The Doubling Cube, Explained
After reading this you can look at your win rate, your gammon rate and who owns the cube, then decide whether to double, take or pass, and know how much margin you have.
The doubling cube looks like a die with the numbers 2, 4, 8, 16, 32 and 64 on it. In a money game it turns backgammon into a betting problem. Either player, on their turn before rolling, may offer to double the stakes. The opponent then chooses: accept (take) and play on at the higher stake with sole ownership of the cube, or decline (pass) and hand over the current stake immediately.
Here is the hook. You are ahead in a straight race with about a 78% chance to win, no gammons in sight. Should you double? Yes. Should your opponent take? No. The reason is a single number: the take point of 25%. If your opponent's winning chances have dropped below 25%, taking loses money faster than passing does. This article shows where that 25% comes from, how it slides to 20% when a live cube is in play, and how to read the full doubling window.
What the cube decides and when this math applies
Every cube decision is really two decisions that meet in the middle. The player on roll asks: does offering the cube gain compared to holding it? The opponent asks: does taking lose less than passing? The answers are governed by probabilities you can estimate at the board.
This advisor covers money play only: each game is scored on its own, doubles and gammons multiply the stake, and there is no target score. Match play is a different animal. Near the end of a match the value of extra points collapses (winning 4 points when you only need 2 buys you nothing), so match take points shift with the score. The Crawford rule and positions like 2-away/2-away change the numbers substantially. None of that is modeled here.
The cube also has a subtle asset value. When you take, you own the cube and can redouble later. That option is worth real equity, and it is exactly why the take point drops from 25% toward 20%. Hold that thought; the formula section makes it precise.
The gammonless model and its four thresholds
Start with the cleanest case: no gammons, no backgammons. A win is worth exactly the cube value, a loss costs exactly the cube value. Let p be the opponent's chance of winning if they take. If they pass, they lose 1 point (the current stake) for certain.
If they take, they play for 2 points. They win 2 with probability p and lose 2 with probability 1-p. Taking is correct when its expected loss is smaller than the guaranteed loss of 1.
Solve for p and you get p \ge 0.25. That is the famous dead-cube take point of 25%. Below 25% winning chances, pass. At or above, take.
The general version accounts for gammons. Let W be the average points won per win and L the average points lost per loss, each equal to 1 + \text{gammon rate} (ignoring backgammons). The dead-cube take point is:
With no gammons, W = L = 1, so \text{TP} = 0.5 / 2 = 0.25. Now add the value of owning the cube. Because the taker can redouble later, their real position is better than the dead-cube math suggests. Classical theory adds roughly half a point of cube-ownership equity, giving the live-cube take point:
Gammonless, that is 0.5 / 2.5 = 0.20. So a fully live cube lets the opponent take down to 20% winning chances rather than 25%.
The doubling window from both sides
Turn it around to the doubler. Offering the cube starts to gain once your winning chances pass the doubling point:
Gammonless that is 1 / 2 = 0.50. Below 50% you should not double in a pure race. Above it, doubling gains, and it keeps gaining until your opponent's take becomes a pass, which happens at their take point measured from your side.
Keep pushing your win rate up and you reach the too-good point. Above it you stop cashing the single point and play on for the gammon, because the extra points from a gammon outweigh the sure point you would collect by doubling the opponent out:
Gammonless that is 2 / 2 = 1.0, meaning you are never too good without gammons: cash whenever you can. Gammons pull it below 1 and open a real too-good region.
A worked example with the demo numbers
78% to win, no gammons, cube in the middle
The demo defaults describe a race: your game-winning probability is 78%, gammon rates are effectively zero, and the cube starts centered (nobody owns it yet). Work the thresholds from your side.
- Set W = 1 and L = 1 because there are no gammons.
- Doubling point: L / (W + L) = 1/2 = 0.50. Your 78% is well above it, so doubling gains.
- Opponent's live take point: (L - 0.5)/(W + L + 0.5) = 0.5/2.5 = 0.20. Their winning chance is 1 - 0.78 = 0.22, which is above 20%.
- Because 22% exceeds their 20% take point, they should take. The verdict is Double, take.
- Check the too-good point: (1 + L)/(W + L) = 2/2 = 1.0. You are not too good, so cashing by doubling is correct rather than playing on.
The margin is thin. Their take is correct by only 2 percentage points (22% versus a 20% requirement). Push your win rate to 81% and their take flips to a pass.
How to read the verdict and the strip chart
The advisor returns a plain sentence and a strip chart of the doubling window with your position marked. Read them together.
- No double, take
- Your win rate is below the doubling point. Hold the cube; there is nothing to gain yet.
- Double, take
- You are inside the window. Offering gains, and the opponent is correct to accept. This is the most common correct double.
- Double, pass
- You have crossed the cash point. Offer the cube and expect the opponent to decline, collecting the current stake.
- Too good, play on
- Your gammon chances are high enough that keeping the cube and going for the gammon beats cashing one point.
The strip chart shows the whole window on one axis: doubling point on the left, cash point on the right, and your marker somewhere along it. When your marker sits near a boundary, the decision is close and small errors in your probability estimate can flip it. When it sits mid-window, you have wide margin and can double confidently.
Try moving the win rate yourself
Common mistakes
The errors below cost the most money over a session.
Do not use 25% as your take point when you own no cube ownership advantage that a redouble could exploit. In most live positions the correct number is closer to 20% because you can recube. Passing takes that are worth 20% to 24% is a slow, steady leak.
Doubling too early. A 55% racing lead is inside the window, but barely. The gain from doubling near the doubling point is tiny, and you hand your opponent cube ownership. Wait until you are clearly past 50% and closing on the cash point before you double a pure race.
Cashing when you are too good. If you have a 75% win rate and a 40% gammon rate among those wins, cashing one point throws away the gammon value. Check the too-good point before you reach for the cube.
Ignoring volatility. These thresholds assume you can double again next turn. In positions that will swing wildly on the next roll (high volatility), you should double earlier because you may lose your chance. The static formulas do not capture that; treat them as the floor, not the ceiling, of when to act.
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Frequently asked questions
Why is the take point 25% and not 50%?
Because a take doubles the stake but does not double your losses relative to passing. If you pass you lose 1 point for sure. If you take you risk 2 to win 2. Break-even is where 2p - 2(1-p) = -1, which solves to p = 0.25. You need only a 25% chance to justify the risk.
What is the difference between the dead-cube and live-cube take points?
The dead-cube point (25% gammonless) assumes the cube never turns again. The live-cube point (20%) credits the taker with the value of being able to redouble later. Most real positions are closer to live, so 20% is the more useful working number when you can recube.
When am I "too good to double"?
When your gammon chances make playing on for the double stake worth more than cashing one point now. With enough loser gammon rate the too-good point falls below 100%, so a very strong position tells you to keep the cube and play on rather than double the opponent out.
Does this work for match play?
No. Match take points depend on the score because extra points above the match target are worthless. The Crawford rule and scores like 2-away/2-away shift the numbers a lot. This advisor models money play only.
How accurate are gammon rate estimates?
They matter most near the thresholds. In a pure race gammons are near zero and the classic 50% to 80% window applies. In a blitz or a heavy prime, a 30% to 40% gammon rate can move the doubling point down several points and open a real too-good region, so estimate them before you trust the gammonless numbers.
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