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Original position | |||||
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Variant 4 ....(Blue owns his 9pt instead of his 10pt) | |||||
If you widen your window, you can view the two diagrams side-by-side.
The left-hand (first) position is the "original" one posted.
The right-hand (second) position is my "Variant 4."
Casper: Very interesting. I would have expected that in the last position 20 and 11 would be clearly right given how effectively White is hemmed in. What do I miss?
You seem to be implying that while you accept that slotting the 7pt is the best play by far (by .053) in the original position, the fact that Blue has improved from 10pt to 9pt should not only nullify slotting's .053 slot-vs-up lead (to .004 the other way), but it should swing it the other way by an amount that is "clear" (say at least .030?).
Well, I'm not confident if I can explain to your satisfaction why the impact of the slot-vs-up margin based on Blue's stronger blocking point is, based on your intuition, "only" .057 rather than .083+. However, for readers who might have found the .057 swing surprisingly large, I would offer this explanation:
The blocking value of Blue's 9pt is optimal against checker(s) on the 22pt. That value is diminished (i.e., it becomes only nominally better than owning the 10pt in a slugfest) when White hits with a 4. In a relative sense, these variations crush Blue when he has slotted in Variant 4.
Leading the pack is 22 (counts as a "4"), which makes the 9pt irrelevant in Variant 4. Moreover, if U is played, 22 hits and escapes in orig whereas it is impotent in Var-4). This roll is somewhat distantly followed by 42, which double hits in Var-4, though 43 64 and 41 are not far behind. The roll 43 is especially cute because in the original position White is induced to break her 8pt (i.e., slotting benefits from extra counterplay that is absent in Var-4).
In sixth place (in the ranking of rolls that support slotting in the original versus coming Up in Var-4) is double 6s! That's a fun reason for staying back on the 21pt in the original position. The swing (difference of differences) on 66 is significant though only about 2/3 that of 43 64 41, and about 40% that of 22.
Granted, a counterargument is that if White neither hits nor escapes, then in Variant 4 there is a greater difference in the number of rolls that Blue can either add a point to his prime or point on the 3pt when he slots than when he does not, than there is making the same comparison in the original position. In Var-4, 62 54 51 43 42 31 22 add a point after the slot play that could not be added after the up play, whereas in the original position only 62 51 44 (total 5 numbers, or 7 numbers if 54 is charitably counted) have the same distinction. (Perhaps this counterargument is the piece you failed to offset, Casper?)
That consideration is valid, but the pitfall is giving it too much emphasis. It is more important to consider White's immediate roll than Blue's next roll, because the latter (in pursuance of a particular factor) represents only a subset of variations. If White hits and/or fully/partially escapes, we're in a different kind of game and the previous paragraph no longer applies. Against Orig $, she does so with all 4s and 2s except 32, plus 31 65 63 66 (25 numbers). Against Var $, it's all 4s and 2s plus 63 31 (24 numbers). Against Orig U, 66 65 63 62 44 22 (9 numbers). Against Var U, 62 54 52 44 (7 numbers). On average, that's 16+ numbers, In other words, the influence of the previous paragraph is only about 55% of what one might otherwise perceive to be its weight (which is already less substantial than it may seem).
In short, it is the swing on 22 42 43 64 41 that primarily explain the nullification of the slotting play's advantage in the Variant 4 position (compared to the original). Other rolls (in descending order of impact, 61 33 53 32 51 52 65 11 in support of that diff-vs-diff paradigm, and 63 31 54 62 55 44 21 in opposition of the paradigm) roughly offset each other.
Nack