March Madness A Math Primer

Transcription

1 University of Kentucky Feb 24, 2016

2 NCAA Tournament Background Single elimination tournament for Men s and Women s NCAA Div. 1 basketball held each March, referred to as March Madness. 64 teams make tournament + 4 play in teams = 68 teams. Four regions, each comprised of 16 teams, make up the four quadrants of the bracket. Each region has teams ranked The 16 seeds in the quadrants with the two highest one seeds are determined by the winner of the play in games. In the first round, 1 plays 16, 2 plays 15, 3 plays 14 and so on.

3 March Madness Vocabulary Brackets are released on the Sunday prior to the NCAA tournament. This day is known as Selection Sunday. Both ESPN and CBS have rights to air specials where the brackets are first made public. The study of seeding teams and making brackets is known as bracketology. The round of 16 teams is called the Sweet Sixteen. The round of 8 teams is called the Elite Eight. The round of 4 teams is called the Final Four.

4 Ranking Teams How are teams ranked prior to the tournament?

5 Ranking Teams How are teams ranked prior to the tournament? 10 member basketball selection committe made up of athletic directors from various conferences. 31 teams get automatic bid for winning their conference tournament + winner of Ivy League Championship (no conference tournament too cool for that)=32 teams with automatic bids Committee must decide remaining (play-in) 32(automatic bid) = 36 teams, referred to as at large teams. Additionally, committe must rank all 68 teams in tournament.

6 Ranking Methods Selection committe uses proprietary rankings. They claim to use ESPN BPI(Basketball Power Index) and Ken Pom s ratings (kenpom.com), as well as the eye test.

7 Ranking Methods Selection committe uses proprietary rankings. They claim to use ESPN BPI(Basketball Power Index) and Ken Pom s ratings (kenpom.com), as well as the eye test. We will look at three ways to rank teams: Winning Percentage, the Massey Method, and the Colley Method.

8 Winning Percentage Let n tot,i be the total number of games for team i, and n win,i be the number of wins for team i. Rank all teams according to r i = n win,i /n tot,i. Works well in professional sports, terrible in college sports.

9 Massey Method Supposes transitivity should hold, i.e that rankings should be able to predict scores. Score differential matters.

10 Example of Massey Method 3 teams Example (UK, LSU, SC. Real games this season.) r LSU r UK = 18 r UK r SC = 27 r SC r LSU = 11 Clearly has no solution, so we force the sum of the ratings to be zero, r LSU + r UK + r SC = 0, and solve by least squares. Least squares: r LSU = 2.33,r UK = 3, and r SC = 5.33.

11 Colley Method I Depends only on wins/losses not point differential. Produces rating for teams with an average value of 1/2. Uses Laplace s Law of Succession.

12 Colley Method I Depends only on wins/losses not point differential. Produces rating for teams with an average value of 1/2. Uses Laplace s Law of Succession. Definition (Adjusted Win Percentage) Let r i = 1+n win,i 2+n tot,i. Compare to usual win percentage, n win,i /n tot,i.

13 Colley Method II Notice that n tot,i = n win,i n loss,i 2 = n win,i n loss,i 2 = n win,i n loss,i 2 + n win,i + n loss,i 2 + n tot,i 2 + 1/2 opponents of i Definition (Effective Wins) Let n eff,i = n win,i n loss,i 2 + j opp. of i r j.

14 Colley Method III Substitute effective wins n eff,i for n win,i into adjusted win percentage, solve. Definition (Colley Matrix) The Colley Matrix is the matrix corresponding to the system: (2 + n tot,i )r i j opp. of i r j = 1 + n win,i n loss,i. 2 In matrix form, C r = b, where r i = r i, b i = 1 + n win,i n loss,i 2 and C is given by: { 2 + n tot,i, if i = j C i,j = number games between i and j, if i j

15 Colley Example Example (UK,LSU, SC) Recall each team played 2 games, with one win and one loss r LSU r UK = r SC 1 This yields all teams with rank 1/2. All teams are average.

16 Hybrid Techniques Suppose we want to weight games closer to the tourney more than games played in November. Give games played from Nov. to Dec. weight 0.5, January games weight 0.75, games in February and March weight 1.

17 Hybrid Techniques Suppose we want to weight games closer to the tourney more than games played in November. Give games played from Nov. to Dec. weight 0.5, January games weight 0.75, games in February and March weight 1. Example (Weighted Massey Method, UK,LSU, SC) r LSU r UK = (0.75) 18 r UK r SC = 27 r SC r LSU = 11 Along with r UK + r SC + r LSU = 0, least squares yields : r LSU = 5/6, r UK = 9/2, r SC = 16/3.

18 Try Yourself Create your own weightings, as well as home vs. away distinctions, blowouts...

19 Works Cited When Life is Linear- from computer graphics to bracketology, Tim Chartier. MAA Colley s Bias Free College Football Ranking Method: The Colley Matrix Explained, Wesley Colley.