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Optimal Doubling for Pure Running Game

Posted By: Curtis Simmons
Date: Tuesday, 29 May 2007, at 12:51 p.m.

Cut-Off Points for the Optimal Strategy1,2

For the pure running game only

Algorithms are based on CPW

by Curtis Simmons

Double:

Minimum difference needed to double, also the percentage to win:

10 20 30 40 50 60 70 80 90 100 110 =x

-2 -1 +2 +3 +4 +5 +6 +7 +8 +9 +10 =y-x

50 64 70 70 70 71 71 72 72 73 73 = %

z=x/10-3 y,z=his extra pips needed to double. x=your total pips. y=x/10-1

Redouble:

Maximum difference at which you can redouble, and you’re winning percentage.

x= 10 20 x= 30 40 50 60 70 80 90 100 110

z= -2 -1 p= +3 +4 +5 +6 +7 +8 +9 +10 +11

% 60 64 %= 67 72 72 72 73 73 74 74 75

z=x/10-3 z,p=his extra pips you need to redouble. x=your total pips. p = x/10

Fold

Minimum pips above yours at which the opponent should fold and his winning percentage.

10 20 30 40 50 60 70 80 90 100 110 =x

2+ 3+ 5+ 6+ 8+ 9 10 11 13+ 14+ 15 =y-x

21 23 24 23 23 23 22 22 21 21% 21 =%

x=your total pips, y=opponents minimum pips to fold. y–x=0.15 x and y=1.15 x.

Necessary Adjustments to “y”, Expressed In Terms Of “x”, For Folding.

1. Subtract one from x if it is 64 to 103.

2. If x is over 103, subtract two.

3. In all cases round up normally at .5 and higher and down at less than .5.

1Calculated from: “On Optimal Doubling in Backgammon” by Norman Zadek and Gary Kobliska. Table 1, pp. 856 and Table 3, pp. 858. Management Science, Vol. 23, No. 8: April 1997.

2And are within 1% or less to: “Optimal Doubling in Backgammon,” by Emmet B. Koeler and Joel Spencer. Table 1, pp. 1068, Table II, pp.1069, and Table III pp. 1069. Operations Research, Vol. 23, No. 5: Nov.-Dec. 1975.

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