One of my favorite two-player games is the double cube, which is often used to play backgammon. The game starts with a bet of one point, but each player has the option of doubling the bet before each turn. If player 1 doubles, player 2 has the option of accepting the double and playing the game for two points, or doubling down, giving up and allowing player 1 to win one point. If player 2 accepts, then player 2 has a “continuing cube” – player 2 can double before each of their turns, while player 1 cannot. If player 1 accepts more than doubles from player 2, the game will be worth four points and player 1 will get the cube.
The choice always has a negative profit, and when you play with the cube, the choice of two bets gives a big profit. We can show this advantage for backgammon by calculating the points (equity) that the player expects and that there are no doubles (“cubic equity” and “cubic equity” respectively) for some simple end game scenarios.
Let’s start with a scenario where, unless they roll doubles, each player needs two rolls to win the game (if either rolls doubles, they win automatically):