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Re: Bearoff cube problem - Rollout
Posted By: Ian Shaw In Response To: Bearoff cube problem (DC-Ohio)
Date: Tuesday, 23 January 2007, at 1:54 p.m.
The score (after 0 games) is: White 0, Blue 0
Move number 9: White on roll, cube decision?
White 37
Blue 53 Position ID: 270NAAB/2wwAAA Match ID: MAEAAAAAAAAA
• White doubles
Cube decision Rollout cubeless equity +0.516 Cubeful equities: 1. Double, take +0.882 2. Double, pass +1.000 +0.118 3. No double +0.785 -0.097 Proper cube action: Double, take Rollout details
Win W g W bg Lose L g L bg Cubeless Cubeful Centered 1-cube 0.758 0.000 0.000 - 0.242 0.000 0.000 +0.516 +0.785 Standard error 0.000 0.000 0.000 - 0.000 0.000 0.000 0.000 0.001 Player Blue owns 2-cube 0.757 0.000 0.000 - 0.243 0.000 0.000 +1.030 +0.882 Standard error 0.000 0.000 0.000 - 0.000 0.000 0.000 0.000 0.001
Here's the rollout, and I can be pleased with my estimate of 24.5% wins for Blue. But did I get there the right way?
The actual epcs are White 60.4, Blue 63.9.
I was a bit too harsh on White; moving two chequers from the five point to the ace IS an improvement, but only by 1.3 pips out of the eight moved. Moving the spares from the two and three points is almost entirely wasted, improving by only 0.3 of the three pips moved. A better estimate would have been 61.7 (not 61.5 as I'd remembered) - 1.3 - 0.3 = 60.1.
I'm happy with my estimate of Blue's position, noting only that stacking a spare from the 5 onto the 4 is worth about 1/2 a pip on its own.
From the correct epcs, my calculation would have been: 60.4 epc - 7.5 wastage = 52.9 pips in an efficient race, and would have given (52.9 - 5) / 7 = 6 pips as the point of last take, and a three-pip lead required for the initial double.
Here is an efficient race with epcs of White 60.5 Blue 63.9, which is about the closest I could construct.
Move number 15: White on roll, cube decision?
White 53
Blue 57 Position ID: uPsAAFDtHgAAAA Match ID: MAEAAAAAAAAA
• White doubles
Cube decision Rollout cubeless equity +0.410 Cubeful equities: 1. Double, take +0.649 2. Double, pass +1.000 +0.351 3. No double +0.640 -0.010 Proper cube action: Double, take Rollout details
Win W g W bg Lose L g L bg Cubeless Cubeful Centered 1-cube 0.705 0.000 0.000 - 0.295 0.000 0.000 +0.410 +0.640 Standard error 0.000 0.000 0.000 - 0.000 0.000 0.000 0.000 0.000 Player Blue owns 2-cube 0.705 0.000 0.000 - 0.295 0.000 0.000 +0.819 +0.649 Standard error 0.000 0.000 0.000 - 0.000 0.000 0.000 0.000 0.000
Rule 62 correctly predicts that in this efficient position, White has an initial double with a 3 pip lead, but not a redouble. (A second rollout had redoubling as a 0.038 error.
Blue wins 29.5% here compared to 24.3% in the original position. Given the same epc, Blue wins 5.2% less when both sides have moderately inefficient positions. I should have penalised Blue a good 2 to 3 pips or more for the inefficiency, because it is both sides that are inefficient; I need to go back and revise Douglas Zare's GammonVillage article on Inefficient Races.
Trailing 67.3 - 60.5 = 6.8 effective pips, the position below shows a borderline pass, suggesting that the rounding-down aspect of Rule 62 needs looking at when dealing with epcs. (Rule 62 would come up with 6.84 pips without rounding down.)
The score (after 0 games) is: White 0, Blue 0
Move number 28: White on roll, cube decision?
White 53
Blue 60 Position ID: sfsBAKDaPQAAAA Match ID: MAEAAAAAAAAA
• White doubles
Cube decision 2-ply cubeless equity +0.568 0.784 0.000 0.000 - 0.216 0.000 0.000 Cubeful equities: 1. Double, pass +1.000 2. Double, take +1.018 +0.018 3. No double +0.849 -0.151 Proper cube action: Double, pass
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