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Janowski's formulae

Posted By: Rick Janowski
Date: Tuesday, 19 January 2010, at 7:39 p.m.

In Response To: Janowski's formulae (NJ)

Thank you very much for the kind words, NJ. It is very much appreciated.

I thought it might be useful if I should attempt to clarify some of the issues raised by NJ and others with regard to my paper “Take-Points in Money Games”.

The main idea underpinning the whole work was the development of a formula which would represent a fully-live cube take-point for games where gammons and backgammons are present. Previous live-cube models were based on single win games (fundamentally races). Once, the take-point expression was derived, it was then trivial to establish relationships for live-cube equities. The fully live-cube and fully dead-cube equity (ie, cubeless) relationships effectively provided an envelope within which real equities are contained, i.e., the real equity must be between the dead-cube and live-cube equities.

Essentially, as explained by NJ and elsewhere by Timothy Chow and others, the process of then establishing cubeful equity and cube-action relationships is essentially one of interpolation within the envelope by considering factors related to cube-efficiency. The basic model considers a single value for cube-efficiency, x, with an estimated value of 2/3 established by calibration with “known” reference positions.

When the work was undertaken, before the widespread availability of neural network backgammon programs, the main purpose was to establish expressions for take-points, initial doubles and redoubles, the “bread and butter” of backgammon as it were. Expressions for beavers, racoons and too good to double-point were included for completeness.

There was some difficulty in establishing relatively straightforward expressions for initial double-points with and without Jacoby, and with and without beavers (again for completeness). Accordingly, the expressions for initial double-points are based on approximations rather than directly on direct interpolation between centred-cube dead and live equities, tests on known race position equities indicated neglible loss of accuracy (less than 0.005 ppg as I recall).

I went wrong however in developing expressions for too good to double-points. The method of interpolation adopted was completely unrealistic. This was pointed out to me by Chuck Bower in the following thread on rec,games.backgammon in early 1997 (see the link below)

http://www.bkgm.com/rgb/rgb.cgi?view+389

I would like to thank Chuck very much for his support of this work and for promoting its dissemination within internet backgammon communities. I completely agree with the approach adopted by NJ and the other developers of Neural Network backgammon software in adopting a straightforward interpolation technique for computing both money and match-score based cubeless equities. I had discussions with Joern Thyssen on this issue in 2000 as reported in the following posting on rec.games.backgammon.

http://www.bkgm.com/rgb/rgb.cgi?view+389

I agree with NJ that the relationship of cubeless equity is not a single linear relationship, but that there are discrete points between which linear relationships apply.

In addition to developing a basic model where there is a single factor, there is a more refined approach considering separate usage factors for the two players. My tests on race positions indicate that the refined model has the potential to estimate actual equities within about 0.005 to 0.010 ppg, whereas the simpler basic model can only expect to limit potential errors to about 4 or 5 times these values. However, the more refined approach is problematic in that equities and cube-action points are very much more sensitive to cube-value assumptions than the basic model with the net result that there is unlikely to be any significant gain following normal analytical approaches.

However, a potential way forward is to set the Neural Networks to learn to make estimates of money-game cube efficiency factors for each position much as they do currently for evaluating cubeless equities. Match score skewing effects could I believe be modelled by suitable algorithms considering key factors (eg, dead-live double window, dead-live take window, distance away from doubling window, etc).

Regards,

Rick Janowski

Messages In This Thread

• Janowski's formulae
  NJ -- Monday, 18 January 2010, at 2:53 a.m.
  • Janowski's formulae
    Rick Janowski -- Tuesday, 19 January 2010, at 7:39 p.m.