The mathematics of ranking sports teams - PowerPoint Presentation

Slide1

The mathematics of ranking sports teamsWho’s #1?

Jonathon PetersonPurdue UniversitySlide2

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.comSlide3

Ivy League Football - 2009What is the best team?

Is Dartmouth better than Yale?Slide4

Ranking MethodsStatistical MethodsGather as much data as possibleCook up a good predicting function

ExamplesJeff SagarinRPIProblemsad-hoc techniques

Dependent on parametersSlide5

Ranking MethodsMathematical methodsRanking based on a mathematical modelMinimize ad-hoc choicesBased on simple principles

ExamplesColley matrixMassey’s methodGeneralized point-difference rankingSlide6

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 scheduleSlide7

Colley Matrix MethodRanking

SOS Adjustment

Keep iterating and hope for convergenceSlide8

Iteration – Simple ExampleTwo teams and one game (team 1 wins)Slide9

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

0.375002Slide10

Colley Matrix - Solution

Two equations:Slide11

Solution – Simple ExampleTwo teams and one game (team 1 wins)

Matrix Form SolutionSlide12

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?Slide13

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?

 Slide14

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

 Slide15

Massey – Least squaresWant to minimize

Try

???

is not invertible

Add condition that

New least squares problem

 Slide16
Slide17

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?Slide18
Slide19

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

.250Slide20

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 YaleSlide21

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-comparisonsSlide22

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

-2.56Slide23

Connection to Linear Algebra

Adjacency Matrix

Dominance VectorSlide24

Limiting DominanceSlide25

Limiting DominanceSlide26

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?Slide27

ConclusionLinear Algebra can be useful!Matrices can make things easier.Complex Rankings, with simple methods.

Methods aren’t perfect.What ranking is “best”?