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Advanced Backgammon #117 -- Rollout and Variants

Posted By: Igor
Date: Saturday, 10 May 2014, at 4:20 p.m.

In Response To: Advanced Backgammon #117 (Igor)

This is a standard one roll action double. We may assume for simplicity that if blue hits he wins and if blue missed he loses. Since there are 23 hitting rolls, it's a clear double. Since white is winning more than 25% it's a clear take.





White is Player 2

score: 0
pip: 44
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----B--a-a-Bbcce-:0:0:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No double Double/Take
Player Winning Chances: 62.68% (G:8.93% B:0.25%) 63.07% (G:10.85% B:0.34%)
Opponent Winning Chances: 37.32% (G:4.99% B:0.01%) 36.93% (G:5.07% B:0.02%)
Cubeless Equities +0.295 +0.645
Cubeful Equities
No double:+0.261 (-0.202)0.003 (+0.258..+0.264)
Double/Take:+0.4630.007 (+0.456..+0.469)
Double/Pass:+1.000 (+0.537)
Best Cube action: Double / Take
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 47187089
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 35.6 seconds

eXtreme Gammon Version: 2.10

David Presser gave an excellent rule of thumb that one needs at least 20 hitting numbers to double in similar positions. My analysis of variants confirms this rule. Below are two positions with exactly 20 hitting numbers:





White is Player 2

score: 0
pip: 42
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----B---a-aBbcce-:0:0:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No double Double/Take
Player Winning Chances: 53.43% (G:7.91% B:0.23%) 54.21% (G:9.77% B:0.30%)
Opponent Winning Chances: 46.57% (G:7.70% B:0.01%) 45.79% (G:7.25% B:0.02%)
Cubeless Equities +0.073 +0.224
Cubeful Equities
No double:+0.101 (-0.074)0.002 (+0.099..+0.102)
Double/Take:+0.1750.006 (+0.169..+0.180)
Double/Pass:+1.000 (+0.825)
Best Cube action: Double / Take
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 63884256
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 38.2 seconds

eXtreme Gammon Version: 2.10





White is Player 2

score: 0
pip: 41
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----B----aaBbcce-:0:0:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No double Double/Take
Player Winning Chances: 51.61% (G:7.25% B:0.20%) 52.48% (G:8.64% B:0.27%)
Opponent Winning Chances: 48.39% (G:8.49% B:0.01%) 47.52% (G:7.78% B:0.01%)
Cubeless Equities +0.022 +0.121
Cubeful Equities
No double:+0.0980.001 (+0.097..+0.099)
Double/Take:+0.091 (-0.007)0.002 (+0.089..+0.093)
Double/Pass:+1.000 (+0.902)
Best Cube action: No double / Take
Percentage of wrong pass needed to make the double decision right: 0.8%
Rollout details
10368 Games rolled with Variance Reduction.
Dice Seed: 63884256
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 4 minutes 22 seconds

eXtreme Gammon Version: 2.10

I think it's not hard to figure out why the second position is a marginal no double: after blue misses white will play on for the gammon, while in the first position white will cash. An average missing roll in the first variant is 10.6 pips and an average missing roll in the second variant is 11.25 pips. So, here are the typical positions after blue misses in each case:





White is Player 2

score: 0
pip: 42
Unlimited Game
Jacoby Beaver
pip: 114
score: 0

Blue is Player 1
XGID=---BBBBCAA-------a-aBbcce-:1:-1:-1:00:0:0:3:0:10
White on roll, cube action?

Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 92.31% (G:13.05% B:0.00%) 92.73% (G:12.98% B:0.00%)
Opponent Winning Chances: 7.69% (G:0.14% B:0.00%) 7.27% (G:0.13% B:0.00%)
Cubeless Equities +0.975 +1.966
Cubeful Equities
No redouble:+0.979 (-0.021)0.002 (+0.977..+0.980)
Redouble/Take:+1.918 (+0.918)0.003 (+1.915..+1.920)
Redouble/Pass:+1.000
Best Cube action: Redouble / Pass
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 63884256
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 19.4 seconds

eXtreme Gammon Version: 2.10





White is Player 2

score: 0
pip: 41
Unlimited Game
Jacoby Beaver
pip: 113
score: 0

Blue is Player 1
XGID=---BBBBCB---------aaBbcce-:1:-1:-1:00:0:0:3:0:10
White on roll, cube action?

Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 95.09% (G:13.36% B:0.00%) 95.19% (G:13.31% B:0.00%)
Opponent Winning Chances: 4.91% (G:0.04% B:0.00%) 4.81% (G:0.05% B:0.00%)
Cubeless Equities +1.035 +2.073
Cubeful Equities
No redouble:+1.0380.001 (+1.037..+1.040)
Redouble/Take:+2.057 (+1.019)0.002 (+2.055..+2.059)
Redouble/Pass:+1.000 (-0.038)
Best Cube action: Too good to redouble / Pass
Percentage of wrong take needed to make the double decision right: 3.6%
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 63884256
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 17.9 seconds

eXtreme Gammon Version: 2.10

Next, when can white still take? She certainly needs to win at least 25%, but counting a certain nontrivial number of gammon wins for blue, white can take as long as blue has fewer than 27 hitting numbers. Here are the variants with 26 and 27 hitting rolls for blue, respectively:





White is Player 2

score: 0
pip: 47
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----B-aa---Bbcce-:0:0:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No double Double/Take
Player Winning Chances: 70.79% (G:11.14% B:0.36%) 71.58% (G:13.98% B:0.43%)
Opponent Winning Chances: 29.21% (G:4.67% B:0.01%) 28.42% (G:4.84% B:0.02%)
Cubeless Equities +0.484 +1.054
Cubeful Equities
No double:+0.422 (-0.359)0.004 (+0.419..+0.426)
Double/Take:+0.7810.008 (+0.773..+0.789)
Double/Pass:+1.000 (+0.219)
Best Cube action: Double / Take
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 47187089
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 46.9 seconds

eXtreme Gammon Version: 2.10





White is Player 2

score: 0
pip: 48
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----Ba-a---Bbcce-:0:0:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No double Double/Take
Player Winning Chances: 75.38% (G:12.75% B:0.42%) 76.09% (G:15.88% B:0.48%)
Opponent Winning Chances: 24.62% (G:3.50% B:0.01%) 23.91% (G:3.80% B:0.02%)
Cubeless Equities +0.604 +1.294
Cubeful Equities
No double:+0.544 (-0.456)0.003 (+0.541..+0.547)
Double/Take:+1.099 (+0.099)0.008 (+1.091..+1.107)
Double/Pass:+1.000
Best Cube action: Double / Pass
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 47187089
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 1 minute 28 seconds

eXtreme Gammon Version: 2.10

Finally, when is blue good enough to redouble? There is a bit of a grey zone between 20 and 22 shots; the decisions are so close that one may use practical and score considerations to make proper choice.

20 shots:





White is Player 2

score: 0
pip: 42
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----B---a-aBbcce-:1:1:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 52.71% (G:10.66% B:0.34%) 54.20% (G:9.71% B:0.29%)
Opponent Winning Chances: 47.29% (G:7.20% B:0.00%) 45.80% (G:7.32% B:0.01%)
Cubeless Equities +0.092 +0.221
Cubeful Equities
No redouble:+0.1840.002 (+0.182..+0.185)
Redouble/Take:+0.174 (-0.010)0.003 (+0.170..+0.177)
Redouble/Pass:+1.000 (+0.816)
Best Cube action: No redouble / Take
Percentage of wrong pass needed to make the double decision right: 1.2%
Rollout details
5184 Games rolled with Variance Reduction.
Dice Seed: 63884256
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 9 minutes 35 seconds

eXtreme Gammon Version: 2.10

21 shots:





White is Player 2

score: 0
pip: 43
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----B--a--aBbcce-:1:1:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 55.55% (G:11.48% B:0.37%) 56.96% (G:10.67% B:0.35%)
Opponent Winning Chances: 44.45% (G:6.55% B:0.01%) 43.04% (G:6.74% B:0.01%)
Cubeless Equities +0.164 +0.364
Cubeful Equities
No redouble:+0.259 (-0.032)0.003 (+0.256..+0.262)
Redouble/Take:+0.2910.006 (+0.284..+0.297)
Redouble/Pass:+1.000 (+0.709)
Best Cube action: Redouble / Take
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 63884256
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 2 minutes 21 seconds

eXtreme Gammon Version: 2.10





White is Player 2

score: 0
pip: 43
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----B---aa-Bbcce-:1:1:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 57.47% (G:11.07% B:0.31%) 59.11% (G:10.35% B:0.29%)
Opponent Winning Chances: 42.53% (G:5.24% B:0.00%) 40.89% (G:5.42% B:0.01%)
Cubeless Equities +0.211 +0.469
Cubeful Equities
No redouble:+0.3100.003 (+0.307..+0.313)
Redouble/Take:+0.287 (-0.023)0.006 (+0.282..+0.293)
Redouble/Pass:+1.000 (+0.690)
Best Cube action: No redouble / Take
Percentage of wrong pass needed to make the double decision right: 3.1%
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 63884256
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 2 minutes 20 seconds

eXtreme Gammon Version: 2.10

22 shots:





White is Player 2

score: 0
pip: 44
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----B-a---aBbcce-:1:1:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 58.15% (G:12.02% B:0.39%) 60.20% (G:10.79% B:0.33%)
Opponent Winning Chances: 41.85% (G:5.96% B:0.01%) 39.80% (G:6.07% B:0.01%)
Cubeless Equities +0.227 +0.509
Cubeful Equities
No redouble:+0.3510.002 (+0.349..+0.353)
Redouble/Take:+0.344 (-0.007)0.003 (+0.341..+0.347)
Redouble/Pass:+1.000 (+0.649)
Best Cube action: No redouble / Take
Percentage of wrong pass needed to make the double decision right: 1.0%
Rollout details
5184 Games rolled with Variance Reduction.
Dice Seed: 63884256
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 8 minutes 15 seconds

eXtreme Gammon Version: 2.10

23 shots:





White is Player 2

score: 0
pip: 44
Unlimited Game
Jacoby Beaver
pip: 124
score: 0

Blue is Player 1
XGID=---BBBBBA----B--a-a-Bbcce-:1:1:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 61.65% (G:11.85% B:0.37%) 63.03% (G:10.99% B:0.33%)
Opponent Winning Chances: 38.35% (G:4.77% B:0.01%) 36.97% (G:5.09% B:0.01%)
Cubeless Equities +0.307 +0.646
Cubeful Equities
No redouble:+0.416 (-0.050)0.004 (+0.413..+0.420)
Redouble/Take:+0.4660.007 (+0.459..+0.473)
Redouble/Pass:+1.000 (+0.534)
Best Cube action: Redouble / Take
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 63884256
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 2 minutes 35 seconds

eXtreme Gammon Version: 2.10

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