Poker and backgammon: the doubling cube and the all-in

The doubling cube is a die with 2, 4, 8, 16, 32 and 64 on its faces. It is never rolled; it records the stake. It was first introduced in New York City in the 1920s.

At the start of the game the cube sits in the middle and either player may double. When one does, the other must take or drop. A drop ends the game, and the player who dropped loses the current stake. A take doubles the stake and gives the taker the cube: from then on, only they may double again. There is no limit on redoubles.

Say the game is worth 1 point and your opponent doubles. If you drop, you lose 1 point. If you take, you play for 2: you win 2 when you win the game and lose 2 when you lose it. With p your chance of winning:

In this simplified game, taking breaks even against dropping at 25%. Cube ownership, gammons and match score can change the take point.

Taking with 30% means losing seven games in ten at double stakes, and it still costs less on average than dropping every time. That is the same point the trainer makes about calls in poker: a good call can lose most of the time.

Picture the backgammon game as a poker pot. Each player has put in 1, so the pot holds 2. A double asks you to put in 1 more to keep playing, and your opponent puts in the same 1. That is a bet of 1 into a pot of 2: half the pot.

The price of calling a half-pot bet, from pot odds:

The two numbers are the same because the two decisions are the same: pay 1 more to play for a pot of 4. A poker player who knows that a half-pot bet asks for 25% already knows the backgammon take point.

The first difference is that owning the cube is worth something. After you take, only you can double next. If the game swings your way, you can double your opponent out at a moment that suits you. That right has value, so in real play the take point is lower than the raw 25%. Ignoring gammons, it is about 22% in money play, and a model of a cube that keeps changing hands puts it at 20%. Poker has no exact match. The nearest idea is implied odds: a call is worth more when it buys the chance to win more later.

The second is that some games are worth more than one stake. A gammon, a win before the loser has borne off a single piece, is a double game, and a backgammon is a triple game. When you might lose a gammon, taking risks more than the doubled stake, and the take point goes up. In poker the most you can lose on a call is the call.

The third is that nothing is hidden. Both backgammon players see the whole board, so each can estimate the same chance of winning. A poker player has to estimate equity against a range of hands they cannot see. That is why bluffing exists in poker. In backgammon, a double you cannot back up is just a bad double, since your opponent can see your position as well as you can.

The cube only doubles, so its nominal stake increase is always the same; the actual take point can still change with the position and match score. Poker lets the bettor choose any size, and the price moves with it: a third of the pot asks for 20%, the pot asks for 33.3%, twice the pot asks for 40%. The simplified 25% take point matches the half-pot row of the poker table.

In both games, work out the price first, then compare it with your chance of winning.