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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 GameJacoby 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: +0.463 ±0.007 (+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: 47187089Moves: 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 GameJacoby 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: +0.175 ±0.006 (+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: 63884256Moves: 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 GameJacoby 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: +0.098 ±0.001 (+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: 63884256Moves: 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 GameJacoby 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: 63884256Moves: 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 GameJacoby 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: +1.038 ±0.001 (+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: 63884256Moves: 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 GameJacoby 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: +0.781 ±0.008 (+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: 47187089Moves: 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 GameJacoby 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: 47187089Moves: 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 GameJacoby 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: +0.184 ±0.002 (+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: 63884256Moves: 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 GameJacoby 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: +0.291 ±0.006 (+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: 63884256Moves: 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 GameJacoby 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: +0.310 ±0.003 (+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: 63884256Moves: 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 GameJacoby 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: +0.351 ±0.002 (+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: 63884256Moves: 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 GameJacoby 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: +0.466 ±0.007 (+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: 63884256Moves: 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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