I was watching episodes of the TV series NUMB3RS, when mathematician Charles Eppes (David Krumholtz) mentioned this problem in game theory. This is an abbreviated version of the problem, one with an quick and easy solution.
There are three shooters, and they are going to have a gun fight. Prior to the fight they spend some time on the shooting range and discover something interesting. At a range of 50 feet shooter A never misses. Shooter B misses 20% of the time. Shooter C misses 50% of the time. Now it’s time for the match.
They stand at points on an equilateral triangle, 50 feet on each side. This is a math problem. A referee pulls a name from a box to determine which shooter goes first. He tells shooter C it’s his turn. The rules are thus:
- Each shooter gets one shot at a time, and the opportunity to shoot passes to the next shooter in turn.
- If a shooter is hit, he is dead and is out of the game. He loses his turn.
- If a shooter misses, the next shooter in turn gets a shot.
- The game is played until only one shooter is left standing.
Shooter C wants to maximize his chances of living to die another day. What does he do?
Post your answer as a comment below. Don’t guess. Justify your answer.
Update and answer
Mike has provided the correct answer on Facebook, so I will post the answer now.
The best option for C is to avoid killing anybody. This seems counterintuitive at first, but look at what happens:
- Suppose C shoots at B. He better not hit B, because if he does he is a dead man. The turn passes to A, who will shoot him dead, 100%.
- If he kills A, then B has a shot at him, and he has a 20% chance of surviving the first round.
- However, if he fires into the air, then it’s the turn of either A or B. A will shoot B for sure. Verify this. Then it’s C’s turn, and C has a 50% chance of killing A and living.
Sometimes the best option in a war is not to fight it.