A Post about Envelopes
I promised a few weeks ago that I’d make another post about “seemingly paradoxical matters”, and I’m finally making good on that promise.1 Be forewarned, however, that this post doesn’t really have anything to do with anything else. It’s just a cool paradox (or “puzzle” or “problem” or whatever other ‘p’-word you want to call it) I learned from my roommate.
The paradox hinges on the notion of expected value, so first we’d better understand what that is. Let’s say I flip a coin, and if it lands heads I give you a dollar. But if it lands tails, you have to give me a dollar. Then your expected value from this game is $0 overall—in any given round you’ll either gain or lose a dollar, but in the long run you’ll break even. Your average expected gain from any single round is $0.
Let’s change the numbers a bit. Now I give you $2 if the coin lands heads, and you give me $1 if it’s tails. Assuming I’m not using a weighted coin, this sounds like a pretty sweet deal for you, right? In any given round you have an equal chance of winning $2 or losing $1. Over many rounds you should average a net gain of 50 cents per round (.5 * 2 + .5 * -1). So we say that the expected value of flipping the coin is $.50.
In general, you can calculate the expected value of any random variable by multiplying the probability of some outcome by the value of that outcome, then summing these over all possible outcomes. For example, the expected value of rolling a die would be (1/6) * (1 + 2 + 3 + 4 + 5 + 6) = 3.5, since every value from 1 to 6 has an equal probability of coming up.
Now that we have this idea of expected value, we can also define what it means to play a game rationally. Quite simply, a rational player is someone who always makes the choice that maximizes their expected value from the game.
Alright. With those preliminaries out of the way, let’s get to the good stuff. Here’s the game: I offer you two envelopes, both of which contain checks written out to you. One of them is worth twice as much as the other. With no other information, you pick one of the envelopes and get to keep whatever’s inside. Not a bad game for you, eh?
So let’s say you’ve selected one of the envelopes. But before you can open it, I interrupt you:
“Hold up! If you’re having regrets, I’ll let you switch to the other envelope, no strings attached.”
“Well that’s kind of a dumb offer, Greg. If I had really wanted the other envelope, I would have chosen it in the first place. What do you take me for? Some kind of flippity-floppity flip-flopper?”
“Of course not! I’d never dream of insinuating such a thing. But here’s the deal: I just so happen to know that the envelope in your hand contains a check for $10. (I know this because I put the check in there myself.) Which means that this other envelope—the one I’m holding right here—has either $5 or $20 in it. It’s a 50-50 chance either way, right?”
“I… guess? Sure.”
“That means the expected value of my envelope is .5 * $5 + .5 * $20 = $12.50. As a rational individual, you should obviously switch envelopes, since the expected value from switching is higher than what you’ve got now.”
“But Greg, that is stupid. If I had chosen that other envelope in the first place, you could have gone through the exact same song and dance, convincing me to switch to this one.”
“That’s true.”
“So why would I switch?”
“Because your expected value is higher, and you’re rational.”
“But I don’t want to switch.”
“But you should.”
“But I like this envelope.”
“So are you some kind of irrational crazy-person, then? Is that what you are?”
“You know what? Screw this.”
And then you punch me in the face and snatch the other envelope from my hand. As you run away, you open the other envelope to find a $5 check inside.
“That sneaky snake! That snakey sneak! I knew he was full of baloney.”
But…. was I? Was there really anything wrong with my expected value argument?
1 You thought I forgot, didn’t you? I didn’t forget! I never forget.
This is a variation on the Monty Hall problem. http://en.wikipedia.org/wiki/Monty_Hall_problem
If you make the assumption that there really was an equal chance of the envelope containing $20 or $5 then the correct solution is to switch. If however we don’t have any assurance of that the math is useless. It really works best if you use three envelopes (see the Monty Hall problem).
When I first heard the problem, I also thought it sounded eerily familiar to the Monty Hall problem. But it’s actually different! The puzzle I’ve presented here is known as the Two Envelopes Problem.
Here’s the difference, as I understand it: in the Monty Hall problem, when the host reveals the goat, he is revealing information about what’s behind the unopened, unselected door. By switching, the player takes advantage of the fact that the host could not reveal a car behind one of the unselected doors.
In the envelope problem, I’m not actually telling you anything about what’s in the other envelope. It’s true that I give you a dollar amount, but that’s arbitrary—I could have said $X instead of $10 and the reasoning would have been exactly the same.
Well, as I see the problem, you need to determine the expected value of your first choice… Which, if I’m not mistaking, is:
$X * 0.5 + 2*$X * 0.5 = 1.5 * $X.
Then, the expected value for switching envelopes is:
$X/2 * 0.5 + 2*$X * 0.5 = 1.25 * $X.
So switching envelopes has a smaller expected value than your first choice.
A less mathematical approach is: my first choice was between winning some money and winning some more money; switching envelopes is a choice between loosing some money and winning some money… So I’ll just stick with the first choice.
I’ll stick to Daniel Answer. I prefer to choose between winning less or more, than between winning some or loosing some…
…but I’m a humanist, I’m crap with the game theory…
@daniel: In your first equation, it looks like X represents the amount of money in the less-valuable envelope, correct? But in your second equation, X represents the amount of money in the envelope you’re holding, which could be either the smaller amount or the larger amount. That is to say, I think the analysis is flawed because X is being used inconsistently.
I’ve thought about the problem a bit, and I think I’ve come up with a variant that helps explain what’s going on here. Let’s change the rules of the game so that one envelope has 100 times as much money in the other. This time you choose an envelope and I tell you that it has $100 in it. Once again I give you the opportunity to switch.
In this case, the other envelope has either $1 or $10,000; now your expected value from switching is just over $5000. Do you switch this time?
While most people tend to be unconvinced by the original statement of the problem, I think this higher-stakes scenario appeals better to people’s intuition. What do you think?
I guess I don’t get this “expected value” stuff. The simple point is that there’s only two envelopes, and there’s a 50-50 chance of getting the one with the the greater amount in it. Without any further information, there’s NO WAY of knowing which one has the greater amount, so why should it be rational to change your mind after selecting one of them? It’s a guess either way.
well it’s exponential isnt it?
so in a mathematically perfect situation, yeah you would swith, because you stand to gain more than to loose. you can always get higher, but the returns (plotted out as a sort of chain of doubling) will be asymptotic to zero, so all in all you’re more likely to gain than louse. you could think of it as a sort of one dimensional version of the drunkard-and-the-lamppost problem.
in a real situation, there are upper limits. if you choose an envelope, and i tell you it has 10 trillion dollars in it (US), then you can be pretty sure that the other envelope does NOT have 20 trillion in it. so you wouldn’t swith. in generaly, you need to make a judgment call. how much money do you thing the host is willing to part with?
That Wikipedia article you linked to has a good explanation of the problem. But I think it gave away the answer to the paradox pretty early in its analysis — one should be a frequentist not a subjectivist!
@macsnafu: Does my 100x variant sway your intuition at all? I wasn’t really convinced by the logic until I looked at it from that angle. I’d also recommend checking out the Wikipedia article if my explication doesn’t do it for you!
@pulsifer: Unless, of course, the host is the U.S. government, in which case a $20 trillion handout isn’t outside the realm of possibility. WOAAAAAH TOPICAL
@JT: Oh man, frequentism vs. subjectivism is a whole ‘nother can of worms. (For what it’s worth, I tend to lean towards subjectivism myself.)
Okay, I see what I missed. You’re told how much is in the selected envelope, and thus know the two possibilities of the other envelope. The larger amount is much larger than the current envelope, while the smaller amount isn’t as great a difference. Therefore, knowing the amount in the current envelope, and what the other envelope might contain, yes, it’s rational to switch–unless you really need that bus fare to get home!
You are actually missing an important fact here which effects the decisions. The person making the offer knows what’s in the envelopes. Thus, they can make up whatever “high” value to make sure that the expected value of switching is better. Thus, there is not a 50-50 chance of the second envelope containing $100 (or $10,000). There is a 0 percent chance. The offer can only result in loss if the second envelope is worth more with no possibility of benefit), thus if the second envelope was worth more, a rational offeror would not make the offer, and thus the rational guessor would not take it if it was made.
That’s true, and it’s an important effect if the game is played exactly as discussed. The player need to be assured that the host is being truthful. A more realistic scenario might be on a game show of some kind (a la Deal or No Deal), where there is a real possibility of winning $10,000.
Another hidden assumption here is that the rational player has a linear utility function for money—that is, every dollar won contributes equally to the player’s happiness. This is hardly the case for real people: the difference between winning $100 and $10,100 is much more significant than the difference between winning $1,000,000 and $1,010,000. People are risk averse, and so at some point a player who knows they have won a significant prize will not want to risk losing it, even if the expected value from switching is greater.
To counteract this effect, we would have to change the prize from dollars to abstract units of happiness (in economic parlance, utils). But this just makes the problem more abstract and difficult to reason about intuitively.
GMS, I thought this was a rational puzzle, not a psychological puzzle. In real life, hardly anybody is going to offer you the choice between two envelopes containing money, except for a game show, prank, or a very eccentric person.
Well, what if you really really need at least $10? What if there’s somebody out to kill you because you owe them $10? What if you have a fatal brain tumor and need $10 to pay for the surgery? Then you wouldn’t want to risk losing that all for a 50% chance of having $20.
Lets consider this a different way. Suppose that instead of telling you that YOUR envelope contains $10, I tell you that ONE of the envelopes contains $10. Now there are 3 outcomes for each envelope. $5 (25%), $10 (50%), $20 (25%). The expected value of each envelope is $11.25. If you then turn around and tell me that MY envelope contains $10, you have lowered the expected value of my envelope (to $10), and raised the expected value of the other one (to $12.50) because the total expected value of the envelopes (.5*15 + .5*30 = $22.50) hasn’t changed by me revealing which one is which. Thus, it is in your interest to switch if you tell me the value of my envelope, because the uncertainty is actually beneficial in this case.
Note that if we change the problem where one envelope has as fixed amount (say $10) more than the other, then the expected value of both envelopes is the same as the value you tell me, so I am ambivalent about switching.