Gun Duel

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

SOLUTION

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

Now if Mick shoots at Nick and kills him, then he will have to face off Rick with chance of survival less than 1/3. Instead, if he decides to shoot in the air, then he will face off Rick or Nick 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.