Two Envelopes Puzzle
Of two indistinguishable envelopes, each containing money, one contains twice as much as the other. The subject may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, the subject gets the chance to take the other envelope instead. What is the optimal rational strategy for maximising the amount of money to be gained?
You might like to think for a bit ….
(I find it helpful to suppose that the two envelopes contain a lot of money and that you have to pay a very small penalty if you want to take the other envelope.)
Two obvious possible answers:
- If your envelope has £u then the other one has either £0.5u or £2u. By the principal of indifference these are equally likely, so the expected amount is £1.25u. This is more than £u, so to maximize expected amount, you should switch.
- Before selecting an envelope, either one is equally likely to contain the most, so it is pointless to switch.
Any more … ?
- If by some argument you have a reason to switch, while there is no argument that switching will harm, you should switch.
- There is no reason to switch, so if you do you will look foolish. Do not switch.
And some more … ?
- It is not clear from the wording if this is a repeated situation under constant rules or a one-off, with the opportunity to switch possibly being conditional on which envelope was chosen. In this case, do we think that we are more likely to be given the opportunity to switch if we have chosen the best envelope or the worst? If in doubt, I would suggest not switching.
Now, what does this have to do with the idea of a numeric probability?
My own thoughts follow …
If I originally chose the envelope deliberately then I (as subject) might see the situation as a kind of game, and consider what game the experimenter was playing, and then seek to identify the appropriate strategy. I might end up tossing a coin to decide whether to switch or I might definitely stick. But I wouldn’t necessarily criticise you if you chose to switch – particularly if you could explain your choice. (Perhaps you are sure that the experimenter is being kind in offering the choice.)
Of course, to simplify things I should have started by looking on the situation as a kind of game and choosing an envelope at random. In this case I would still analyse the remaining situation as a game, but would stick. (Again, if the experimenter happens to be a friend, you might be justified in switching.)
Now, you might argue that if my envelope contains £x then the other contains £x/2 or £2.x, so the ‘expected’ contents are £1.25.x and I should definitely switch. If I allow your argument then we have a paradox. The general argument is that I should choose the option with the greatest value. But why is this? If I am faced with very many similar decisions, then in the long-run I would expect to obtain very close to £x or £1.25x, and so will be better off if I always switch envelopes. But in the context of the question this is not open to me. The relevant strategic decision point is before I choose the first time, not half-way through the process.
If we try to fix the argument based on expectation, suppose that the experimenter decides on an amount y and then puts y/a into one envelope, a.y into the other. If we stick then we expect to get (y/a+a.y)/2. If we switch having x in the envelope, we subjectively ‘expect’ to get (x/a2+a2.x)/2. But if the envelope contains y/a we get a.y, and vice-versa, so objectively we ‘should’ expect to get the same as if we stick, as I assumed above.
The problem is that the subjective expectation part-way through the process is misleading. The straightforward rule of thumb of choosing the option with the best expected return assumes that you start before the random events and can relate utility to some time after them. Otherwise things can get messy.