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Mec26: a new matchequity table

 

By Albert Silver

 

 

 

Any player serious about reaching the upperechelons of backgammon knows that one requirement is to know the double andtake Point in match play especially as the number of points to finish shrinks.Naturally one will want to use the best data available, and different sourceshave published METs, or Match Equity Tables, over the years. The most popularto date is that of Kit Woolsey and HalHeinrich, published in 1993 in Woolsey's seminal work, How to Play Tournament Backgammon and available at Gammonline.Others have appeared since then, such as the highly regarded one by Jake Jacobsand Walter Trice in their 1996 work, Cana Fish Taste Twice as Good?, and the Snowie 2.1 MET in the 1999 version ofthe program Snowie. How different are they? And is it worth the effortswitching to another? The answer to that last question is entirely up to thereader.

 

Sometime about a month ago, there was adiscussion among the GNU Backgammondevelopers on the default MET used in the program. The program allows the userto choose the match equity table of their choice, an option that is probablyused very little, and by default uses the table by Zadeh published in 1977.This default choice seemed very poor so a discussion took place on which shouldbe the new default. Joseph Heled, the developer of the neural nets,mentioned he used none of them, and instead used Mec26. No one knew what this was, so he explained that it was basedon a small program published in 1996(!) by Claes Thornberg, and made freelyavailable for download. News of it was published in the newsgroup rec.games.backgammonwith a link (no longer working by the way) to share with others. The reasonsgiven were errors spotted with someone else's match equity table work, and thusa program was written, to generate a new one using selected gammon values andwinning chances. Having never seen it mentioned in either literature, or byexpert players, I can only presume it went unnoticed or unheeded for the mostpart.

 

The table in question is the result of the program usingvalues of 26% gammons, and 50% winning chances. The post-crawford values weredone by Joseph Heled himself using GNUBG's rollouts. Before going anyfurther, the first question is whether it is verifiably better, and the answeris yes. It was tested against the Woolsey table and after 500,000 7-pointmatches, alternating the sides while retaining the same rolls so that each sidegot the same rolls once, the result was 50.06% against 49.94% for a total of0.12% difference. This is despite the fact that differences of close to 2% canbe seen between the two tables as shown below. It was also tested against theJacobs/Trice table winning by a slightly lesser margin, and against the Snowietable as well. The very small overall effect on the score is explained byHeled, the effect of MET is not that bigsince, again, many decisions are relative and will be right as long as the METis consistent.

 

Still,it is worth mentioning that the table covers matches up to 25 points, extends todecimal points, and does bear some striking differences at some scores comparedto Woolsey's table. Far more than 0.12%. Comparing the Pre-Crawford 7-pointmatch scores from the two tables, some of the more notable differences are:

 

Woolsey Mec26

 

1-away 2-away: 70% / 30% 68.5%/ 31.5%

2-away4-away: 68% / 32% 66.4% / 33.6%

2-away6-away: 81% / 19% 79.5% / 20.5%

2-away7-away: 85% / 15% 83.5% / 16.5%

3-away4-away: 59% / 41% 57.1% / 42.9%

3-away5-away: 66% / 34% 64.6% / 35.4%

 

And now of course for the table itself, but onelast point. I am providing the table exactly twice below. The reason is simplythat if you plan to print this out, the full table will easily go right offyour screen, not to mention paper. The first table only covers 15 pointmatches, and should be easily printable, whereas the second includes the entirething up to 25-away 25-away.

 

Pre-Crawford table (15-point)

 

1-away

2-away

3-away

4-away

5-away

6-away

7-away

8-away

9-away

10-away

11-away

12-away

13-away

14-away

15-away

1-away

50

68.5

75

81.8

84.2

89.2

90.9

93.6

94.6

96.2

96.8

97.8

98.1

98.7

98.9

2-away

31.5

50

59.5

66.4

73.7

79.5

83.5

86.9

89.6

91.9

93.5

94.9

95.9

96.9

97.5

3-away

25

40.5

50

57.1

64.6

71

75.8

80

83.5

86.7

89

91.1

92.7

94.2

95.2

4-away

18.2

33.6

42.9

50

57.5

64

69.4

73.9

78.1

81.7

84.7

87.2

89.4

91.2

92.7

5-away

15.8

26.3

35.4

42.5

50

56.7

62.3

67.3

72

76.1

79.6

82.7

85.3

87.6

89.5

6-away

10.8

20.5

29

36

43.3

50

55.9

61.2

66.2

70.6

74.6

78

81.1

83.8

86.1

7-away

9.1

16.5

24.2

30.6

37.7

44.1

50

55.3

60.5

65.2

69.4

73.2

76.6

79.7

82.4

8-away

6.4

13.1

20

26.1

32.7

38.8

44.7

50

55.2

60

64.5

68.5

72.2

75.5

78.5

9-away

5.4

10.4

16.5

21.9

28

33.8

39.5

44.8

50

54.9

59.5

63.7

67.6

71.2

74.5

10-away

3.8

8.1

13.3

18.3

23.9

29.4

34.8

40

45.1

50

54.7

59

63.1

66.9

70.3

11-away

3.2

6.5

11

15.3

20.4

25.4

30.6

35.5

40.5

45.3

50

54.4

58.6

62.5

66.2

12-away

2.2

5.1

8.9

12.8

17.3

22

26.8

31.5

36.3

41

45.6

50

54.2

58.2

62

13-away

1.9

4.1

7.3

10.6

14.7

18.9

23.4

27.8

32.4

36.9

41.4

45.8

50

54

57.9

14-away

1.3

3.1

5.8

8.8

12.4

16.2

20.3

24.5

28.8

33.1

37.5

41.8

46

50

53.9

15-away

1.1

2.5

4.8

7.3

10.5

13.9

17.6

21.5

25.5

29.7

33.8

38

42.1

46.1

50

 

Post-Crawford table (15-point)

 

1-away

2-away

3-away

4-away

5-away

6-away

7-away

8-away

9-away

10-away

11-away

12-away

13-away

14-away

15-away

1-away

50

48.5

31.95

30.2

18.9

17.5

11.4

10.5

6.7

6.2

4.0

3.7

2.4

2.2

1.4

---------------------------------------------

 

Pre-Crawford table (25-point)

 

1-away

2-away

3-away

4-away

5-away

6-away

7-away

8-away

9-away

10-away

11-away

12-away

13-away

14-away

15-away

16-away

17-away

18-away

19-away

20-away

21-away

22-away

23-away

24-away

25-away

1-away

50

68.5

75

81.8

84.2

89.2

90.9

93.6

94.6

96.2

96.8

97.8

98.1

98.7

98.9

99.2

99.3

99.5

99.6

99.7

99.8

99.8

99.9

99.9

99.9

2-away

31.5

50

59.5

66.4

73.7

79.5

83.5

86.9

89.6

91.9

93.5

94.9

95.9

96.9

97.5

98.1

98.4

98.8

99

99.3

99.4

99.5

99.6

99.7

99.8

3-away

25

40.5

50

57.1

64.6

71

75.8

80

83.5

86.7

89

91.1

92.7

94.2

95.2

96.2

96.9

97.5

98

98.4

98.7

99

99.2

99.3

99.5

4-away

18.2

33.6

42.9

50

57.5

64

69.4

73.9

78.1

81.7

84.7

87.2

89.4

91.2

92.7

94

95

96

96.7

97.3

97.8

98.2

98.5

98.8

99

5-away

15.8

26.3

35.4

42.5

50

56.7

62.3

67.3

72

76.1

79.6

82.7

85.3

87.6

89.5

91.2

92.6

93.9

94.9

95.8

96.5

97.1

97.6

98

98.4

6-away

10.8

20.5

29

36

43.3

50

55.9

61.2

66.2

70.6

74.6

78

81.1

83.8

86.1

88.2

90

91.5

92.8

93.9

94.9

95.7

96.4

97

97.5

7-away

9.1

16.5

24.2

30.6

37.7

44.1

50

55.3

60.5

65.2

69.4

73.2

76.6

79.7

82.4

84.8

86.9

88.8

90.4

91.8

93

94

94.9

95.7

96.4

8-away

6.4

13.1

20

26.1

32.7

38.8

44.7

50

55.2

60

64.5

68.5

72.2

75.5

78.5

81.2

83.6

85.8

87.7

89.4

90.8

92.1

93.2

94.2

95.1

9-away

5.4

10.4

16.5

21.9

28

33.8

39.5

44.8

50

54.9

59.5

63.7

67.6

71.2

74.5

77.4

80.1

82.5

84.7

86.7

88.4

89.9

91.3

92.4

93.5

10-away

3.8

8.1

13.3

18.3

23.9

29.4

34.8

40

45.1

50

54.7

59

63.1

66.9

70.3

73.5

76.5

79.1

81.6

83.8

85.7

87.5

89.1

90.5

91.7

11-away

3.2

6.5

11

15.3

20.4

25.4

30.6

35.5

40.5

45.3

50

54.4

58.6

62.5

66.2

69.5

72.7

75.6

78.2

80.7

82.9

84.8

86.6

88.3

89.7

12-away

2.2

5.1

8.9

12.8

17.3

22

26.8

31.5

36.3

41

45.6

50

54.2

58.2

62

65.6

68.9

72

74.8

77.4

79.8

82

84

85.9

87.5

13-away

1.9

4.1

7.3

10.6

14.7

18.9

23.4

27.8

32.4

36.9

41.4

45.8

50

54

57.9

61.6

65

68.2

71.3

74.1

76.7

79.1

81.3

83.3

85.1

14-away

1.3

3.1

5.8

8.8

12.4

16.2

20.3

24.5

28.8

33.1

37.5

41.8

46

50

53.9

57.6

61.2

64.5

67.7

70.6

73.4

76

78.4

80.5

82.6

15-away

1.1

2.5

4.8

7.3

10.5

13.9

17.6

21.5

25.5

29.7

33.8

38

42.1

46.1

50

53.8

57.4

60.8

64.1

67.2

70.1

72.8

75.3

77.7

79.9

16-away

0.8

1.9

3.8

6

8.8

11.8

15.2

18.8

22.6

26.5

30.5

34.4

38.4

42.4

46.2

50

53.6

57.1

60.5

63.7

66.7

69.5

72.2

74.7

77.1

17-away

0.7

1.6

3.1

5

7.4

10

13.1

16.4

19.9

23.5

27.3

31.1

35

38.8

42.6

46.4

50

53.5

56.9

60.2

63.3

66.3

69.1

71.7

74.2

18-away

0.5

1.2

2.5

4

6.1

8.5

11.2

14.2

17.5

20.9

24.4

28

31.8

35.5

39.2

42.9

46.5

50

53.4

56.7

59.9

63

65.9

68.6

71.2

19-away

0.4

1

2

3.3

5.1

7.2

9.6

12.3

15.3

18.4

21.8

25.2

28.7

32.3

35.9

39.5

43.1

46.6

50

53.3

56.6

59.7

62.6

65.5

68.2

20-away

0.3

0.7

1.6

2.7

4.2

6.1

8.2

10.6

13.3

16.2

19.3

22.6

25.9

29.4

32.8

36.3

39.8

43.3

46.7

50

53.2

56.4

59.4

62.3

65.1

21-away

0.2

0.6

1.3

2.2

3.5

5.1

7

9.2

11.6

14.3

17.1

20.2

23.3

26.6

29.9

33.3

36.7

40.1

43.4

46.8

50

53.2

56.2

59.2

62.1

22-away

0.2

0.5

1

1.8

2.9

4.3

6

7.9

10.1

12.5

15.2

18

20.9

24

27.2

30.5

33.7

37

40.3

43.6

46.8

50

53.1

56.1

59

23-away

0.1

0.4

0.8

1.5

2.4

3.6

5.1

6.8

8.7

10.9

13.4

16

18.7

21.6

24.7

27.8

30.9

34.1

37.4

40.6

43.8

46.9

50

53

56

24-away

0.1

0.3

0.7

1.2

2

3

4.3

5.8

7.6

9.5

11.7

14.1

16.7

19.5

22.3

25.3

28.3

31.4

34.5

37.7

40.8

43.9

47

50

53

25-away

0.1

0.2

0.5

1

1.6

2.5

3.6

4.9

6.5

8.3

10.3

12.5

14.9

17.4

20.1

22.9

25.8

28.8

31.8

34.9

37.9

41

44

47

50

 

Post-Crawford table (25-point)

 

1-away

2-away

3-away

4-away

5-away

6-away

7-away

8-away

9-away

10-away

11-away

12-away

13-away

14-away

15-away

16-away

17-away

18-away

19-away

20-away

21-away

22-away

23-away

24-away

1-away

50

48.5

31.95

30.2

18.9

17.5

11.4

10.5

6.7

6.2

4.0

3.7

2.4

2.2

1.4

1.3

0.85

0.75

0.51

0.46

0.31

0.27

0.18

0.16