Doubling Cube Quiz
Rick Durrett, 2003

## The doubling cube

The doubling cube is common in backgammon but could be used in any game. The cube has 1, 2, 4, 8, 16, and 32 on its six sides. At the beginning of the game the cube sits in the middle and is on 1. A player who thinks they are ahead can double: i.e., tell the other player to continue and play for 2 points, or give up 1 point right then. A player accepting the double has possession of the cube and only they can redouble.

## When should you double?

We begin with an unrelated problem:

1. John and Jenny are playing a game of flipping pennies. Each flips one of their coins. If the coins are the same, John gets them both. If they are different, Jenny gets them both. At the start of the game John has 8 pennies, Jenny has 12. They quit when someone has all of the money. What is the probability John will win?

Hint: This is a fair game so the average amount of money John will have at the end is the same as what he starts with.

2. Consider a flipping pennies game with 100 coins total. How many pennies should you have to offer a double? When should you accept?

Hint: Suppose the optimal threshold for offering a double is when you have a fraction p of the coins. At this point the two decisions: continue to play or give up will have the same average payoff.

## What does flipping pennies have to do with backgammon?

The decision to offer a double or in the other direction to accept one is based only on the probability that you will win the game. That is hard to calculate in backgammon but, if we could, then the threshold would be the same as for flipping pennies or any other game.