Thanks for sharing this. Obviously, these positions are rare enough that they do not warrant too much study. Nevertheless, they carry a fascination for me.
When TerryPin posted the following Kauder Paradox at TwoPlusTwo, Bill Robertie wrote the following:
Is this a Beaver?
Positions that satisfy the Kauder Paradox usually fit these conditions:
1) Cube in the middle, Jacoby rule in effect.
2) Black on roll wins 30%-35% of the time. (Black has a single direct shot to win.)
3) When Black wins, he mostly wins a gammon.
4) When Black misses his immediate shot, he gets doubled out.
Bill Robertie
Source: TwoPlusTwo
While I do not agree with all of that, I think the main points are often correct. More generally, Black needs winning chances that are somewhere in the range 30% to 50%, and enough gammons wins to make it worthwhile to activate them. The position TerryPin posted is interesting because the rollout shows him winning a majority of all games, and yet his equity is still negative.
                       |
| ||||
| XGID=-BBBBCB-A-----a-b--bad-cbA:0:0:1:00:0:0:3:0:10 | |||||
| Blue on roll, cube action? | |||||
| Analyzed in Rollout | No double | Double/Take |
| Player Winning Chances: | 50.443% (G:14.570% B:0.216%) | 50.326% (G:17.609% B:0.308%) |
| Opponent Winning Chances: | 49.557% (G:0.746% B:0.006%) | 49.674% (G:0.734% B:0.006%) |
| Cubeless Equities | +0.1492 | +0.3566 |
| Cubeful Equities | ||
| No double: | -0.1823 (-0.0125) | ±0.0005 (-0.1828..-0.1818) |
| Double/Beaver: | -0.1697 | ±0.0021 (-0.1718..-0.1677) |
| Double/Pass: | +1.0000 (+1.1697) | |
| Best Cube action: Double / Beaver | ||
| Rollout details | ||
| 31104 Games rolled with Variance Reduction. Dice Seed: 29924256 Moves and cube decisions: 4-ply Search interval: Large | ||
| Double Decision confidence: | 100.0% | |
| Take Decision confidence: | 100.0% | |
| Duration: 59 minutes 45 seconds | ||
Rollout by Taper_Mike |
eXtreme Gammon Version: 2.10