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Let's say I get paid by my boss. There are 2 envelopes, A and B. I know for a fact that one envelope contains twice more money than the other one.

1) If I pick an envelope A, and the boss asks me if I want to change to envelope B. Is it better to change?

-> Since I do not know which one has more money, the expected value for both envelope is $\frac{x+2x}{2}=1.5x$ each. So it does not matter if I change or not.

2) If I pick an envelope A and see that it has \$100 in it, and the boss asks me if I want to change to envelope B. Is it better to change?

-> Envelope B will have either \$50 or \$200. The expected value of envelope B is now $\frac{50+200}{2}=\$125$. So it is better to change.

What makes these two situations different? While I think that some assumptions or logic of the problem might be flawed, I just cannot figure out why these two situations lead me to a different result. Thank you in advance :)

Xander Henderson
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    This is known as the two envelopes paradox – lulu Jun 07 '20 at 18:08
  • In reality , it would depend on the boss : Perhaps he wants to minimize what you get, then he offers the switch only if you have chosen the envelope with the larger amount. Vice versa, if he wants that you get the evenelope with the larger amount. If he offers no matter which envelope you have chosen, in fact, concerning the expectation, it is better to change (assuming that the amount you read can be divided by $2$) – Peter Jun 07 '20 at 18:22

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