I had assumed that if one were doing a live-cube rollout, then (ignoring variance reduction) a double/pass would count as a simple win, rather than some estimated breakdown into win/gammon/backgammon.
I think we're talking past each other, but again the rollout game breakdown in a cubeful rollout is "cubeless." Suppose in a rollout Black doubles White out at a point where (if the game were to continue) Black would win 40% single games, 30% gammons, 5% backgammons, and lose 25% single games, and I want to know the cubeless equity, what makes more sense:
(a) Black's cubeless equity is 1.000.
(b) Black's cubeless equity is 0.400 + 2*0.300 +3*0.050 - 0.250 = 0.900.
I imagine (b) must make more sense since you write Anyway, the point I was making is that you shouldn't expect to be able to compute anything other than the cubeless equity from the 6 probabilities.
How would a bot report the results of a rollout of the hypothetical cube decision described above? As:
W 0.750 G 0.350 BG 0.050 L 0.250 LG 0.000 LBG 0.000 = Cubeless equity 0.900
even though with Double/Pass all games are a single win and the cubeful equity is obviously just 1.000 for double/pass and (assuming for simplicity that White has no recube vig whatsoever) 1.800 for double/take.
Tim, you also wrote: If the bot were to report how many games ended with a single win ...
Gnubg does have this feature which I view infrequently and may not understand completely but seems to be only of limited value. Click the "View Statistics" tab on the rollout window when your rollout is finished. The feature does not work if you interrupt and resume a rollout, and does not work if the rollout is truncated at the bearoff database.
For example, For a 1296-trial rollout of this position:
The score (after 0 games) is: White 0, Blue 0
Move number 6: Blue on roll, cube decision?
| White | 30 |
|---|---|
| |
| Blue | 30 |
Position ID: 2zYAANi2AQAAAA Match ID: cAkAAAAAAAAA | |
• Blue doubles
Alert: double decision marked doubtful
| Cube decision | |||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Rollout cubeless equity | +0.3598 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| Cubeful equities: | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1. | Double, take | +0.5760 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 2. | Double, pass | +1.0000 | +0.4240 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 3. | No double | +0.5759 | -0.0002 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| Proper cube action: | Double, take | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| Rollout details | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Full cubeful rollout with var.redn. | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1296 games, Mersenne Twister dice gen. with seed 846112396 and quasi-random dice | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Player 0: | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Play: supremo 2-ply cubeful prune [world class] | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| keep the first 0 0-ply moves and up to 16 more moves within equity 0.32 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skip pruning for 1-ply moves. | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Cube: 0-ply cubeful [expert] | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Player 1: | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Play: supremo 2-ply cubeful prune [world class] | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| keep the first 0 0-ply moves and up to 16 more moves within equity 0.32 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skip pruning for 1-ply moves. | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Cube: 2-ply cubeful prune [world class] | |||||||||||||||||||||||||||||||||||||||||||||||||||||
the "View Statistics" shows that after double/take Blue wins 832 2-cubes, 38 4-cubes, and six 8-cubes, and loses 350 2-cubes, 69 4-cubes, and one 8-cube -- 1296 results in all.
That's (2 * 832 + 2 * 38 + 6 * 8) - (2 * 350 + 69 * 4 + 1 * 8)
But if I add that up (the sum is 880) and divide by 1296, I don't get 0.576, the cubeful equity; instead I get 0.679, the cubeless win %.