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# Two envelopes, with money and a twist

## Question

Two envelopes, given one each to you and a friend, contain an amount of money between \$5 and \$160 inclusive, and one amount is twice the other amount. You look inside your envelope to see the amount you have received, but you do not disclose the amount. Your friend does the same. You see that you have received \$x.

You may suggest a swap (a one-time exchange of envelopes) but you don't converse directly with your friend. Instead, a facilitator comes to each of you, in turn, and asks what you want to do. If and only if you both ask for a swap the envelopes are exchanged. You and your friend make your decisions independently and in ignorance of the decision made by the other. For what values of x will you favor swapping envelopes? Assume your friend acts reasonably and in his/her own best interest.

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If someone has 80 (and more), he will not (propose to) change. (If I have 120, as you cannot have 240, you must have 60).

If someone has 40-79, he will not change. (If I have 60: if you have 120, you will not agree to change; if you have 30, I am loosing).

If someone has 20-39, he will not change. (Same reason, you will refuse if you have 40-79.)

Recursivity comes to my mind. But what if I have i.e. 17?

Edited by harey
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normal distribution curve, one should refer to swap if his (x) value was within the range of least 3%.
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If someone has 80 (and more), he will not (propose to) change. (If I have 120, as you cannot have 240, you must have 60).

If someone has 40-79, he will not change. (If I have 60: if you have 120, you will not agree to change; if you have 30, I am loosing).

If someone has 20-39, he will not change. (Same reason, you will refuse if you have 40-79.)

Recursivity comes to my mind. But what if I have i.e. 17?

Good start.

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If I have more than \$80, I must have the larger amount and therefore do not want to trade.

If I have more than \$40 but less than \$81, I know that if I have the smaller amount then my friend will know this and not want to trade. Therefore I can only lose, so I still don't want to trade.

If I have more than \$20 but less than \$41, I know that if I have the smaller sum then my friend will not want to trade, by the above reasoning. So I can still only lose. I still won't trade.

If I have more than \$10 but less than \$21, I know that if my friend has the higher sum he will not want to trade, by the above reasoning. Therefore I can only lose, and still do not want to trade.

If I had exactly \$10, I also can only lose because my friend will not trade with me if he has the higher value, by above reasoning.

I'd be willing to trade if I had less than \$10, even though my friend wouldn't take me up on it.

Edited by ThunderCloud

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