doubling in races (long, first of several parts)
Michael J Zehr
have been batted around, some of them contradictory. There are some
area of analysis here that I've been meaning to tackle for a bit anyway,
and since I have a bit of spare time on my hands, it seems like now
would be a good time.
I intend this to be the first of several parts, slowly building on
previous work, with postings over the next week or so.
To set the stage a bit, the topic is strictly discussing races, not
bearoffs. In a race, one assumes that by the time one starts bearing
off, one will have a decent distribution. Thus one can ignore issues of
gaps and stacks and look just at the pip count. Of course this is a
simplication and in a real game this simplification is going to be more
and more wrong as one gets closer to bearing off. Nonetheless, it's
worthwhile doing because one can quickly develop rules of thumb and
there will be many real positions for which these rules are close enough
to use without fear of a major blunder.
This article will be limited to cubeless games and will look at percent
chance of winning. Ignoring the cube isn't quite as bad as it sounds
for racing positions. Deviations between cubeless and cubeful results
are greater in bearoffs than in races, and with only a few games to go in
a match you need to look at cubeless chances anyway.
I'll start by looking at two formulas and a model.
The first formula is the 8/9/12 formula: double when ahead 8%, redouble
when ahead 9%, and take when down no more than 12%. The best feature of
this formula is the ease of remembering and using it. Keep in mind one
of the meta-rules of backgammon: remember and applying an inaccurate rule is
often better than not remembering and not applying an accurate rule.
The rule is simple and fairly intuitive, but misses out on one of the
key factors of statistics and randomness: under most modeled random
processes, variation from expected values is proportional to the square
root of the number of trials/sample-size/problem-size, not proportional
to the trials/size itself.
This is where the Kleinman formula comes into play. This formula looks
at the pip lead squared divided by the pip sum. (It first applies a 4
pip reduction to the side on roll.) In other words, it's looking at the
size of the lead compared to the square root of the length of the race
instead of the ration of the lead to the plain length of the race.
The advantage of this formula is that it predicts cubeless winning
percent (CWP) by cross-referenceing the value obtained above with a
table of values. The disadvantage is that it's a bit tricky to
calculate across the board, but there are some tricks that allows one to
do this anyway.
The final item to look at is a model rather than a formula. One can
model an n-pip race by a single checker n pips away from bearing off.
Of course it's a gross simplification, but it's one that allows exact
calculations of CWP and equity for this simpler game. Provided one can
understand and predict the variation between this simple model and a
real backgammon race, the results from the simple model are applicable.
[For those who are interested, a program to do these calculations is
fairly simple to write, and when properly written it can take less time
to calculate the table than to read it from disk, so there's no need to
store a database for it. E-mail me if you're interested in the
programming aspects of this type of analysis.]
We'll look at a 50pt race, a 100pt race, and a 150pt race, and then draw
some conclusions from the results (See Appendix I for the full table.)
Leader Trailer Klein. Model
50 56 75.80 74.48
50 57 77.82 76.42
50 58 79.72 78.26
100 110 75.47 74.17
100 111 76.95 75.59
100 112 78.35 76.96
100 113 79.70 78.27
150 163 75.29 74.01
150 164 76.50 75.18
150 165 77.68 76.32
150 166 78.82 77.43
150 167 79.93 78.50
150 168 81.00 79.55
Note that the 8/9/12 rule says to drop at 50-56/57, 100-112/113, and
150-168/169. (The split values are because I don't know if the rule says
take when behind less than 12% or less than or equal to 12%.)
The Kleinman formula is consistenly 1-1.5% better for the leader than
the model is in the neighborhood of the cashpoint (75-79%). (They are
closer when the race is closer -- see the appendix for details.)
How are these results likely to compare to real postitions? Since I
know all the details of the model, I can make a direct comparison there,
and use that as the basis for the comparison with the two formulas.
One obvious discrepancy is that in the one-checker model, pips are never
wasted. This affects both sides, so an N-pip race in the one-checker
model is over sooner than a real N-pip race. The longer race tends to
favor the trailer usually, but to win, the trailer needs to win high
doubles. In the one-checker model, the trailer can make full use of
them, whereas in a real game, the trailer can't. (Also when bearing in
to set up for bearing off, the doubles make it harder to smooth one's
distribution.) These factors hurt the trailer.
As a fixed data point to compare all of these race versions, I'll look
at the 000456 and 000357 positions. According to Walter Trice (and I'll
take his word for it rather than verifying it myself *grin*), these are
the two bearoff positions with the least wastage as a percentage of
total pips. Hence they should be closest of any bearoff position to the
one-checker model. According to a highly accurate (but not exact)
bearoff database of mine, the 000456 symmetric position (77-77 pips)
yields a CWP of 59.38 to the side on roll. The 000357 (79 pip) position
yields a 59.2 CWP.
The Kleinman yields 59.11 and 59.00 respectively and the one-checker
model yields 58.70 and 58.58 respectively.
As a comparison in a "more interesting" range of races, let's look at
the 000346 vs 000456 positions. This is a 68-77 pip race and yet should
still be relatively wastage-free. The bearoff database says 78.63,
Kleinman says 78.05, and the one-checker model 76.64.
Conclusions:
(You've been waiting for me to get this far, haven't you? *grin*) The
one-checker model consistenly overrates the trailer's chances due to the
trailer needing high doubles (which will be partially wasted in a real
game).
The Kleinman formula looks pretty accurate compared to the one-checker
model when you take into account the bias in the one-checker model.
At the few points at which I'm able to check the racing models against
real bearoff values, the Kleinman still overestimates the trailer's
chances. This overestimation seems to range from about .2% for close
races to .5% near the drop point.
The 12% rule looks like it gets worse and worse the longer the race is.
-michael j zehr
(upcoming topic: one-checker model with cube in play.)
Appendix I
Cubeless Win Percent with leader at roll at given pip counts. The
predictive value using the Kleinman formula and the calculated value for
the one-checker model are given.
Leader Trailer Klein. Model
50 50 61.34 60.84
50 51 64.02 63.33
50 52 66.58 65.74
50 53 69.05 68.07
50 54 71.41 70.30
50 55 73.66 72.44
50 56 75.80 74.48
50 57 77.82 76.42
50 58 79.72 78.26
50 59 81.51 80.00
100 100 57.98 57.63
100 101 59.95 59.45
100 102 61.83 61.24
100 103 63.71 62.99
100 104 65.53 64.71
100 105 67.32 66.40
100 106 69.05 68.04
100 107 70.74 69.64
100 108 72.36 71.20
100 109 73.95 72.71
100 110 75.47 74.17
100 111 76.95 75.59
100 112 78.35 76.96
100 113 79.70 78.27
100 114 81.01 79.54
150 150 56.50 56.24
150 151 58.10 57.73
150 152 59.70 59.22
150 153 61.25 60.68
150 154 62.80 62.13
150 155 64.30 63.55
150 156 65.79 64.95
150 157 67.25 66.33
150 158 68.67 67.68
150 159 70.07 69.00
150 160 71.42 70.30
150 161 72.75 71.57
150 162 74.04 72.80
150 163 75.29 74.01
150 164 76.50 75.18
150 165 77.68 76.32
150 166 78.82 77.43
150 167 79.93 78.50
150 168 81.00 79.55
150 169 82.02 80.56
Michael J Zehr
As a quick addendum to my earlier post:
After writing the conclusions, I was speculating on how much the pip
wastage from the high doubles required for the trailer to win affected
the CWP. I reran the model treating 44 as 15 pips, 55 as 19 pips, and
66 as 22 pips, and the one-checker model then produced almost eactly the
same values as the Kleinman formula.
I've worked the cube into the model so I'll wait a few days in case
people want to follow up on this article and then post another
addressing the cube.
-mjz