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Reconciliation example
Posted By: Nack Ballard
Date: Monday, 25 January 2016, at 5:25 a.m.
In Response To: 51SXXP62 mystery (Tenland)
It's easier to explain a detailed comparison of two positions rather than of three, and the two I'll choose are the 5pt position vs the 4pt position (ignoring the 3pt position).
As you noted, 51S42P62S is a .051 error, but the actual difference to account for conceptually is a bit larger. If we assume that White runs with 24/16 (not quite best against the 5pt, but it's more rigorous to compare apples vs apples) rather than 23/15 (which is blocked/illegal against the 4pt), the XGR evaluations (for Blue, after White's thirdroll play) are: 51S31P62S = .307, 52S42P62S = .281, 51S31P62r = .318, and 52S42P62R = .230.
Summary: Against the 5pt, White's S beats r by .318  .307 = .011. Against the 4pt, R beats S by .051. For a proper understanding, then, the amount to reconcile is not just the latter .051 error, but rather that margin plus the margin by which r (24/16) beats S against the 5pt (which is .011), for a total of .051 + .011 = .062.
As I see it, to reconcile the full swing of .062 that no doubt surprises you even more than the .051 (and evidently still surprises you even after the explanations given), it is necessary not only to find the relevant factors that cause the swing, but also to make some attempt to MEASURE them.
Encouraged by Neil's answer, you counted shots and concluded that White has 1 fewer hitting number back as a result, and that another 2 of those numbers (53) are duplicated. Okay, good, so far. In fact, White should anchor (not hit) with 53, and therefore the simplest way to look at it is that the shot difference is 3 numbers. [If White should hit instead of anchor, a partial adjustment would still be appropriate.]
I recommend you widen your window to view the diagrams below in a 2by2 grid.
In the "a" column (1 and 2), Blue made his 5pt (with 31). In the "b" column, Blue made his 4pt (with 42).
In the position 1 row (a and b), White split with 62. In the position 2 row, White ran to her 16pt with 62.
Let's see what happens when Blue hits loose on his 7pt after White splits (1a and 1b). As an example, Blue has a 51 to play. His equity (according to XGR++ evaluation) after hitting in 1a is .301. His equity after hitting in 1b is .273 (difference of only .028).
If we give Blue the same roll of 51 on the 2row, his parallel equities are .144 and .033 (difference of .111). This comparison illuminates a more usual advantage of the 5pt over the 4pt.
The edge of 5pt over 4pt is (.111.028) = .083 less on the 1row than on the 2row. Presumably, this is because of White's 3 extra return hits (as counted earlier). As 51 is rolled 1/18 of the time, multiplying that by the .083 gives us the overall size of the effect, which is .0046.
But 51 is not the only hitting roll! A similar effect occurs with 42, 61 and 63, and to a lesser extent 62 (where Blue lifts the 2 to his 5pt). Adding the effects of these five rolls (including 51) together comes to (let's say) .021, which is already a third of the overall amount to be reconciled.
In the same vein, consider rolls of 21 and 31. With blots on the 8pt and 7pt (well, not quite in the case of 21), White has many more shots, but there are again 3 fewer (including the anchoring 53) when Blue owns his 4pt instead of 5pt. Applying the same 2by2 grid concept (this time not diagrammed) to 31 yields equities of .111 and .104, and .387 and .292, where the difference of differences is .078. Multiply by 1/18 gives .0043. If a similar amount is estimated for the roll of 21, then that's .0086 you can add to the .021 from the total effect of the other hitting rolls (51 61 62 63). Now you're up to .030  nearly halfway to the .062 target.
In the other post I mentioned double 1s, which hurts much worse if White splits against the 4pt. Let's apply the grid to it: .341 .715, .280 .330. Wow, the diff of diffs is .324. Multiply by 1/36 = .009. I also mentioned double 2s, where the grid is .425 .355, .834 .476, yielding .288, multiplied by 1/36 = .008. Now you're up to (.030 + .009 + .008) = .047.
Evidently, to get to .062, the remaining rolls (41 32 43 53 54 33 44 55 66) add another .015 (though to be thorough, one should check these as well) and/or there is some slop in my estimate.
When you said, "I am in the process of doing a highly technical analysis of the position," this is the sort of thing I expected to see, and as a bonus it would have saved me the trouble!
Now that you know how to do such a study, I recommend you try your own in the future. Account for all the rolls individually (until you learn shortcuts to lump them)  you can do it. [I'm not sure if XG has a a crosstable of all twentyone rolls/equities, but Gnu has one called a "temperature map" under the analysis tab. Otherwise, punch up each position separately.] Alternatively, you can humbly accept the general explanations offered, but you won't learn as much. :)
Nack
White is Player 2
score: 0
pip: 153Unlimited Game
Jacoby Beaverpip: 163
score: 0
Blue is Player 1XGID=aBDaBcEadeB:0:0:1:51:0:0:3:0:10 Position 1a ...51S31P62S51
1. XG Roller++ 13/7* eq: +0.301
Player:
Opponent:56.51% (G:18.68% B:0.88%)
43.49% (G:11.53% B:0.63%)
White is Player 2
score: 0
pip: 153Unlimited Game
Jacoby Beaverpip: 161
score: 0
Blue is Player 1XGID=aBDaBcEadeB:0:0:1:51:0:0:3:0:10 Position 1b ...51S42P62S51
1. XG Roller++ 13/7* eq: +0.273
Player:
Opponent:55.43% (G:18.75% B:0.90%)
44.57% (G:11.74% B:0.61%)
White is Player 2
score: 0
pip: 153Unlimited Game
Jacoby Beaverpip: 163
score: 0
Blue is Player 1XGID=aBDBadEdeB:0:0:1:51:0:0:3:0:10 Position 2a ...51S31P62r51
1. XG Roller++ 13/8 6/5 eq: +0.144
Player:
Opponent:52.28% (G:16.31% B:0.48%)
47.72% (G:10.87% B:0.53%)
White is Player 2
score: 0
pip: 153Unlimited Game
Jacoby Beaverpip: 161
score: 0
Blue is Player 1XGID=aBDBadEdeB:0:0:1:51:0:0:3:0:10 Position 2b ...51S42P62R51
1. XG Roller++ 13/8 6/5 eq: +0.033
Player:
Opponent:49.99% (G:14.87% B:0.47%)
50.01% (G:12.56% B:0.71%)

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