You are playing backgammon (wlog as white). (The rules of backgammon are explained more clearly here.) At any given point during a round, there are six possible outcomes of the round, with corresponding values:

- white wins by a backgammon ()
- white wins by a gammon ()
- white wins ()
- red wins ()
- red wins by a gammon ()
- red wins by a backgammon ()

At the start of a round, and both players have control of the *doubling cube*.

When it is Alice’s turn, she may offer a *double* if and only if she controls the doubling cube. If Alice offers a double, then Bob must accept or decline. If Bob declines, then Bob forfeits the round and Alice gains points. If Bob accepts, then Alice ceases to control the cube, Bob gains control of the cube, and is multiplied by 2.

(There are other rules.)

## The problem

Suppose that both you and your opponent can perfectly calculate the probability of each of the above outcomes.

- It is your turn. Should you offer a double?
- It is your opponent’s turn, and they have offered a double. Should you accept?

For a harder problem, suppose that you assign probabilities to each of the above outcomes, and your opponent assigns different probabilities , but you know both the and the .