The option to double the stake
You and an opponent flip a fair coin repeatedly: heads gives you a point, tails gives your opponent a point. The first to reach 2 points wins 1 dollar from the other.
You hold a one-time option, which you may use at any score before the game ends, to raise the stake to 2 dollars. What is this option worth?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
At which scores is the game in your favour?
At 1 to 0 for you, how much is the game worth per dollar of stake?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Doubling a game doubles its value. Use the option only when the game is in your favour; when it is not, waiting costs nothing, because you can always exercise later.
Step 1: the value per dollar at each score.
- At 0 to 0 and at 1 to 1, the game is symmetric: worth 0.
- At 1 to 0 for you, you win unless the next two flips are both tails: probability . The game is worth per dollar.
- At 0 to 1 it is worth .
Step 2: when to exercise. At 1 to 0, doubling adds dollar. Waiting does not help: the next flip either ends the game in your favour (too late to double) or brings it back to 1 to 1, where doubling is worth nothing. At other scores, doubling adds 0 or loses money.
Step 3: the value. You reach 1 to 0 with probability (the first flip is heads), and then the option adds :
The timing matters. If the option had to be used before the first flip, it would be worth nothing, since the game is fair at the start. The value comes from being able to wait and see.