First I'd start with my opponent's take point so let's do that. (even though we know him dropping is a joke)
- Double/Pass gives 6 away 9 away or 67% match equity
- Double/Take/Lose gives 6 away 5 away or 43% match equity
- Double/Take/Win gives him the match, 100% match equity
I figured the match equity out for 6 away 9 away using Neil's Numbers. The other I had memorized. Our opponent would be risking 24% to gain 33%. So using risk/(risk+gain) we get our opponent's take point of 24/(24+33) = 42%. This is way high and would be the first clue if we otherwise didn't know that maybe we should be thinking of cubing.
Next I would move on to figuring out our estimated win percentage. We win 2/36 immediately by throwing double 5s or double 6s. Then how many of the remaining games will we win when our opponent throws any roll containing an ace that isn't double aces? The rolls we aren't automatically off with if we get the chance for that next turn are [21 31 41 51 61 22 32 42 52 62 43] Ouch, that's 21 rolls and you notice how some of those didn't even get a checker off? It looks like we'll win approximately 2/3rds of those 10 games bringing our total to 6.67 + 2 or 8.67 games. That is just above 24% as our win percentage.
If we lose as is we'll be down 2 away 13 away which is ~4% ME. So doubling risks 4% to gain 24% meaning our doubling window of a last roll position opens at about 14%. This isn't a pure last roll position but damn it's close! It will be a last roll position in at least 26 of the games. The remaining games as we saw above will be scattered into games where we're almost gin to where we still need doubles to win. Some of these games will fall into d/t category and others into d/p. (they will all still be a double because it will be a true last roll and the worst win % we will have at that point is 13.89% after the 42 for us/not off for him sequence.)
I think it's clear we're within' the window. Note how even changing the score to 7 away 13 away instead of 6 away 13 away will have a big effect on whether or not you should cube. I point this out because some people looked at this problem and basically said "Well of course we double, we're down so much of course I'm doubling." If that's your analysis I highly doubt you would have looked at 6 away 13 away and thought differently.
                 |
| ||||
| XGID=-A---B-----------------b--:2:1:1:00:0:7:0:13:10 | |||||
| Blue on roll, cube action? | |||||
| Analyzed in Rollout | No redouble | Redouble/Take |
| Player Winning Chances: | 24.78% (G:0.00% B:0.00%) | 24.78% (G:0.00% B:0.00%) |
| Opponent Winning Chances: | 75.22% (G:0.00% B:0.00%) | 75.22% (G:0.00% B:0.00%) |
| Cubeless Equities | -0.504 | -0.302 |
| Cubeful Equities | ||
| No redouble: | -0.445 (-0.143) | ±0.000 (-0.445..-0.445) |
| Redouble/Take: | -0.302 | ±0.000 (-0.302..-0.302) |
| Redouble/Pass: | +1.000 (+1.302) | |
| Best Cube action: Redouble / Take | ||
| Rollout details | ||
| 1296 Games rolled with Variance Reduction. Dice Seed: 82510205 Moves: 3-ply, cube decisions: XG Roller | ||
| Double Decision confidence: | 100.0% | |
| Take Decision confidence: | 100.0% | |
| Duration: 0.1 second | ||
Stick