The Two Envelopes Problem

Thanks to Nick S. from London, UK for this week’s topic suggestion.

Let’s play a game. I give you a $20 bill. In my hand, I have an envelope, and I tell you that there’s money in the envelope—either half ($10) or double ($40) what you have in your hand. I give you two options:

  1. Keep the $20
  2. Give me back the $20 and take whatever’s in the envelope

What do you do?

Well simple math tells you to take the envelope. If you played this game 10 times and kept the $20 each time, you’d end up with $200 at the end. If you took the envelope each time, you’d on average end up with $10 five times and $40 five times, which totals to $250. The envelope will average having $25 in it, or 5/4 of the $20 in your hand. So you switch. No brainer.

Okay new game. There are two envelopes on the table and I tell you that they both have money in them, but one of them has double the amount the other has. I tell you to pick one, and after you do I tell you to look inside. You find $20. I tell you that you can either take your winnings or switch to the other envelope. What do you do?

Same game, right? The other envelope is either the bigger of the two, in which case it has $40 in it, or the smaller of the two, in which case it has $10 in it. So by switching you have a 50% chance of losing $10 and a 50% chance of winning $20. Good move to switch.

Okay last game. There are two envelopes on the table and I tell you that they both have money in them, but one of them has double the amount the other has. I tell you to pick one, but this time you can’t look inside. Then I tell you that you can either keep whatever’s in the envelope or switch to the other envelope. What do you do?

Well now you don’t know the amount, so let’s call it X. You don’t know if you picked the envelope with the larger or smaller amount. If you picked the larger amount, the other envelope has 1/2 X in it. If you picked the smaller amount, the other envelope has 2X in it. If you switched 10 times in a row, you’d on average end up with 2X five times and 1/2 X five times for a total of 12.5X. If you never switched you’d end up with 10X. So good move to switch. Right?

But then after you switch, I tell you I’m going to let you switch again if you want. You look at the envelope in your hand and the other envelope on the table—the one you were holding just a minute ago. Whatever you have in your hand, switching to the envelope on the table gives you a 50% chance of losing half of that amount and a 50% chance of doubling up and gaining 100% of it. Good move to switch—so you do. And now you’re back where you started. Remembering that logic earlier told you to switch when you were holding that envelope, you switch again. Then you remember what happened when you were holding that envelope, so you switch back. This continues forever and instead of making any money, you just switch envelopes for the rest of your life until you die.

What the hell is going on?

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