三人决斗：先手该打谁
Mr.10
题目详情
概率题：三人决斗：先手该打谁？。
英文原题
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 chance of killing him. Mr. 30 is a better shot; he has a chance of killing whomever he shoots at. Mr. 60 is even better; he has a 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：。
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。
英文解析
This is a tough "game theory" problem. Early editions of my book included a full and formal solution to this problem. It was more than five pages long and far too detailed. I have now cut my answer down to bare bones key issues only.10
Mr. 30 and Mr. 60 are going to shoot at each other because they do not see you as an immediate threat; you do not die first because Mr. 60 and Mr. 30 are shooting it out; you do not want to be put into a shoot- out where your opponent is a very good shot and gets to shoot first; if Mr. 30 gets to shoot before Mr.60, it is less likely that you end up facing Mr. 60 than if Mr. 60 gets to shoot first, so you shoot in the air; if the direction of play is reversed, and Mr. 60 gets to shoot before Mr. 30, then you should help out Mr. 30 (and yourself) by shooting at Mr. 60 also, otherwise, leave it to Mr. 30; the cost of stepping in and shooting at Mr.60 is that if you hit Mr. 60, you lose your chance to shoot first in the final shootout with Mr. 30; the benefit of stepping in and shooting at Mr. 60 is that you increase the likelihood of your facing
Mr.30 rather than Mr. 60 in the final shoot- out; there is a delicate balance between leaving it to Mr. 30 and stepping in to help him out, and it changes with the direction of play. Finally, there is a slim chance that everyone shoots at the sky, but this requires some sort of cooperation.