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Theory
Jump Model for money game cube decisions

I've been developing a new model for calculating cube decision points in
money games. Link here:
http://arxiv.org/abs/1203.5692
It's an extension of Janowski's model. His model accounts for the odds of
losing your market by interpolating between the dead cube model, where the
cube has no value, and the live cube model, where the cubeless win
probability diffuses continuously and you can always wait until right
before the cash point to offer your double.
My model also models the cubeless win probability, but assumes it jumps
from step to step instead of changing continuously. The advantage of this
approach is that the model parameters can be calculated in a local game
state with a 2ply evaluation, which leads to crisper doubling decisions.
Keeler and Spencer's approach was really quite cool in that it abstracted
everything about the game into the dynamics of one variable: the cubeless
probability of winning the game. The weakness with their model is that it
assumes the gamewinning probability diffuses continuously, so you can
never get a great roll and jump past the cash point, losing your market.
That means it tends to overestimate the cubeful equity.
Janowski's model does account for those jumps with his cube efficiency
parameter. He notes that cube efficiency is a function of game state and
depends on volatility of the position, distance from the optimal double
point, and width of the doubling window. But in practice it was difficult
to calculate it in a specific game state, so people tend to use a constant
cube efficiency, or perhaps different values for different broad game
states.
My model is another take on the same thing: I model game evolution through
jumps in gamewinning probability each turn, and the size of those jumps I
call the "jump volatility". If you assume a constant jump volatility it's
basically equivalent to Janowski's model with a constant cube efficiency.
The main benefit, I think, is that jump volatility is a welldefined
statistical quantity that you can estimate in your current game state. If
you are a bot you do it with a 2ply lookahead. But humans can use bot
selfplay to get a feel for jump volatility in benchmark game states and
estimate from there.
Janowski also developed a more general model where each player has a
different cube life index. It seems like there should be some congruence
between my model with a local and average jump volatility, and Janowski's
twoindex model, but I've yet to find it. Rick and I are having some
interesting email conversations about this one.


Rick Janowski writes:
I have been in close correspondence with Mark since the initial drafts of
his paper, who I think has done splendid work in expanding theoretical
knowledge into this area of backgammon theory. Key features of his work
include:
* Thorough understanding and practical knowledge of previous work by
Keeler, Zadeh. Kleinman and myself.
* Independent verification of the live cube model extended to gammons and
backgammons originally developed in my paper, "Takepoints in money
games".
* Development of alternative "discontinuous" theoretical methods of
analysis introducing the original concept of "jump volatility".
* Proof and elaboration of a rigorous solution employing both "local" and
remote" jump volatilities.
* The development of a simplified and practical approach where a single
composite jump volatility is considered (including the introduction of
optimised nonlinear and linear approximations).
* Comparison of the single composite jump volatility approach with the
basic approach from my paper  demonstrating close correlation between
both nonlinear and linear approximations and the basic method from my
paper (utilising an optimal average value of 0.7 established from bot
simulation/rollouts).
* The close correlation between the simplified jump volatility approach and
the algebraic/interpolative cubeefficiency approach tends to provide
complimentary confidence that both approaches are practically valid.
* The equity plots resulting from the rigorous single composite jump
volatility approach (and its nonlinear approximation) show remarkable
similarity from actual equity plots I derived from the Sconyers Bearoff
Database (particularly in the zone covering optimal double and cash
points) a few years ago. In my opinion this strong evidence that the jump
volatility model is consistent with the nitty gritty mechanics of the
cube in the vicinity of too optimal doubling points.
I would like to say that I was very pleased to be associated with this work
in providing a checking/advisory role. Moreover, I fully support the
validity of this work which I believe has made a significant contribution
to the science and understanding of the doubling cube in backgammon. I look
forward to further work calibrating/optimising the more rigorous approach
utilising both local and remote jump probabilities.
Excellent work, Mark!




Theory
 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 rateWhy count forced moves? (Ian Shaw+, Apr 2009)
 Error ratesRepeated ND errors (Joe Russell+, July 2009)
 Inconsistencies in how EMG equity is calculated (Jeremy Bagai, Nov 2007)
 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 nocontact positions (Darse Billings+, Mar 2004)
 Optimal strategy? (Gary Wong, July 1998)
 Proof that backgammon terminates (Robert Koca+, May 1994)
 Solvability of backgammon (Gary Wong, June 1998)
 Undefined equity (Paul Tanenbaum+, Aug 1997)
 Underdoubling dice (Bill Taylor, Dec 1997)
 Variance reduction (Oliver Riordan, July 2003)
From GammOnLine
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