System explanation

 

Overview

 

All new players enter the rating system with a score of 50.00.  During the first few sessions, the score will change quickly, depending on performance.  After playing several sessions, the score is more accurate, so further games are scored less and less.  Each session increases the weight on the previous score.  However, each week this weight is reduced so that players who take a break for several weeks will find their score more volatile when they return.  For consistent players, the effect still applies, discounting the influence of old games.  To determine the expected matchpoint percentage for each player/pair, the player’s score (or the average of both partners for pairs events) is compared to the strength of the field excluding that player/pair’s contribution.  This is just the average score of all other players in the direction.  After the event, the inverse calculation is performed (i.e. what level of player would be expected to receive the matchpoint percentage earned).  Depending on the prior activity level, the updated score will be a weighted average of the old score and the new score.

(11/27/02) Added support to handle IMP Pairs game as well. IMP/board is converted to a percentage using the same multiplier as Lehman: (IMP/board * 20/3) + 50.

 

Conversion table

 

Except for extreme rankings, the expected score for a player/pair is 1% more/less than 50% for each ranking point higher/lower than the field’s average. These values come from 1/(1+e^(diff/25)).

 

Ranking score difference vs. Expected matchpoint percentage

-22

29.32%

-13

37.29%

-4

46.01%

5

54.98%

14

63.65%

-21

30.15%

-12

38.23%

-3

47.00%

6

55.97%

15

64.57%

-20

31.00%

-11

39.17%

-2

48.00%

7

56.95%

16

65.48%

-19

31.86%

-10

40.13%

-1

49.00%

8

57.93%

17

66.37%

-18

32.74%

-9

41.10%

0

50.00%

9

58.90%

18

67.26%

-17

33.63%

-8

42.07%

1

51.00%

10

59.87%

19

68.14%

-16

34.52%

-7

43.05%

2

52.00%

11

60.83%

20

69.00%

-15

35.43%

-6

44.03%

3

53.00%

12

61.77%

21

69.85%

-14

36.35%

-5

45.02%

4

53.99%

13

62.71%

22

70.68%

 

Comparison to OKbridge Lehman System

 

The system as described is based on score differences.  This allows field strength to be determined with a simple arithmetic average.  The Lehman system is based on ratios instead.  Thus a 50-level pair facing a 45-level pair would be expected to get 50/95 or 52.63%.  A 55-pair facing a 50-pair would only be expected to earn 52.38%.  Under the system above it would be 54.98% in both instances.  The scores are much tighter to provide a 1:1 relationship between point difference and matchpoint difference, rather than trying to approximate the Lehman scale.  Overall there is a 2:1 ratio between Lehmans and the ranking points described above, stretching out low scores and collapsing high ones.

 

Ranking score vs. Lehman score (assumes 50=50)

28

20.74

37

29.73

46

42.61

55

61.07

64

87.53

29

21.59

38

30.94

47

44.35

56

63.56

65

91.11

30

22.47

39

32.20

48

46.16

57

66.16

66

94.82

31

23.38

40

33.52

49

48.04

58

68.86

67

98.69

32

24.34

41

34.88

50

50.00

59

71.67

68

102.72

33

25.33

42

36.31

51

52.04

60

74.59

69

106.91

34

26.36

43

37.79

52

54.16

61

77.64

70

111.28

35

27.44

44

39.33

53

56.37

62

80.80

71

115.82

36

28.56

45

40.94

54

58.68

63

84.10

72

120.54

 

Note that 68+ maps to 100+, defying the idea that Lehmans are on a percentage scale.  Reading the specification carefully, there is no reason why a super-expert could not exceed 100 Lehman points.  The proposed system is a symmetric alternative.  Also, the board decay rate is slower (0.950 instead of 0.933).  Every player has the additional weight of 2.8 weeks of boards so a new player should move 26% toward the performance level of their first session (assuming it is a 24 board session).  These values were reached by experimentation, minimizing the root-mean-squared error.

 

Some statistics from the MIT Draper Lab Bridge Club (as of May 2000)

 

These statistics include only people considered active (>30 boards after decay):

Active players: 153

Mean: 50.93

Median: 50.93

Standard deviation: 4.904

RMS-error: 7.633

 

Is the distribution of scores “normal”?  Judge for yourself:

 

Send your questions and comments to: qubits@hotmail.com
Copyright © 2000 Jason Woolever
Last modified: 5/25/2000