FOUR SCORE Positions -- Consolidation

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FOUR SCORE Positions -- Consolidation

Posted By: Nack Ballard
Date: Tuesday, 19 January 2016, at 2:01 a.m.

A "four-score position" is one (based on the highest quality rollouts that today can be performed practically) for which a different play is best at each of the four classically recognized main scores: m, d, s, and g [which respectively stand for Money (center cube strongly preferred), DMP, gammon save, and gammon go]. These positions are subdivided into two types: "T4" (technical four-score) and "R4" (real four-score).
For a "T4," it is necessary only for each best play to "win" its high quality rollout (i.e., with strong settings, and with enough trials to achieve 100% confidence), no matter by how tiny a margin. Only one T4 has been found as early as the second roll, which is 63S-32, where the best plays at m, d, s and g win the long 4-ply rollouts (over their respective second best moves) by .0018, .0003, .0054 and .0013.
The requirements for an "R4" are more stringent. At each score, the best play cannot be "tied" with any play that is best at any of the other three scores, as defined by the B/W (Ballard/Weaver) error scale published on the final page of Backgammon Openings. The B/W scale states that any play within .01 is tied for Money. According to my empirical research, that translates to .0045 at DMP, and to .007 at gammon save and gammon go. If all four of these equity-difference cusps (.01, .0045, .007, .007) are (equalled or) surpassed, the position qualifies as an R4.
[There is no such thing as a T5 or R5. The number immediately following the T or R is the number of MAIN scores (m, d, s, g) that have a different best play, and that number is capped at four. Following the plus sign is the number of other plays that are best at at least one other score. For example, if play A is clearly best at m and g, play B is clearly best at d and s, play C is tied/best at –2–5, and play D is clearly best at –3–7, –3–8 and –3–9 (no extra credit for these second and third scores), then I would call such a relatively unimpressive position merely a R2+1+T1 (or T2+2).]
In 2007 (prior to XG coming out), the pioneer R4 was published on pages 98–99 of Backgammon Openings (see link above). One of the GnuBG rollouts (done by version 14.3 devel, which I later discovered to be corrupt) was overturned by XG. Fortunately, I was able to restore R4 integrity by adjusting two spares (Blue 13pt to 3pt, White 9pt to 6pt).
Over the last year or so, I've posted eight positions that qualify as R4's (real four-score positions). These are consolidated below (by dice roll rather than by chronology). For each position, the best play (nactated in red) is listed for m, d, s and g, respectively. [Because many plays are complex doublets, unusually high-level Nactation is necessary, so feel free to click on "Details" where you'll also find rollouts and other information.]

11 - Blitz Position................ m I ... d e... s @... g I ........Details.....More Details

White is Player 2

score: 0
pip: 155

Unlimited Game
Jacoby Beaver

pip: 147
score: 0

Blue is Player 1

XGID=-aBaBaB-B-Bb-C--aabbbb--B-:0:0:1:11:0:0:3:0:10
65R-64P-21D-33S-32@-52Z-43H-53Z-51H-43U-
64H-64R-61Z-31P-31D-65R-42D-43H-11Y-64S-
43H-F-64R-31U-54P-21H-51K-11e-53-11
(perfectly played money sequence)........................Nactation tutorial

11 - Prime-v-Prime......... m y ... d D... s y... g I ...–2–4 E ......Details.....More Details

White is Player 2

score: 0
pip: 119

Unlimited Game
Jacoby Beaver

pip: 150
score: 0

Blue is Player 1

XGID=--aaBBBBA---BAA-aacbbbbAA-:0:0:1:11:0:0:3:0:10
32S-43H-22E-22E-54@-44S-41U-51r-52R-11B-
31P-32N-31D-65S-F-66D-63H-F-11D-426-
63r-21K-53-11A.-21-54J-528-21U-11
(perfectly played money sequence)........................Nactation tutorial

22 - 12pt Variant............... m E... d Z... s B... g Y ......Details

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
64S-61P-65R-52D-41P-42P-51S-54R-22N-54R-
63H-21H-31H-F-61S-21H-65R-43D-F-31P-33M-22
(perfectly played money sequence)........................Nactation tutorial

22 - 10pt Variant............... m E... d Z... s B... g Y .....Details.(second half of post)

White is Player 2

score: 0
pip: 131

Unlimited Game
Jacoby Beaver

pip: 138
score: 0

Blue is Player 1

XGID=-a-aBBCBAA-AA--aabbcbb--B-:0:0:1:22:0:0:3:0:10
51S-63R-21H-42S-11P-32@-63H-51Z-21P-44C-
329-61P-64R-43H-54R-31H-61H-11Y-33M-22
(perfectly played money sequence)........................Nactation tutorial

33 - Original/Adj. ........... m V... d @... s B... g K .....Details.....More Details

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
53P-55P-64P-33C-33B-53N-66P-31H-51H-41H-
52H-61H-52P-61H-31U-33
(perfectly played money sequence)........................Nactation tutorial

44 - Early Game................. m B... d R... s b... g C ......Details

White is Player 2

score: 0
pip: 163

Unlimited Game
Jacoby Beaver

pip: 169
score: 0

Blue is Player 1

XGID=-a-a--D-D---dE--ad-d----B-:0:0:1:44:0:0:3:0:10
65R-54S-41K-31-53Z-32K-41-65r-64X-43H-21R-43Z-44
(perfectly played money sequence)........................Nactation tutorial

55 - Two Anchors........... m P... d S... s R... g Y... c6 R ......Details

White is Player 2

score: 0
pip: 165

Unlimited Game
Jacoby Beaver

pip: 183
score: 0

Blue is Player 1

XGID=-a---aD-D--acCb--b-e-B-B--:0:0:1:55:0:0:3:0:10
51S-21U-41X-65R-65H-54Z-41H-32H-52P-41H-
54K-42-42R-55
(perfectly played money sequence)........................Nactation tutorial

66 - Bar Points.................. m F... d B... s S... g P .......Details

White is Player 2

score: 0
pip: 137

Unlimited Game
Jacoby Beaver

pip: 153
score: 0

Blue is Player 1

XGID=-a-BBBC-a---cC---b-ccb--C-:0:0:1:66:0:0:3:0:10
54S-44P-33M-41H-52C-51A-51H-11P-11P-66
(perfectly played money sequence)........................Nactation tutorial

53 - Non-doublet.............. m J... d R... s D... g C ......Details

White is Player 2

score: 0
pip: 117

Unlimited Game
Jacoby Beaver

pip: 121
score: 0

Blue is Player 1

XGID=---CBBBbB---bB----beBb-aa-:0:0:1:53:0:0:3:0:10
31P-66B-44B-117-65&-54$-64P-21P-32I-64&-53
(perfectly played money sequence)........................Nactation tutorial

Below is a summary of (and links for) the eight R4 positions and the margins by which the best plays distinguish themselves at each of the four main scores. In the far right column is a measure of overall strength based on a formulaic composite of the four margins: [sqrt m + sqrt 20d/9 + sqrt 10s/7 + sqrt 10g/7] = Power Score.
..............Position.................... m..... d....... s...... g.......... Power Score
......11 - Blitz Position ............32......15......48.......33...........26.58
......11 - Prime-v-Prime ..........23......11........8.......13...........17.43....(R4+1?)
......22 - 12pt Variant.............101........8......25.......48...........28.52
......22 - 10pt Variant...............11........8........9.......61...........20.45
......33 - Original/Adj. ............13......16.......35......39...........24.10
......44 - Early Game...............18........8.......31.........9...........18.70
......55 - Two Anchors........... 12........6.......37.......33...........21.25....(R4+1)
......66 - Bar Points................. 16......21.......47.......15...........23.65
......53 - Non-doublet..............12........7.......43.......11...........19.21
The two R4+1 positions deserve a bonus for having yet another clearly best play at a fifth score, though I've been treating that as a separate issue. There may be other factors as well to consider, such as the likelihood of the position arising, doublet vs. non-doublet, etc. A determination of the "best" R4 may largely come down to a matter of taste.
Nack