People Can Be Weird Sometimes

So, following a series of links from a random xkcd comic, I just spent a few minutes perusing the Wikipedia article about the so-called "two-envelopes problem." The idea is this: suppose I give you two identical envelopes, each of which has some money in it, and allow you to pick one of those envelopes and take whatever money is in it. Now suppose that after you pick up one of the envelopes, I give you the chance to switch your choice to the other envelopes. If we call the amount of money in the envelope you originally chose "A," and assume it's equally likely that A is the higher amount or the lower amount, then there's a 50% chance that by switching you'll gain A dollars, and a 50% chance that you'll lose A/2 dollars, so on average you gain money by switching. But now, once you've switched, suppose I offer you the chance to switch envelopes again. Well, calling the amount of money in your new envelope "B," I can reason exactly as before, and convince you to switch. In fact, if I keep offering switches and making this argument, I can get you to switch forever!


This sounds really silly, and of course it is. The Wikipedia article is full of people talking about how to resolve the apparent paradox, which says you should switch from envelope #1 to envelope #2, and then other people talking about how to foil the resolutions and revive the paradox. But it's obvious that this is uninteresting, right? Suppose that before you pick either envelope, we just call one of the amounts "1" and one of them "2," allowing the logic of real/nominal to remove the units. Then there's a 50% chance you've picked 1, and would switch to 2, gaining a dollar, and a 50% chance you've picked 2, and would switch to 1, losing a dollar. This is just a true way of representing the situation, and anything else must perforce be some kind of wordplay designed to create a problem that isn't there by kooky notation.


This is not the "stick-or-switch" problem I grew up with, and the difference is crucial. In that problem, you are presented with three choices, one of which contains the treasure. After you make your initial selection, the game host reveals that one of the two options you didn't choose was empty, and gives you the option to stick with your original selection or switch to the other remaining option. Which should you choose? The answer is that you should switch. There was a one-third chance you picked the treasure initially, and because the host knows where the treasure is, his revelation doesn't chance that chance. He'll successfully reveal an empty door whether you hit the treasure or not. All the revelation does, in effect, is make the "switch" option equivalent to taking both of the options you didn't pick originally, which is clearly a win.

In this problem, however, there is no new information, and the choices don't chance. At every step you are choosing between two identical envelopes, one of which has X dollars, and one of which has 2X dollars. That's all you know. If, originally, you had no basis for preferring one over the other, then you continue to lack such a basis when given a subsequent opportunity to choose again under the exact same circumstances. You don't need a lot of fancy algebra to figure this out. No basis on which to choose = no basis on which to choose.

I'm not exactly complaining about people's tendency to spin words around obvious non-problems like this. It's fun, and really, what else are bored math people to do? But I find the tendency kind of weird and fascinating. I wonder, has anyone ever actually thought that this is a real problem, and that one must always switch between the two envelopes and that one can thereby accumulate infinite wealth? Or has everyone who's ever published a paper defending the paradox known that it's not a real problem, and just been being kind of mischievous?

In any event, it's a break from my usual politics-centric blogging. I am, after all, a math major too!