Cube Handling in Races

 Kleinman count

 From: Chuck Bower Address: bower@bigbang.astro.indiana.edu Date: 25 March 1998 Subject: Re: Is this a take? Forum: rec.games.backgammon Google: 6fbr4v\$q98\$1@jetsam.uits.indiana.edu

```Dan Scoones wrote:
> Last night I was playing a "money" session with JellyFish Level 7 and
> got to the following position.
>
>   +24-23-22-21-20-19-+---+18-17-16-15-14-13-+
>   |    O  O  O  O  O |   | O     O     O    |
>   |    O  O     O  O |   |             O    |
>   |                O |   |                  |
>   |                  |   |                  |
>   |                  |   |                  |
>   |                  |   |                  |[64]
>   |                  |   |                  |
>   |                  |   |                  |
>   |                  |   |    X             |
>   |                X |   |    X             |
>   |    X  X     X  X |   | X  X             |
>   |    X  X     X  X |   | X  X  O          |
>   +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+
>
> X (rook) on roll.  Cube action?
>
> The pip count is 84-96 in X's favour.  X's lead is just over 14% of
> his count.
>
> In Advanced Backgammon, Bill Robertie states that, in long races, the
> trailer has a pass if the leader's racing advantage exceeds 12% of his
> pip count.  That is the case here.
(snip)
> Applying the Thorp count yields a different picture.  Here, the
> 0, which indicates a take.
(snip)
> ...One of them must be yielding an incorrect estimate.  The question is:
> which one, and why?

Good question.  My experience with races has led me to use the
percentage method and the Thorp Count to complement each other.  That is,
usually they don't both work for the same position.  Thorp is best when
both players have ALL their checkers borne in (and/or off), in other
words-- no checkers in the outfield.  The percentage method tends to work
well in these longer, smoothly distributed races.

As most of you know, I've converted both of these formulas into
cubeless game winning chances.  I repeat those modifications:

W = 0.54 + 2*p     (where 'p' is the percentage lead of player on
roll as described by Dan above)

W = 0.74 + 2*T     (where 'T' is the Thorp Count)

Here we have p = 1/7 and T = 0, as Dan has already said, giving 82.5%
(I have a hangup rounding 0.5!) and 74% respectively.  Clearly a
disagreement (again as Dan observed).

There is another slightly more complicated method which, I believe when
fully implemented, works over the full range of both the percentage and
Thorp Counts.  That is the Kleinman Count.  I have yet to learn the FULL
Kleinman Count.  (Anybody want to loan me some time?  I promise to pay
you back in my next lifetime....)  In it's simple form (for long races
like this one), K = (D + 4)^2 / (S - 4) where D is the pip count
difference and S is the pip count sum.  If you plug the above pip counts
into this formula, you find K = 1.45.  So what do you do with THAT??

Danny (Kleinman, that is) created a table based up the standard
normal distribution.  If you memorize the table, then you can convert K
to cubeless winning chances.  Some people just memorize the "key" money
values of 68% (or 70%), 72%, 78%.  In tournament backgammon, though,
you often want to know winning chances over a MUCH wider range.  How
do you get THAT??

I've developed an algorithm (which isn't really THAT hard to use,
especially if you're willing to learn a couple chapters of Art Benjamin's
book "Mathemagics").  I "reveal" the algorithm here for the first time.
(A roll of the drums, please!)

if K < 1:     W = 0.50 + 0.267*sqrt(K)

This reproduces the Kleinman Table to 0.5% or better over the range
0.50 < W < 0.75.  If you don't like 0.267, then 0.27 works even better
except between 73% and 76%, but even there it is off less than 1%.

Now, what about K >= 1?  Well, I derived a logarithmic fit there
(which works up to 95%) but even Art doesn't normally do logarithms
in his head!  (...though he could if he saw the need, I'm sure.)
I then saw that it's not THAT hard to memorize the table if you see
the following pattern:

(approx.)
W            K         delta K

0.76    *     1.0          ---
0.77          1.1          0.1
0.78          1.2          0.1
0.79          1.3          0.1
0.80    *     1.4          0.1
0.81          1.55         0.15
0.82          1.7          0.15
0.83          1.85         0.15
0.84    *     2.0          0.15
0.85          2.2          0.2
0.86          2.4          0.2
0.87          2.6          0.2
0.88          2.8          0.2
0.89    *     3.0          0.2
0.90          3.33         0.33
0.91          3.67         0.33
0.92    *     4.0          0.33
0.93          4.33         0.33
0.94          4.67         0.33
0.95    *     5.0          0.33
0.96    *     6.0          1.0
0.97    *     7.0          1.0
0.98    *     8.0          1.0
0.99     (does it matter???)

(Note:  * are "key" values of K in my algorithm.)  As above, this part
of the algorithm matches Kleinman's table to 0.5% or better.

Notice that in the above problem the Kleinman Count says X will win
just over 80%.  I did 180 level-6 cubeless JF3.0 rollouts and it got X
winning 79.7% with an S.D. of 0.2%.  Are you impressed with the Kleinman
Count?  One problem doesn't mean a whole lot, but I've been amazed
at how accurate it is in general.

So where do you go from here?  Well, as I mentioned, this is the
original Kleinman Count.  There are advances (which don't involved any
more square roots!) which make adjustments to the Count for distributional
anomolies (similar to the adjustments made in the Thorp Count).  That is
the part I still need to learn.  Kleinman's many books (which are valuable
to the serious student) cover the extensions.  You can get them (and about
everything else on the BG market) for Carol Cole ( cjc@flint.org ).

Special thanks to Marc Gray (FlashGammon on FIBS) for urging me to
learn the Kleinman Count.

Chuck
bower@bigbang.astro.indiana.edu
c_ray on FIBS
```

### 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)
Doubling formulas  (Michael J. Zehr, Jan 1995)
Doubling in a long race  (Brian Sheppard, Feb 1998)
EPC example: stack and straggler  (neilkaz+, Jan 2009)
EPC examples: stack and straggler  (Carlo Melzi+, Dec 2008)
Effective pipcount  (Douglas Zare, Sept 2003)
Effective pipcount and type of position  (Douglas Zare, Jan 2004)
Kleinman count  (Øystein Johansen+, Feb 2001)
Kleinman count  (André Nicoulin, Sept 1998)
Kleinman count  (Chuck Bower, Mar 1998)
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)
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)