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World's incorrect 14-away/6-away pass: Analysis
Posted By: Timothy Chow
Date: Tuesday, 13 April 2010, at 11:03 p.m.
There has been some interesting discussion of the World's incorrect pass but since it's buried in an old thread, I thought I would start a new thread so that more people could see it.
A critical question in evaluating this position is the World's recube vig. This is difficult to do, but I want to point out that it is at least possible to extract some information about recube vig from GNU that we might be able to use to improve our understanding of the position ex post facto. Hopefully this understanding can then be applied in the future if a similar position comes up.
The information I'm referring to is hidden away in GNU's "View Statistics" feature. This information has been recently posted by Daniel Murphy. At first glance it might seem hopeless to extract any useful information out of such a messy collection of random numbers. However, one thing we can extract that I think is quite valuable is the number of effective gammons won and the number of effective gammons lost (assuming the World takes the cube).
What do I mean by "effective gammons"? With the cube on 4, an "effective gammon" is simply an 8-pointer. You can win an 8-pointer in two ways: either you can win a gammon with the cube on 4, or you can win a single game with the cube on 8. Similarly you can lose an 8-pointer in two ways: either you can lose a gammon with the cube on 4, or you can lose a single game with the cube on 8. As far as any calculation of your equity goes, these are mathematically identical. Therefore you can lump them together. The lumped-together number is what I call your effective gammons.
(The alert reader will note that in principle, one could win or lose a gammon with the cube on 8; this is a gammon that nets 16 points. In situations where 16-pointers weigh heavily, then my term "effective gammons" is not such a good one and I confess that I have not come up with a good way to analyze such situations. Fortunately, in the position at hand, as we can see from Daniel's statistics, 16-pointers are basically negligible, and so are backgammons, so I will ignore them.)
In this particular position, we see from Daniel's statistics that the World wins 5.1% effective gammons (all of which are single wins with the cube on 8) and loses 8.6% effective gammons (3.0% of which are gammon losses with the cube on 4 and 5.6% of which are single losses with the cube on 8).
I'll come back in a moment to the question of how you might estimate these numbers OTB, but first let me explain what to do with these numbers once you have them. To decide whether to take, you simply do the usual take-point calculation, but using effective gammons: start with your wins (including your recubes that your opponent passes), add in your effective gammon wins times the appropriate gammon price, and subtract off your effective gammon losses times the appropriate gammon price. Finally, compare this number with your raw takepoint to arrive at your take decision. The punchline is that you don't need to do any further calculations to take into account recube vig. No live-cube or dead-cube calculations are necessary. The recube vig is automatically taken into account by the effective gammon rates.
Now let me come back to the critical question. What I've just said is all well and good, but how are you supposed to come up with an estimate of the effective gammons over the board? Well, there is no easy shortcut here; you just have to be able to use your backgammon knowledge and experience to judge that about 11% of the time you will be able to make a takable redouble, and that you'll be able to win almost half of those 8-cube games. This is admittedly not an easy judgment call to make, but I venture to say that it is really not that much harder a task than the task of estimating that you will lose a gammon about 3% of the time. Estimating gammon losses is something that top players have gotten pretty good at with practice, so I don't see why estimating the frequency of takable redoubles, and the win/loss breakdown afterwards, can't also be something one can get good at. The payoff for this extra training will be more accurate estimates of recube vig, so it seems worth it if you really want to hone your cube skill.
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