I came up with a new model for estimating cubeful equity in backgammon money games that seems like an interesting way to think about the Janowski cube life index in specific game states:
A variation on Janowski’s cubeful equity model is proposed for cube handling in backgammon money games. Instead of approximating the cubeful take point as an interpolation between the dead and live cube limits, a new model is developed where the cubeless probability of win evolves through a series of random jumps instead of continuous diffusion. Jumps occur when a Poisson process fires, and each jump is drawn from a normal distribution with zero mean and a standard deviation called the “jump volatility” that can be a function of win probability but is assumed to be small compared to the market window.
Simple closed form approximations for the market window and doubling points are developed as a function of local jump volatility. The jump volatility can be calculated for specific game states, leading to crisper doubling decisions.
All cube decision points under this model match Janowski’s if his cube life index is implied from this model’s take point, so these results can be viewed as a framework for estimating the Janowski cube life index more accurately for specific game states.
Abstract
A variation on Janowski’s cubeful equity model is proposed for cube handling in backgammon money games. Instead of approximating the cubeful take point as an interpolation between the dead and live cube limits, a new model is developed where the cubeless probability of win evolves through a series of random jumps instead of continuous diffusion. Jumps occur when a Poisson process fires, and each jump is drawn from a normal distribution with zero mean and a standard deviation called the “jump volatility” that can be a function of win probability but is assumed to be small compared to the market window.
Simple closed form approximations for the market window and doubling points are developed as a function of local jump volatility. The jump volatility can be calculated for specific game states, leading to crisper doubling decisions.
All cube decision points under this model match Janowski’s if his cube life index is implied from this model’s take point, so these results can be viewed as a framework for estimating the Janowski cube life index more accurately for specific game states.
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