The money in 2 evelopes "paradox"

[CityFan]

The correct answer is the second one: always switching and always sticking are of equal value.

Rather than repeating what's been said already, I give you a SIMILAR but subtly DIFFERENT situation in which always switching would be the better strategy.

Suppose you are allowed to choose an envelope with a cheque in it from a range, and the donor promises that AFTER YOU HAVE CHOSEN IT he will flip a coin and if it comes up heads write out a cheque for twice the amount, but if it comes up tails write out a cheque for half the amount.

(Obviously, you don't get to see the coin flip etc.)

Then you are allowed, if you wish, to swap the cheque you have for his new cheque.

In this situation, you should always switch. The reason is that whatever amount you see on the first cheque, there is an evens chance that the second cheque has twice that amount vs half that amount.

That is not (CANNOT BE) true in the "paradox" situation, which is why argument A is flawed.

[CityFan]

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This can be shown mathematically and has been confirmed by computer simulation.

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[img]/images/graemlins/mad.gif[/img] Give the mathematicians some credit eh?

[LinusKS]

Here's my attempt.

Suppose somebody puts $50 and $100 in two envelopes, without telling me the amounts.

If I choose one, the odds that the other has twice as much are 1:1. The odds that it has half as much are also 1:1.

The argument that it's +EV to switch says:

By switching you either gain 1x, or lose .5x, with equal probability. Since you gain twice as much when you win as you lose, and the odds are equal, it's profitable to switch. (-.5x + 1x)/2 = + .25x.

The problem is, whenever you lose, x is always twice as much as when you win.

For example, if you draw $100, and switch, you lose $50. But when you draw $50, and switch, you only gain $50.

So it's EV neutral.

Unless, as several others have pointed out, you have some prior knowledge of the likely range of numbers that were have been put in the envelopes in the first place. In that case, the game would be easy. Every time you drew a number that was on the high end of the range, you hold it. When it was on the low end, you switch.

[LinusKS]

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For people who think it's clearly irrational to switch, I offer the following variant. To keep it simple, assume one pound sterling is worth US $2. I'm an American, and I bet $1,000 against 500 pounds sterling with my English friend that the US dollar will get weaker (that is, it will take more than $2 to buy 1 pound) in one year. Assume the odds are 50/50 of this happening.

If I lose, I send my friend $1,000. If I win, he sends me 500 pounds, which will be worth more than $1,000. So clearly this bet is +EV to me. If he loses, he sends me 500 pounds. If he wins, he gets $1,000, which are worth more than 500 pounds. So it's clearly +EV to him as well.

If that's not enough for you, I can go to the foreign exchange option markets and get $60 in cash today for agreeing to exchange the 500 pounds if I win for $1,000, with no obligation if I lose the bet. So I get $60 today to take a 50/50 $1,000 bet; and my friend can get 30 pounds to take the other side of the bet.

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Aaron, I don't understand either of these examples.

In the first case, the dollar either appreciates or depreciates. When it goes up, and you "lose," that means the $1k that you lost is actually worth more than $1k - because the value of your dollars has gone up.

Conversely, when your British friend loses, he loses more than 500 GBP - because those 500 pounds are worth more at that point.

The extra dollars he could have gotten on the open market - whatever they happen to be - represent the extra value he's lost on the bet.

Put it this way - suppose you were talking about carrots and oranges, instead of currencies. You have a bunch of carrots, and your friend has oranges. Carrots and oranges are all worth $1.

When you "lose," because the value of your carrots goes up, you send your friend a bunch of carrots at a discounted rate. Say carrots, after a year, are worth twice as much as oranges. So you send your friend 1000 carrots. Only, those carrots are worth $2 each, now, not $1. So the value of what you've lost is $2000, not $1000.

It doesn't make any more sense to talk of losing "only" 1000 carrots, when the value of carrots has gone up, than it does to talk of losing "only" $1000, when the value of those dollars has increased.


On the option example, I'm not quite clear about what kind of option you're talking about. But it sounds like you're talking about selling an option to someone to exchange dollars for pounds at a specific rate. What I think you're leaving out of the equation, is that whenever the market moves against your position, you'll be forced to cover the difference - and it could be much more than $60. In other words, you're selling a kind of insurance, where the most you can win is $60, but you can lose much more than that. It might be a good play, and it might not, but I don't think it's free money.