The three-way duel where the worst shot should fire in the air
Alice, Bob and Cheri fight a three-way duel. Alice hits her target with probability 1/3, Bob with probability 2/3, and Cheri always hits. They take turns in the order Alice, Bob, Cheri, Alice, ..., each firing one shot at a target of their choice (or deliberately into the air), until only one person is alive. Everyone plays optimally and knows the others' accuracies.
What is Alice's best first move, and what is her probability of surviving if she makes it?
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Bob and Cheri each regard the other as the bigger threat and will shoot at each other first while both are alive. Alice's real question is whether she wants to be the one who breaks that standoff.
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