# Truel

A truel is a duel among three people. Here is a truel puzzle popular in game theory.

Three opponents, Red, Green and Blue, have agreed to a truel with pistols. You are Red, and you are a poor shot: you are only able to hit your target once out of every three tries. Green is better, and can hit the target two out of every three tries. Blue never misses. In the interest of fairness, you are allowed to shoot first, firing one shot, then Green (if still alive) fires one shot, then Blue (if still alive) fires one shot, and the turns continue until only one remains alive. Clearly you have made a bad decision agreeing to this contest. Now the question is, where should you fire your first shot in order to maximize your chances for survival?

What if you fire at Green first? If you kill Green, Blue has the next shot and can only fire at you, with a 100% chance of killing you. If you chose to fire at Green, you would be better off if you missed, because then at least Green would probably try to eliminate Blue, with a 2/3 chance of succeeding. If Green eliminates Blue, you still must face Green, who is a better shot, but at least you get the first try. If Green fails to eliminate Blue, then Blue will eliminate Green, leaving you to face Blue, but again, at least you get to shoot first.

Let’s examine what happens if you choose to fire at Blue first. If you eliminate Blue (which you have a 1/3 chance of doing), then Green will fire at you, and there is a 2/3 chance you will die. Those are not good odds. On the other hand, if you fire at Blue and miss, then there is a 2/3 chance that Green will kill Blue, putting you in the same position as above.

In both scenarios, firing at Green or firing at Blue, you are better off if you miss. Therefore, you should fire your first shot in the air.

This puzzle and other brain teasers are presented in Simon Singh’s excellent book Fermat’s Enigma, which tells the story of the centuries-long quest to solve Fermat’s Last Theorem.

The Good, the Bad and the Ugly

A famous cinematic truel is contained in the film The Good, the Bad and the Ugly, directed by Sergio Leone and starring Clint Eastwood. The three titular characters face off in the quick-draw manner of duels in the Wild West. However, unlike a duel between two gunslingers, in which the fastest to draw is likely to win, in a three-person Mexican standoff, with each participant having only one gun available, the best course of action is unclear. If the Good fires at the Bad, the Ugly may take the opportunity to fire at the Good. You can watch how the truel plays out in the movie below.