The Two-Envelope Paradox: Stated but Not Resolved
Introduction
Suppose you are offered two envelopes labelled Alpha and Beta, each containing a sum of money. One of them contains twice as much money as the other. Which do you pick? Suppose after you pick one envelope, you are allowed to exchange it for the other?
It seems obvious that you are indifferent between Alpha and Beta, since you don’t know which is the big-money envelope. And if you picked Alpha first, you would be indifferent about switching to Beta, since you still don’t know whether Alpha is the big-money envelope. This seems easy. But there is a paradox, which I will next lay out.
First, though, there is an ambiguity in my description of the situation which gives us two variants on the paradox, the Open-Envelope Variant and the Closed-Envelope Variant. Do you get to see what’s in the first envelope before you are allowed to switch, or not?
The Open-Envelope Variant
In the Open-Envelope Variant, you open envelope Alpha and find it contains $100. Do you switch to Beta, or not?
Envelope Beta contains either $50 or $200. You don’t know anything about whether Alpha is the big-money envelope or not, and there are just two possibilities, so you must use a 50% probability that Beta has $50 and a 50% probability it has $200. But since
E(Beta) = .50($50) + .50($200) = $25 + $100 = $125,
you should definitely switch to Beta. But earlier we saw it was obvious you should be indifferent between Alpha and Beta!
But the paradox can be even worse.
The Closed-Envelope Variant
In the Closed-Envelope Variant, you pick envelope Alpha, but you don’t get to open it. You are simply asked whether you want to switch to Beta. Do you switch or not?
Suppose we label the amount in Alpha as 1 currency unit, 1C. Envelope Beta contains either .5C or 2C. You don’t know anything about whether Alpha is the big-money envelope or not, and there are just two possibilities, so you must use a 50% probability that Beta has .5C and a 50% probability it has 2C. But since
E(Beta) = .50(1C) + .50(2C) = .25C + C = 1.25C,
you should definitely switch to Beta. But earlier we saw it was obvious you should be indifferent between Alpha and Beta!
So you switch to Beta. But now you are offered a new choice, in this variant. You are asked if you would like to switch back to Alpha.
Suppose we label the amount in Beta as 1 “New Currency” unit, 1N. Envelope Alpha contains either .5N or 2N. You don’t know anything about whether Beta is the big-money envelope or not, and there are just two possibilities, so you must use a 50% probability that Alpha has .5N and a 50% probability it has 2N. But since
E(Alpha) = .50(1N) + .50(2N) = .25N + N = 1.25C,
you should definitely switch back to Alpha But earlier we saw it was obvious you should be indifferent between Alpha and Beta, and then we saw it was clear you should definitely prefer Beta! And of course once you switch back to Alpha, you should want to reswitch to Beta again, and so forth forever.
What we must do to resolve the paradox
Wikipedia says there is no satisfactory solution:
There have been many solutions proposed, and commonly one writer proposes a solution to the problem as stated, after which another writer shows that altering the problem slightly revives the paradox. Such sequences of discussions have produced a family of closely related formulations of the problem, resulting in voluminous literature on the subject.[2]
No proposed solution is widely accepted as definitive;[3] despite this, it is common for authors to claim that the solution to the problem is easy, even elementary.[4] However, when investigating these elementary solutions they often differ from one author to the next.
I think I have a solution, and it seems different from the many proposed in the Wikipedia article, though I have only skimmed them to see if theirs is the same as mine or just as good. It may be that the “Proposed resolutions through mathematical economics” is the same as mine, and it is definitely similar. I at least can hope to explain it more simply. I thought about this for a couple of weeks before I figured it out, though, and though it seems obvious to me ex post, it was not at all obvious ex ante.
The original intuition that since we know of nothing to distinguish the two envelopes we shouldn’t switch is correct for the closed-envelope variant (for the open-envelope variant we *do* learn something that might be relevant: that envelope Alpha has $100). The resolution to the paradox comes from explaining why the expected value calculation is incorrect. More specifically, we’ll see that using 50%-50% is incorrect, and that sometimes you should switch from Alpha to Beta and sometimes not, depending on the background implicit assumptions you make.
Resolving the Paradox: The Simplest Cases
I will build up a general resolution to the paradox starting from the simplest case and moving to the most general. In all cases, we must start by specifying what are the possible values for the amount in the big-money envelope and how probable each of those values is. In the simplest case, the only possible value is $100. In the second-simplest case, it is $100 or $200. In the third-simplest case, all values between $0 and $100 are equally likely. In the fourth-simplest case, the value is between $0 and $100, but not all values are equally likely. In the fifth-simplest case, the value is $0 or more, and we don’t know the probabilities, but we do know that E(Alpha) is finite. In the 6th-simplest case, which is the most difficult, the value is $0 or more and E(Alpha) might be infinite.
To Be Continued
At the point, I need to go to church. So I will publish this unfinished, as an exercise to the reader. Later I will publish a continuation.
If you liked this Substack, you might like “The Two-Envelope Paradox Resolved (Part II)” and "I Am the Very Model of a Modern Major-General," Part I”.