In the position diagrammed above (originally posted here), bgonliners were asked:
"What would you play for Blue with double 2s, at each of the four (classically recognized) main scores?"
m (Money)...(as shown)
The best move is E (Each) = 24/22(2) 5/3*(2), selected by Sean Garber. The other supporter of E was ChanningW, who articulated at length: "[With 24/22(2)]...priming me in is nearly impossible. Holding this anchor means even White's great rolls still aren't great enough to offer a cube in the next couple rolls... The other twos I'd use to hit 5/3(2)*... After hitting, white has 6 great [numbers from] the bar: 53, 54, 56. But as long as I have an anchor to land on, I don't feel like white has much of a cube. If he does it's a huge take that has a ton of recube vig. The other rolls where white comes in but doesn't hit really don't help his overall game much. However, if white fans next roll blue has a huge double. Seems to me the reward from hitting far outweighs the risks of getting hit."
Casper (who found Money to be the score that offered the most difficult choice), hedged with: "I think OTB I'd make the same play as at DMP [which is Z, see the next paragraph of notes], but I can imagine that E is best, because it seems to trade some wins for a good combination of net gammons, and maybe Jacoby does its magic on some adverse sequences."
Actually, the decision is a "wipeout" (as termed by the B/W error scale). Here is the rollout:
........m (Money) 3-ply

              
 
              
 | | White is Player 2
score: 0 pip: 133 | Unlimited Game Jacoby Beaver | pip: 138 score: 0
Blue is Player 1 | |
| XGID=-a-aBBCBAA-AAa--abbcbb--B-:0:0:1:22:0:0:3:0:10 |
| Blue to play 22 |
| 1. | Rollout1 | 24/22(2) 5/3*(2) | eq: +0.202 |
| Player: Opponent: | 52.51% (G:19.95% B:1.57%) 47.49% (G:13.39% B:0.79%) | Conf.: ± 0.008 (+0.193...+0.210) - [100.0%] Duration: 2 hours 12 minutes |
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| 2. | Rollout1 | 24/22 12/8 11/9 | eq: +0.100 (-0.101) |
| Player: Opponent: | 52.69% (G:15.33% B:0.82%) 47.31% (G:14.04% B:0.85%) | Conf.: ± 0.008 (+0.092...+0.108) - [0.0%] Duration: 2 hours 01 minute |
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| 3. | Rollout1 | 24/22(2) 12/8 | eq: +0.086 (-0.115) |
| Player: Opponent: | 51.91% (G:13.96% B:0.67%) 48.09% (G:12.26% B:0.44%) | Conf.: ± 0.007 (+0.079...+0.094) - [0.0%] Duration: 2 hours 01 minute |
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| 4. | Rollout1 | 24/22(2) 11/9 8/6 | eq: +0.053 (-0.149) |
| Player: Opponent: | 50.62% (G:15.00% B:0.79%) 49.38% (G:11.51% B:0.45%) | Conf.: ± 0.006 (+0.046...+0.059) - [0.0%] Duration: 2 hours 14 minutes |
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1 5184 Games rolled with Variance Reduction. Dice Seed: 53420863 Moves: 3-ply, cube decisions: XG Roller
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d (DMP)...
The best move is Z (reverse split 1:3) = 24/22 12/8 11/9, backed by Christian, Karl, ah_clem... and Casper, who commented: "6-primes are tremendously strong, and a good way to win the game. You will not win that many gammons, and may lose a few gammons yourself, so that is why you'd look for other plays at the other scores."
Below are rollouts, 3-ply and 4-ply:
........d (DMP) 3-ply

              
 
              
 | | White is Player 2
score: 0 pip: 133 | | 1 point match | pip: 138 score: 0
Blue is Player 1 | |
| XGID=-a-aBBCBAA-AAa--abbcbb--B-:0:0:1:22:0:0:0:1:10 |
| Blue to play 22 |
| 1. | Rollout1 | 24/22 12/8 11/9 | eq: +0.058 |
| Player: Opponent: | 52.91% (G:16.31% B:2.32%) 47.09% (G:15.43% B:3.08%) | Conf.: ± 0.001 (+0.057...+0.060) - [100.0%] Duration: 5 hours 08 minutes |
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| 2. | Rollout1 | 24/22(2) 5/3*(2) | eq: +0.050 (-0.008) |
| Player: Opponent: | 52.50% (G:21.78% B:3.94%) 47.50% (G:15.21% B:1.99%) | Conf.: ± 0.001 (+0.049...+0.051) - [0.0%] Duration: 5 hours 41 minutes |
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| 3. | Rollout2 | 24/22(2) 12/8 | eq: +0.044 (-0.014) |
| Player: Opponent: | 52.22% (G:14.74% B:1.84%) 47.78% (G:14.88% B:1.56%) | Conf.: ± 0.002 (+0.043...+0.046) - [0.0%] Duration: 2 hours 24 minutes |
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| 4. | Rollout3 | 24/22(2) 11/9 8/6 | eq: +0.013 (-0.046) |
| Player: Opponent: | 50.63% (G:16.33% B:2.16%) 49.37% (G:13.26% B:1.05%) | Conf.: ± 0.003 (+0.010...+0.015) - [0.0%] Duration: 1 hour 01 minute |
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1 31104 Games rolled with Variance Reduction. Dice Seed: 27727012 Moves: 3-ply, cube decisions: XG Roller
2 15552 Games rolled with Variance Reduction. Dice Seed: 27727012 Moves: 3-ply, cube decisions: XG Roller
3 5184 Games rolled with Variance Reduction. Dice Seed: 27727012 Moves: 3-ply, cube decisions: XG Roller
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........d (DMP) 4-ply

              
 
              
 | | White is Player 2
score: 0 pip: 133 | | 1 point match | pip: 138 score: 0
Blue is Player 1 | |
| XGID=-a-aBBCBAA-AAa--abbcbb--B-:0:0:1:22:0:0:0:1:10 |
| Blue to play 22 |
| 1. | Rollout1 | 24/22 12/8 11/9 | eq: +0.057 |
| Player: Opponent: | 52.84% (G:16.68% B:2.44%) 47.16% (G:15.25% B:3.15%) | Conf.: ± 0.001 (+0.056...+0.058) - [100.0%] Duration: 3 days 12 hours 49 minutes |
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| 2. | Rollout1 | 24/22(2) 5/3*(2) | eq: +0.050 (-0.007) |
| Player: Opponent: | 52.50% (G:21.71% B:3.87%) 47.50% (G:14.82% B:1.90%) | Conf.: ± 0.001 (+0.049...+0.051) - [0.0%] Duration: 3 days 21 hours 37 minutes |
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1 20736 Games rolled with Variance Reduction. Dice Seed: 93687480 Moves and cube decisions: 4-ply Search interval: Large
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s (Gammon Save)...
The best move is B (Both) = 24/22(2) 12/8, supported by Christian, ChanningW ("...locking up the 5point prime for the rest of the game"), and by Casper, who emphasized anchoring (24/22(2)) then with the other two deuces he weighed three options: "(1) 12/8 makes a 5 prime... but leaves a direct number; (2) 11/9 8/6 makes a broken 5 prime, and leaves no direct shots, and (3) 5/3(2)* leaves no directs, but a few indirects and a lot of blots, and temporarily breaks the prime, while also making it difficult for white to extend the prime... [but] looks wrong at GS. I probably would play pure [with B], but not very sure."
Casper implied (correctly) that the very close second best move is F (Field) = 24/22(2) 11/9 8/6. (This score-thematic shotless play was the first choice of ah_clem and Karl.) By the way, this is not the best move at any of the other three scores either; in the arena of four-score contemplation, F should be regarded as a fifth play that happens to be a strong gammon save alternative.
Below are rollouts, 3-ply and 4-ply:
........s (Gammon Save) 3-ply

              
 
              
 | | White is Player 2
score: 0 pip: 133 | 2 point match Crawford | pip: 138 score: 1
Blue is Player 1 | |
| XGID=-a-aBBCBAA-AAa--abbcbb--B-:0:0:1:22:1:0:1:2:10 |
| Blue to play 22 |
| 1. | Rollout1 | 24/22(2) 12/8 | eq: -0.203 |
| Player: Opponent: | 51.45% (G:14.24% B:1.82%) 48.55% (G:11.62% B:1.23%) | Conf.: ± 0.001 (-0.204...-0.203) - [100.0%] Duration: 14 hours 57 minutes |
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| 2. | Rollout1 | 24/22(2) 11/9 8/6 | eq: -0.211 (-0.007) |
| Player: Opponent: | 50.25% (G:15.39% B:1.73%) 49.75% (G:10.77% B:0.90%) | Conf.: ± 0.001 (-0.211...-0.210) - [0.0%] Duration: 13 hours 57 minutes |
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| 3. | Rollout2 | 24/22 12/8 11/9 | eq: -0.229 (-0.025) |
| Player: Opponent: | 52.52% (G:16.10% B:2.17%) 47.48% (G:13.95% B:2.25%) | Conf.: ± 0.004 (-0.233...-0.224) - [0.0%] Duration: 46 minutes 41 seconds |
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| 4. | Rollout2 | 24/22(2) 5/3*(2) | eq: -0.236 (-0.032) |
| Player: Opponent: | 52.23% (G:22.26% B:4.19%) 47.77% (G:14.02% B:2.06%) | Conf.: ± 0.004 (-0.240...-0.232) - [0.0%] Duration: 51 minutes 15 seconds |
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1 103760 Games rolled with Variance Reduction. Dice Seed: 27727012 Moves: 3-ply, cube decisions: XG Roller
2 5184 Games rolled with Variance Reduction. Dice Seed: 27727012 Moves: 3-ply, cube decisions: XG Roller
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........s (Gammon Save) 4-ply

              
 
              
 | | White is Player 2
score: 0 pip: 133 | 2 point match Crawford | pip: 138 score: 1
Blue is Player 1 | |
| XGID=-a-aBBCBAA-AAa--abbcbb--B-:0:0:1:22:1:0:1:2:10 |
| Blue to play 22 |
| 1. | Rollout1 | 24/22(2) 12/8 | eq: -0.204 |
| Player: Opponent: | 51.53% (G:14.28% B:1.77%) 48.47% (G:11.71% B:1.24%) | Conf.: ± 0.001 (-0.205...-0.203) - [100.0%] Duration: 5 days 01 hour 05 minutes |
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| 2. | Rollout1 | 24/22(2) 11/9 8/6 | eq: -0.210 (-0.007) |
| Player: Opponent: | 50.26% (G:15.08% B:1.72%) 49.74% (G:10.77% B:0.89%) | Conf.: ± 0.001 (-0.211...-0.209) - [0.0%] Duration: 4 days 18 hours 15 minutes |
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1 36288 Games rolled with Variance Reduction. Dice Seed: 53420863 Moves and cube decisions: 4-ply Search interval: Large
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g (Gammon Go)...
The best move is Y (Yosemite) = 24/22 7/3*/1*, turning more than half of Blue's checkers into blots! Casper nailed this one: "This ought to win the most gammons by far, but also loses the most gammons. For the escape of the back checkers, 2s and 6s work at the moment, and we have quite a bit of time to blitz and escape. The hitting escaping 6 is also strong." ah_clem also had the nerve to hit twice with 7/3*/1* but elected to remake his four-prime with 9/7. (This leaves only four blots instead of eight, but fails to threaten the vital 22/16*.)
Here is the rollout:
........g (Gammon Go) 3-ply

              
 
              
 | | White is Player 2
score: 1 pip: 133 | 2 point match Crawford | pip: 138 score: 0
Blue is Player 1 | |
| XGID=-a-aBBCBAA-AAa--abbcbb--B-:0:0:1:22:0:1:1:2:10 |
| Blue to play 22 |
| 1. | Rollout1 | 24/22 7/3* 3/1* | eq: +0.506 |
| Player: Opponent: | 48.68% (G:26.63% B:4.41%) 51.32% (G:21.39% B:6.68%) | Conf.: ± 0.005 (+0.501...+0.511) - [100.0%] Duration: 1 hour 55 minutes |
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| 2. | Rollout1 | 24/22(2) 5/3*(2) | eq: +0.458 (-0.048) |
| Player: Opponent: | 53.29% (G:19.63% B:2.91%) 46.71% (G:14.54% B:1.75%) | Conf.: ± 0.004 (+0.455...+0.462) - [0.0%] Duration: 1 hour 38 minutes |
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1 5184 Games rolled with Variance Reduction. Dice Seed: 53420863 Moves: 3-ply, cube decisions: XG Roller
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The nacbracs summary below lists the best play at each score followed by successively weaker plays (with error sizes), according to the rollouts (diagrammed above). For example, for Money (3-ply), E is best by far, then relatively Z is –.101, B is –.115, and E is –.149.
.................................12pt Variant
...............3-ply........................................4-ply
..... m[E Z101 B115 F149] "<=5
...... d[Z E8 B14 F46] "<=31*15 5 .... d[Z E7] "^20
...... s[B F7 Z25 E32] "<=103*5 ........ s[B F7] "^36
...... g[Y E48] "<=5
Key: B (Both) = 24/22 12/8, b (both) = 24/22(2) 12/10 11/9, E (Each) = 24/22(2) 5/3*(2), F (Field) = 24/22(2) 11/9 8/6, Y (Yosemite) = 24/22 7/3*/1*, Z = (reverse split 1:3) = 24/22 12/8 11/9.
Nactation tutorial • explanation of nacbracs
In the above summary, a quick look down the left column reveals that each of the four scores (m, d, s, g) champions a different best play (E, Z, B, Y). Moreover, the margins of .101, .008, .025 and .048 clearly exceed their respective error-scale cusps of .010, .0045, .007 and .007.
Note that the fifth play, F, is ignored for purposes of four-score appraisal. Hence, the power score (of this double 2s position) is [sqrt 101 + sqrt 8*20/9 + sqrt 25*10/7 + sqrt 48*10/7] = 28.52, the highest R4 power score to date. For more information on four-score positions, error-scale cusps and power scores, click mainly here, also here.
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The above position has a closely related variant, in which White's position differs by 2 pips. In the original position (which I call the "12pt variant"), White's trailing builder/blot is on the 12pt. In the alternate position (the "10pt variant"), that blot has been nudged forward to the 10pt. The nacbracs summary appears below:
.................................10pt Variant
...............3-ply........................................4-ply
..... m[E Z11 B29 F48] "<=62*5 ....... m[E Z10.3] "^103
...... d[Z B8 F32 E40] "<=25*5 .......... d[Z B7] "^41
...... s[B F9 Z32 b61 E83] "<=62*5 .... s[B F9] "^36
...... g[Y E61 Z77] "<=5
This position's power score is [sqrt 11 + sqrt 8*20/9 + sqrt 9*10/7 + sqrt 61*10/7] = 20.45. At least one of the margins is too close to the error-scale cusps for me to feel confident declaring this 10pt-variant position to be a definite R4 (real four-score position), though for the time being it does appear to qualify. (Rollouts of future bots will be needed to confirm that its status is indeed elevated above T4.) Currently, the money margin of .011 (3-ply) or .0103 (4-ply) exceeds its cusp of .010, and the gammon save margin of .009 exceeds its cusp of .007.
Allow me to make a final observation (in case you haven't already noticed). In the original position, E is a "wipeout" (best by .101). By contrast, in this 10pt variant position E is "barely correct" -- best by only .011 (or .010 in the 4-ply rollout). Those 2 pips that reduce White's timing really boost the value of Blue's priming plays!
Detailed rollouts of the 10pt variant position appear below.
Nack
........m (Money) 3-ply
| 1. | Rollout1 | 24/22(2) 5/3*(2) | eq: +0.191 |
| Player: Opponent: | 52.76% (G:18.94% B:1.38%) 47.24% (G:13.61% B:0.81%) | Conf.: ± 0.002 (+0.189...+0.193) - [100.0%] Duration: 14 hours 46 minutes |
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| 2. | Rollout1 | 24/22 12/8 11/9 | eq: +0.180 (-0.011) |
| Player: Opponent: | 54.66% (G:16.05% B:0.96%) 45.34% (G:13.57% B:0.87%) | Conf.: ± 0.002 (+0.178...+0.183) - [0.0%] Duration: 14 hours 11 minutes |
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| 3. | Rollout2 | 24/22(2) 12/8 | eq: +0.162 (-0.029) |
| Player: Opponent: | 54.11% (G:13.69% B:0.62%) 45.89% (G:11.86% B:0.45%) | Conf.: ± 0.007 (+0.155...+0.169) - [0.0%] Duration: 1 hour 13 minutes |
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| 4. | Rollout2 | 24/22(2) 11/9 8/6 | eq: +0.143 (-0.048) |
| Player: Opponent: | 53.05% (G:15.52% B:0.81%) 46.95% (G:11.14% B:0.43%) | Conf.: ± 0.007 (+0.136...+0.150) - [0.0%] Duration: 1 hour 11 minutes |
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1 62208 Games rolled with Variance Reduction. Dice Seed: 53420863 Moves: 3-ply, cube decisions: XG Roller
2 5184 Games rolled with Variance Reduction. Dice Seed: 53420863 Moves: 3-ply, cube decisions: XG Roller
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........m (Money) 4-ply...
(extra decimal place)