Cube Handling in Races

Forum Archive : Cube Handling in Races

Doubling formulas

From:   Michael J. Zehr
Address:   tada@ATHENA.MIT.EDU
Date:   30 January 1995
Subject:   doubling in races (long, first of several parts)
Google:   3gjpeo$858@senator-bedfellow.MIT.EDU

The topic of doubling in a race has come up recently, and a few ideas
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.

(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

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
Did you find the information in this article useful?          

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Cube Handling in Races

Bower's modified Thorp count  (Chuck Bower, July 1997) 
Calculating winning chances  (Raccoon, Jan 2007) 
Calculating winning chances  (OpenWheel+, Nov 2005)  [GammOnLine forum]
Doubling formulas  (Michael J. Zehr, Jan 1995) 
Doubling in a long race  (Brian Sheppard, Feb 1998) 
EPC example  (adambulldog+, Jan 2011) 
EPC example: stack and straggler  (neilkaz+, Jan 2009) 
EPC examples: stack and straggler  (Carlo Melzi+, Dec 2008) 
Effective pipcount  (Douglas Zare, Sept 2003)  [GammOnLine forum]
Effective pipcount and type of position  (Douglas Zare, Jan 2004)  [GammOnLine forum]
Kleinman count  (Øystein Johansen+, Feb 2001) 
Kleinman count  (André Nicoulin, Sept 1998) 
Kleinman count  (Chuck Bower, Mar 1998)  [Recommended reading]
Lamford's race forumla  (Michael Schell, Aug 2001) 
N-roll vs n-roll bearoff  (David Rubin+, July 2008) 
N-roll vs n-roll bearoff  (Gregg Cattanach, Nov 2002) 
N-roll vs n-roll bearoff  (Chuck Bower+, Dec 1997) 
Near end of game  (Daniel Murphy, Mar 1997) 
Near end of game  (David Montgomery, Feb 1997) 
Near end of game  (Ron Karr, Feb 1997) 
One checker model  (Kit Woolsey+, Feb 1998) 
Pip count percentage  (Jeff Mogath+, Feb 2001) 
Pip-count formulas  (Tom Keith+, June 2004)  [GammOnLine forum]
Thorp count  (Chuck Bower, Jan 1997) 
Thorp count  (Simon Woodhead, Sept 1991) 
Thorp count questions  (Chuck Bower, Sept 1999) 
Value of a pip  (Tom Keith, June 2004) 
Ward's racing formula  (Marty Storer, Jan 1992) 
What's your favorite formula?  (Timothy Chow+, Aug 2012) 

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