#The 25% rule
When your opponent doubles, you choose between two losses. Pass, and you lose one point now. Take, and you play on for two points. Which is better depends only on how often you would win from here.
Suppose you win a fraction p of the games and every game is a single one. Taking is worth 2 × (p - (1 - p)) = 4p - 2 points on average, and passing is worth -1. They are equal when 4p - 2 = -1, which is p = 1/4. So in a money game with no gammons, you can take if you win at least 25% of the time. Below that, pass.
That number is the take point. It assumes the cube is dead once you take it: that you never get to use it. You do, which moves the number.
#The live cube: why real take points are lower
After a take, the cube is yours. When the game turns your way, you can redouble to 4, and your opponent may have to pass, so you collect games you would otherwise have had to play out and might have lost. That is worth something, and the more of it there is, the fewer games you need to win to take.
Janowski's model captures this with one number, the cube efficiency x: 0 for a cube that is never usable, 1 for one used perfectly. The money take point becomes:
take point = (L - 0.5) / (W + L + 0.5 × x)
where W is the average value of the games you win and L the average value of the games you lose, 1 each when there are no gammons. With x = 0 this is the dead-cube 25%. With a perfectly live cube it is 0.5 / 2.5 = 20%. HedgeHog uses x = 0.68 for a money game, a typical value for the middle game, which gives 21.4%.
#Gammons move the take point
Gammons change W and L, and the formula shows by how much. Say a quarter of the games you lose are gammons, and you win only single games. Then L = 1.25 and W = 1:
| Cube efficiency | No gammons | Gammons in a quarter of losses |
|---|---|---|
| 0 (dead) | 25.0% | 33.3% |
| 0.68 (HedgeHog) | 21.4% | 29.0% |
| 1 (perfectly live) | 20.0% | 27.3% |
Gammons against you raise the take point sharply: a position where you win 25% of the games can be a clear take when it is a pure race and a clear pass when you are facing a blitz. Gammons of your own work the other way. If a quarter of your wins are gammons, W = 1.25 and the dead-cube take point falls to 0.5 / 2.25 = 22.2%.
#The cash point and the doubling window
The take point read from the doubler's side is the cash point: when the doubler's chances are above it, the opponent should pass, and doubling cashes the game. With the numbers above, a taker's 21.4% take point is the doubler's 78.6% cash point.
Between the point where a double first becomes right and the cash point lies the doubling window. Below it, doubling gives away the cube too early; above it, the opponent passes and the double collects one point. The best doubles come near the top of the window, just before you lose your market: when enough of your next rolls would take the position past the cash point (the market losers), waiting means you will be doubling a position your opponent can simply pass.
Far above the cash point is a third region, too good to double. There the opponent would pass, but you win so many gammons that playing on for them at the current stake is worth more than the single point a double would collect. How HedgeHog shows all of this on a real position is in The doubling cube, and the words are in the glossary.
#Take points in a match
At a match score, points are worth what they do to your chance of winning the match, and the take point is worked out from match winning chances (MWC) instead. You take when:
your chances ≥ (MWC after passing - MWC after losing the doubled game) / (MWC after winning the doubled game - MWC after losing the doubled game)
The take points for the first double, from HedgeHog's match equity table, with the cube dead once taken and gammons left out:
| You (taker) | Doubler 2-away | Doubler 3-away | Doubler 4-away | Doubler 5-away |
|---|---|---|---|---|
| 2-away | 32.3% | 26.1% | 19.8% | 17.4% |
| 3-away | 36.8% | 30.3% | 24.0% | 20.9% |
| 4-away | 37.2% | 35.2% | 28.8% | 22.7% |
| 5-away | 39.4% | 28.8% | 30.4% | 23.8% |
Two scores show where the numbers come from.
2-away, doubled by a 4-away opponent: 19.8%. Take and win, and you have won the match. Take and lose, and it is 2-away all, an even match. Pass, and you lead 2-away to 3-away with a 59.9% chance. Passing gains you little over the worst case of taking, so a small chance of winning the match outright is enough.
3-away, doubled by a 2-away opponent: 36.8%. Take and lose, and the match is over. Pass, and your opponent is 1-away in the Crawford game, where you still win the match 24.9% of the time. Take and win, and you lead 1-away to 2-away, worth 67.7%. Passing keeps a real chance alive and taking risks all of it, so you need far more than a quarter of the games.
A double from an opponent who is 2-away is the classic case: whatever your own score, it asks for more than 30%. After the Crawford game, the player who is behind usually doubles at the first chance, since the leader's cube can never be used against them.
The match equity table has the MWC at every score and a calculator that works out the dead-cube take point for any score and cube.