This will be my attempt to manually analyse the moves first discussed in:
http://www.bgonline.org/forums/webbbs_config.pl?noframes;read=198354
Position discussed --->
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| XGID=-I-----------------AdbcDeA:1:1:-1:11:12:10:0:13:10 | |||||
First, let me give a brief recap.
"Paul" had thought a certain XG play 5/4 1/off(3) was a bug vs 5/4(4) and started a discussion in RGB:
https://groups.google.com/d/msg/rec.games.backgammon/X9ypTmJ1IBo/axXHgN0pCwAJ
"BlueDice" said "Yes, very strange, gnubg rollout agrees with XG rollout and by a similar equity difference! {5/4 1/off(3) vs 5/4(4)}"
"Michael" said "..the only way to check if it's in fact the best move or not is to actually count rolls in depth of possibly 5 plies. Who would care do such a boring thing anyway so let's rely on bot's results."
"Stick" said "5/4 1/off(3) is the right play and that's coming from me, not XG" and forcefully insisted he was right and wouldn't waste time explaining.
"Paul" then submitted but Stick was nice enough to explain some anyway saying "Notice XG's play wins more games and more gammons. (which sounded like "Stick knows so, because XG says so") and further added "when you throw a two you see... the next deuce will either create a gap itself or leave a shot. your play also fails to do is remove checkers from the board so that if you are ever hit you'll have a lot more men off..."
"Chow" said his "standard thing to do is to look at the distance between my highest point and the anchor and see how those numbers play. ...How do my 2's play? ... A couple of these and I could be looking at a blot."
"Murat" said "What does it matter if some people think rolling a 2 twice in a row may leave a blot, etc...? With XG's play, a 65 on the very next roll leaves a blot. Snowie's play (also mine and your initial choice) opens up the 5 point increasing the odds for the opponent to enter sooner and thus reduce his hitting combinations, etc. etc."
Now, let me start my analysis.
Position after 5/4(4) --->
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| XGID=-I-----------------A-fcDeA:1:1:-1:00:12:10:0:13:10 | |||||
Not all X2's are bad. Only 62, 52, 42, 32 force plays a 2 from the 3 point (8/36 = 0.2222), 22 creates a gap safe for now, 12 is safe.
Let's note that blue enters 27/36 = 0.7500, dances 9/36 = 0.2500
Position after first X2 --->
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| XGID=-I-----------------A-ebDeA:1:1:-1:00:12:10:0:13:10 | |||||
This time only 62, 52, 42 force plays another 2 from the 3 point (6/36 = 0.1667), 32, 22 create a gap still safe, 12 is safe
Chances of leaving a blot in 2 turns is 8/36x6/36 = 0.0370
Position-A after second X2 --->
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| XGID=-I-----------------A-daDeA:1:1:-1:00:12:10:0:13:10 | |||||
Blue had twice 27/36 chance of entering so most likely it will only hit with any 1 (11/36 = 0.3056)
Combined chances of White leaving a blot and Blue hitting is 8/36x6/36x11/36 = 0.0113
If Blue is still on the bar, it hits with any 3 and 61, 51, 21 (17/36 = 0.4722) and combined chances of White leaving a blot and Blue hitting becomes 8/36x6/36x17/36 = 0.0175
Let's note that at this stage White has 5 men off.
Position-B after second X2 --->
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| XGID=-I-----------------A-e-DfA:1:1:-1:00:12:10:0:13:10 | |||||
After rolling two X2's in a row, white can still be quite safe.
Position after 5/4 1/off(3) --->
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| XGID=-I-----------------AcccDbA:1:1:-1:00:12:10:0:13:10 | |||||
A 65 immediate leaves a blot on 1st roll, 2/36 = 0.0556
Let's note that blue enters 20/36 = 0.5556, dances 16/36 = 0.4444
Position after rolling 65 --->
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| XGID=-I-----------------AaccDbA:1:1:-1:00:12:10:0:13:10 | |||||
If Blue had entered on his previous one roll, it only hits with any 3 (11/36 = 0.3056)
Combined chances of White leaving a blot and Blue hitting is 2/36x11/36 = 0.0170
This is in just one turn vs two turns in the Snowie case above and 1.5 times the 0.0113!
If Blue is still on the bar, almost 50/50 in this case, it hits with any 5 and also 63, 53 (15/36 = 0.4167) and combined chances of White leaving a blot and Blue hitting becomes 2/36x15/36 = 0.0231
Again, this is in just one turn vs two turns in the Snowie case above and 1.32 times the 0.0175 comparing apples to apples.
If we also consider that Blue is 7 times more likely to be on the bar in this case than the Snowie case above, we should compare 0.0231 to 0.0113 which is more than twice!
Let's also note that at this stage White has 6 men off. Only 1 more than the Snowie case above.
Position after rolling X2 --->
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| XGID=-I-----------------AbcbDcA:1:1:-1:00:12:10:0:13:10 | |||||
Let's forget about XG rolling a 65 for a moment and make an alternative argument.
Since it was argued that XG's play was right because it avoided the dreaded X2's, let's grant their wish and make it roll a 62 or 52 instead.
This is the most likely position it will end up in (with a couple of similar possibilities only slightly better).
Now on the second roll 63, 53, 43, 66, 55, 44, 33 will leave a blot 10/36 = 0.2778
If Blue is not on bar,
a) and a 63, 53, 43 left a blot on the 5 point, any 3 hits 11/36= 0.3056
b) and a 66, 55, 44, 43 left a blot on the 4 point, any 2 hits 11/36= 0.3056
c) and a 44 left a blot on the 5 point, any 3 and 21 hit 13/36 = 0.3611
d) and a 33 left a blot on the 4 point, any 2 and 11 hit = 12/36 = 0.3333
e) and a 33 left a blot on the 3 point, any 1 hits 11/36 = 0.3056
Combined chances of White leaving a blot and Blue hitting in these cases range from 0.0848 to 0.1003
Compared to 0.0113 they are 7.5 to 8.9 times!
Since White kept its 4-point board during Blue's last 2 rolls, Blue is still more likely to be on the bar with more possibilities to hit.
If Blue is on bar,
a) and a 63, 53, 43 left a blot on the 5 point, any 5 and 63, 53 hit 15/36 = 0.4167
b) and a 43 left a blot on the 4 point, any 4 and 62, 22 hit 14/36 = 0.3888
c) and a 66, 55, 44 left a blot on the 4 point, any 4 and 62, 52, 22 hit 16/36 = 0.4444
d) and a 44 left a blot on the 5 point, any 5 and 63, 43, 23, 41 hit 19/36 = 0.5278
e) and a 33 left a blot on the 4 point, any 4 and 62, 32, 22, 31 hit 18/36 = 0.5
f) and a 33 left a blot on the 3 point, any 3 and 61, 41, 21 hit 17/36 = 0.4722
Combined chances of White leaving a blot and Blue hitting in these cases range from 0.1080 to 0.1467
Compared to 0.0175 they are 6.2 to 8.4 times!
Once more, if we also consider that after two turns for each, Blue is 3.2 times more likely to be on the bar in this case than the Snowie case above, thus compared to 0.0113 they are 9.6 to 13 times!!
My conclusions:
Whether Snowie is still relevant or not, and to the extent of Stick's/Chow's arguments regarding rolling two X2's in a row, etc. 5/4(4) is the correct play and 5/4 1/off(3) is the wrong/"buggy" play.
XG/GNUBG and Stick/Chow being wrong means, for XG, that its bug is inherited from GNUBG GPL code (or vice-versa:), for Stick/Chow, that they will say anything to defend that XG is right.
Even if I accept the possibility that XG/GNUBG's play may be right for some other reasons, Stick/Chow wouldn't know it unless the bots told them what it was.
If I made any calculation errors, I'm sure someone will enjoy pointing them out to me ;), as I enjoyed writing this tediously long piece... :)