Forum Archive : Theory

Double/take/drop rates

From:   Gary Wong
Date:   20 June 1999
Subject:   Re: cubeful distribution
Google:   wt7loyxm5b.fsf@brigantine.CS.Arizona.EDU

( writes:
> I rolled out the starting position 50.000 times using Jellyfish
> Level 5 to produce the table below to show cubeful distribution.
> [table edited]
> Cube              1       2      4       8      16      32    Total
> Cashes        37.83   13.54   1.78    0.28    0.03    0.01    53.47
> Single         0.00   26.63   5.03    0.59    0.06            32.31
> Gammon         1.45   10.90   1.17    0.10    0.01            13.63
> Backgammon     0.07    0.46   0.04    0.01    0.01             0.59
>               -----   -----   ----    ----    ----    ----   ------
>               39.35   51.53   8.02    0.98    0.11    0.01   100.00

Thanks very much for that data!

> I'm not technically minded enough to make use of these numbers, but
> would they maybe affect match equity tables etc or could they shed
> any sort of light on recube vig?

Unfortunately the results are probably inapplicable to match play (where
cube behaviour is likely to vary drastically from money play depending
on the score) but they ought to help us model certain features of money

I have attempted to produce a simple Markov process which would produce
a distribution similar to that above.  It is:

If the cube is centred:
 - it is offered and dropped with probability 0.378 (terminal)
 - it is offered and taken with probability 0.606 (go to the state below)
 - a player wins a gammon with probability 0.015 (terminal)
 - a player wins a backgammon with probability 0.07 (terminal)

If the cube is owned:
 - it is offered and dropped with probability 0.220 (terminal)
 - it is offered and taken with probability 0.152 (remain in this state)
 - a player wins a single game with probability 0.455 (terminal)
   (These wins can reasonably expected to belong to the player who last
   turned the cube; we presume the cube holder can always reach a point
   where she has a correct double before she wins a single game).
 - a player wins a gammon with probability 0.172 (terminal)
 - a player wins a backgammon with probability 0.001 (terminal)
   (It's hard to determine who the gammon and backgammon wins belong to.
   Certainly the vast majority will be the player who turned the cube, but
   there will be occasions where the cube owner becomes too good to double
   and goes on to win a gammon or backgammon without having had a correct

If two players happen to play with cube behaviour described by the above
model, then we can make the following observations about the results of
their games:

 - Most (62%) initial doubles are takes.  This value agrees reasonably
   closely to an earlier result at:

 - Most (59%) redoubles are drops.

 - The final cube value follows the following (geometric-like)

     p(X=1) = 0.394
     p(X=2^n) = 0.514 x 0.152^{n-1}    for n = 1, 2, 3...

 - The cube is turned and accepted 0.71 times per game on average; the
   average final value of the cube is 1.87.

 - A single game is worth 2.2 points on average; the standard deviation is
   3.0.  I have previously assumed 2 and 3 respectively for these
   parameters in my own games.

   Those statistics do not change a significant amount if the Jacoby rule
   is used in interpreting the score (naturally, the parameters of the
   model might change depending on the use of the rule).

 - Assuming all gammon and backgammon wins belong to the player who last
   cubed, a double/take leaves the taker with an average normalised cubeful
   equity of -0.446.  (If all doubles were perfectly efficient, this figure
   would be -0.5.)

   With a few more assumptions it may be possible to come up with other
   statistics on cube efficiencies, recube vig, etc.  I don't have time
   to think about it right now, though -- more another day perhaps.

        Gary Wong, Department of Computer Science, University of Arizona
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Derivation of drop points  (Michael J. Zehr, Apr 1998) 
Double/take/drop rates  (Gary Wong, June 1999) 
Drop rate on initial doubles  (Gary Wong, July 1998) 
Error rate--Why count forced moves?  (Ian Shaw+, Apr 2009) 
Error rates--Repeated ND errors  (Joe Russell+, July 2009) 
Inconsistencies in how EMG equity is calculated  (Jeremy Bagai, Nov 2007)  [GammOnLine forum]
Janowski's formulas  (Joern Thyssen+, Aug 2000) 
Janowski's formulas  (Stig Eide, Sept 1999) 
Jump Model for money game cube decisions  (Mark Higgins+, Mar 2012) 
Number of distinct positions  (Walter Trice, June 1997) 
Number of no-contact positions  (Darse Billings+, Mar 2004) 
Optimal strategy?  (Gary Wong, July 1998) 
Proof that backgammon terminates  (Robert Koca+, May 1994)  [Recommended reading]
Solvability of backgammon  (Gary Wong, June 1998) 
Undefined equity  (Paul Tanenbaum+, Aug 1997)  [Recommended reading]
Under-doubling dice  (Bill Taylor, Dec 1997)  [Recommended reading]
Variance reduction  (Oliver Riordan, July 2003)  [Long message]

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