The Three-Way Duel

You are Mr. 10 in a three-way duel with Mr. 30 and Mr. 60. On a hit you kill your target with probability $0.1$ , Mr. 30 with probability $0.3$ , and Mr. 60 with probability $0.6$ . You shoot first, then Mr. 30, then Mr. 60, cycling until one survivor remains. At whom should you shoot?

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#Test the obvious shots

Suppose you fire at Mr. 60 and connect, which happens with probability $0.1$ . Only you and Mr. 30 remain, and it is his turn, so he shoots at you with a $0.3$ chance of killing you. You have handed yourself a duel that the other side opens.

Suppose instead you fire at Mr. 30 and connect. Now you face Mr. 60 with his turn next, and he kills you with probability $0.6$ . This is worse still. In both cases a successful shot only promotes you to the next gunman's target.

#Miss on purpose

So fire into the air and kill no one. Mr. 30 and Mr. 60 are each other's greatest danger, so on their turns each aims at the other, never at you. They duel while you wait, costing you nothing.

Mr. 30 and Mr. 60 are each other's deadliest rival, so they shoot at one another and never waste a turn on the harmless Mr. 10. By firing into the air you stay off the target list and inherit the first shot once only one rival is left standing.

Eventually one of them falls. Then it is your turn against a single survivor, and you take the first shot in that final duel. Every alternative left you shooting second against a live opponent, so abstaining is strictly better.

#Does the order change it

If the cycle were you, then Mr. 60, then Mr. 30, nothing changes. Each of the strong pair still sees the other as the bigger threat and fires at him, not at the harmless Mr. 10. You still gain by holding fire and inheriting the first shot once only one rival is left.

#Result

Shoot at no one. Deliberately missing maximises your survival, because killing either rival only makes you the next target, whereas waiting hands you the opening shot in the duel that remains.