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"Nack 57" rule -- new race formula (for both traditional and Naccel)
Posted By: Nack Ballard In Response To: "Nack 57" rule -- new race formula (for both traditional and Naccel) (David Rockwell)
Date: Wednesday, 3 February 2010, at 11:13 p.m.
David: I may have missed this in other posts - where can I find the "gold standard" table?
Nack: I couldn't find it searching online (though, admittedly, I'm not so good at that). Ian Shaw, who did some analysis/work on the gold standard table some years ago, sent it to me. (Thanks, Ian!)
David: How was the gold standard table derived?
Nack: This is an explanation sent to me in an e-mail from Paul Weaver, who writes better than I do:
More than thirty years ago, while I was working as a computer programmer, I did the single checker model. I was neither the first nor the last to do this.
Here is what I mean by the single checker model: Assume that each player has only one checker and a very long backgammon board, consisting of 150 pips and no contact.
All of the racing probabilities and doubling decisions can be computed recursively. Start with (1,1). Each side has just 1 pip to go to win. The guy on shake is 100% to win and the cube actions are initial double, redouble, and pass.
Walter actually took this concept a step further and programmed the single checker model not only for money (as I had done) but also for all match scores, giving the user the option to enter his own MET.
How does the single checker model differ by the real backgammon race of 15 checkers? Walter said that there are two forces acting in opposite directions that cancel each other out.
The two factors: (1) The 15-checker race has more wastage than the single checker model and (2) the 15-checker race has greater variance. It was Walter's belief that these two forces negated each other.
In other words, Walter believed that the results of the single checker model applied to the real 15-checker backgammon race (assuming positions of minimal wastage).
David: All pip counts can be verified on a spreadsheet with about 30 seconds work once I have the table.
I'll e-mail you a copy of the table along with how various formulae match up. I'm also encapsulating just the key information below, for others.
I'm not sure if the first part of this table is considered to be gold standard. I derived it simply from Trice's "subt 5, div by 7, round down" formula, which (of course) matches perfectly. (You probably shouldn't pay much attention to the very low end; I just wasn't sure exactly where I should stop extrapolating, so I went all the way down.) Points of last take (differences) are on the left; corresponding leader count ranges are on the right.
0 -- 5 to 11
1 -- 12 to 18
2 -- 19 to 25
3 -- 26 to 32
4 -- 33 to 39
5 -- 40 to 46
6 -- 47 to 53
7 -- 53 to 60Walter included 61 in his table even though he acknowledged it as a miss, because the upper part of his "Rule 62" (10%, round UP, add 1) also misses 61 (as does Zare/Weaver/Shaw).
[Other than for Nack 57, Robertie's 8/9/12% rule is the only one that hits 61's take point. However, Robertie's rule creates way more misses than Trice or Z/W/S on other parts of the table; not that we would expect otherwise -- it was a much earlier work.]
I'll pick up the gold standard table from there, overlapping the pipcounts that work for 7 with the Nack 57 rule:
7 -- 55 to 61
8 -- 62 to 69
9 -- 70 to 78
10 -- 79 to 88
11 -- 89 to 99
12 -- 100 to 110
13 -- 111 to 122As I mentioned earlier, the Nack 57 rule (subt 33, double, find nearest square root) misses only at 111. To get a perfect match, add the corollary: "At 111, add 1."
Also, above the (admittedly, relatively unimportant) leader counts of 122, I believe Nack 57 creates fewer misses than Trice or Zare/Weaver/Shaw, by any reasonable extrapolation.
Nack
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