三人决斗：先手该打谁？

Mr.10

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Your name is Mr. 10. You are standing in a field with two opponents: Mr. 30 and Mr. 60. Each of you has a gun and plenty of ammunition. Each of you is in clear sight of the others and well within firing range. The goal is to maximize the probability of survival. Unfortunately, you are not a very good shot. If you take a shot at one of your opponents, you have only a $10\%$ chance of killing him. Mr. 30 is a better shot; he has a $30\%$ chance of killing whomever he shoots at. Mr. 60 is even better; he has a $60\%$ chance of killing his target. You take turns shooting in a pre- arranged order: first you, then Mr. 30, then Mr. 60, and then through this cycle again and again until only one person remains.

You get to shoot first. At whom do you shoot?

解析

最优策略是： $\boxed{故意打空（朝天开枪）}$ 。

理由要点：你命中率最低（10%）。如果你第一枪杀掉某人（小概率），会立刻把局面变成“你 vs 更强者”，对方下一枪通常会优先瞄准你，你的生存概率会变差。

相反，若你故意打空，接下来 Mr.30 最合理会优先射击威胁最大的 Mr.60；Mr.60 存活时也更倾向回击 Mr.30。让两位强者先互相消耗，你更有机会在后续轮到自己时进入更有利的局面。