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BGonline.org Forums
Will the real takepoint please stand up?
Posted By: Timothy Chow In Response To: Will the real takepoint please stand up? (bill calton)
Date: Monday, 6 October 2014, at 10:23 p.m.
It may help to take a step back and clarify what the exact criterion for a take/pass decision is, before jumping into the zoo of approximations that people have come up with.
Let W be the fraction of the time that you win. For any number n > 1, let Gn be the fraction of the time that you win exactly n points. So for example, G2 is the fraction of the time that you win an undoubled gammon, plus the fraction of the time that you win a single game with the cube on 2. Similarly, let gn be the fraction of the time that you lose exactly n points.
Next, let Pn denote the price of winning n points, and let pn denote the price of losing n points. In more common parlance, P2 is your "gammon price" and p2 is your opponent's gammon price.
Your raw take point (a.k.a. dead cube take point) is calculated by ignoring gammons and assuming that the cube will not be turned again. It's given by the ratio
(equity if you drop – equity if you take and lose) / (equity if you take and win – equity if you take and lose)
Now we can state:
TAKE/PASS CRITERION. Take an initial double if and only if W + Σn Pn Gn – Σn pn gn exceeds your raw take point.
(I've stated this for an initial double, but everything holds for a redouble or a reredouble if you just scale everything up appropriately.)
This formula is not that hard to derive, and if you're mathematically inclined, you might want to rederive it yourself. I believe it will greatly improve your understanding of this topic if you do so.
Everything else that people have come up with is an attempt to approximate this complicated formula with more practical formulas that can be used over the board.
Note that "recube vig" does not appear as a separate term in this formula. The cube is implicitly taken care of by various terms in the sum. For example, the G12 term tracks all the games in which you win a backgammon with the cube on 4, and g4 tracks your single losses with the cube on 4 plus your gammon losses with the cube on 2. Of course it is all but impossible to come up with good estimates of all these quantities OTB, let alone do the arithmetic to combine them all, but I think that it is still good to be aware of the exact formula when you're trying to make sense of approximations.
Note also that Pn and pn depend only on the score and are independent of the position, while Gn and gn depend on the particular position. We're all familiar with the fact that gammon wins and losses vary with position; what people tend to gloss over is the fact that recube vig also varies with position. The practical recommendations that I've seen have usually suggested ignoring the fact that the recube vig varies with position, and they offer some approximate formula for estimating them. This may be the best thing to do in practice, but there is no reason in principle why one could not get good at estimating (for example) how often I win a single after sending back the cube in this type of position, and thereby getting a better estimate of G2 directly instead of using a generic estimate of recube vig. After all, it's possible to get better at estimating gammon wins and losses, so why couldn't one get better at estimating Gn and gn?
In this post, I'm not going to bother analyzing all the different approximations that people have come up with and arguing their pros and cons. Others can do that much better than I can. But I did want to answer your question about the "real takepoint." Mathematically speaking, the real takepoint is the one I've described above. Beyond that, it's up to you to pick whatever approximation you personally find strikes the right balance between accuracy and usability.
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