## Triple Duel

A three-way standoff in the film Reservoir Dogs between (from left) Nice Guy Eddie, Joe and Mr. White. Such situation is known as a "Mexican standoff".

You and two of your “friends” get into a dispute and decide to solve it with a three-way duel. Pistols are selected.

Friend A is a crack shot and never misses. Friend B hits the target 2/3 of the time. You are not quite as good as the others and hit your target 1/3 of the time. It is decided that you will take the fist shot, Friend B (2/3) will take the second shot (if alive) and Friend A (perfect shot) will go last (again if still alive). This rotation will continue until there is one person left alive. On each turn, one shot only is allowed. As stated, you are going first.

Where should you take your first shot at to have the greatest chance of winning the duel?

(I might add that all the people involved in the duel are extremely rational and selfish — they always perform the actions that maximize their own chances. -SR)

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### 2 Responses to Triple Duel

1. jamesoswald says:

A’s best response would be not to shoot the first round. If A hit B, A’d be killed by C in round 1, so A will shoot at C round 1. If A must shoot, A should shoot at C first.

B will shoot at C, because if B does not kill C, C will kill B in round 1. C would rather go to round 2 with A than with B, because A is a worse shot and both A and B shoot before C.

C, if still surviving will shoot at B, because B is a better shot.

Round 2:
If A hits C, A has a 1/6 chance of survival. If A misses C and B hits C, A has a 50% chance of survival. If both A and B miss C, A has a 1/3 chance of survival. A’s odds of survival given a perfectly played game is 19/54 (I think).

• jamesoswald says:

I flipped it. C should be A, and I got the probability wrong. Oh well.