The Two-Envelope Paradox

The Paradox

two_envelopes

You just finished some yard work for your eccentric neighbor, Mr. Grimes. He prepares some lemonade for you while you rest at his kitchen table.

“Thank you for your help today,” he says as he places two envelopes on the table in front of you. “Both envelopes contain money. Pick one and you can keep the money inside.” As you reach for one of the envelopes he adds, “One envelope has exactly twice the amount of money as the other.”

You pick one of the envelopes and open it. Inside you find a $20 bill. You are about to pocket the money when Mr. Grimes interjects, “I’ll give you a chance to switch envelopes if you’d like.”

Should you switch envelopes?

Since one envelope contains twice as much as the other, this means that the other envelope has either $10 or $40. If you switch you will either lose $10 or gain $20, so it makes sense to switch. However, if you had originally chosen the other envelope, the same reasoning would lead you to switch to the envelope you’re holding now. Hence the paradox.

What’s going on?

Spoiler Alert

If you want to chew on this puzzle for a while, then don’t read any further.

This is a famous paradox and it’s one of my favorites. You can find quite a bit of discussion about the two-envelope paradox on the web. Most solutions resolve this paradox with statistical equations, but I’d like to take a different approach that will hopefully be more intuitively satisfying.

Useless Information

You’ve been a victim of useless information. Let’s say that instead of saying, “One envelope has exactly twice the amount of money as the other,” Mr. Grimes said, “One envelope has money from 2006 and the other has money from 1995.” Should you switch envelopes? Obviously, it doesn’t matter! The date the money was minted is irrelevant. It also happens that the fact that one envelope has twice as much money as the other is irrelevant, too. It’s useless information masquerading as useful information.

20_Gold_Certificate

When you first opened the envelope and saw the $20, you still had no idea what was in the other envelope. It could have been a penny or a check for a million dollars. When Mr. Grimes tipped you off that one envelope was worth twice the other, all of a sudden the practically infinite possibilities shrunk down to only two: $10 and $40. Because you suddenly got more information about what’s in the other envelope, you intuitively felt like you made progress in solving the problem.

However, as far as your decision to switch is concerned, the only useful information is information that helps you determine whether the other envelope has more or less than your current envelope. The exact amount of the other envelope doesn’t matter. If it’s more, you switch; otherwise, you don’t. By narrowing down the possibilities to $10 and $40, you still don’t know whether the other envelope has more or less so you are no closer to the solution than you were before.

Rolling an Eight

Perhaps that explanation didn’t satisfy your intuition enough. After all, if by switching you stand to either lose $10 or gain $20, doesn’t that favor switching? If you went into a casino and they had a game that gave you an equal chance of losing $10 or gaining $20, wouldn’t you play all day?

Even though you know that the other envelope contains either $10 or $40, it is, in fact, incorrect to think that both outcomes are equally likely. Consider the following twist:

dice

You are playing a dice game with your friend Kyle. Mary prepares the game for you by rolling a die, writing down the number, and putting it in an envelope. Then she does the same for a second envelope. She gives one envelope to you and the other to Kyle. You both open your envelopes; yours contains the number 4. The objective of the game is to try to guess the other person’s number. You are about to guess Kyle’s number when Mary says, “One number is exactly twice the other.” You smile because you now know Kyle’s number.

Before Mary gave you a clue, the set of possible outcomes for Kyle’s envelope was {1, 2, 3, 4, 5, 6}. Applying Mary’s clue to the number 4 gave you {2, 8} as the set of possible values, but since you know that dice are six-sided, 2 was the only real choice.

So it’s important to distinguish between actually possible outcomes (a six-sided die can only physically land on one of six numbers) versus potentially possible outcomes. We consider 2 and 8 as potentially possible outcomes because they fit the “exactly twice the other” rule, but then we have to further restrict these outcomes to the set of actually possible outcomes.

Now let’s return to you, Mr. Grimes, and the $20 bill. At first it seems like the other envelope has two actually possible outcomes: {$10, $40}. However, this is incorrect. You are in one of two scenarios. Scenario A has two envelopes, $10 and $20, and you’ve chosen the $20. Scenario B has two envelopes, $20 and $40, and you’ve chosen the $20. In either scenario there are only two actually possible outcomes, because there are only ever two amounts of money. If you have an envelope with $20 and are thinking that the other envelope contains either $10 or $40, then you are dealing with three amounts of money. Since there are only two actual amounts of money, only one of {$10, $40} is an actually possible outcome. The other is just a potentially possible outcome. And since you don’t know which is which, you don’t have any reason to switch.

The Fictional Third Envelope

alice20a

Our intuition leads us astray because it invents a fictional third envelope.

Let’s say that after you opened the envelope and took out the $20, Mr. Grimes produced a third envelope and said, “One of these unopened envelopes contains $10 and the other contains $40. I’ll give you a chance to pick one of the unopened envelopes instead of keeping your $20.” In that case, it really does make sense to try your luck at another envelope. But this only works because there are three envelopes. The paradox relies on us turning the two-envelope situation into this fictional three-envelope situation.

Because the third envelope is completely fictional, it turns out we can make it contain whatever value we want. Let’s say Mr. Grimes is a stingy man and wants to use the two-envelope paradox to his advantage by forcing you to choose the lesser envelope. He places $3 in one envelope and $27 in the other.

“Thank you for your help today,” he says as he places two envelopes on the table in front of you. “Both envelopes contain money. Pick one and you can keep the money inside.”

You choose an envelope and open it to find three bucks inside. “Better luck next time,” smirks Mr. Grimes as he sends you on your way.

The next time you go to work for Mr. Grimes (because you have anterograde amnesia) he offers you the same deal. This time you choose the $27 envelope. Before you pocket the money, Mr. Grimes interjects, “I’ll give you a chance to switch envelopes if you’d like. One envelope contains exactly nine times the amount of the other.” You reason that the other envelope has either $3 or $243 and so the prospect of gaining more than $200 causes you to readily switch.

Mr. Grimes could have said instead, “One envelope is the cube root of the other.” In that case you would have reasoned that the other envelope contained either $3 or $19,683, which makes switching seem like a no-brainer.

Mr. Grimes could also have said, “If you take the amount of one envelope, subtract $1.07, raise that amount to the 5th power, then add $0.22, you’ll have the value of the other envelope.” From that you reason that the other envelope contains either $3 or $11,722,293.53, and you’ll switch just at the thought of becoming an instant millionaire.

Summary

So the paradox works because Mr. Grimes tells you useless information about the relationship between the two amounts and you use this useless information to construct a fictional third envelope that you treat as real.

I hope you had as much fun with this paradox as I did! :)

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46 Responses to “The Two-Envelope Paradox”

  1. Zack says:

    I don’t feel that this is explained well enough:

    “Since there are only two actual amounts of money, only one of {$10, $40} is an actually possible outcome. The other is just a potentially possible outcome. And since you don’t know which is which, you don’t have any reason to switch.”

    We don’t know which one is possible, but we do know that ONE OF THEM is possible. And as you said numerous times, we stand to either lose $10.00 or gain $20.00. Everything else is irrelevant, no? Why should we not switch?

  2. Colleen says:

    Mr. Grimes is an asshole

  3. techiferous says:

    Zack, I agree, I didn’t explain that part well enough. There’s an interesting discussion of this paradox on Hacker News. Perhaps hearing it explained differently will help! :)

    http://news.ycombinator.com/item?id=1582219

  4. zach says:

    I don’t think this is a paradox. You don’t understand them

  5. techiferous says:

    Zach, this is a well-known riddle and I didn’t come up with the name for it. Here’s a brief history of it: http://en.wikipedia.org/wiki/Two_envelopes_problem#History_of_the_paradox

  6. Zoel says:

    This is definitely not a paradox, this is a game of chance, a gamble. Same as a game of “heads or tails”. you have all the relevant information, and “one envelope has double the other” is important, it tells you you have twice as much to gain then to loose…now the real question is, did he tell you that so you would take that chance?

  7. Jonas says:

    I fear you, the author, have a fundamental misunderstanding of this two-envelope paradox. I researched it briefly after reading your article and am bewildered to say the least. My advice to you: don’t over analyze.

  8. The Kid says:

    That bastard Mister Grimes played me like a fool, alluring me with the CHANCE at more money as I had decidedly reached for one envelope already. So I picked the other one. Fucker.

  9. Charlie says:

    I’m afraid you are all making the wrong decision.

    “You pick one of the envelopes and open it. Inside you find a $20 bill. You are about to pocket the money when Mr. Grimes interjects, “I’ll give you a chance to switch envelopes if you’d like.””

    If you are about to pocket the money you have taken the money out of the envelope. At this point it would be in your best interest to switch the envelopes because there is nothing left in the one you are holding.

  10. S says:

    I think the point is that if you think switching once is helpful than switching again should be helpful as well.
    It could be infinite, like the gag in spy movies “If he knows we know he knows that we know that he knows…” Following that line you could never give a “final answer” you would just keep switching because the envelopes are actually identical. Reasoning that applies to one applies to the other as well.

  11. Shaun says:

    I like Charlie’s answer.

  12. leftfoot says:

    If you want to follow Schrodinger’s logic, the envelope contains both $10 and $40, since you don’t know the correct amount until you open the 2nd envelope. (the cat is simultaneously alive and dead)

    However, this isn’t a riddle. Riddles have a definite answer. Curiosity (and greed) is the only is the only thing that makes this “problem” interesting. You can either settle with what you have, which is more than you began or you can risk it for something else. Kind of like “Let’s make a deal”. Where’s the statistical analysis on that as it’s essentially the same problem. Sure, you can even break blackjack into odds, but that’s based on specific numbers of cards, shuffling accuracy, number of players and so on. Here, you don’t know the motivation of Grimes. Is he a generous man? Is he a chronic liar (because it’s statistically possible he’s just screwing with you and there’s nothing in the envelope).

    Technically it’s a paradox because there is no answer. It cannot be solved, just given probabilities for each outcome. Even then, statistics aren’t answers.

  13. Havenspire says:

    It’s not a paradox: in the end, you still gain more than you started with, which was $0.00

    In this particular case, by averages of human nature, you’re going to lose:

    Let’s say the envelopes had $10 and $20… you pick up the $20, and Mr Grimes will make the offer to switch envelopes.

    In the other scenario, the envelopes contain $20 and $40… you pick up the $20. Mr Grimes knows the amounts in each envelope, so he doesn’t even offer the chance to switch.

    In other words, why would he interject in a way that would potentially cause him to lose an additional $20? It’s illogical. The point of there being a lesser or greater value in this situation would be to take advantage of it since you have the upper hand of prior knowledge. If you picked the 20, and the other envelope contained 40, it would make no sense for Mr Grimes to interject since you’ve already selected the lesser amount and therefore Mr Grimes has already won the ‘gamble’.

    Thus, the simple conclusion is that if you are ever offered to switch envelopes, you already have the greater amount. The only way this would differ is on a gameshow where the motives are different (they make money from sponsors and therefore having someone win every once in a while will improve ratings).

  14. techiferous says:

    You’re right Havenspire.

    This puzzle, like most mathematical and logical puzzles, is a bit contrived. It’s really a mathematical puzzle that’s expressed in the form of a story to make it more interesting. The point is the mathematical nature of the situation, so you really need to ignore other factors like people’s motivations. It’s kind of like when a physics professor asks you to ignore friction. :)

  15. Saggingpeach says:

    It’s only a paradox if you don’t get to see what’s in the envelope before you are given a chance to switch, (like “s” said).
    In your examples it is not a paradox, but a question of “are you willing to risk losing half this amount for the chance of receiving double this amount?”
    Mr. Grimes probably just has a gambling addiction.

  16. steve says:

    this isn’t a paradox; this is the person giving you information being an asshole. if someone has 2 envelopes, gives you one, and says “one of the envelopes has twice the amount of money than the other one,” you’re either holding the one that is half of the other or double the other. there’s no reason to bring a third envelope into it. if he says “one envelope is worth 20 million times more than the other” and he means it, if you’re holding 20 dollars, you’d switch. this isn’t a paradox, this is an asshole trying to scam you. you’re stupid.

  17. steve says:

    let me explain it further; if a robot randomly fills two envelopes with money, and somehow calculates how much is in each envelope, and you pick one, and it offers you to switch (regardless of what you pick) and says “you can switch if you want. one envelope contains 200 times more than the other,” switching would make sense. say you’re holding 20 dollars, you’re either going to have 4,000 dollars or 20 cents. the gain outweighs the loss.

  18. steve says:

    what about two $10 bills? It would be double the amount of money, just the same value.

  19. JoJo says:

    Your description of the paradox misses a crucial point. In the actual paradox you are not allowed to open the letter you receive. All you know is that it contains $X and the other envelope has 2X or 1/2X. Thus statistically it contains 5/4X (slightly more than X). Thus you swap. Once you have swapped (and again not opened the envelope) you can apply the same logic to swap back. And so on and so fourth infinitely.

    If on the other hand you open the envelopes, as is the case in your description the whole thing becomes moot. i.e. you open envelope 1 and see you have $20, if you then swap and get $40 you would not swap back as you know you will loose. If you swap and get $10 you will swap back as you know you will gain. thus the problem is solved in two turns.

  20. jayman says:

    I think the prospect of doubling your money or halving your money is a no-brainer. The basic means of reasoning “risk vs. reward” makes switching an obvious decision. If he had said “one envelope has $10 more than the other then the above explanation would have been correct.

  21. Birco says:

    JoJo is correct, the only way this is a paradox is if you don’t open the first envelope before the option to swap is offered.

  22. tom says:

    Why do people insist that this is not a paradox based solely on the claim that you stand to lose nothing (having started with $0.00), and only can gain something?

    Money is nothing more then a certificate of proof of work, trading the work you preform (entertainment, manual labor, producing goods, services, etc) for that of another. e.g. the farmer harvests the grain, the baker bakes the bread, the butcher cuts the meat, the dairy farmer makes the cheese, and together they can make a sandwich.. Same principle of bartering applies today, only greatly compounded in the range and complexities of the work being done and it’s value. So yes, you do start with something; your labor and services, which you expended working for this man under the pretense of fair payment.

    Besides the details of the story are unimportant to the overall question, so no need picking it apart in order to come off as superior in knowledge.

  23. Anon says:

    No paradox. Simple probability. Switch.

  24. matt says:

    Logic is not one of your strong points. “Fictional third envelopes” have nothing to do with this “paradox”. If you find $20 in one envelope, there is only one other envelope, not two, and it contains either $10 or $40. Perhaps this confuses you because you do not understand statistics. No matter what, there is a 50-50 chance of either halving or doubling your money. Let’s remove numbers from the situation. You open the envelope to find it contains x dollars. If you choose to switch, you will end up with x/2 dollars or 2x dollars. Now, why you think this is paradoxical is unclear. No matter what, there is a 50-50 chance. The ratios of possible gains or losses are irrelevant. So, do you want guaranteed money, or equal chances of either doubling or halving it. Actual numbers are irrelevant. Which envelope you have picked is also irrelevant, since you cannot know whether you have the greater or lesser of the two values. So this should not be considered in your evaluation. No matter what, you have a 50-50 chance, so there is no paradox.

  25. techiferous says:

    Hi Matt,

    Here’s why the two envelopes problem is referred to as a paradox. Let’s say you have two envelopes labeled A and B. You pick A. You learn that one of the envelopes has twice as much money as the other. Based on your assertion that “there is a 50-50 chance of either halving or doubling your money”, it would make sense to switch to envelope B. So you switch to envelope B.

    However, what if you had initially chosen envelope B? By the same reasoning you should switch to envelope A. In other words, you’ve come to the conclusion that A is the better envelope and also that B is the better envelope. These are contradictory statements, hence the paradox.

    Part of what makes this problem seem so paradoxical is that people assume that the two outcomes of halving or doubling your money are equally likely to occur. Just because there are two outcomes to a situation doesn’t mean that the probability that either will happen is equal. For example, a lottery ticket doesn’t have an equal chance of winning and losing. So you can’t just say that there is a 50-50 chance, you have to demonstrate why.

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