The envelope paradox

I think of a number at random and write you a cheque for that amount. I put it in an envelope.

I write you a cheque for twice the amount and put it into another envelope.

I swirl the envelopes round a bit so you don’t know which is which, and invite you to take one of them. Whichever cheque is in that envelope, it is yours.

You pick an envelope but just before you open it, I offer you the chance to change your mind and take the other one instead.

There is a paradoxical argument which seems to say that you will always profit by changing your mind. Which is nonsense – but where is the hole in the argument?

The argument for changing your mind and taking the other envelope is as follows.

  1. Let’s say that the cheque in the envelope you’re holding is for £X.
  2. That cheque has a 50-50 chance of being the larger one.
  3. If you change your mind and the cheque was the smaller one, you go from £X to £2X, gaining £X.
  4. If you change your mind and the cheque was the larger one, you go from £X to £½X, losing £½X.
  5. Gaining £X half the time and losing £½X half the time means that on average you gain £¼X by changing envelopes.
  6. So it is worth changing to the other envelope.

So you put down the envelope you had chosen and pick up the other one. But before you open it, you think. You wonder whether you ought to change your mind again.

And exactly the same argument applies. So you change envelopes again, back to the first one.

Opening the second envelope is better than opening the first envelope, and opening the first envelope is better than opening the second envelope.

Either you go on swapping envelopes for ever, because each time swapping is better than opening the one you have in your hand, or you work out where the paradox is coming from..

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