I have found it interesting that splitting is correct after 31P-32Z but running is correct after the similar 31P-32S. I did some 36ing on this once and it appeared that the differences were very subtle. Such is the difficulty of Backgammon.
Yes, S is clearly correct for 31P-32Z-65. Is R clearly correct for 31P-32S-65 -- do you have a rollout for that? I had guessed the latter to be a close decision. Regardless, the point that the conditions for 65S are a lot friendlier against 32Z than against 32S is valid, and I'll discuss the specific third roll positions above at the end of this post.
Three similar position pairs, from the second roll of the game, are diagrammed below. The first is 32Z-51 (left-hand) versus 32S-51 (right-hand). On the left, going to the 18pt (24/18) is exactly tied with hanging back (24/23 13/8). On the right, hanging back is better by .032. (More concisely, bringing down a fourth checker to the 8pt has equal value to coming out on the left, whereas 13/8 gains a lot more than 23/18 on the right.)
In a similar vein: At DMP, going to the 18pt is best by .008 on the left, whereas it is wrong by .009 on the right.
That is the first example. As Paul Weaver and I discovered long ago, it is more efficient to go to (or stay on) the 18pt against a 10pt builder than an 11pt (or 9pt, or even more so extra 8pt) builder.
The second position pair (below) in this second roll series is 32Z-62 versus 32S-62. On the left, staying on the 18pt with S (split, 24/18 13/11) is better than R (running to the 16pt, 24/16) by .052. On the right, S beats R by only .028. (We're ignoring the hitting alternatives and focusing on S vs R.) The efficiency of staying on the 18pt is higher by .024 on the left.
The third second-roll position pair (below) is 32Z-65 versus 32S-65. On the left, R (Run, 24/13) is better than S (Split, 24/18 13/8) by .009. On the right, however, R is better by .068. It's a difference of .059!
Recap: With replies of 51, 62 and 65, it is relatively better or less worse by .032, .024 and .059, respectively, to go to or stay on the 18pt in reply to opening 32Z (24/22 13/10, with a 10pt builder) than in reply to opening 32S (24/21 13/11, with an 11pt builder).
How much of this has to do with the placement of White's lead back checker? Not much. Consider the two variant positions below. In the first, 31S-65, R beats S by .009, which is identical to the R/S margin for 32Z-65 (except with White's back checker on the 23pt instead of 22pt). This reinforces the concept that it is the exact placement of White's 10pt builder (and not the placement of her back checker) that is primarily responsible for Blue's greater incentive to stay on the 18pt.
In the second variant position (below), 33S-65 (with 33 illegally played as a non-doublet), S actually beats R (by .006). Granted, there is a contrived degree of 3 duplication in this case. That said, though, the point is that nudging the back checker back onto the 21pt (while keeping the 10pt builder in the same place) doesn't cause Blue's 65 to become any less S-ish; rather, it is nudging White's 10pt builder to the 11pt that causes it.
Finally, that leads me back to the comparison of 31P-32Z-65 and 31P-32S-65 (the pair of positions shown at the top of this post and re-diagrammed below).
In the left-hand position, S (Split, 24/18 13/8) is better than R (Run, 24/13) by .027 (Snowie full, though no live cube). I have no rollout for the right-hand position, but I guarantee that S will either lose the rollout by a much smaller margin, or it will win the rollout (as David seems to imply).
If Blue splits, he is immune to a hit if White rolls 64 or 41 on the left, and he deflects White from making the 18pt or 5pt if she rolls 63 or 61 on the right. The effects of those rolls mostly cancel out (though to the extent they don't, there is a pull towards splitting on the left and running on the right).
The swing on 54 is large. S embarrasses White in the left-hand position, whereas S permits White to salvage a double-hit in the right-hand position.
Other rolls argue that splitting on the left works relatively better than splitting on the right: 51 62 43, and to a lesser extent 22 52 11 55. Arguing the opposite to any meaningful degree seem to be only rolls of 21 53.
It is paramount to remember that in the early game having a tall midpoint and stripped 8pt is mostly a prerequisite for 65S being a serious candidate, and that having the opponent outboarded (preferably with the 5pt, which is better than the 4pt, which is better than the 3pt, etc.) increases the motivation to split to (or stay on) the 18pt in general. However, the main wrinkle I'm conveying in this post that it is substantially more efficient to be on the 18pt against a 10pt builder than against a builder on some other point. That principle extends to rolls other than 51, 62 and 65, and to 18pt scenarios other than those touched on here.
Nack
