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Four score and seven years ago...

Posted By: Nack Ballard
Date: Saturday, 24 September 2011, at 5:49 a.m.

About seven years ago, I started paying serious attention to positions that might have a different best play at each of the four main scores (money, dmp, save, and go). I call these "technical four-score positions" or "T4's."

With some experimentation, a T4 is not too hard to find according to evaluations, but the rollouts always seem to contradict, with two (or more) of the scores invariably sharing one play. T4's do exist, though, just as the planets do sometimes align.

More difficult yet to find is an "R4," a "real four-score position." This is one in which no best play is even a "tied" candidate at any of the other three scores. [The term "tied" is based on the Ballard/Weaver error scale, referring to a play that is closer in equity (compared to the best play) than .01 for money, or closer than .007 at (gammon) save or (gammon) go, or closer than .0045 at dmp.]

In 2006, I found what I believed to be an R4, and Paul and I published it in Backgammon Openings a year later. This alleged R4 was verified by Stick's GnuBG rollouts -- which were unfortunately done with 14.3 devel, a version that was discovered years later to be corrupt. Stick's XG2 rollout and his new uncorrupted GnuBG rollout demonstrate that in fact the position is not even a T4, let alone an R4. In an effort to accommodate, Tim (who originally discovered the bust) tried an XG2 rollout with the cube turned, which might cause the position to be a T4 with an extension or higher-ply rollout, but it has always been my intention to have the cube centered. Besides, I prefer the greater challenge of an R4.

Does an R4 exist? Yes. Last year I created one with double 1s, sitting under my hat for now. In addition, a couple of days ago I found a variant of the published double 3s position that works. Below are the supporting rollouts for the latter position (with a nacbrac summary before and afterwards).

..[V B13 @23 K74] "<=62*15 5
d[@ V16 B43 K49] "<=5
s[B V35 @168 K263] "<=5
g[K @39 V52 B63] "<=5

Key: V = Variant-up (24/21(2) 24/18), B = Both (24/21(2) 13/10(2)), @ = anchor (24/18(2)), K = Kill (13/4*/1*). Margins are in thousandths. Brackets are followed by "<= (referring to XG2 3-ply checker, roller cube) and the number of trials in thousands, rounded down. (Asterisk repeats a number.)

money

		White is Player 2 score: 0 pip: 127 Unlimited Game Jacoby Beaver pip: 143 score: 0 Blue is Player 1
XGID=-aACaBD------B-bbb-c--bbC-:0:0:1:33:0:0:3:0:10
money

1.	Rollout1	24/21(2) 24/18	eq: -0.305
	Player: Opponent: 42.45% (G:12.68% B:0.60%) 57.55% (G:17.72% B:0.72%) Conf.: ± 0.002 (-0.307...-0.303) - [100.0%] Duration: 6 hours 14 minutes
2.	Rollout1	24/21(2) 13/10(2)	eq: -0.317 (-0.013)
	Player: Opponent: 41.17% (G:12.84% B:0.45%) 58.83% (G:14.88% B:0.65%) Conf.: ± 0.002 (-0.320...-0.315) - [0.0%] Duration: 6 hours 51 minutes
3.	Rollout2	24/18(2)	eq: -0.327 (-0.023)
	Player: Opponent: 43.35% (G:13.24% B:0.62%) 56.65% (G:23.90% B:1.03%) Conf.: ± 0.004 (-0.331...-0.324) - [0.0%] Duration: 1 hour 23 minutes
4.	Rollout3	13/4* 4/1*	eq: -0.378 (-0.074)
	Player: Opponent: 40.44% (G:16.84% B:0.42%) 59.56% (G:28.33% B:4.79%) Conf.: ± 0.012 (-0.390...-0.367) - [0.0%] Duration: 26 minutes 20 seconds
1 62208 Games rolled with Variance Reduction. Dice Seed: 724 Moves: 3-ply, cube decisions: XG Roller 2 15552 Games rolled with Variance Reduction. Dice Seed: 724 Moves: 3-ply, cube decisions: XG Roller 3 5184 Games rolled with Variance Reduction. Dice Seed: 724 Moves: 3-ply, cube decisions: XG Roller

eXtreme Gammon Version: 2.00

dmp

		White is Player 2 score: 0 pip: 127 1 point match pip: 143 score: 0 Blue is Player 1
XGID=-aACaBD------B-bbb-c--bbC-:0:0:1:33:0:0:0:1:10
dmp

1.	Rollout1	24/18(2)	eq: -0.128
	Player: Opponent: 43.62% (G:11.90% B:0.86%) 56.38% (G:23.09% B:2.26%) Conf.: ± 0.002 (-0.130...-0.125) - [100.0%] Duration: 26 minutes 36 seconds
2.	Rollout1	24/21(2) 24/18	eq: -0.143 (-0.016)
	Player: Opponent: 42.83% (G:12.92% B:1.01%) 57.17% (G:22.57% B:1.78%) Conf.: ± 0.003 (-0.146...-0.141) - [0.0%] Duration: 29 minutes 29 seconds
3.	Rollout1	24/21(2) 13/10(2)	eq: -0.171 (-0.043)
	Player: Opponent: 41.47% (G:12.64% B:0.88%) 58.53% (G:18.38% B:1.62%) Conf.: ± 0.003 (-0.173...-0.168) - [0.0%] Duration: 27 minutes 03 seconds
4.	Rollout1	13/4* 4/1*	eq: -0.176 (-0.049)
	Player: Opponent: 41.18% (G:16.58% B:0.85%) 58.82% (G:29.92% B:13.40%) Conf.: ± 0.003 (-0.179...-0.173) - [0.0%] Duration: 27 minutes 13 seconds
1 5184 Games rolled with Variance Reduction. Dice Seed: 724 Moves: 3-ply, cube decisions: XG Roller

eXtreme Gammon Version: 2.00, MET: Kazaross XG2

(gammon) save

		White is Player 2 score: 0 pip: 127 2 point match Crawford pip: 143 score: 1 Blue is Player 1
XGID=-aACaBD------B-bbb-c--bbC-:0:0:1:33:1:0:1:2:10
(gammon) save

1.	Rollout1	24/21(2) 13/10(2)	eq: -0.474
	Player: Opponent: 41.38% (G:13.55% B:1.38%) 58.62% (G:15.09% B:1.39%) Conf.: ± 0.004 (-0.478...-0.471) - [100.0%] Duration: 24 minutes 50 seconds
2.	Rollout1	24/21(2) 24/18	eq: -0.509 (-0.035)
	Player: Opponent: 42.24% (G:13.83% B:1.32%) 57.76% (G:17.67% B:1.81%) Conf.: ± 0.003 (-0.512...-0.505) - [0.0%] Duration: 26 minutes 42 seconds
3.	Rollout1	24/18(2)	eq: -0.643 (-0.168)
	Player: Opponent: 43.69% (G:14.67% B:1.69%) 56.31% (G:25.82% B:1.45%) Conf.: ± 0.004 (-0.646...-0.639) - [0.0%] Duration: 23 minutes 05 seconds
4.	Rollout1	13/4* 4/1*	eq: -0.737 (-0.263)
	Player: Opponent: 40.37% (G:16.69% B:0.81%) 59.63% (G:27.21% B:12.13%) Conf.: ± 0.005 (-0.741...-0.732) - [0.0%] Duration: 23 minutes 17 seconds
1 5184 Games rolled with Variance Reduction. Dice Seed: 724 Moves: 3-ply, cube decisions: XG Roller

eXtreme Gammon Version: 2.00, MET: Kazaross XG2

(gammon) go

		White is Player 2 score: 1 pip: 127 2 point match Crawford pip: 143 score: 0 Blue is Player 1
XGID=-aACaBD------B-bbb-c--bbC-:0:0:1:33:0:1:1:2:10
Blue to play 33

1.	Rollout1	13/4* 4/1*	eq: +0.160
	Player: Opponent: 41.13% (G:16.86% B:0.82%) 58.87% (G:29.68% B:12.90%) Conf.: ± 0.004 (+0.156...+0.164) - [100.0%] Duration: 24 minutes 22 seconds
2.	Rollout1	24/18(2)	eq: +0.120 (-0.039)
	Player: Opponent: 43.61% (G:12.42% B:0.87%) 56.39% (G:23.17% B:2.58%) Conf.: ± 0.003 (+0.117...+0.124) - [0.0%] Duration: 24 minutes 22 seconds
3.	Rollout1	24/21(2) 24/18	eq: +0.107 (-0.052)
	Player: Opponent: 43.09% (G:12.28% B:0.96%) 56.91% (G:21.28% B:1.64%) Conf.: ± 0.004 (+0.104...+0.111) - [0.0%] Duration: 26 minutes 26 seconds
4.	Rollout1	24/21(2) 13/10(2)	eq: +0.097 (-0.063)
	Player: Opponent: 41.96% (G:12.87% B:0.67%) 58.04% (G:19.77% B:2.20%) Conf.: ± 0.004 (+0.093...+0.100) - [0.0%] Duration: 23 minutes 53 seconds
1 5184 Games rolled with Variance Reduction. Dice Seed: 724 Moves: 3-ply, cube decisions: XG Roller

eXtreme Gammon Version: 2.00, MET: Kazaross XG2

Here, again, is a nacbrac summary of the rollouts:

..[V B13 @23 K74] "<=62*15 5
d[@ V16 B43 K49] "<=5
s[B V35 @168 K263] "<=5
g[K @39 V52 B63] "<=5

Key: V = Variant-up (24/21(2) 24/18), B = Both (24/21(2) 13/10(2)), @ = anchor (24/18(2)), K = Kill (13/4*/1*).

Nack

Messages In This Thread

• Four score and seven years ago...
  Nack Ballard -- Saturday, 24 September 2011, at 5:49 a.m.
  • Four score and seven years ago...
    Timothy Chow -- Sunday, 25 September 2011, at 6:02 p.m.
    • Four score and seven years ago...
      Nack Ballard -- Monday, 26 September 2011, at 8:57 p.m.
      • Four score and seven years ago...
        Timothy Chow -- Monday, 26 September 2011, at 10:53 p.m.
        • Four score and seven years ago...
          Nack Ballard -- Tuesday, 27 September 2011, at 12:41 a.m.
          • Four score and seven years ago...
            Timothy Chow -- Tuesday, 27 September 2011, at 4:56 p.m.
            • Four score and seven years ago...
              Nack Ballard -- Tuesday, 27 September 2011, at 5:53 p.m.