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Solution To A More General Question

Posted By: Paul Weaver
Date: Saturday, 1 February 2014, at 4:12 p.m.

In Response To: Advanced Backgammon #34 (Igor)

 White is Player 2 score: 0 pip: 135 Unlimited GameJacoby Beaver pip: 126 score: 0 Blue is Player 1
Blue on roll, cube action?

eXtreme Gammon Version: 2.10

Igor has posted the above cube-action position. I have found the solution to a more general question: Let's keep the checkers on Blue's side fixed, let's give White a 5-prime, and then, let's find the pip-count differences that will give a marginal pass (Position 1), a marginal take (Position 2), a marginal double (Position 3), a marginal no double (Position 4), a marginal redouble (Position 5) and a marginal redouble (Position 6). If you were a really committed student of BG, you might also construct a 9 x 9 grid and determine the cube action at all scores in a 9pt match, including all possible redoubles. I was too lazy to do that.

Position 1. Marginal Pass, 12-pip difference.

 White is Player 2 score: 0 pip: 138 Unlimited GameJacoby Beaver pip: 126 score: 0 Blue is Player 1
Blue on roll, cube action?

 Analyzed in Rollout No double Double/Take Player Winning Chances: 75.37% (G:9.31% B:0.06%) 75.53% (G:10.16% B:0.06%) Opponent Winning Chances: 24.63% (G:1.49% B:0.04%) 24.47% (G:1.52% B:0.05%) Cubeless Equities +0.586 +1.194 Cubeful Equities No double: +0.902 (-0.098) ±0.007 (+0.896..+0.909) Double/Take: +1.030 (+0.030) ±0.011 (+1.019..+1.042) Double/Pass: +1.000 Best Cube action: Double / Pass Rollout details 1296 Games rolled with Variance Reduction.Moves: 3-ply, cube decisions: XG Roller Double Decision confidence: 100.0% Take Decision confidence: 100.0% Duration: 5 minutes 16 seconds

eXtreme Gammon Version: 2.10

Position 2. Marginal Take, 11-pip difference.

 White is Player 2 score: 0 pip: 137 Unlimited GameJacoby Beaver pip: 126 score: 0 Blue is Player 1
Blue on roll, cube action?

 Analyzed in Rollout No double Double/Take Player Winning Chances: 74.39% (G:9.04% B:0.06%) 74.54% (G:10.35% B:0.07%) Opponent Winning Chances: 25.61% (G:1.53% B:0.04%) 25.46% (G:1.77% B:0.06%) Cubeless Equities +0.563 +1.154 Cubeful Equities No double: +0.873 (-0.122) ±0.007 (+0.866..+0.881) Double/Take: +0.995 ±0.012 (+0.983..+1.007) Double/Pass: +1.000 (+0.005) Best Cube action: Double / Take Rollout details 1296 Games rolled with Variance Reduction.Moves: 3-ply, cube decisions: XG Roller Double Decision confidence: 100.0% Take Decision confidence: 80.0% Duration: 5 minutes 40 seconds

eXtreme Gammon Version: 2.10

Position 3. Marginal double, 5-pip difference.

 White is Player 2 score: 0 pip: 131 Unlimited GameJacoby Beaver pip: 126 score: 0 Blue is Player 1
Blue on roll, cube action?

 Analyzed in Rollout No double Double/Take Player Winning Chances: 69.97% (G:6.94% B:0.08%) 70.08% (G:7.40% B:0.09%) Opponent Winning Chances: 30.03% (G:1.80% B:0.06%) 29.92% (G:1.82% B:0.07%) Cubeless Equities +0.451 +0.915 Cubeful Equities No double: +0.714 (-0.006) ±0.011 (+0.703..+0.725) Double/Take: +0.720 ±0.014 (+0.706..+0.734) Double/Pass: +1.000 (+0.280) Best Cube action: Double / Take Rollout details 857 Games rolled with Variance Reduction.Moves: 3-ply, cube decisions: XG Roller Double Decision confidence: 76.1% Take Decision confidence: 100.0% Duration: 4 minutes 10 seconds

eXtreme Gammon Version: 2.10

Position 4. Marginal no double, 4-pip difference.

 White is Player 2 score: 0 pip: 130 Unlimited GameJacoby Beaver pip: 126 score: 0 Blue is Player 1
Blue on roll, cube action?

 Analyzed in Rollout No double Double/Take Player Winning Chances: 68.97% (G:6.47% B:0.07%) 69.16% (G:6.84% B:0.07%) Opponent Winning Chances: 31.03% (G:1.79% B:0.05%) 30.84% (G:1.77% B:0.05%) Cubeless Equities +0.426 +0.868 Cubeful Equities No double: +0.671 ±0.009 (+0.663..+0.680) Double/Take: +0.661 (-0.010) ±0.011 (+0.651..+0.672) Double/Pass: +1.000 (+0.329) Best Cube action: No double / Take Percentage of wrong pass needed to make the double decision right: 2.8% Rollout details 1296 Games rolled with Variance Reduction.Moves: 3-ply, cube decisions: XG Roller Double Decision confidence: 91.6% Take Decision confidence: 100.0% Duration: 6 minutes 29 seconds

eXtreme Gammon Version: 2.10

Position 5. Marginal redouble, 7-pip difference.

 White is Player 2 score: 0 pip: 133 Unlimited GameJacoby Beaver pip: 126 score: 0 Blue is Player 1
Blue on roll, cube action?

 Analyzed in Rollout No redouble Redouble/Take Player Winning Chances: 71.43% (G:7.85% B:0.08%) 71.84% (G:7.74% B:0.08%) Opponent Winning Chances: 28.57% (G:1.76% B:0.05%) 28.16% (G:1.75% B:0.05%) Cubeless Equities +0.490 +0.994 Cubeful Equities No redouble: +0.809 (-0.002) ±0.008 (+0.801..+0.818) Redouble/Take: +0.811 ±0.012 (+0.799..+0.824) Redouble/Pass: +1.000 (+0.189) Best Cube action: Redouble / Take Rollout details 1296 Games rolled with Variance Reduction.Moves: 3-ply, cube decisions: XG Roller Double Decision confidence: 60.5% Take Decision confidence: 100.0% Duration: 6 minutes 05 seconds

eXtreme Gammon Version: 2.10

Position 6. Marginal no redouble, 6-pip difference.

 White is Player 2 score: 0 pip: 132 Unlimited GameJacoby Beaver pip: 126 score: 0 Blue is Player 1
Blue on roll, cube action?

 Analyzed in Rollout No redouble Redouble/Take Player Winning Chances: 70.43% (G:7.59% B:0.07%) 70.72% (G:7.78% B:0.05%) Opponent Winning Chances: 29.57% (G:1.73% B:0.04%) 29.28% (G:1.81% B:0.04%) Cubeless Equities +0.467 +0.948 Cubeful Equities No redouble: +0.775 ±0.008 (+0.766..+0.783) Redouble/Take: +0.758 (-0.017) ±0.011 (+0.747..+0.769) Redouble/Pass: +1.000 (+0.225) Best Cube action: No redouble / Take Percentage of wrong pass needed to make the double decision right: 6.5% Rollout details 1296 Games rolled with Variance Reduction.Moves: 3-ply, cube decisions: XG Roller Double Decision confidence: 99.2% Take Decision confidence: 100.0% Duration: 6 minutes 17 seconds

eXtreme Gammon Version: 2.10

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