In digesting your beautifully framed explanations and brilliant demonstrations, I'm convinced that undefined equities have some sort of effect on all contact positions (I'm not sure there are any exceptions). It seems to me that non-contact positions are immune.
However, I'm concerned that bgonliners might get the impression that undefined equities have a larger effect than they actually do, and that therefore rollouts are not nearly as reliable as they thought. On any positions (at least the overwhelming majority of them) that they or you or I will encounter in play, or in any positions that arose in the Smith-and-Jones game, I believe the effect is negligible, and the definition of perfect play, for all practical purposes, remains intact. While I realize you primarily intended to convey a theoretical point and made a sincere effort to put your points into context, I nevertheless feel that your initial characterization of "out the window" could be misleading to many readers.
Even for the FARE position reached by Smith and Jones, I do not believe that an effect of undefined equity could overturn either the redouble or the take. As each side has only a one-point board, the chances of a consecutive string of fans is calculated by powers of 36, which dwarf powers of 2. By the same token, the rollout itself should be pretty much as reliable as XG can be, as the chance of hitting the 2^10 cube cap is only 1 / 36^10 (which is 3.6 trillion times less likely than 1 / 1024)! In short, I feel comfortable with the claim that Smith and Jones played perfectly.
[I'm adding this note right before posting. I see that Casper, in his second paragraph here stated something similar (and probably more elegantly than I did.)]
Your position (scroll to third diagram there) and Bob Floyd's position (see below) manifest an enormous difference in magnitude compared to the Smith-Jones position, because the players had FIVE-point boards instead of one-point boards. The chances of fanning are 25/36 instead of 1/36. The effect of undefined equities -- or (sort of in lieu of that) of the XG cube cap -- is profoundly greater. This time, the odds of hitting the cube cap are 1/192, which is more than five times as likely as 1/1024 -- and monkey business can play a role.
FWIW, for Bob Floyd's position, detailed rollout results for Blue redoubling, and for White redoubling, respectively, can be seen below. (If anyone would like to extend the files, let me know and I'll e-mail them to you.) Note the size of the confidence intervals even after 10k trials.
Nack
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| XGID=aBBBBBa--AA-aAaa---AbbbbbA:1:1:1:00:0:0:3:0:10 | |||||
Blue on roll | |||||
| Analyzed in Rollout | No redouble | Redouble/Take |
| Player Winning Chances: | 56.54% (G:43.71% B:9.10%) | 56.56% (G:43.74% B:9.05%) |
| Opponent Winning Chances: | 43.46% (G:31.31% B:5.93%) | 43.44% (G:31.31% B:5.94%) |
| Cubeless Equities | +0.286 | +0.573 |
| Cubeful Equities | ||
| No redouble: | +0.788 (-0.212) | ±0.123 (+0.665..+0.910) |
| Redouble/Take: | +1.268 (+0.268) | ±0.183 (+1.086..+1.451) |
| Redouble/Pass: | +1.000 | |
| Best Cube action: Redouble / Pass | ||
| Rollout details | ||
| 10368 Games rolled with Variance Reduction. Moves: 3-ply, cube decisions: XG Roller | ||
| Double Decision confidence: | 100.0% | |
| Take Decision confidence: | 99.8% | |
| Duration: 1 hour 21 minutes | ||
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| XGID=aBBBBBa--AA-aAaa---AbbbbbA:1:-1:-1:00:0:0:3:0:10 | |||||
White on roll | |||||
| Analyzed in Rollout | No redouble | Redouble/Take |
| Player Winning Chances: | 56.56% (G:42.99% B:8.71%) | 56.59% (G:43.03% B:8.57%) |
| Opponent Winning Chances: | 43.44% (G:31.53% B:6.33%) | 43.41% (G:31.50% B:6.33%) |
| Cubeless Equities | +0.270 | +0.539 |
| Cubeful Equities | ||
| No redouble: | +0.511 (-0.199) | ±0.109 (+0.402..+0.620) |
| Redouble/Take: | +0.710 | ±0.161 (+0.549..+0.870) |
| Redouble/Pass: | +1.000 (+0.290) | |
| Best Cube action: Redouble / Take | ||
| Rollout details | ||
| 10368 Games rolled with Variance Reduction. Moves: 3-ply, cube decisions: XG Roller | ||
| Double Decision confidence: | 97.7% | |
| Take Decision confidence: | 100.0% | |
| Duration: 2 hours 01 minute | ||