Our paradox resolved!

• Mike Caro   Exit

This article first appeared in Card Player magazine.

Last issue I posed a paradox, or what seemed to be one. Actually, it wasn’t I who posed it, rather it was Mike Morell who wrote a thoughtful letter, making readers ponder.

Publishing is a funny business. You write one column and – even if you wait till the last minute – you need to write another column just after the last issue of Card Player hits the stands. That means I might be writing this before you have had the chance to ponder the paradox. Despite this, I’ve already received a very heavy sampling of responses via e-mail, fax, and in person. If you don’t know what I’m talking about, let me summarize the paradox.

Johnny Moss’ envelopes revisited. Morell imagined a situation in which the ghost of Johnny Moss appeared and handed you two sealed envelopes. Inside each is a check made out to you. Because we’re dealing with checks, not cash, it’s impossible to tell anything about the amount of money by the thickness or weight of the envelope. And, of course, there are no other tricks you can use that would spoil the purity of this paradox. For instance, we can’t deduce anything from checks ending in odd pennies which can’t be halved.

Paradox? Where’s the paradox? I’m just getting to it. Johnny says you can choose either envelope, open it, and decide whether to keep it or trade for the other envelope. He gives you some further information, and this extra information seems very important to you, and you are grateful.

He explains that one envelope contains a check for exactly twice the amount of the first envelope. you open an envelope and see that the check is written for $100. Should you switch?

Half of your mind says, no, it doesn’t matter because you had an even chance of choosing the better envelope to begin with, and switching is a waste of energy. (And this is what Morell believed should be the answer, despite interesting arguments to switch.) But the second half of your mind screams, wait just a minute! This check is for $100, so since one check is twice the other, the other must be either $50 or $200. So, by switching, I can lose $50 or gain $100. Obviously I should switch.

Then the third half of your brain argues… What? You’re right, we’re talking thirds, not halves. Anyway, a portion of your brain thinks there’s something wrong here. How can it possibly be profitable to switch. If it is, you should always switch. And if you should always switch, then you might as well just always choose the other envelope to begin with.

The solution. Well, Morell was right. Switching makes no sense. It’s a waste of your time and effort. I’m going to give you two explanations. One is slightly sophisticated, the other is simple and has powerful poker applications.

Sophisticated comes first. Let’s suppose Johnny Moss asked me to write the checks for him, and then present the envelopes to you. The first thing I’d ask is, “How much should I write these checks for?”

Well, nobody’s bankroll is unlimited. Besides, Johnny decides that a range of $50 to $800 is about right for this experiment. No need to give away big money, just to prove a point.

Now it’s up to me to write the checks. I elect to start with the $50 minimum and double the sum to make a companion envelope for each pair. So, I prepare the following sets of envelopes: $50/$100, $100/$200, $200/$400, and $400/$800. Having started with my instructed minimum and reached my maximum, I stop writing checks. (Obviously, I could write other pairs of checks, such as $75/$150, but that would just complicate things without changing the explanation.)

When I’m finished, I’ve created four pairs of envelopes. Now I randomly select one pair. I’m not going to make this difficult by breaking it down into what percent of the time you’ll get which envelope and what you gain or lose by switching, but if you’re so inclined, you should be able to map this out for yourself in a few minutes.

What I’m going to tell you is this: If you choose a $100, $200, or $400 envelope, you always gain by switching, because half the time you’ll lose half the amount, and half the time you’ll double. In the case of a $200 envelope, this means half the time you’ll lose $100 (ending up with $100) and half the time you’ll win $200 (ending up with $400). Since each is equally likely, you average a $100 gain for every two times you switch, so trading envelopes is worth $50.

This is the crux of the paradox. It previously seemed as if you should always trade. But now that we know the secret size limits of the checks, we can look at it differently. Now we see that there are two exceptions to your lose-half-or-double expectation. If you open a $50 envelope, there’s only one thing that can happen by switching: You gain $50. And if you open an $800 envelope, there’s only one thing that can happen by switching: You lose $400.

So, by switching in both those “extreme” cases, you lose $350. And, wouldn’t you know it, that exactly balances out all the gains from all the other choices. So if your strategy is to always switch, you gain nothing. If your strategy is to never switch, you lose nothing. Since you don’t know what range Johnny Moss decided on for the checks (but that obviously he had to use some range), it does you no good (or no harm) to switch. If you had information letting you know when you were at the high or low end, then you could beat the system by always switching when low, never switching when high. But you don’t have this information.

By the way, this explanation holds true no matter how long the sequence of choices you devise, how small the minimum, or how large the maximum.

A simpler explanation. In poker, you should always take your opponent’s point of view into account before you react. Let’s say someone has opened the pot in an early position. The important thing now is what, from his point of view, is required to open. You must relate to that and make your decisions accordingly. If someone bets, you’ve got to put yourself in his mind and guess what he would require to bet. How often would he bluff? Well, to determine this, you need to know the pot odds he faced when he bet, not the pot odds now if you call.

When you think about switching envelopes, don’t try to figure out all the mathematical implications. Just ask yourself if your opponent wants you to switch. Imagine you offered the envelopes. You shuffled these until you couldn’t remember which was which. At that point, would you care which envelope was opened? Of course not! That’s the point; look no further.