In case anyone is collecting Kauders that turn up in actual play, here's one from a recent game in Boston. Over the board I thought it might be one but couldn't quite bring myself to pull the trigger (I was in a strong enough chouette that I knew I'd get 4 beavers). So, in order to help me get it right, can we elucidate the elements of a proper Kauder Double? (I haven't found this spelled out in previous threads, but sorry if this has been covered -- feel free to just point me elsewhere)
Am I right in figuring that as a benchmark if you have a position where you win 12 gammons and lose the other 24, that would be an Optional Kauder -- definitely right to double and the beaver is optional because:
Undoubled: +12 -24 = -12/36 Doubled: +48 - 48 = 0/36 Beavered: +96 - -96 = 0/36 (Doubling zero is still zero obviously)
So then if you're winning fewer gammons you have to compensate with extra single wins -- and if you're looking at getting yourself gammoned much at all you probably aren't in the Kauder zone since you're activating your opponent's gammons as well and would need a commensurate number of extra gammons to balance those out?
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| XGID=-CBBBCB-----aa------abccdA:0:0:1:00:0:0:3:0:10 | |||||
| Blue on roll, cube action? | |||||
| Analyzed in Rollout | No double | Double/Take |
| Player Winning Chances: | 52.60% (G:15.57% B:0.40%) | 52.63% (G:15.63% B:0.42%) |
| Opponent Winning Chances: | 47.40% (G:0.21% B:0.00%) | 47.37% (G:0.21% B:0.00%) |
| Cubeless Equities | +0.209 | +0.422 |
| Cubeful Equities | ||
| No double: | -0.143 (-0.050) | ±0.006 (-0.148..-0.137) |
| Double/Beaver: | -0.093 | ±0.023 (-0.116..-0.070) |
| Double/Pass: | +1.000 (+1.093) | |
| Best Cube action: Double / Beaver | ||
| Rollout details | ||
| 1296 Games rolled with Variance Reduction. Moves: 3-ply, cube decisions: XG Roller | ||
| Double Decision confidence: | 100.0% | |
| Take Decision confidence: | 100.0% | |
| Duration: 20.4 seconds | ||
eXtreme Gammon Version: 2.10
In this example I'm only winning about 6 gammons -- just about half the time when I roll one of my 12 hitters. Over the board I didn't think this would be enough, and I figured I'd have to take a strong re-cube if I missed, so I balked. The rollout shows I also win another 13 of the remaining 30 games to his 17 -- so I'm netting +6 gammons to his +4 games, which sounds great, but yeah, around 3/4 his 17 wins turn out to be after a recube.
So, compared to other Kauder's I've seen here, this one seems to have unusually low gammon wins along with unusually high single wins and an efficient re-cube in the mix. Any suggestions for efficient ways of thinking this through over the board?
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| XGID=-CBBBCB-----aa------abccdA:1:-1:-1:00:0:0:3:0:10 | |||||
| White on roll, cube action? | |||||
| Analyzed in Rollout | No redouble | Redouble/Take |
| Player Winning Chances: | 75.03% (G:0.49% B:0.00%) | 75.01% (G:0.39% B:0.00%) |
| Opponent Winning Chances: | 24.97% (G:1.45% B:0.03%) | 24.99% (G:1.56% B:0.03%) |
| Cubeless Equities | +0.491 | +0.977 |
| Cubeful Equities | ||
| No redouble: | +0.745 (-0.158) | ±0.004 (+0.741..+0.749) |
| Redouble/Take: | +0.903 | ±0.005 (+0.898..+0.908) |
| Redouble/Pass: | +1.000 (+0.097) | |
| Best Cube action: Redouble / Take | ||
| Rollout details | ||
| 1296 Games rolled with Variance Reduction. Moves: 3-ply, cube decisions: XG Roller | ||
| Double Decision confidence: | 100.0% | |
| Take Decision confidence: | 100.0% | |
| Duration: 13.0 seconds | ||
eXtreme Gammon Version: 2.10