Today’s post is going to be a riddle; I figured it was about time for another one.
You and I both have envelopes filled with money. My envelope contains either double or half the amount of money that’s in yours. If you want, I’m going to let you switch envelopes. Should you stay, switch, or does it not matter?
The riddle itself isn’t actually the riddle (you’ll see what I mean in a minute). I’m going to solve this riddle for you in two different ways.
Suppose your envelope contains $100. That means my envelope contains either $200 or $50. If you switch, half the time you will gain $100, and half the time you will lose $50. The expected value that you will end up with is $125. It is therefore in your favor to switch envelopes because, on average, you will come out $25 richer. Assuming that little x is the amount of money in your envelope, and big X is the amount of money you get when you switch, below is the simple calculation:
Suppose the amounts of money in the two envelopes are $100 and $200. That means that half the time when you switch you will gain $100 (going from $100 to $200), and the other half of the time you will lose $100 (going the other way, from $200 to $100). Therefore your gains and losses cancel out on average and it doesn’t matter if you stay or switch because the expected value is $150 either way. Assuming the values in the envelopes are x and 2x, and that big X is the expected value of your envelope whether you switch or not, the simple calculation is below:
Obviously, the two solutions are contradictory and can’t both be right. The first solution proves that it’s better to switch envelopes, and the second solution proves that it’s not.
Here is the real riddle: which is the correct solution? Or, alternatively, is there a third solution that I neglected to mention? Also, just determining the correct solution isn’t enough, you must also explain why the other solution(s) is/are wrong.
By the way, there is a good discussion about this on Reddit. For those of you who are stumped, the answer can be found below in my first comment, encoded so as not to spoil it.