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BGonline.org Forums
A few questions for Mark and Rick
Posted By: Rick Janowski In Response To: A few questions for Mark and Rick (neilkaz)
Date: Thursday, 26 April 2012, at 7:48 p.m.
Thanks very much Neil! It’s very interesting and reassuring that we have all derived similar figures independently of each other. Of course, we could each have blundered in our analyses but I doubt it (lol). As you say, your figures are completely consistent with a cube efficiency of 70.5%. When I had written my paper about 20 years ago, I estimated the typical cube efficiency value to be of the order of 0.6 to 0.75, from back-calculating from “known” cubeless equities from a range of borderline double/no double and take/pass positions. To some extent the value of 2/3 was a compromise value, as I realised it would tend to slightly underestimate double points but slightly overestimate cash points. To get around this compromise I also developed a more rigorous approach allowing for different cube efficiencies for the two players (covered in Appendix 2 of my paper). To use this method, I had estimated typical values of 0.75 for the player close to doubling and 0.6 for the player close to being doubled (generally the closer the player is the double point the greater cube efficiency). With these values I estimate typical values of 68.9%, 71.9% and 78.3% for initial doubles, redoubles and cash points respectively. Again very close to your estimates.
Interestingly, about 10 years after you I also struggled with finding a means to accommodate gammons and backgammons but couldn’t deal with the unwieldy algebra until I hit on the notion of representing average win values by W and average loss values by L. Then it was possible to derive appropriate expressions by employing simple risk/benefit analysis. Moreover, similar to your experience I tried but failed to develop relatively simple equations for cube action at match scores because of the multitude of factors, and the finite nature of the cube action. The methodology I subsequently developed with the writers of Snowie basically defines dead and live cube models (in the latter case by considering all possible future redoubles until the cube is dead). I understand that BGBlitz, GNU and XG all follow a similar process. Interestingly, Mark has determined from multiple simulations that the optimal cube efficiency for matches is very close to that for money, ie 0.7. A not unexpected result but reassuring nevertheless.
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