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Very nice, but just a tad conservative for raw pip counts
Posted By: Ian Shaw In Response To: Very nice, but just a tad conservative for raw pip counts (Timothy Chow)
Date: Wednesday, 3 February 2010, at 11:47 p.m.
I can clear up a few points here.
Nack, I've added your new formulae to the spreadsheet and can confirm that they do indeed work for all counts except 111. But if you were awake at 4 am doing square roots in your head to check all this then you must be quietly bonkers!
David, with reference to the term "Gold Standard", it's a phrase Nack and I started using as a shorthand to refer to the table published by Walter Trice, (firstly in the Gammon Village article "From Pip Count to Cube Action" in Dec 2001, and then in the chapter of the same name in "Backgammon Boot Camp", p. 136.)
As I wrote to Nack, "we've referred to Trice's table as the "Gold Standard", but he didn't document how the table is derived, so we can't assess it's accuracy. I assume he collated it from data produced by his various race and bearoff programs, taking averages of various positions for each pip count and lead." If anyone actually knows how it was derived, I'd love to hear it.
Timothy, Trice discusses the "one checker model" in the GV article "Cube Value" (Dec 2005). He states that, "the assumption of a single checker implies higher variance than what one generally finds in real positions, and as a result the model produces results that are just slightly optimistic for the trailer in the race."
If you compare the two tables, you will see that the table he gives for cube action is more cautious than the one checker model. For example, for a pip count of 75 he states the point of last take is 9 pips for backgammon, but 10 pips for the one checker model. Somehow, Walter adjusted for real backgammon in some unstated way.
If I were going to investigate individual positions, I would start with an existing database. Sconyer's is well known, but I don't have it. The gnubg database is freely available and can be interrogated - instructions are available somewhere in the documentation.
Bob, your match adjustment is intriguing. You ask, "Would anyone have that 78% value more accurately?" Have you checked the graph published by Tom Keith at http://www.bkgm.com/articles/CubeHandlingInRaces/#probability_of_winning ?
Neil, I don't think Walter ever claimed that his table and formula were spot on for every position. However, he did say they applied to low wastage positions, and the three you present certainly qualify. I'm going to speculate here: if Walter's rule is supposed to generally cover low wastage positions in general, his analysis might have included some positions with slightly more wastage then the very efficient positions you rolled out. Again according to Trice, the doubling window is narrower when at least one side has somewhat higher wastage. I think Douglas Zare also discusses this in one of his GV articles. Therefore, if Trice's rules reflect the mean of low-wastage positions, and your examples are at the extreme low-wastage end of the valid range, then it is possible that the trailer can legitimately take a pip lower when both positions are as efficient as those you offered. And as Bob pointed out, the equity losses for the wrong decisions in your rollouts are very small.
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