Cube Efficiency, Part II
by Rob Adams – July, 2005
This article is a continuation of Cube Efficiencey, Part I.
How about an example to see how Cube Efficiency affects match play take points:
This position with X on roll is a borderline take for money. Below are the exact numbers from the Sconyers database:
- Blue on Roll
- White (O) Pipcount: 79
- Blue (X) Pipcount: 70
- CPW: .77731
| Equity | |
|---|---|
| Cubeless: | .554620 |
| Roller’s Cube: | .862392 |
| Centered Cube: | .842545 |
| Opponent’s Cube: | .488394 |
- Fair settlement = Equity × Cube
- Fair settlement if: Cube = 2
- Roller’s cube: 1.724783
- Opponent’s cube: .976788
Redouble → Take
OK, so that is the story for money play. But what about in a match to 5 with X ahead 2-0? Using the information from part one, try to work this out yourself before reading on.
There are many problems in trying to calculate match play take points, but let’s give it a try. First we need some match equities so we can figure the naïve take point. I’ll round Snowie’s MET off to the nearest whole percent since I don’t like some of the figures that Snowie uses. But no matter what numbers/guesses you use, it is important to use some… otherwise how do you even begin to approach this problem?
Double/Drop will leave 3-0 or −2,−5 for about 74%; Double/Take/Win leaves −1,−5 Crawford for 84%; and Double/Take/Lose leaves −3,−3 for 50%. So taking risks 10% to try to gain 24% for a take point for O of 10/34 or about 29.4% (without the rounding, Snowie gets about 2% higher, which just doesn’t seem right to me. But, of course, it is probably me, not the bot, who is wrong.) This is quite a bit higher than the 25% money naïve take point, but it would be a mistake just to stop here and call this a drop.
On the coming 4 cube, X could Drop for −3,−3 and 50%; Take/Win for 100%; or Take/Lose for −3,−1 Crawford and 25%. So taking would risk 25% to try to gain 50% for a take point of 1/3. This take point is much higher than the money take point of 25%. Plus, O will be redoubling aggressively with his doubling window opening earlier than the 50% for money. Using a similar risk versus gain formula, O’s redoubling window would open at 16/41 or about 39%.
So, for our formula (presented in Part I), we have the Naïve Take point and the Take point on the coming 4 cube. We just need the Recube Efficiency. For money in a long race such as this, a Recube Efficiency of about 1/2 works. That would make the take point just under 22%, and since in the above position O wins just over 22%, he can take. But, at this match score, O will be redoubling more often and X’s take point in the match is higher than for money. O will be turning the game around and giving efficient redoubles more often. So the figure to use for the Recube Efficiency should be higher than 1/2. Let’s plug in some guesses and see what we get:
- 10/34 − [10/34] × [1/3] × [1/2] ≈ 24.5%
- 10/34 − [10/34] × [1/3] × [3/5] ≈ 23.5%
- 10/34 − [10/34] × [1/3] × [2/3] ≈ 22.9%
- 10/34 − [10/34] × [1/3] × [3/4] ≈ 22.1%
So it would take a Recube Efficiency of almost 3/4 to justify taking in this position at that match score of −5,−3. The GNU backgammon program seems to use a Recube Efficiency of about 2/3 here, but I’m not sure why they use that figure. Still, it seems reasonable. And so, with a guess of 22.9% as a take point and O winning only 22.3% in the given position, it would seem to be a close drop.
More generally, it seems that at −5,−3 in a long race match, your take point (assuming equally skilled players) is very similar to that of money play (just a little higher) despite the much higher Naïve Take point because of your greater value from cube ownership. To see just how difficult it can be to figure the Take point here, I looked at some different match equity tables that come with the gnubg program and assumed that 2/3 guess at the Recube Efficiency for each. I came up with the following Take points:
| Snowie: | 24.5% |
|---|---|
| Mec26: | 23.9% |
| Ortega: | 23.0% |
| G11: | 22.4% |
| Zadeh: | 22.3% |
| Woolsey: | 22.2% |
| Dunstan: | 22.2% |
| Jacobs: | 20.6% |
Each of these assumed equally skilled players, but their slightly different assumptions on the match winning chances from the various relevant match scores result in the different take points. And, of course, if we assumed a different value for the Cube Efficiency, the take points would be somewhat different as well. Adjusting for different skill levels is pretty straightforward, though difficult to quantify. As the stronger player, you would have a higher take point, and as a weaker player, a lower one. This is because you are happy to play a race on a higher cube as the weaker player, but as the stronger player, you’d rather let your skill come into play later in a new game than let the luck of the dice decide this race. The “Jacobs” table resulting in the 20.6% take point above was from the Jacobs/Trice book on skill difference in backgammon, Can a Fish Taste Twice as Good. Using their different skill tables results in the following take points (again assuming the 2/3 Recube Efficiency):
| Jacobs +100: | 22.7% |
|---|---|
| Jacobs +50: | 21.7% |
| Jacobs: | 20.6% |
| Jacobs −50: | 17.8% |
| Jacobs −100: | 15.4% |
The differences listed above are in FIBS ratings system points. Even the 100-point difference is quite small really assuming only 52% chances at DMP or 56% chances in a 5-point match. A greater difference in rating would, of course, amplify the differences in the take points. So, clearly, skill difference is another factor that needs to be considered when calculating a match play take point.
So is that original position a take or a drop? That really is a matter of opinion. My guesses with equally skilled players resulted in a close drop. But if I’m off by just a little bit in the value of the cube ownership, or if there were even a tiny difference in skill, or if the match equities I guessed at weren’t really accurate, then it could easily be a proper take. Backgammon is a complicated game, even in a simple 5-point match, race position.