Backgammon Theory

Theory

Jump Model for money game cube decisions

From:   Mark Higgins
Address:   mghiggins@yahoo.com
Date:   27 March 2012
Subject:   New model for money game cube decision points
Forum:   BGonline.org Forums

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 2-ply 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 game-winning 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 game-winning 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 well-defined statistical quantity that you can estimate in your current game state. If you are a bot you do it with a 2-ply lookahead. But humans can use bot self-play 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 two-index 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, "Take-points 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 non-linear and linear approximations). * Comparison of the single composite jump volatility approach with the basic approach from my paper - demonstrating close correlation between both non-linear 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 cube-efficiency 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 non-linear 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!

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