Two Envelope Problem

This is a problem like the MontyHallProblem. (Perhaps it is just another one of the InevitableIllusions.) For a detailed analysis of both problems, see (search for envelope).

OK, you have won a game show, and you have a choice of two prizes, Box A and Box B. All you know is that one box contains twice as much money as the other.

So you choose one at random (say A). The presenter shows you that Box A contains (as it turns out) £100, and offers you the chance to switch.

What do you do?

A. There are even chances that B contains £50 or £200, so you switch with an expected gain of £25?

B. Your choice was random, and knowing the amount makes no difference, so it makes no difference whether you switch?

{or C. Decide according to a randomizing procedure - see the link above for details.]

Stop reading now if you want to figure this out.

There are two boxes, x and 2x. You don't know which box you picked, so there's a 50% chance it was x and a 50% chance it was 2x. If you picked x and then switch, you will get 2x. If you picked 2x and then switch, you will get x. So if you switch, half the time you will get x and half the time you will get 2x, so your expectation is 1.5x. If you don't switch, then you will be randomly choosing between two boxes, so your expectation is again 1.5x. Staying or switching makes no difference.

The trick is that there are only two values, x and 2x, not three, 0.5x, x, and 2x. The other box is not chosen randomly from 0.5x and 2x. Instead it is already determined from either x or 2x. Since you don't know whether your box is the x or the 2x, then by switching you are essentially still making a random choice between x and 2x. Same choice, same expectation.

From my experience trying to figure out this problem, I would say it is also another Inevitable Illusion, since it took me three different methods of simulating it with Excel before I figured it out. I still have this nagging doubt that I'm wrong, but I'm trusting my math rather than my intuition.

I think the above answer is correct, but maybe not sufficiently explicit. The problem is that the statement that there are even chances that the second box contains £50 or £200 is not right. That depends upon the relative probabilities of how the boxes are initially configured, and we have no information about that. If we assume that either 50/100 or 100/200 was put into the boxes with equal probability, then it would be correct to switch to maximize expected value, but we don't know that. You can't really "simulate" this without making some assumption about how the initial state is generated.

Any information you obtain about the probability density of the amount X can be used to increase your chances. Such information can be obtained by watching earlier shows, or if it is the first show, by making an analysis of what the producer of the show can afford to give as prize money. (E.g. compare with the prize money people get on shows with similar challenges.) -- StephanHouben

You guys are pretty good. Spot on in fact. The unstated assumption is that all amounts are equally likely. This is known in Bayesian statistics as an Improper Prior. There is no probability distribution that makes all amounts equally likely, and this is the source of the potential error.

Oddly enough, Improper Priors can be used in most other statistical or decision theoretic situations without causing any problems.

You are right to say that in practice you will know something about the prize levels. However, even if you haven't seen the show before, and it is taking place on another planet, you would be wrong to assume that all amounts are equally likely. All probability distributions have to tail off eventually.

-- JoeOtten

From a psychological point of view, you could approach the problem like this: If you look at the prize and are disappointed by how small it is, switch. If it looks good enough to you, don't switch.

Looks small: If you stick with the initial prize, you're already disappointed. If you switch and it's smaller, well, you were already disappointed, so nothing has changed really. If you switch and it's bigger, bonus!

Happy with it: If you stick with the initial prize, you're already happy. If you switch and it's bigger, well, you would have been happy with the initial prize anyway, so nothing has really changed. If you switch and it's smaller, bummer!

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