I thought I'd say a little more about the topic of undefined equity, using a variant of an old position from rec.games.backgammon.
Consider the position below. It's the first game of a sevenpoint
match. Note that the position is symmetrical. What's the cube action?
  White is Player 2
score: 0 pip: 90  7 point match  pip: 90 score: 0
Blue is Player 1  
XGID=aBBBBABAAAaaababbbbA:0:0:1:00:0:0:0:7:10 
Blue on roll, cube action? 
Let's start by using our normal backgammon intuition to assess the
position. Since Blue has 25 dancing rolls, it doesn't seem possible
that it could be a pass for White. What about the double? Every
sequence that doesn't start with a dance loses Blue's market, unless
Blue rolls 51 or 52 (lifting his 5pt blot) and White enters with 55.
Furthermore these sequences are hugely gammonish. It's hard to weigh
this against the whopping 25 dancing rolls, but it seems plausible that
this could be a double, maybe even a big double.
Let's see what an XG rollout says.
Analyzed in Rollout  No double  Double/Take 
Player Winning Chances:  58.19% (G:48.35% B:8.22%)  58.02% (G:48.31% B:7.89%) 
Opponent Winning Chances:  41.81% (G:33.71% B:5.16%)  41.98% (G:33.86% B:5.49%) 
Cubeless Equities  +0.348  +0.693 
Cubeful Equities 
No double:  +0.328 (0.023)  ±0.016 (+0.312..+0.343) 
Double/Take:  +0.351  ±0.019 (+0.332..+0.369) 
Double/Pass:  +1.000 (+0.649) 

Best Cube action: Double / Take 
Rollout details 
1296 Games rolled with Variance Reduction. Dice Seed: 271828 Moves: 3ply, cube decisions: XG Roller


Double Decision confidence:  96.9% 
Take Decision confidence:  100.0% 
Duration: 10 minutes 27 seconds 
eXtreme Gammon Version: 2.19.208.prerelease, MET: Kazaross XG2
O.K., so this accords with our intuition. Maybe a small double, certainly a huge take.
But let's look ahead a little more. What happens if Blue dances? Then White is faced with exactly the same decision
that Blue is faced with in the original position. If the original
position is D/T, then the position after a dance must also be D/T.
Seems like the cube could get up to a high value pretty quickly. Of
course, since it's a sevenpoint match, the cube can't get higher than
8. So what happens if we change the score to 15a15a? Let's see what an
XG rollout says:
  White is Player 2
score: 0 pip: 90  15 point match  pip: 90 score: 0
Blue is Player 1  
XGID=aBBBBABAAAaaababbbbA:0:0:1:00:0:0:0:15:10 
Blue on roll, cube action? 
Analyzed in Rollout  No double  Double/Take 
Player Winning Chances:  58.03% (G:48.30% B:7.46%)  58.17% (G:48.20% B:7.36%) 
Opponent Winning Chances:  41.97% (G:33.60% B:4.94%)  41.83% (G:33.72% B:4.77%) 
Cubeless Equities  +0.338  +0.688 
Cubeful Equities 
No double:  +0.221 (0.266)  ±0.014 (+0.208..+0.235) 
Double/Take:  +0.488  ±0.016 (+0.471..+0.504) 
Double/Pass:  +1.000 (+0.512) 

Best Cube action: Double / Take 
Rollout details 
2592 Games rolled with Variance Reduction. Dice Seed: 271828 Moves: 3ply, cube decisions: XG Roller


Double Decision confidence:  100.0% 
Take Decision confidence:  100.0% 
Duration: 19 minutes 17 seconds 
eXtreme Gammon Version: 2.19.208.prerelease, MET: Kazaross XG2
Whoa! That made a big difference. Now the rollout says it's a huge
double. Those big cubes are evidently having a significant effect.
O.K., now comes the obvious question. What happens if we try a moneygame rollout?
  White is Player 2
score: 0 pip: 90  Unlimited Game  pip: 90 score: 0
Blue is Player 1  
XGID=aBBBBABAAAaaababbbbA:1:1:1:00:0:0:0:0:10 
Blue on roll, cube action? 
Analyzed in Rollout  No redouble  Redouble/Take 
Player Winning Chances:  58.06% (G:48.55% B:7.71%)  58.13% (G:48.56% B:7.52%) 
Opponent Winning Chances:  41.94% (G:33.77% B:5.24%)  41.87% (G:33.79% B:5.25%) 
Cubeless Equities  +0.334  +0.666 
Cubeful Equities 
No redouble:  +0.958 (0.042)  ±0.041 (+0.917..+0.998) 
Redouble/Take:  +1.582 (+0.582)  ±0.058 (+1.525..+1.640) 
Redouble/Pass:  +1.000 

Best Cube action: Redouble / Pass 
Rollout details 
46656 Games rolled with Variance Reduction. Dice Seed: 271828 Moves: 3ply, cube decisions: XG Roller


Double Decision confidence:  98.0% 
Take Decision confidence:  100.0% 
Duration: 2 hours 22 minutes 
eXtreme Gammon Version: 2.19.208.prerelease
You've got to be kidding me!! A huge pass? No way! Could this possibly be correct?
The answer is no. This is a rare case where we can actually analyze
a complex contact position "by hand" as it were. Let us hypothesize,
for the sake of argument, that the "perfect play" cube action in this
position, is D/P, and let's draw out the logical consequences. A pass
of course gives White an equity of –1. On the other hand, suppose that
White takes. Then, 25/36 of the time, Blue dances, and then by hypothesis, the correct cube action is D/P so White's equity is +2.
So the contribution to White's equity from these sequences is 50/36.
Now suppose that Blue hits. Then we can't analyze exactly what happens,
but we can say this: White has the option of adopting the
strategy of refusing to redouble again under any circumstances. This
may not be White's best strategy, but it is one strategy that is available to her. If she adopts this strategy, then at worst she loses 6 points (if she gets backgammoned). Therefore, White's equity from these sequences is at worst 11/36 × (–6) = –66/36. If we add this to 50/36 then we get a grand total of –16/36. This is not White's equity, but it's a lower bound
on White's equity. Whatever White's true equity is, it can't be any
worse than this. And since –16/36 > –1, it follows that White does
better by taking than by passing. So it can't be a pass. This
contradicts our initial hypothesis that the perfect cube action is D/P.
Therefore, our initial hypothesis must have been incorrect, and the
perfect play cube action cannot be D/P.
On the other hand, if the perfect play cube action is D/T, then an
argument along the lines of the one presented by Gary Wong in that old rec.games.backgammon article applies to show that the equity is unbounded.
What does this say about the concept of "perfect play"? If there is
such a thing as perfect play, then White's perfect play after a double
by Blue must be either "take" or "pass" since those are her only two
options. But we've just shown that "pass" can't be the right answer,
and "take" leads to undefined equities. Therefore "perfect play"
doesn't make sense in this position.
As a final comment, note that even though the above moneygame
rollout involves 46656 trials, the confidence interval is still huge!
This is on account of the large cubes. Below, I've reproduced XG's
"Rollout Statistics" to show how often the cube gets up to some gigantic
value. Note that XG has a builtin cap of 1024 on the cube value, or
else the numbers would have been even more crazy.
XGID=aBBBBABAAAaaababbbbA:1:1:1:00:0:0:0:0:10
Redouble/Take Non VR Equity: +1.957 (Cost: +3.913)
Cube Win BG Win G Win S Cash Pass Lose S Lose G Lose BG D/T D/P Take % D/T D/P Take %
2
4 1,785 9,923 1,658 461 7 13 1 32,808 461 98.61%
8 1 16 19 396 1,407 6,884 1,258 22,827 396 98.29%
16 883 4,768 1,047 249 9 14 15,857 249 98.45%
32 10 8 204 699 3,357 561 11,018 204 98.18%
64 430 2,318 533 125 9 7,603 125 98.38%
128 6 8 107 330 1,575 287 5,290 107 98.02%
256 419 2,098 460 406 6 1,443 263 195 406 32.45%
512+ 1 37 29 104 23 1 49 29 62.82%