Two Envelopes

[SteveA]

I read something like this on another site, but I'll change it slightly to make the strangeness more obvious:

Imagine there are two envelopes on a table that contain money. One envelope has 1000 times as much money as the other.

You get to select to look in one first and then can decide to swap it for the unknown contents of the other. Is there a best strategy?

Let's say you open one and find a thousand dollars in it. Should you swap it for the other envelope?

[Meem]

My luck, I would be forced to think there is only a dollar in the other one. The glass is half full or empty ... almost, but this particular glass isn't visible. Why risk emptying it ... so to speak.

Reminds me of another saying as well, "don't look a gift horse in the mouth."

I think if the other envelope had 1,000,000 "dollars" in it ... it wouldn't exactly be hidden. Perhaps to make it even trickier, you could say, "You find a check for 1000 in the first."

[SteveA]

My luck, I would be forced to think there is only a dollar in the other one. The glass is half full or empty ... almost, but this particular glass isn't visible. Why risk emptying it ... so to speak.

Reminds me of another saying as well, "don't look a gift horse in the mouth."

I think if the other envelope had 1,000,000 "dollars" in it ... it wouldn't exactly be hidden. Perhaps to make it even trickier, you could say, "You find a check for 1000 in the first."

Yes, maybe a check would be better.

I'm still trying to figure out how this works though. It seems like it should always be better to swap, at least mathematically.

If we assumed you had a 50% chance of picking the lower valued envelope, then you should have a 50% chance of getting a million dollars by swapping, which averages to 500,000 payoff, whereas holding the current envelope would just get you the $1000.

It's seems like you should always swap, but that should be even true before you open it ... wierd.

[Meem]

My thought is, some money is better than no money. One envelope contains a thousand times more or less than the other, depending on what you pick up. If the first one I picked up only had a dollar, I'd definitely swap. If the first one I picked up had a thousand ... more than when I started the contest to begin with ... no contest, I have 1000 more dollars than I did when I approached. I wouldn't risk ending up with only a dollar for hopes of the million.

"Go on take the money and run."

[SteveA]

My thought is, some money is better than no money. One envelope contains a thousand times more or less than the other, depending on what you pick up. If the first one I picked up only had a dollar, I'd definitely swap. If the first one I picked up had a thousand ... more than when I started the contest to begin with ... no contest, I have 1000 more dollars than I did when I approached. I wouldn't risk ending up with only a dollar for hopes of the million.

"Go on take the money and run."

Ok, let's say it's just a 10 to 1 ratio and you open your envelope and see $10, that would appear to say that the other envelope has either $1 or $100 dollars.

So you should, on average gain ($100+$1)/2-$10=$45 by swapping.

That's so wierd ... it shouldn't be the case. You didn't know what was in the first envelope and so why should it appear to matter if you swapped, yet probability appears to say you'd be gaining by swapping. What am I missing?

The amount of money in the envelopes isn't relevant. We could say the envelopes contained checks for charities of your choice, if you prefer. I'm just trying to figure out what's wrong with the logic and why it appears to show you'd be gaining by swapping.

[Graybeard]

.... has either $1 or $100 dollars.

So you should, on average gain ($100+$1)/2-$10=$45 by swapping.

That's so wierd ... it shouldn't be the case.

Its not really the case. Your presuming that the unopened envelope has $100 in it ... but it has a 50/50 chance of having only $1 in it.

You can't divide its 2 possible alternatives by 2 ... as only one exists. Which one exists is unknown. Therefore your choice is always 50/50 no matter the $10 in your hand ?

cool bananas ... greg

[SteveA]

Its not really the case. Your presuming that the unopened envelope has $100 in it ... but it has a 50/50 chance of having only $1 in it.

You can't divide its 2 possible alternatives by 2 ... as only one exists. Which one exists is unknown. Therefore your choice is always 50/50 no matter the $10 in your hand ?

cool bananas ... greg

Yes, it's 50/50 but half the time you get $100. It's not a 50/50 double or nothing, which would be even.

So, yes, you might trade your 10 dollar bill for a 1 dollar bill and lose 9 dollars half the time, but the other half of the time you trade 10 for 100, which gains you 90.

So half the time by swapping you lose 9, but the other half you gain 90. So it seems like you should always swap because the payoff is in your favor, but I can't figure out how it works ... it shouldn't matter which envelope. It seems like it shouldn't matter if you swap, but the numbers appear to say you should - what am I missing?

All you know is that the envelope you picked could have been either the lesser of the two or the larger of the two, but you don't know whether you picked the larger or smaller amount. So if you swap you have a 50% chance of losing a little money but also a 50% chance of making much more than what currently have and so it's biased toward swapping (again, it makes no sense to me though - maybe one of you have already explained it and I didn't get it).

[SteveA]

I think I finally figured it out. Though you gain more of a percent if you swap for the larger quantity, this larger percent is of a smaller quantity and if you swap the other way, where you swap for the lower amount, then you lose less of a percent, but that percent is of a larger amount, so it comes out to a wash.

[labelwench]

May I suggest that the real issue here is the human need to know what's in the other envelope.

No matter the odds of increasing or decreasing your instant windfall,

....you have 100% chance to know what's in both envelopes...

Additionally, the odds are 100% that you will have more than before you opened either envelope.

The more interesting aspect to me, is what amount in the first envelope would deter most persons from further exchange?

Regards,

Labelwench

[Meem]

Unlimited attempts? Or just one? If I had only one ... I'd be happy with what I had. I can pick one up and look, the second I can't look, but I can exchange for the first. Could be more, could be less ... why risk loss when you have already gained? The only reason there seems to be a need to risk it is if there is only a small value in the first. This is easy to do with money, but harder to with other things that one could place in side envelopes or ... just imagine people ...