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 The grass is always greener... (Posted on 2004-05-06)
You are told there are two envelopes. One contains twice as much money as the other one. You pick one but are allowed to change your mind after picking it. (You are equally likely to pick the one with less money as the one with more money.)

To figure out how much on average the other envelope should contain, one might average x/2 and 2x because one is equally likely to pick one as picking the other. Since this comes out to 5x/4, one might always change his or her mind. But wouldn't this end up with one never making up his or her mind?

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 Solution | Comment 40 of 46 |
Too many solutions here, and I apologize if I'm repeating other people (I tried to read through what's here already, but might have missed a post or two):

The logic presented above is correct for a slightly modified version of the game.  It is not correct for the game as described above.  Let's consider the two games:

Game 1:  As described in the problem.

Game 2:  You are given a certain amount of money, and shown two envelopes on the table--one that contains 1/2 the amount of money you are initially given and one that contains twice the amount.  You can keep what's already been given to you, or switch for what's in one of the envelopes.

The two games may seem the same to you, but really they are quite different!  In game 1, it makes no difference whether you switch or not, while in game 2, you should always switch!  This can be proven via a thought experiment:  Let's suppose there are two dummies, Dummy X and Dummy Y, who play Game 1 100 times.  The amounts in the cups are always the same (Envelope A has \$1, Envelope B has \$2, but because X and Y are dummies, they don't know this even after playing 100 times).  Dummy X plays the game always switching (but randomly choosing Envelope A or B) while Dummy Y plays the game never switching (but randomly choosing Envelope A or B).  Who gets more money?  Near as I can tell, both should end up with about \$150 each.  So switching makes no difference in this game.

Now let's say Dummies X and Y play Game 2 100 times.  Each game, they are shown \$1, and are given the opportunity to switch for what's in one of the envelopes on the table (50/50 chance of being either \$0.50 or \$1.00).  Dummy X always chooses an envelope, Dummy Y never chooses to switch.  Dummy X should get about \$125, while Dummy Y gets only \$100, so switching does make a difference in the game.

To reiterate what's already been said by others, the correct way to analyze Game 1 is this:

You are given Envelope A and shown it's contents.  You now know that what you are looking at is either \$X or \$2X (EV=\$1.5X).  You are given the chance to switch over to Envelope B, which contains either \$X or \$2X (EV=\$1.5X).  Since the expected values are the same, it makes no difference whether or not you switch envelopes.

 Posted by ron on 2006-02-13 14:12:06

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