Three way duel (puzzle)

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The three way duel is a logic puzzle proposed by Martin Gardner. [1]

Three men are in a pistol duel. Each man will shoot in turn. The three men are identified as A, B, and C. A is a poor shot, and hits only 50% of the time. B is an expert marksman, and always hits. C is a moderate shot, and hits 80% of the time. (In some variants of the problem, A's probability is 25% and C's is 50%; in practice this makes little difference as long as A's is lower than C's.)

The exact order in which the three men will take turns shooting is variable (although in some variants it is stated as being A, B, C).

As the problem is based on classical dueling it is assumed that a hit target is "killed", but this is not necessary and some variants of the problem alter the theme to a game where targets are unharmed, but required to withdraw, when hit.

The question is, what is the best strategy for A to follow to win the duel?

[edit] Solutions

The most common solution [2] is that A should deliberately fire into the ground until one of the other two men is dead, then shoot at him.

The reasoning for this is as follows: B and C are greater threats to each other than A is to either of them, and thus rationally should target each other first. If B fires first, he will certainly kill C; if C fires first, he may kill B, and if he does not, B will certainly kill him. In the first "round" of the duel one of these two interactions will occur between B and C and it is not in A's interest to disrupt it. If A kills one of the men, the other man will target him, probably killing him; if A fires but does not kill either man, he makes no difference. After either of these interactions completes, it will become A's turn with A never having been targeted and having a chance to kill the one remaining man and win the duel, and this is the best possible position for him.

[edit] Paradox

This solution is considered by some observers to expose a paradox in reasoning. They argue that, if A is permitted to fire into the ground, then B and C could do so too. Based on that, if survival is a duelist's only priority, the best strategy is for all three men to fire into the ground until the ammo runs out and then walk away. Intuitively and emotionally we tend to believe that if A dies by the end of the duel, it makes no difference how many people he killed before he died; but using this heuristic to judge duel strategies may lead to this strategy being considered optimal. If "winning the duel" has additional requirements other than survival (eg, killing the opponents), A's strategy above might no longer be considered the best because it fails to meet those other requirements. Although since B is going to be the first one to get shot anyway, he might as well shoot someone(C if he is still there), therefore C should shoot him, so this strategy does work only for A.