Cash game, center cube.
Should White double? If White doubles, what should Black do?
By convention, most cash games use several auxiliary rules beyond the basic rules of backgammon, intended both to speed up the games and increase the stakes in an exciting way. The most common cash game rules are these:
The Jacoby Rule: You can’t win a gammon unless the cube has been turned. This rule eliminates long, dull games where one side gets a huge early edge and plays for the gammon while leaving the cube in the middle.
Automatic Doubles: If both sides roll the same number to start the game, the cube starts at 2. This rule doubles the stakes in one game out of six. Since weaker players tend to play more cautiously with bigger cubes, this rule strongly favors the better player.
Carryovers: Carryovers are just an extension of automatic doubles. If the players have started with an automatic double and then they both roll the same number again, the automatic double is carried over to the next game. If automatic doubles are good, you certainly wouldn’t want to let one go to waste!
Beavers: When your opponent doubles you, you aren’t required to just take or drop. You can beaver, meaning you turn the cube an extra notch and keep possession. A beaver is a good way of punishing an especially bad double.
Many players get confused and think that if your opponent doubles incorrectly, you must have a beaver, but that’s not true. An incorrect double, by definition, lowers the doubler’s equity in the position. Mostly, however, incorrect doubles lower the equity from one positive number to a smaller positive number. You should only beaver if your opponent’s bad double has actually lowered his equity to a negative number!
Here’s a quick numerical example. Suppose your opponent is White and he’s on roll. If he leaves the cube in the center, his equity is, let’s say, +0.4 points per game. (Numbers like this can be calculated by bots using estimates or rollouts, but good players can often make reasonable estimates over the board.) Suppose that if he doubles and you take, however, his equity drops to +0.25 points per game. Clearly, his double is a mistake, because his equity dropped as a result. But it’s not a mistake you can exploit by beavering, because beavering just doubles his equity from +0.25 points per game up to +0.50 points per game. In fact, if he knew you were going to beaver, his double would be correct, because his equity would actually have risen from +0.4 to +0.5 points per game!
In actual play between good players plenty of bad doubles occur, but positions where a beaver is correct are very rare. When they do occur, the player on roll will have some obvious threats, but the defender will have plenty of long-term resources which the doubler has overlooked or discounted. (Fatigue, liquor, and bad luck are often accessories to the crime.)
This position shows an example of a beaver position from actual play. White was stuck and steaming, and when he saw his 6-prime and some shots at a second checker, he reflexively turned the cube. Black beavered after a little thought and turned the cube back a couple of moves later. White has only one really good roll here (6-3) and because Black has so much timing, White’s prime usually won’t hold up. Rollouts showed that White’s equity with a center cube was about -0.20 points per game, and it drops to -0.66 if he doubles and Black takes. Beavering is extremely profitable, increasing White’s loss to -1.32 points per game.
