Cube Handling in Races

 Thorp count

 From: Chuck Bower Address: bower@bigbang.astro.indiana.edu Date: 23 January 1997 Subject: Re: Thorpe count, was Re: Cube action? Forum: rec.games.backgammon Google: 5c6eth\$sf2@dismay.ucs.indiana.edu

```Robert-Jan Veldhuizen wrote:
> I saw the Thorpe count explained here a while ago, but the terms used
> in that explanation were so ambiguous that I didn't understand it.
> Could someone explain it to me please ?
> (snip)
> For instance, where does the 74 value come from ?

The Thorp Count was popularized in Robertie's "Advanced Backgammon".  This
method was developed by Edward O. Thorp who is famous for his pioneering
work "Beat the Dealer" which explained how to become a favorite at
blackjack.  Note that the CORRECT spelling is "Thorp" without an "e" at
the end (even though Robertie misspelled it in both the 1st and 2nd
editions of his classic).

The method as described there is for money play only.  I have since
modified it for match play (and THIS is where the "74 value" comes from):

Here is how I perform the Thorp Count (and basically how it is written up

2) Add 2 for each checker remaining on the board.
3) Add an additional 1 for each checker on the 1 pt.
4) Subtract 1 for each home board point covered.
5) If the result up to now is 30 or greater, multiply by 1.1.
(Call this final result "R" for roller)

Repeat steps 1-4 for the NON-roller, BUT NOT STEP 5.
(Call this result "N" for NON-roller)

Now, subtract R from N ( = T) and compare the result as follows:

N - R .ge. -2, roller has an INITIAL double (centered cube);
N - R .ge. -1, roller has a REdouble;
N - R .le. +2, NON-roller has a take.

(here ".ge." means "greater than or equal to", and ".le."--you figure it
out!) As far as how to convert this to game winning chances, I noticed the
following correlation between the Thorp Count (above) and the percentage
cubeless game winning chances required to double/redouble/take as published
in the classic (but hard to find) article "Optimal Doubling in BG" by
Emmitt B. Keeler and Joel Spencer in OPERATIONS RESEARCH, vol 23 p.
1063(1975).

Assuming Roller's pip count is 70, K & S say roller should do the
following:

Roller's cubeless         Cube             Thorp
game winning chances       Action            Count

70%               Double, Take          -2
72%              REdouble, Take         -1
78%                  PASS               +2

Now if you overlook the "singularity" at 78% (Thorp says "barely a take"
and K & S say "barely a pass") you see that there is a linear relation
between the cubeless game winning chances and the Thorp Count:

W = 74 + 2*T   (in percentage).

Finally I assumed that you could "extrapolate" the result outside
the 70-78% range.  A study of bearoffs showed that this "Extened Thorp
Method" works rather well (IN GENERAL) for winning chances in the range
55-85%.  Happy (Thorp) counting!

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)