| Match length | Value of a 4 point win | Value of a 8 point win | Ratio |
|---|---|---|---|
| 9 | 22.5 | 44.4 | 1.97 |
| 10 | 21.3 | 42.3 | 1.99 |
| 11 | 20.0 | 39.4 | 1.97 |
| 12 | 19.1 | 37.9 | 1.98 |
| 13 | 18.2 | 36.0 | 1.98 |
| 14 | 17.5 | 34.7 | 1.98 |
| 15 | 16.8 | 33.3 | 1.98 |
| 16 | 16.2 | 32.2 | 1.99 |
| 17 | 15.6 | 31.0 | 1.99 |
| 18 | 15.1 | 30.1 | 1.99 |
| 19 | 14.7 | 29.2 | 1.99 |
| 20 | 14.2 | 28.4 | 1.99 |
| 21 | 13.9 | 27.6 | 1.99 |
Big Cubes
Why this?
This is to investigate take points when the cube level gets high and the match score is tied.
Example
Here’s one example from a recent match:
Analyzed in Rollout
Redouble/Take
Player Winning Chances: 76,10% (G:0,00% B:0,00%)
Opponent Winning Chances: 23,90% (G:0,00% B:0,00%)
Cubeful Equities:
No redouble: +0,809 (-0,168)
Redouble/Take: +0,977
Redouble/Pass: +1,000 (+0,023)In the match, I was playing White and passed this cube, which was a very small mistake. I expected the take point to be a bit higher than for money, but apparently that is not the case.
My thinking was, that points get progressively more valuable the closer you get to the end of a match: At double match point you have 50% chance of winning, an actual win get you to 100%, so a point is worth 50 percentage points. At 17-away, 17-away a single point win get you from 50% to 54% so is worth just 4 percentage points. At 5-away, 5-away a point is worth about 8 percentage points.
So in a sense doubling more than doubles the stakes. A bit like ‘doubling’ from 4 to not 8 but perhaps 9 points. That should make you more reluctant to take, I thought.
An actual calculation shows that the take point when doubled from 4 to 8 at 13-away, 13-away is 24.8% – practically the same as for money, if anything a bit lower.
That agrees with the numbers in the position above; White can take with just 23.9%, since he gains a bit from being able to double Black out efficiently when they get to a two-roll position.
So exactly where did my logic go wrong?
The thing is, as the cube get bigger, the value of the points currently at stake also increases, so doubling does not necessarily “more that doubles” the stakes as I thought. Here’s a table to illustrate:
We see that the value of both a 4-point win and an 8-point win increases as the match gets shorter, but the ratio between the two remains approximately the same, just about 2. So doubling from 4 to 8 does in fact double the stakes, not more than that.
Let’s plot the take points for more scores and more cube values:
Near the end of the match the take points can be a bit funny, as we know. When both players need 4 points, it’s hard to take a 4-cube (take point 33%). Here, doubling from 2 to 4 does indeed “more that doubles” the stakes.
At the other extreme, at 5-away, 5-away, it’s easy to take an 8-cube, because the alternative, dropping, would leave you with just 15.8% winning chances.
At longer matches, when the score is tied, cubes at the 1-, 2- or 4-levels behaves just like for money, as we might expect.
8- and 16-cubes are different, however. These take-point are quite a bit lower that 25% (unless the match is very long). There intuition here is that conceding 8 or 16 points would leave you in a pretty desperate situaition anyway, so it’s not something one should do easily.