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. Meteo says:

    I get it! But I had to do a test it out in a spreadsheet to fully understand…

    My understanding:
    The reason you don’t gain an advantage by always switching is: there is always an upper limit to how much money you can get. Imagine the lowest amounts Mr. Grimes offers is $1/$2 in the two envelopes and the highest combination he uses is $256/$512 with all of the factors of 2 in between (so the amounts of every envelope can be doubled or halved into eachother). You don’t know this information, only the rules of his game. The average expected result for switching is always positive (meaning you should always switch) UNLESS you happen to get the maximum amount. If you switch when holding the unknown maximum $512, the $256 you lose… exactly equals the sum of the average winnings from all of the other scenarios. Therefore the strategy of always switching is no better that always staying in this case. If you think Mr. Grimes would offer more than is in your envelope, only then is switching better. However, you could be wrong and Mr, Grimes could be a dick and do things like only make the offer if you chose the higher amount.

  2. Ben says:

    As opposed to a paradox, i believe this is more a comment on the avariciousness of man. Without the prior knowledge of the second variable, there is an inclination to switch because either way, your prerogative will benefit you given that choice>0 (being the beginning). Therefore switching is the wisest of the choices.

    Now if Mr. Grimes is a douchebag and is out to spite humanity because kids just won’t stop skateboarding on his lawn, the choice for switching will only be presented when you have selected the lesser value. But as I stated above, your earnings will always outweigh the starting value so even if you switch to the lesser value, so what?

  3. Gerhard says:

    To apply probability theory you have to specify the events under consideration and assign probabilities to them. In this case there are exactly two events (x,2x) and (2x,x) each occurring with probability 0.5 assuming that there is no preference whatsoever for choosing one envelope or the other. Now switching or not switching is your decision (not other events!), or, to put it mathematically, you can perform two functions on two events with
    F1((x,2x)=2x and F1((2x,x))=x,
    F2((x,2x))=x and F2((2x,x))=2x.
    Should you perfom F1 or F2? To decide you calculate the expectations
    E(F1)=0.5*2x+0.5*x=1.5*x,
    E(F2)=0.5*x+0.5*2x=1.5*x.
    So considering expectations in a risc neutral world it doesn’t matter what you do. But if there is a small switching fee as it would typically occur in the real world you should stick to your initial choice.

  4. techiferous says:

    Hi Gerhard,

    That’s a great way to model it. I can’t see anything wrong with your explanation and it’s nice and succinct. Thanks for contributing this to the discussion!

  5. Peter Flom says:

    I think this is a different problem…. Nothing to do with probability

    You open the envelope. It has $20. The other envelope has either half or twice $10 or $40. So far, no paradox. He hasn’t told you if they are equally likely, but, if they are, you should switch.

  6. mike says:

    This IS a paradox because the actual outcome is contrary to what most people would expect. To me, this particular paradox isn’t as striking as the Monty Hall paradox in that upon my first reading it, I didn’t think there was any advantage to switching and, as it turns out, there isn’t.

  7. rahul says:

    Good article..after a lot of reading on this paradox, found out Useless information’ is the key to understand why the problem we are facing to switch or not,that shouldn’t be there. The examples you put out on cube and powers of relation between two envelopes helped to think about the actual situation… made tens of cases ,this and that in mind and got what you are trying to say by ‘useless information’

  8. TeQ says:

    Hey dorkmunch, you misunderstood the actual paradox. It only works when you don’t know what’s in either envelope. As soon as you identify an amount, you know what the other outcomes might be and can make an educated guess. Without either value being known it is an actual paradox. Because you changed the actual math you are basically making people stupid for reading your blog post.

  9. techiferous says:

    @TeQ: I sincerely appreciate you calling me dorkmunch; it added some humor to my day.

  10. WackJob says:

    This is mathematically impossible, is it not? You cannot have a probability density function with a sum of infinity, (with a sum of anything other than 1, actually) and that is surely what you get if you deem it 50/50 between the two values. It is effectively saying that any two values of money, up to infinity are equally plausible to be in the envelope. This is obviously not the case.

  11. Ben says:

    This paradox is not resolved.

    If the two envelopes are prepared prior to the one being chosen, and are entirely random, then the two outcomes of halving or doubling your money are equally likely, despite what you claim.

    How else could it not be so? What magic could occur so that you pick one envelope at random and automatically the other envelope is more likely to be something else?

    The paradox rears its head fully when Mr. Grimes gives you 2 envelopes, telling you beforehand one contains twice the other. He also tells you you can switch. You have all the knowledge you’re going to get at the start.

    You pick envelope A… and you know that envelope B has either double or half the money of envelope A. If you switch, you either lose half your money, or double your money. Seems like a good choice, until you realise the exact same thing applies the other way around.

    If you think you have fully solved this paradox, contact the statistical journals, because no one else has. Most effort has been centered on the problems with infinity, and concentrated on bounded solutions.

  12. zach says:

    The paradox doesn’t lie in what the envelopes contain, but in the fact that no matter the envelope you choose, you’re faced with the same dilemma.

  13. Shelbik09 says:

    The 20’s from 2006 are different than the ones in 1995

  14. Nate says:

    This should not be called a paradox. The story itself is not because there is a good risk/reward ratio. There is no third envelope but since you do not know what is in the second envelope it has equal chance of being half your amount or double your amount. You stand to gain twice what you stand to lose. This is almost like schrodinger’s envelope in that the envelope (to you) is equally likely to have either amount of money in it (only if you do not take human behavior into account because a person would only offer you to switch envelopes if you had taken the higher sum). The overall point is that in a purely theoretical sense this is a paradox, but when you attach the story to it it just becomes switching is the better choice. You stand to lose half of what you stand to gain and it is a fifty-fifty chance in your eyes.

  15. jrad says:

    I feel like this is not a paradox to be analyzed statistically but logically. No one is going to pay 40$ for their lawn to be mowed. Some may argue that we do not know the area mowed, but here you will have to further use more contextual clues. Mr. Grimes is an eccentric old man, anyone old and eccentric can be considered a little odd and typical to play mind games such as this one, so in any case of either the envelopes being {20,40$} or {10,20$} we know that the set will be {10,20$} because Mr. Grimes’ eccentric nature he would humor himself on your going through the above statistical evaluations and deliberations of gaining 20$ or losing 10$ dollars, while in the end he would make it so if you switched you would lose 10$ and if you didn’t you would live with the “what if” problem. Some might say we don’t know Mr. Grimes economic background and that he may have the dividends to humor his odd games, but through evidence such as the lemonade he serves we gain some understanding of his socio-political standing. It establishes him as a typical eccentric old man in a middle class standing in society, where he doesn’t have the funds to frequently entertain his eccentric nature. So the other envelope had 10$ in it, and if you switched Mr. Grimes would laugh in your face.

  16. Steven says:

    Game theory suggests that you switch. It’s too long for me to care to spell it out here, but look it up. I take the switch all day!

  17. Trevor says:

    Wow, reading these comments made me realize that people understand probability a lot less than I thought. That actually says a lot because my expectations were extremely low in the first place. Almost everyone on here is saying that you need to switch because you will have a net gain. The problem is, this is assuming that each option is equally likely. You have no reason to think this. Just because there are 2 options does not make them equally likely.

    For example, if I hand you an envelope with 20 bucks in it and give you the option of switching to another envelope that I tell you has either zero or a million dollars in it, why would you assume that they have the same probability of being true? I saw some comments trying to break this down to “ok, if a robot randomly decides to fill the envelope with zero or a million dollars, then you should obviously switch”, but that is an over-simplification because you know that there is a 50% chance of each outcome. There is no reason for you to think that in the first place. If some random person came up to you in the street with an envelope and said to you “this envelope has either nothing or a million dollars in it”, would you pay him 100 bucks for it? If so, then I think I discovered a new get-rich-quick scheme: buy a bunch of empty envelopes and walk around doing that.

  18. Mystic says:

    I had once believed that there was too much thought put into it. After all, either your money will be halved or doubled. Just chose since the odds are 50:50. But then you have to consider variables not mentioned. Can I see through the envelope? Does one envelope appear to have multiple bills in it? Is there loose change in it? The only way for this to be paradoxical is if both envelopes were opaque, held the same number of bills, and you weren’t the kind of person that would take them both anyway. If these were true and one envelope held two $5’s while the other held two $20’s, then one could consider this scenario a paradox. Without those being true, your senses would likely deduce which is the best choice.

  19. Gary says:

    Wow, cotradictory conclusions do NOT make a paradox. You have to be presented with two contradictory FACTS to have a paradox. Paradoxi are actually very very rare in reality. Usually what is represented as a paradox turns out to be two SEEMINGLY contradictory facts. Which is NOT a paradox it just looks like one. Why the word paradox is even mentioned in relation to this puzzle is beyond me.

  20. Gary says:

    par·a·dox
    [-doks]
    – noun 1. statement that seems self-contradictory
    – Related Forms    un·par·a·dox·i·cal- adjective    un·par·a·dox·i·cal·ly- adverb    non·par·a·dox·i·cal·ness- noun
    Origin:
    1530–40; < L paradoxum < Gk parádoxon, n. use of neut. of parádoxos unbelievable, lit., beyond belief. See para-1, orthodox

  21. Colin says:

    I have come to believe that in all “paradoxes” there exists a faulty premise or faulty reasoning. Language fails us in these situations because it is invented by us. Without it, paradoxes wouldn’t exist. If you look at this situation simply without trying to find contradictions, it is quite straight forward. Suppose you run an experiment where you always switch and compare the findings to a run where you never switch. Your average “winnings” will be the same. Most of the time, it is best to solve problems simply because when human invention (such as language) is added, solutions (or the lack of them) are more likely to be faulty and lose all credibility.

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