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Formlas for Doubling, Redoubling and Folding.

Posted By: Curtis Simmons
Date: Wednesday, 23 May 2007, at 9:06 a.m.

Cut-Off Points for the Optimal Strategy1,2 For the pure running game only Algorithms based on CPW by Curtis Simmons

Double: Minimum difference needed to double, also the percentage to win: x= 10 20 30 40 50 60 70 80 90 100 110

-2 -1=z y= +2 +3 +4 +5 +6 +7 +8 +9 +10 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 30 40 50 60 70 80 90 100 110

-2 -1=z 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. x= 10 20 30 40 50 60 70 80 90 100 110 y-x= +2 +3 +5 +6 +8 +9 +10 +11 +13 +14 +15

21% 23% 24% 23% 23% 23% 22% 22% 21% 21% 21% x=your total pips, y=opponents minimum pips to fold. y–x=0.15x and y=1.15x.

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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