Despite this blog’s title, there really aren’t enough rants here. Let’s make up for that.

I was just reading another less than interesting paper on the two-envelope paradox when I started thinking, “Why is anyone still writing about the two-envelope paradox? Surely everything that needs to be said about it has been said.” And I was right. But perhaps everything that needs to be said about it has not been said in the one place. So I’ll say it all here. Note that nothing that I’ll say here is even close to being original – the real message of this rant is that this is a puzzle that’s well past its use-by date. (And I use lots of italics when I’m ranting.)

The argument for the paradoxical conclusion, that you’re better off switching no matter which envelope you get, relies crucially on an inference like the following.

The amount of money X in your envelope is from the set {x₁, x₂, …, x_n, …}. (Note this is a countably infinite set. There are probably versions of the paradox where the set is uncountable. The same things can be said about that version of the paradox.) Call this set S.

For all x in S, the conditional expected utility of swapping given X=x is positive.

Therefore, it is in your interest to swap.

Call the inference here (CC). (CC) is a kind of conglomerability principle – it says if something is good according to every member of a particular partition, then it is good simpliciter. Given some standard Bayesian assumptions, (CC) is equivalent to the following principle.

Let Y and Z be bets. For any proposition p, and bet W, let W & p be the bet that pays what W pays if p, and nothing otherwise. (I assume bets can have negative ‘payouts’, so all choices are bets.) Let (p₁, p₂, …, p_n, …} be a countable partition of possibility space. Then if for all i, Y & p_i is preferable to Z & p_i, then Y is preferable to Z.

It’s really important to keep in mind here that (CC), or something very much like it, is just essential to the paradoxical reasoning. There’s simply no argument that you should swap that doesn’t use as a premise the principle that you should swap whatever is in your envelope. And this premise doesn’t get you to the conclusion without (CC), or something stronger than it.

Now (CC) in either its intuitive or formal versions is a very plausible principle. To prove this, just note how many people have tacitly appealed to it in setting up the two-envelope paradox. But unfortunately it is inconsistent. Vann McGee showed this “An Airtight Dutch Book” Analysis 1999. The only agents that can satisfy (CC) are those that have either (a) bounded utility curves or (b) ‘opinionated’ belief states – more precisely, there are only finitely many propositions about which their credence is neither 0 nor 1. And, as has been known since at least John Broome’s 1995 Analysis paper, the two-envelope paradox only gets going if you assume the agent in question satisfies neither condition.

Now as far as I can tell, that’s all one needs to say about the paradox. The paradoxical conclusion is only reached by taking an inconsistent principle of reasoning, and applying it in just the case where we know on independent grounds that it cannot safely be applied. But is that all people say? Well, no.

If you want to find out what they do say, Google is your friend. Note that some of the links you’ll find, such as the first one, do say reasonable things – i.e. something similar to what I say. But not many I’m afraid.

To be sure, there still is something odd here. Principles like (CC) are very intuitive. It’s hard to know what to do when faced with a situation where you know your preferences will change as soon as you find out something, and you know you’re about to find it out. As McGee says, it looks like you face an Airtight Dutch Book in that situation. But that’s all the counterintuitiveness – there’s simply nothing special about the two-envelopes here, because there’s simply no argument from consistent premises that says you should switch.

Happy holidays!