Consider the following (reference?) position from Robertie's Advanced Backgammon, 2nd ed. vol. 1, position 144, page 198:
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| XGID=aBBBBBB-----------AccccbB-:1:1:1:00:0:0:1:0:10 | |||||
| Blue on roll, cube action? | |||||
| Analyzed in Rollout | No redouble | Redouble/Take |
| Player Winning Chances: | 62.89% (G:2.08% B:0.02%) | 62.37% (G:1.56% B:0.02%) |
| Opponent Winning Chances: | 37.11% (G:6.15% B:0.20%) | 37.63% (G:6.11% B:0.17%) |
| Cubeless Equities | +0.215 | +0.401 |
| Cubeful Equities | ||
| No redouble: | +0.407 | ±0.001 (+0.406..+0.409) |
| Redouble/Take: | +0.186 (-0.221) | ±0.002 (+0.184..+0.188) |
| Redouble/Pass: | +1.000 (+0.593) | |
| Best Cube action: No redouble / Take | ||
| Percentage of wrong pass needed to make the double decision right: 21.4% | ||
| Rollout details | ||
| 10368 Games rolled with Variance Reduction. Dice Seed: 29834315 Moves: 3-ply, cube decisions: XG Roller | ||
| Double Decision confidence: | 100.0% | |
| Take Decision confidence: | 100.0% | |
| Duration: 2 minutes 52 seconds | ||
eXtreme Gammon Version: 2.10
Here there are few gammon wins for Blue. In the problem position, let's assume 1/3 of Blue's wins and 1/6 of White's wins will be gammons. Further, let's assume that the cubeless equity at the drop/take point is 0.55. Finally we'll ignore backgammons. Let 'W' represent Blue's winning chances at the drop/take point. Then:
W + W/3 - (7/6)*(1 - W) > 0.55.
Solving for W gives the Drop/Take point at 69% wins for Blue. Comparing to the reference position, can Blue win an extra ~6.5% games? Note Blue has about 6 more pips of timing in the problem position (non-trivial plus) and that most of the time White will have two checkers on the bar if/when Blue's board starts to go South. White's homeboard spares aren't as nicely placed in the problem position, and if Blue can prime White's (eventual) 24-pt anchor Blue may have time to force White to crunch. Do all those add up to 6.5% more wins? Probably close but I'm going to say 'yes'.
R+P.
P.S. For ~30 years since I read this book I've remembered a quote which happens to accompany Robertie's analysis of this reference position: In prime vsersus prime positions, it is often correct to redouble your opponent out of the game, rather than redouble him in. I wonder if in the bot era he still feels that way. FWIW, IMO this book was the best backgammon book written in the pre-bot era, which was in its death throes when Advanced BG first published (1991). TD-Gammon was already in the works and Jellyfish's release ~4 years away.