Value Bets in Backgammon and Poker

In backgammon, the doubling cube allows a player to double the stakes at any point in the game. The opponent can either immediately forfeit the original stakes or accept the cube and keep playing for the doubled stakes. What is the mathematically correct way of using the cube? My analysis follows. Caveat: I'm not actually a good backgammon player.

Let the probability of winning be p. Assume for the moment that you and the opponent make the same assessment of p. If p > 0.5, you should offer the doubling cube. Your opponent should accept the cube if p <, i.e., the opponent has at least a 25% chance of winning. Consider two examples:

• p = 0.51. You should double. If the opponent (correctly) accepts, you expect to win 0.51 * 2 - 0.49 *2 = $0.04, which is greater than the expected value ($0.02) if you didn't double. But if your opponent incorrectly refuses the cube, you do even better: $1.
• p = 0.99. You should double. If the opponent (correctly) refuses, you immediately win $1, which is still greater than the expected value ($0.98) if you didn't double. But if the opponent incorrectly accepts, your expected value is even higher: 0.99 * 2 - 0.01 * 2 = 1.96.

To use poker terminology, the first case (0.5 < p < 0.75) is betting to protect your hand. You're hoping your opponent folds, but you're still ahead if he doesn't. The second case (p > 0.75) is betting for value. Here you're hoping your opponent calls. A third case is bluffing: p < 0.5.

Of course estimating p is not easy. This is particularly true in poker where there is hidden information. The better your estimate of p--and the worse your opponent's--the more you win.