Gun Duel

Mick, Nick, and Rick arrange a three-person gun duel. Mick hits his target 1 out of every 3 times, Nick hits his target 2 out of every 3 times, and Rick hits his target every time. If the three are taking turns shooting at each other, with Mick starting first and Nick second, what should be Rick’s strategy?

Clearly, Mick should not aim for Nick, because if he kills him, then he will be killed by Rick. Similarly, Nick should not aim for Mick, because if he kills him, then he also will be killed by Rick. Therefore, if Rick ends up against alive Mick and Nick, he will aim at Nick, because he would prefer to face off a weaker opponent afterward. This means that if Nick is alive after Mick shoots, he will shoot at Rick.

Thus, if Mick shoots at Rick and kills him, then he will have to face off Nick with chance of survival less than 1/3. Instead, if he decides to shoot in the air, then he will face off Nick or Rick with chance of survival at least 1/3. Therefore, Mick’s strategy is to keep shooting in the air, until he ends up alone against one of his opponents.

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