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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 Game
Jacoby Beaver
pip: 126
score: 0

Blue is Player 1
XGID=---B-aD-C-B-dD---baeb-----:0:0:1:00:0:0:3:0:10
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 Game
Jacoby Beaver
pip: 126
score: 0

Blue is Player 1
XGID=---B-aD-C-B-cD--cbbbb-----:0:0:1:00:0:0:3:0:10
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 Game
Jacoby Beaver
pip: 126
score: 0

Blue is Player 1
XGID=---B-aD-C-B-cD--bcbbb-----:0:0:1:00:0:0:3:0:10
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:+0.9950.012 (+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 Game
Jacoby Beaver
pip: 126
score: 0

Blue is Player 1
XGID=---B-aD-C-B-bD--cbbcb-----:0:0:1:00:0:0:3:0:10
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:+0.7200.014 (+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 Game
Jacoby Beaver
pip: 126
score: 0

Blue is Player 1
XGID=---B-aD-C-B-bD--bcbcb-----:0:0:1:00:0:0:3:0:10
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:+0.6710.009 (+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 Game
Jacoby Beaver
pip: 126
score: 0

Blue is Player 1
XGID=---B-aD-C-B-bDa-bbbcb-----:1:1:1:00:0:0:3:0:10
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:+0.8110.012 (+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 Game
Jacoby Beaver
pip: 126
score: 0

Blue is Player 1
XGID=---B-aD-C-B-bD-abbbcb-----:1:1:1:00:0:0:3:0:10
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:+0.7750.008 (+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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