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BGonline.org Forums
Cubeful equities: how to derive from cubeless?
Posted By: NJ In Response To: Cubeful equities: how to derive from cubeless? (Matt Ryder)
Date: Saturday, 9 January 2010, at 11:21 a.m.
In early contact positions, is x always 6.8?
Well this is somewhat subjective. Janowski originally estimated x to be 2/3. GNU refined this estimation to be a function of the current board position, with 0.68 as the value for contact positions. I'm not sure what basis they used for that. Personally, I just use x=2/3 because it's easier that way. The value of x could certainly an area for future study. In fact, the neural net could be made to output x if trained to do so. Also, Janowski mentions that x can be different for each side, which can be handled in the various formulas.
I'm assuming W = (Win + Gammon + BG)/Win and L = -(Opp Win + Opp Gammon + Opp BG)/Opp Win?
Yes, W and L are exactly how Janowski described. Your formula works if Win includes gammons and backgammons, and Gammon includes backgammons.
Can you explain how GNU does this in a little more detail?
Suppose the cube is centered. First, GNU computes Ec as I described earlier. This is the equity for no double. The equity for double/pass is trivial to compute. The equity for double/take is computed using the formula for Eu (equity of unavailable cube), with the cube at 2 owned by the opponent. The graphs for Eo and Eu (cube owned equities) are two-line-segment graphs.
For example, let's go over how the Eu would be computed. First, your take point for the opponent owning a 2-cube is computed. The graph for Eu (live) then looks like two line segments with the three endpoints of (0,2L) (TP, TPE) (1,2W), where TP is your take point at cube=2 and TPE is the take point equity. The take point equity is simply the equity of a redouble/pass from the opponent when he is holding a 2-cube. You then find Elive by interpolating on the two segmented graph, and find Edead by interpolating from (0,2L) to (1,2W). Then Eu = Elive * x + Edead * (1-x).
Now that you have three equities. Ec = equity of no double. Eu = equity of double take. Ep = equity of double pass. First, you figure out whether the opponent will take or pass. If Eu > Ep, opponent will pass. If Eu < Ep, opponent will take. Call Ed the equity of doubling, and set Ed to min(Eu, Ep). Now compare Ec with Ed. If Ed > Ec, you should double. If Ed = Ec, it's an optional double. If Ed < Ec, you should not double.
At this point, the final cubeful equity E should be: E = max(Ed, Ec).
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