Comments on: A Post about Envelopes

By: Brad

Brad — Thu, 23 Apr 2009 00:36:26 +0000

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.

By: Brady Kj

Brady Kj — Sat, 18 Apr 2009 16:51:04 +0000

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.

By: macsnafu

macsnafu — Wed, 08 Apr 2009 20:41:49 +0000

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.

By: Greg

Greg — Wed, 08 Apr 2009 19:05:19 +0000

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.

By: GMS

GMS — Wed, 08 Apr 2009 18:35:41 +0000

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.

By: macsnafu

macsnafu — Wed, 08 Apr 2009 12:21:34 +0000

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!

By: Greg

Greg — Wed, 08 Apr 2009 07:18:15 +0000

@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.)

By: JT

JT — Tue, 07 Apr 2009 20:00:06 +0000

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!

By: pulsifer

pulsifer — Tue, 07 Apr 2009 15:39:32 +0000

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?

By: macsnafu

macsnafu — Tue, 07 Apr 2009 12:56:19 +0000

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.