Thursday, April 5, 2012

Jump model revisited

After conversations with Rick Janowski I realized that I'd approached the jump model incorrectly. I've since revamped the whole thing and reposted the corrected paper on arxiv.org - the same link as before, overwriting the old (incorrect) version.

I feel pretty confident about this second cut. The implied Janowski cube life index is right around the optimal value I found, and jump volatility found by optimizing the model parameter in bot self-play ties out well with statistical estimates of jump volatility.

I put the python code I used to generate cubeful equities onto github, released under the GPL. It contains functions for calculating cubeful equity in the three cube states (player-owned cube, unavailable cube, and centered cube). Three methods for calculating the equity: numerically, solving the exact model; a linear approximation; and a nonlinear approximation that improves on the linear one. They are all developed in the paper.

Cube Handling In Backgammon Money Games Under a Jump Model

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. Each jump is drawn from a distribution with zero mean and an expected absolute jump size called the “jump volatility” that can be a function of game state but is assumed to be small compared to the market window.

Closed form approximations for cubeful equities and cube decision points are developed as a function of local and average jump volatility. The local jump volatility can be calculated for specific game states, leading to crisper doubling decisions.