Question

一道概率题，请解答

You are a risk-neutral investor and you have a fair six-sided dice (numbered 1-6) and you are asked to participate in the following gamble. You are allowed to throw the dice a maximum of three times, after each throw you have two choices:

A: To throw the dice again
B: To stop and receive $10 per point showing on the dice.

Obviously, if you throw the dice for the third time you must choose B.

What is your optimal strategy and what is the fair price to pay to enter into this particular gamble?

Answer 1

有答案么，是不是第一步得到的骰子大于等于5则停止，否则，继续第二步，如果第二步大于等于4，停止，否则，继续第三步并停止。此游戏应该付25/6。对嘛？

追问

讲明白一点阿谢谢

追答

这里所谓策略，其实是一种决定什么时候继续，什么情况停止的策略。目标是使得获得收益的期望最大。用数学一点的语言来讲。假设有一列独立同分布随机变量X(1),...,X(n)，是{1,2,3,4,5,6}上的均匀分布。寻找停时TX(k)时，选择继续，因为继续的话，用最优策略能获得更多收益(平均意义下的)(这里用到了独立同分布这个条件)。如上就是策略啦。具体的a(1)=3.5,a(2)=我忘了，挺复杂的，a(3)=25/6。所以公平起见，应该付25/6，如果我没算错的话。你还希望我写的更严格嘛？比如你需要我证明X(T)>=a(n-T)嘛？

Answer 2

三次掷骰子间是相互独立的，每掷一次都有25/6的期望收益，对于风险中性的人可接受的价格恰为25/6，但是现在最多可掷三次就要triple为25/2

追问

25/6?