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End Game Double - Beaver - Take Action?

Posted By: Chris Yep
Date: Friday, 28 June 2013, at 11:40 a.m.

In Response To: End Game Double - Beaver - Take Action? (Daniel Murphy)

By my calculations blue should pass white's beaver. In fact, if the original position were a 7-roll vs. 6-roll position blue should still pass white's beaver (even with 36.1% cubeless winning chances). Compared to a "standard" take/pass decision that blue would normally face, after a double/beaver/take white still owns the cube. Thus blue needs much more than 25% cubeless winning chances to take the beaver.

In your "Doubling when you're a roll behind" table, blue actually wins less than "16.7% cubeful" in the 3-roll vs. 2-roll and 4-roll vs. 3-roll cases. I.e. 5/6 of the time he immediately loses, while 1/6 of the time he's winning but white is still in the game (since white owns the cube).

In the (6 rolls vs. 5 rolls) case, while it's true that 5/6 of the time blue retains 28% cubeless winning chances, white owns the cube and is close to a redouble.

Here's a table of normalized equity for a player on roll owning the cube (for more than 3 decimal places see the HTML source):

Player owns cube, on roll, normalized equity
Opponent Rolls ¨
Player Rolls 1 2 3 4 5 6 7 8
1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2 -0.667 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3 -1.000 -0.491 1.000 1.000 1.000 1.000 1.000 1.000
4 -1.000 -0.954 -0.359 0.917 1.000 1.000 1.000 1.000
5 -1.000 -1.000 -0.897 -0.257 0.753 1.000 1.000 1.000
6 -1.000 -1.000 -0.994 -0.838 -0.186 0.652 1.000 1.000
7 -1.000 -1.000 -1.000 -0.981 -0.781 -0.148 0.587 1.000
8 -1.000 -1.000 -1.000 -0.999 -0.964 -0.728 -0.126 0.544
Key: Black = Redouble/Pass, Red = Redouble/take, Blue = No Redouble

In the original position (6 rolls vs. 5 rolls), white's equity after a double/beaver/take is 4 * (5/6 * 0.753 + 1/6 * -0.257) = 2.340. Since this is more than 2.000, blue should pass white's beaver.

For (7 rolls vs. 6 rolls), white's equity after a double/beaver/take is 4 * (5/6 * 0.652 + 1/6 * -0.186) = 2.049. So again blue should pass white's beaver (but it's close).

For (8 rolls vs. 7 rolls), white's equity after a double/beaver/take is 4 * (5/6 * 0.587 + 1/6 * -0.148) = 1.857. In this case blue should take white's beaver.

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