88What 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”?