A Paradox From A Poker-Playing Reader
This article first appeared in Card Player magazine.
Sometimes I open mail that’s so good I need to share it with you, even if it isn’t specifically related to poker. This column will stretch your mind and force you to ponder.
After you read today’s letter from Mike Morell, try to explain the seeming paradox presented. Should you switch envelopes or shouldn’t you? Think about it for two weeks, and next time I’ll respond with an analysis and an answer.
While choosing the better strategy discussed in Morell’s letter won’t directly help your poker game, this exercise might help you win indirectly by building your mental muscles.
Here’s the letter. Dear Mike, I’ve been confronted with a problem which I have not been able to solve. I’ve posed it to some of my brightest colleagues, both during my studies at Harvard University and at my workplace on Wall Street. It has generated hours of discussion, leading recently to lower productivity here at the office. No one has been able to give a satisfactory answer to the problem.
So, I said to myself: "Self, you’re a pretty bright guy – particularly when it comes to probability theory. Yet you can’t solve this problem. Neither can some of your brightest friends. You’ll never rest until you find the answer. The only logical course of action is to ask the man with all the answers — the undisputed leader in cutting-edge thought in matters of probability."
Who is that man? America’s Mad Genius — Mike Caro, of course. [You can see why I like this letter, right? - MC] So, Mike, here’s the problem:
Johnny Moss’ envelopes. As you’re strolling through the poker room at Binion’s late one night, the ghost of Johnny Moss comes up to you with two sealed envelopes. He informs you that inside of each envelope is a check, payable to you. [We have to stretch our imagination here to assume the legendary Johnny Moss is offering checks and not cash. - MC] The amount of each check is a positive real number. The amount of one check is exactly twice the amount of the other check. You may choose either envelope. You are permitted to open it, and then decide whether to keep that check, or switch envelopes and take the other check. Since dead gamblers don’t lie [Morell adds a footnote explaining that this phrase comes from the book Caro on Gambling. - MC], you assume that everything Mr. Moss has told you is true. You think of two possible strategies to maximize your expected profit from this situation.
Strategy A: Pick an envelope and open it. Take the money, and proceed immediately to the nearest poker table. Don’t waste time switching.
Strategy B: Pick an envelope and open it. Regardless of the contents, switch, and take the money in the other envelope. Then play poker.
Pick a strategy. Which strategy do you choose? Strategy A seems intuitively correct – there can be no gain from switching, since you picked the first envelope at random. Yet there is a powerful argument for Strategy B. Define the amount of money you find in the first envelope you choose as x. Since there’s a 50% chance that your first choice was the envelope with the higher amount, there’s a 50% chance that you will lose [half] by switching. Similarly, there’s a 50% chance that your first choice was the envelope with the lower amount, implying a gain of x if you switch [you'll double].
[I have omitted Morell's carefully presented formulae defining the mathematical expectations of the two choices, because they go beyond what most readers want to wrestle with, and we can ponder this seeming paradox without them. - MC]
My sincere belief is that Strategy A is correct – it is a waste of time to switch envelopes. The fact that Strategy B recommends switching no matter what x is makes it extremely suspicious. Without ever opening the first envelope, I should switch, and then switch back (based on the same argument, defining the amount in the other envelope as y), and keep switching infinitely. Nevertheless, I can’t find a flaw in the logic of Strategy B. Even more disturbing are the implications of Strategy A. [Morell's equation and related comments are again omitted; they are mathematically interesting, but we don't need them to proceed. - MC]
You can assign values to the variables [representing money in the envelopes] and not make the problem any easier. Suppose you find $50 in the first envelope. The expected value of the other envelope becomes: 50% x ($25) + 50% x ($1OO) = $62.50 leading to an expected gain of $12.50 from switching. Note this is not analogous to the "Monty Hall" problem, where the argument for switching is correct, and not too difficult to discover. This problem is sinister. It plays tricks on the mind. I had to turn to someone as clear-headed as yourself for guidance.
Here are some thoughts I’ve had in trying to find the solution:
(1) Perhaps I can’t take an expectation of the other envelope’s value because it isn’t really a random variable – it’s a parameter.
(2) Statistically speaking, the amount in the first envelope should be an unbiased estimate of the amount in the second envelope. That suggests no gain from switching.
(3) I’ve assigned the 50% probabilities to having the higher and lower values arbitrarily, with insufficient information. This is really a Bayesian problem, and since Moss didn’t tell me the prior distribution on the amounts in the envelopes, I can’t solve it.
I would be very interested in your thoughts on this problem. Perhaps the readers of Card Player would be, as well. Particularly, how does this relate to your maxim that "in the beginning, all bets were even money"? What’s the bet here? Should each envelope be assumed to have an equal amount, leading to Strategy A? Or should all possible pairs of legitimate amounts (e.g. $25 & $50, $50 & $1OO) be considered equally likely, leading to Strategy B?
I’m not sure if this problem has any direct application to poker, but it’s interesting nonetheless. Please let me know if you have a solution to offer. Thank you for your time. Warmest personal regards, Mike Morell.
It’s me again. I thank Mike Morell for a provocative letter, and apologize for omitting portions that enhanced his excellent analysis of the strategic choices.
However, the question is easy to summarize: There are two envelopes. One contains twice as much as the other. So, if the envelope you choose contains $100, the other will contain either $50 or $200. Should you switch? In my next column, I’ll explain the correct strategy.