London Colin said:
I surely can't be the only person to have cheated and looked up the history of this problem?
In case I am ...
Meistro said:
There are two envelopes, one has twice the money as the other. You open the first envelope and it has $100 in it. Do you switch?
With the question as it is phrased above, the situation is entirely symmetrical between the envelopes.
There can be no possible reason to switch; there is no information on which to base a switching decision. (As Sonny intimated.)
This is hopefully the common-sense, obvious answer that is everyone's starting point. But it's hard to pin down the flaw in the logic that seems to support switching -
ZenKinG said:
Regarding EV, you could switch and net your imaginary $25 in EV that you've just generated. You can receive a potential of $50 or $200 instead of the original guaranteed $100. $125 average from both - $100 = +$25.
It seems like a paradox, and you can begin to convince yourself that common sense must be wrong. But, for once, common sense is entirely correct.
There's a wealth of reading material on this problem and variations of it -
https://en.wikipedia.org/wiki/Two_envelopes_problem
I found reference no. 10 from the Wikipedia page quite helpful -
https://arxiv.org/abs/1411.2823 (sections 1.3 and 2.3, in particular)
Reading that paper, the only thing I am still struggling with is the rationale behind the stated need to -
weigh the return derived from each event with the average fixed amount by which the game is played in this event
Without a proper grasp of that, I feel like I'm still leaning more on common sense than on mathematical proof.
But the interesting thing I take away from that paper is that you can make switching favourable if you recast the problem slightly -
Two envelopes are labelled A and B. An amount is placed in envelope A, and a coin is tossed to determine whether to place half or twice that amount in envelope B.
Now you have an asymmetry. If you are offered envelope A, you should ask to switch to envelope B.