Rating Table Tennis Players

An application of Bayesian inference

Ratings

• The USATT rates all members

• A rating is an integer between 0 and 3000

Fan Yi Yong 2774

Example

Lee Bahlman 2045

Dell Sweeris 2080

Todd Sweeris

Old System

Rating Difference Expected Result Upset0-12 8 8

13-37 7 1038-62 6 1363-87 5 1688-112 4 20

113-137 3 25138-162 2 30163-187 2 35188-212 1 40213-237 1 45

238- 0 50

Example

Difference Expected Upset0-12 8 813-37 7 1038-62 6 1363-87 5 16

88-112 4 20113-137 3 25138-162 2 30163-187 2 35188-212 1 40213-237 1 45

238- 0 50

Lee Bahlman (2045)Dell Sweeris (2080)

If Lee winsBahlman (2055)Sweeris (2070)

If Dell winsBahlman (2038)Sweeris (2087)

Complications

• Unrated Players

• Underrated or Overrated Players

Processing a Tournament

• First Pass - Assign Initial Ratings– Rate unrated players

• Second Pass - Adjust Ratings– The “fifty point change” rule

• Third Pass - Compute Final Ratings– Using the table of points

Problems

Arbitrary Numbers (table of points, fifty-point rule)

Rating Difference Expected Result Upset0-12 8 8

13-37 7 1038-62 6 1363-87 5 1688-112 4 20

113-137 3 25138-162 2 30163-187 2 35188-212 1 40213-237 1 45

238- 0 50

Problems

Arbitrary Numbers (table of points, fifty-point rule)

Human Intervention Necessary

Manipulable

A New Rating System?

• USATT commissioned a study

• David Marcus (Ph.D., MIT, Statistics) developed a new method

• Under review by USATT • May or may not be adopted

Proposed New Method

Based on three mathematical ideas– Either player may win a match (probability)– Ratings have some uncertainty (probability)– Tournaments are data to update ratings (statistics)

11

( ) ce

rating difference between A and B

( ) probability of win for A

0.0148540595817432c

Player A rating

Player B rating

( )c

c c

e

e e

-200 -100 100 200

0.2

0.4

0.6

0.8

What is a rating?

• Classical statistical model –– a rating is a parameter that is possibly unknown– We need to estimate the parameter

• Bayesian model -− our uncertainty about the parameter is reflected

in a probability distribution, the probability is subjective probability

What is a rating?

• A rating is a probability distribution

• The distributions used are discrete versions of the normal distribution

• The mass function is nonzero on ratings 0, 10, 20, … , 3590, 3600

Dell Sweeris 2096 (84)

0.000

0.010

0.020

0.030

0.040

0.050

1800 1900 2000 2100 2200 2300 2400

Rating

Pro

babili

ty

Dell and Lee

0.00

0.01

0.02

0.03

0.04

0.05

0.06

1700 1800 1900 2000 2100 2200 2300 2400

Rating

Pro

bab

ilit

y

Lee

Dell

Unrated Players

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

200 700 1200 1700 2200 2700

Rating

Pro

babili

ty

Unrated

Dell

Unrated Players 1400 (450)

Rating Change with Time

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1760 1860 1960 2060 2160 2260 2360

Dell in a year

Dell now

Updating RatingsDell and Lee

0

0.1

0.2

0.3

0.4

0.5

0.6

2000 2050 2100 2150 2200

Rating

Pro

bab

ilit

y

Lee

Dell

Lee's Outcomes

Win 20006%

Win 205018%

Win 210013%

Lose 200019%

Lose 205031%

Lose 210013%

Example

.5 (.25 .32 .5 .67 .25 .90)

Probability that Lee is rated 2050 and loses

Lee Rated 2050 Dell Rated 2000

Probability Lee loses if rated 2050 and Dell rated 2000

Lee's Outcomes

Win 20006%

Win 205018%

Win 210013%

Lose 200019%

Lose 205031%

Lose 210013%

Lee Wins

Loss63%

Win 200016%

Win 210035%

Win 205049%

Win36%

Lee’s Rating

Before After

2000 .25 .16

2050 .50 .49

2100

Average

.25

2050

.35

2059

Dell’s Rating

Before After

2000 .25 .46

2100 .50 .46

2200

Average

.25

2100

.08

2062

Bayes’ Theorem

1 2

1

Given probabilities of events , ,...,

and event with conditional probabilities ( | )

the probabilities of events given are

( and ) ( and )( | )

( ) ( | ) ... (

k

i

i

i ii

prior B B B

E P E B

posterior B E

P B E P B EP B E

P E P E B P E

| )kB

Updating Ratings

• Each player has an initial rating

• The results of the tournament are the data

• Bayes Theorem is used to update the ratings

• Computationally intense - hundreds of players and hundreds of possible ratings per player