Norman Samish writes:

Assume an eccentric millionaire offers you your choice of either of two
sealed envelopes, A or B, both containing money.  One envelope contains
twice as much as the other.  After you choose an envelope you will have the
option of trading it for the other envelope.

Suppose you pick envelope A.  You open it and see that it contains $100.
Now you have to decide if you will keep the $100, or will you trade it for
whatever is in envelope B?

You might reason as follows: since one envelope has twice what the other one
has, envelope B either has 200 dollars or 50 dollars, with equal
probability.  If you switch, you stand to either win $100 or to lose $50.
Since you stand to win more than you stand to lose, you should switch.

The problem is that you are reasoning as if the amount in each envelope can vary during the game, whereas in fact it is fixed. Suppose envelope A contains $100 and envelope B contains $50. You open A, see the $100, and then reason that B may contain either $50 or $200, each being equally likely. In fact, B cannot contain $200, even though you don't know this yet. It is easy enough for an external observer (who does know the contents of each envelope) to calculate the probabilities: if you keep the first envelope, your expected gain is 0.5*$100 + 0.5*$50 = $75. If you switch, your expected gain is 0.5*$100 (if you open B first) + 0.5*$50 (if you open A first) = $75, as before.

Ignorance of the actual amounts may lead you to speculate that one of the envelopes may contain $200, but it won't make the money magically materialise! And even if you don't know the actual amounts, the above analysis should convince you that nothing is to be gained by switching envelopes.

If the game changes so that, once you have opened the first envelope, the millionaire decides by flipping a coin whether he will put half or double that amount in the second envelope, then you are actually better off switching.

Stathis Papaioannou

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