White is Player 2 score: 0 pip: 137 Unlimited Game Jacoby Beaver pip: 164 score: 0 Blue is Player 1
XGID=---aaCBB-Cb--B--ab-cc-bBA-:0:0:1:65:0:0:3:0:10
Blue to play 65

You have the Blue checkers. How do you play 65, for...

.....(1) Money ?

.....(2) DMP ?

.....(3) Gammon Save ?

.....(4) Gammon Go ?

Thanks to Christian for his insightful comments. He nailed the correct plays at all four scores.

Below is a nacbrac summary of the rollouts (one line for each score).

.........m[@ X33 K68 R96] "<=5
..........d[X @5 K25 R67] <130*5,.....d[X @11] ^5
..........s[R @31 X70 K176] "<5
..........g[K X12 @71 R141] "<28*5

Key:..@ (anchor) = 24/18 23/18, X (hit-and-split) = 24/18 9/4*, K (Kill) = 9/4* 9/3*, R (Run) = 24/13.
Nactation tutorial ....nacbracs explanation

Each of the four main scores has a different best play. In fact, this is not only a T4 (technical four-score position). It is also a "R4" (real four-score position), meaning that for each of money, DMP, gammon save and gammon go, the best play clears its respective .010, .0045, .007 and .007 "tied" margin (compared to any play that is best at some other score).

The power rating for this R4 is [sqrt33 + sqrt 5*20/9 + sqrt31*10/7 + sqrt 12*10/7] = 19.87. To compare to the power ratings of other R4's, click here (scroll to the bottom). Also not yet included in that consolidation are R4's with rolls of 41, 43, 64, 44, 65, 21, 54, 52 and 31.

Detailed rollout results follow.

Money

		White is Player 2 score: 0 pip: 137 Unlimited Game Jacoby Beaver pip: 164 score: 0 Blue is Player 1
XGID=---aaCBB-Cb--B--ab-cc-bBA-:0:0:1:65:0:0:3:0:10
.43D-55P-11N-41S-43O-31P-41K-11E_54H-41H- -54H-41H-64R-63H-42Z-62R-21B-32U-65

1.	Rollout1	24/18 23/18	eq: -0.163
	Player: Opponent: 47.21% (G:12.21% B:0.55%) 52.79% (G:20.74% B:0.75%) Conf.: ± 0.009 (-0.172...-0.154) - [100.0%] Duration: 1 hour 11 minutes
2.	Rollout1	24/18 9/4*	eq: -0.196 (-0.033)
	Player: Opponent: 47.15% (G:14.88% B:0.85%) 52.85% (G:23.40% B:1.76%) Conf.: ± 0.009 (-0.205...-0.187) - [0.0%] Duration: 1 hour 04 minutes
3.	Rollout1	9/4* 9/3*	eq: -0.232 (-0.068)
	Player: Opponent: 46.25% (G:17.13% B:1.00%) 53.75% (G:27.25% B:3.08%) Conf.: ± 0.010 (-0.241...-0.222) - [0.0%] Duration: 55 minutes 24 seconds
4.	Rollout1	24/13	eq: -0.260 (-0.096)
	Player: Opponent: 43.68% (G:11.72% B:0.50%) 56.32% (G:16.03% B:0.79%) Conf.: ± 0.008 (-0.267...-0.252) - [0.0%] Duration: 1 hour 09 minutes
1 5184 Games rolled with Variance Reduction. Dice Seed: 6991078 Moves: 3-ply, cube decisions: XG Roller

DMP, 3-ply

		White is Player 2 score: 0 pip: 137 1 point match pip: 164 score: 0 Blue is Player 1
XGID=---aaCBB-Cb--B--ab-cc-bBA-:0:0:1:65:0:0:0:1:10
Blue to play 65

1.	Rollout1	24/18 9/4*	eq: -0.037
	Player: Opponent: 48.13% (G:15.13% B:2.10%) 51.87% (G:28.15% B:7.22%) Conf.: ± 0.001 (-0.038...-0.037) - [100.0%] Duration: 1 day 00 hour 08 minutes
2.	Rollout1	24/18 23/18	eq: -0.042 (-0.005)
	Player: Opponent: 47.90% (G:11.84% B:1.20%) 52.10% (G:23.13% B:2.24%) Conf.: ± 0.001 (-0.043...-0.041) - [0.0%] Duration: 20 hours 51 minutes
3.	Rollout2	9/4* 9/3*	eq: -0.063 (-0.025)
	Player: Opponent: 46.85% (G:17.62% B:2.97%) 53.15% (G:30.81% B:10.03%) Conf.: ± 0.004 (-0.067...-0.059) - [0.0%] Duration: 53 minutes 16 seconds
4.	Rollout2	24/13	eq: -0.105 (-0.067)
	Player: Opponent: 44.77% (G:12.04% B:1.16%) 55.23% (G:22.41% B:3.01%) Conf.: ± 0.003 (-0.107...-0.102) - [0.0%] Duration: 43 minutes 26 seconds
1 130896 Games rolled with Variance Reduction. Dice Seed: 66734220 Moves: 3-ply, cube decisions: XG Roller 2 5184 Games rolled with Variance Reduction. Dice Seed: 66734220 Moves: 3-ply, cube decisions: XG Roller

DMP, 4-ply

		White is Player 2 score: 0 pip: 137 1 point match pip: 164 score: 0 Blue is Player 1
XGID=---aaCBB-Cb--B--ab-cc-bBA-:0:0:1:65:0:0:0:1:10
Blue to play 65

1.	Rollout1	24/18 9/4*	eq: -0.032
	Player: Opponent: 48.42% (G:15.35% B:2.07%) 51.58% (G:27.96% B:7.63%) Conf.: ± 0.003 (-0.034...-0.029) - [100.0%] Duration: 11 hours 00 minute
2.	Rollout1	24/18 23/18	eq: -0.042 (-0.011)
	Player: Opponent: 47.89% (G:11.68% B:1.14%) 52.11% (G:23.49% B:2.07%) Conf.: ± 0.002 (-0.044...-0.040) - [0.0%] Duration: 10 hours 25 minutes
1 5184 Games rolled with Variance Reduction. Dice Seed: 54253448 Moves and cube decisions: 4-ply

Gammon Save

		White is Player 2 score: 0 pip: 137 2 point match Crawford pip: 164 score: 1 Blue is Player 1
XGID=---aaCBB-Cb--B--ab-cc-bBA-:0:0:1:65:1:0:1:2:10
Blue to play 65

1.	Rollout1	24/13	eq: -0.421
	Player: Opponent: 41.96% (G:10.64% B:1.09%) 58.04% (G:12.99% B:2.08%) Conf.: ± 0.004 (-0.424...-0.417) - [100.0%] Duration: 28 minutes 57 seconds
2.	Rollout1	24/18 23/18	eq: -0.452 (-0.031)
	Player: Opponent: 47.04% (G:11.02% B:1.10%) 52.96% (G:19.61% B:1.21%) Conf.: ± 0.004 (-0.456...-0.447) - [0.0%] Duration: 31 minutes 00 second
3.	Rollout1	24/18 9/4*	eq: -0.491 (-0.070)
	Player: Opponent: 46.16% (G:14.39% B:1.91%) 53.84% (G:20.70% B:6.40%) Conf.: ± 0.005 (-0.495...-0.486) - [0.0%] Duration: 34 minutes 25 seconds
4.	Rollout1	9/4* 9/3*	eq: -0.597 (-0.176)
	Player: Opponent: 45.66% (G:16.78% B:2.50%) 54.34% (G:25.51% B:11.13%) Conf.: ± 0.005 (-0.602...-0.592) - [0.0%] Duration: 34 minutes 48 seconds
1 5184 Games rolled with Variance Reduction. Dice Seed: 54253446 Moves: 3-ply, cube decisions: XG Roller

Gammon Go

		White is Player 2 score: 1 pip: 137 2 point match Crawford pip: 164 score: 0 Blue is Player 1
XGID=---aaCBB-Cb--B--ab-cc-bBA-:0:0:1:65:0:1:1:2:10
Blue to play 65

1.	Rollout1	9/4* 9/3*	eq: +0.287
	Player: Opponent: 46.66% (G:17.67% B:2.32%) 53.34% (G:30.84% B:10.55%) Conf.: ± 0.002 (+0.285...+0.289) - [100.0%] Duration: 4 hours 08 minutes
2.	Rollout1	24/18 9/4*	eq: +0.275 (-0.012)
	Player: Opponent: 47.96% (G:15.80% B:1.88%) 52.04% (G:28.55% B:7.29%) Conf.: ± 0.002 (+0.273...+0.277) - [0.0%] Duration: 4 hours 02 minutes
3.	Rollout2	24/18 23/18	eq: +0.216 (-0.071)
	Player: Opponent: 47.38% (G:13.40% B:1.06%) 52.62% (G:24.09% B:2.34%) Conf.: ± 0.004 (+0.211...+0.220) - [0.0%] Duration: 51 minutes 47 seconds
4.	Rollout2	24/13	eq: +0.146 (-0.141)
	Player: Opponent: 44.35% (G:12.93% B:1.10%) 55.65% (G:23.32% B:3.74%) Conf.: ± 0.004 (+0.142...+0.150) - [0.0%] Duration: 50 minutes 10 seconds
1 28512 Games rolled with Variance Reduction. Dice Seed: 6991078 Moves: 3-ply, cube decisions: XG Roller 2 5184 Games rolled with Variance Reduction. Dice Seed: 6991078 Moves: 3-ply, cube decisions: XG Roller

-----------------------------------------------------------------------------------------------------------------------

		White is Player 2 score: 0 pip: 136 Unlimited Game Jacoby Beaver pip: 164 score: 0 Blue is Player 1
XGID=---aaCBB-Cb--B-a-b-cb-cBA-:0:0:1:65:0:0:3:0:10
Money, 3-ply

1.	Rollout1	24/18 23/18	eq: -0.0489
	Player: Opponent: 49.98% (G:13.32% B:0.60%) 50.02% (G:19.49% B:0.63%) Conf.: ±0.0017 (-0.0506...-0.0472) - [100.0%] Duration: 1 day 01 hour 42 minutes
2.	Rollout1	24/18 9/4*	eq: -0.0604 (-0.0115)
	Player: Opponent: 50.23% (G:16.40% B:0.96%) 49.77% (G:22.17% B:1.55%) Conf.: ±0.0020 (-0.0624...-0.0584) - [0.0%] Duration: 1 day 01 hour 20 minutes
3.	Rollout2	9/4* 9/3*	eq: -0.0992 (-0.0502)
	Player: Opponent: 49.33% (G:18.50% B:1.13%) 50.67% (G:25.96% B:2.96%) Conf.: ±0.0095 (-0.1087...-0.0896) - [0.0%] Duration: 44 minutes 57 seconds
4.	Rollout2	24/13	eq: -0.1298 (-0.0809)
	Player: Opponent: 46.93% (G:12.55% B:0.54%) 53.07% (G:15.22% B:0.71%) Conf.: ±0.0075 (-0.1373...-0.1222) - [0.0%] Duration: 59 minutes 14 seconds
1 130896 Games rolled with Variance Reduction. Dice Seed: 66900508 Moves: 3-ply, cube decisions: XG Roller 2 5184 Games rolled with Variance Reduction. Dice Seed: 66900508 Moves: 3-ply, cube decisions: XG Roller

		White is Player 2 score: 0 pip: 136 Unlimited Game Jacoby Beaver pip: 164 score: 0 Blue is Player 1
XGID=---aaCBB-Cb--B-a-b-cb-cBA-:0:0:1:65:0:0:3:0:10
Money, 4-ply

1.	Rollout1	24/18 23/18	eq: -0.0462
	Player: Opponent: 50.14% (G:13.56% B:0.62%) 49.86% (G:20.02% B:0.67%) Conf.: ±0.0012 (-0.0474...-0.0450) - [100.0%] Duration: 3 days 15 hours 53 minutes
2.	Rollout1	24/18 9/4*	eq: -0.0532 (-0.0070)
	Player: Opponent: 50.33% (G:16.48% B:0.95%) 49.67% (G:22.12% B:1.52%) Conf.: ±0.0015 (-0.0547...-0.0517) - [0.0%] Duration: 3 days 14 hours 25 minutes
1 156816 Games rolled with Variance Reduction. Dice Seed: 96983194 Moves and cube decisions: 4-ply

Above is a companion position (reachable by perfect play for money with the sequence 43D-55P-11N-41S-43O-31P-41K-31-31U-41H-31H-41@-61B-63X-64H-33F-21@-42R-65) that does not quite qualify as an R4, at least not yet. The two rollouts (side-by-side, if your window is wide) are for Money. With 3-ply, the .010 threshold appears to be met, though there is enough variance to perhaps reverse it. With 4-ply, the threshold is not yet met, though enough variance remains that there is a glimmer of hope. As time permits, I may further extend the rollouts.

Below is a nacbrac summary of the rollouts of this variant at all four scores (one line for each score).

.........m[@ 11.5 K50 R81] "<=130*5, .....m[@ X7.1] ^156
..........d[X @9 K16 R65] <10*5
..........s[R @26 X68 K164] "<5
..........g[K X18 @88 R164] "<5

Key:..@ (anchor) = 24/18 23/18, X (hit-and-split) = 24/18 9/4*, K (Kill) = 9/4* 9/3*, R (Run) = 24/13.
Nactation tutorial ....nacbracs explanation

-----------------------------------------------------------------------------------------------------------------------

A few months ago, Timothy Chow ran across the position below and shared it with me. He noted that according to XGR++ evaluation, three scores are evaluated as technically diverse. That is, by evaluation at least, the position is a T3. This is fodder I can sometimes, with perseverance, tweak and elevate to a R4 (real four-score position).

		White is Player 2 score: 0 pip: 110 Unlimited Game Jacoby Beaver pip: 161 score: 0 Blue is Player 1
XGID=----BBBBA--b-C-ad--bbbbBA-:1:1:1:65:0:0:3:0:10
Blue to play 65

For money, there is no cube position that generates a different best play from the other three scores (DMP, gammon save, gammon go). Even if there were, a basic qualification I set for T4 and R4 positions is that the cube is still in the middle. Hence, the first thing I did was to center the cube. However, Blue will be unable to accept the cube here regardless of his play.

To bring the money score into the fold, my next step was to weaken White's board. Eventually, with multiple adjustments, I found a few positions that qualify as a T4, but none that roll out to a R4. The fundamental obstacle is lack of separation of DMP from Gammon Go; their incentives are too similar here.

Reviewing my collection, I saw no R4 in which the player off roll has zero checkers behind the 8pt of the player on roll (the closest being here and here). I stubbornly decided to challenge myself with this new condition, sometimes setting aside the project for weeks at a time. Ultimately I gave up, and resorted to putting a couple of White checkers behind Blue's offense, which led to the R4 featured in this post.

I have yet to construct R4's with rolls of 32, 42, 51, 61, 62 and 63. If you run across what appears to be a T3 or T4 position with one of these rolls, please let me know and perhaps I can adjust from there.

Nack

Messages In This Thread

• FOUR SCORE Position, Rollouts for Shesh Besh (65)
  Nack Ballard -- Friday, 7 September 2018, at 6:06 p.m.
  • FOUR SCORE Position, Rollouts for Shesh Besh (65)
    Joe Russell -- Monday, 10 September 2018, at 9:16 a.m.
    • FOUR SCORE Position, Rollouts for Shesh Besh (65)
      Nack Ballard -- Thursday, 13 September 2018, at 12:37 a.m.

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