Exchange Paradox

专题: Strategy / 策略
难度: L4
来源: QuantQuestion

题目详情

有两个密封信封。你拿到其中一个，竞争对手拿到另一个。你知道一个信封中有 $m$ 美元，另一个信封中有 $2m$ 美元，但 $m$ 的值未知。

如果你偷看自己的信封，看到里面有 $X$ ，你不知道对方信封里是 $2X$ 还是 $X/2$ 。如果不偷看，换信封的期望收益是多少？假设对方看到的是 $Y$ ，对方换信封的期望收益是多少？你应该换吗？如果换了，是否会基于同样理由再次交换（假设双方都没有继续偷看）？
假设你们一开始都偷看了各自信封。此时交换的收益是多少？应该交换吗？如果交换了，是否会基于同样理由再次交换？

Two sealed envelopes are handed out. You get one and your competitor gets the other. You understand that one envelope contains $m$ dollars, and the other contains $2m$ dollars, where $m$ is unstated.

If you peek into your envelope and see $X$ , you do not know whether your opponent has $2X$ or $X/2$ . Without peeking, what is your expected benefit to switching envelopes? What is your opponent's expected benefit to switching envelopes, assuming your opponent sees $Y$ ? Should you switch? If you do, do you do it again for the same reason, assuming neither of you peeked?
Suppose that you both peek into your envelopes initially. What is the payoff to switching? Should you switch? If you do, do you do it again for the same reason?

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