Position 2:
The score is: White 0, Blue 4 (match to 11 points). Blue on roll
| gnubg | 61 | ||
|---|---|---|---|
                       | |||
| Blue | 52 | ||
| Position ID: 23YnAAC7tQcAAA Match ID: UQlgAQAAIAAE | |||
I noticed your GWC estimates and match equity calculations were pretty good, according to Tim's results, but I also noticed something missing.
(let’s look it up : a score of 5a/11a gives 20% chance, 3/11 gives 11% chance, so this is a risk of 9%, winning gives 50% chance (7a/7a), TP = 9/39, about 20%, so white can take until 80%)
The match equities are good, rounded, and your R/(R+G) = 9/39 is right, but 9/39 is actually 23%, not 20%! So, cubeless, this is a slightly larger pass than your estimate, since White wins 18% cubeless, or 5% short of a take (you had estimated 16% wins and a 20% takepoint).
What's missing is the value of holding a 4-cube in a not-cubeless race trailing 11a7a. Holding a 4-cube, White can (1) force Blue to drop an 8-cube or (2) give Blue a takable 8-cube. Hard to estimate, easier to see that it must be important.
Recall that in a money game, the doubling window is 50-75% in a last roll position, or typically up to ~78% if the cube is live. White must get to at least 50% GWC to consider doubling, and more than 75% or ~78% to force Blue to pass. But at the score, White only needs to get to 66% GWC to force Blue to drop an 8-cube, and only 30% GWC to be able to give Blue a takable 8-cube.
The top of White's redoubling window:
Pass -> 7a7a = 50% MWC
Take/win -> 0a11a= 100% MWC
Gain from taking = 50%
Take/lose -> trail 7a3a = 24% MWC
Risk of taking = 26%
Blue's takepoint = R/(R+G) = 26/(26+50) = 26/76 = 34%
Top of White's redoubling window is 100-34 = 66%.
The bottom of White's redoubling window:
No double and win 4 points -> 7a7a = 50% MWC
Double and win 8 points -> 3a7a = 76% MWC
Doubling gains 26% MWC
No double and lose 4 point -> 11a3a = 11% MWC
Double and lose 8 points -> 0% MWC
Doubling risks 11%.
Bottom of White's redoubling window is R/(R+G) = 11/(11+26) = 11/37 = 30%.
It must be more likely that White will improve from 18% to 30% (possible redouble/take) or to 66% (redouble/pass) than from 18% to 50% (possible money game redouble/take) or to 75-78% (money game redouble/pass). I verified this with Gnubg's "show statistics" rollout feature.
Here are the actual results of a rollout at the score after double/take:
| Cube level | All games | White wins | Blue wins |
| 4 | 83.41% | 11.34% | 72.07% |
| 8 | 16.59% | 8.64% | 7.95% |
| Any | 100.00% | 19.98% | 80.02% |
| 8-cubes | |
|---|---|
| Taken | 16.59% |
| Passed | 9.33% |
| Given | 25.92% |
With those results, White's MWC after taking the 4-cube is:
| Result | freq. * ME | MWC |
|---|---|---|
| Wins 8 points | 0.0864 * 0.76 = | 6.56% |
| Wins 4 points | 0.1134 * 0.50 = | 5.67% |
| Loses 4 points | 0.7207 * 0.11 = | 7.92% |
| Loses 8 points | 0.0795 * 0 = | 0.00% |
| Total | 20.16% |
... a little better than White's 20% MWC -- precisely, 19.72% -- from passing.
In the ATS rollout, White went from 18% GWC to at least 66% GWC, forcing a pass, in 9.33% of all games. Unfortunately we do not know how often White would have won those games if he had not been able to redouble Blue out. White went from 18% GWC to at least 30% GWC, allowing a takable 8-cube, in only 16.59% of all games. But these games were worth 3.25% match equity to White, compared to passing the 4-cube:
| White's MWC gained in games where there was an 8-cube taken | |||||
|---|---|---|---|---|---|
| Result | Frequency | * | MWC over/under Pass MWC | = | MWC |
| Win | .0864 | * | (76-20 = 56%) | = | 4.84% |
| Lose | .0795 | * | (20-0 = 20%) | = | -1.59% |
| Total | = | 3.25% | |||
For comparison, here are the actual results in a money game rollout:
| Cube level | All games | White wins | Blue wins |
| 4 | 96.66% | 14.35% | 77.31% |
| 8 | 7.94% | 5.47% | 2.47% |
| 16 | 0.38% | 0.08% | 0.30% |
| Any | 100.00% | 19.91% | 80.09% |
In only 3.34% of money games did the cube reach 8 or higher, compared to 16.59% in the ATS rollout.
| 8-cubes | |
|---|---|
| Taken | 8.33% (instead of 16.59% ATS) |
| Passed | 11.11% (instead of 9.33% ATS) |
| Given | 19.44% (instead of 25.92% ATS) |