rating table tennis players an application of bayesian inference
TRANSCRIPT
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