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11.5.5.4 n-ply Cubeful equities

The previous section concerned the calculation of 0-ply cubeful equities, so how so GNU Backgammon calculate cubeful 2-ply equities? The answer is: by simple recursion:

Equity=0

Loop over 21 dice rolls

Find best move for given roll

Equity = Equity + Evaluate n-1 ply equity for resulting position

End Loop

Equity = Equity/36

Note that evaluating the n-1 ply equity involves a cube decision, since the opponent may double, so GNU Backgammon will actually calculate the two n-1 ply equities: (a) assuming no double, and (b) assuming double, take. These two equities are combined with the equity for a pass, and the optimum of these three is added to the resulting equity. For a cubeful 2-ply evaluation GNU Backgammon will end up calculating the following cubeful 0-ply equities: centered 1-cube, opponent owns 2-cube, owned 4-cube, and opponent owns 8-cube.

Note that the 2-ply level does not use the cube efficiency, it's not used until at the 0-ply level, but it's possible to calculate an effective one by isolating x in the basic cube formula:

x(eff) = (E(2-ply cubeful) - E(2-ply dead))/(E(2-ply live)-E(2-ply dead)).