Solution to Challenge 1, Fall '04

The Problem: After spending an evening at a saloon, three cowboys get into an argument and decide to settle it with a "three-way duel." Cowboy 1 will take a shot, possibly killing one of the other two. Cowboy 2, if he is still alive, will then take a shot. Cowboy 3 will then take a shot if he is still alive, after which it will be Cowboy 1's turn again, if he is still alive. The turn goes round and round the circle until only one cowboy is standing.

Cowboy 3 is a good shot and can hit his target every time. Cowboy 2 can only hit his target with a probability of 2/3. Cowboy 1 drank a lot of whiskey, and can only hit his target with a probability of 1/3. All three cowboys know these things about each other.

You are Cowboy 1 and the duel is starting. What can you do to maximize your chances of survival?

Hint: There is a way to ensure that your chance of survival is strictly greater than 1/3.

Solution: The best course of action is to shoot up in the air.

Here's the main insight: as long as they're both alive, the other two cowboys will shoot at each other, not at you. It's best to let one of them shoot the other, whereupon it will be your turn again. If you shoot one of them, it will then be the other one's turn, and he might shoot you before you get another turn.

For calculating your odds, observe the following: If you're in a duel with just one of the other two cowboys, your odds of survival are at least 1/3 if you have the first shot, and strictly less than 1/3 if the other cowboy has the first shot.

Let's apply this observation to the case of two other cowboys. If you hit one of them, then what remains of the contest is a two-way duel where the other guy has the first shot. Your odds of survival are less than 1/3. If you shoot up in the air, the other cowboys, who are also motivated to maximize their chance of survival, will shoot at each other before they turn their attention to you. Since Cowboy 3 is a perfect shot, after the first round, one of them will be dead, and it will be your turn again. What remains of the contest is a two-way duel where you have the first shot.

For a precise analysis of the odds, and other commentary, click here.

(I once told the problem to a businessman who isn't very good at math and didn't get this solution. However, when he heard it, he offered the following improvement that I hadn't thought of: Offer to forgo your first turn in exchange for two shots on your second turn.)

People who got this solution: Nicholas Rohrbacke, Nick Krier, Tim O'Connor, Onyx Mueller, Troy Butler, Saravanan Sellappa, Doug Beeman, Andres Alvarez. In a process reminiscent of the duel, I used coin flips to randomly eliminate people from this list until one of them was left standing: Troy Butler. Troy wins the ice cream.