h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

OK, I've rephrased things to avoid using win/lose for the diffent outcomes, inserted a table for the alternative solution and removed the reference to the second players envelope being random.

I also tidied up the references to the impossibility of the stranger having infinite funds, and qualified the references to playing once or many times.

How's that?

Zube

Entry: The Two Envelope Paradox - A19012961
Author: Zubeneschamali - U721452

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While the overall fact that switching cannot be beneficial is correct, I don't think your arguments are particularly convincing. I think the upper limit proof has a serious flaw in that it would seem that you can simply offer two amounts which are less than the maximum each time - while this may not be the fact (as we are trying to keep the punter confused as to which envelope has the greater amount), it makes the argument very unconvincing. You should make a greater point of the fact that if two players are involved, they cannot both be gaining over time from switching, hence proving by contradiction that there is no benefit from switching. Another line of reasoning is that, given 20 pounds, it is not reasonable to assume the possibility of both 10 and 40 pounds. That's the crux of the problem - you have to look at the envelopes as containing x and 2x before opening them, as opposed to considering one to contain 20 and the other either 10 or 40.

Alex

I also think the idea of swapping and swapping back changing the result confuses things. While I was an advocate of the "better bet" I felt that both this approach (and the "all the money in the world scenario") simply clouded the issue, since the player had no knowledge of what the upper limit was.

Again it might confuse rather than clarify, but if the game is not "twice or half", but "five times or a fifth" the odds of making the right choice stay the same (1 in 2), but the risk against return changes.

I've been lurking this with interest. I think it's an excellent paradox, and well-worth a guide entry.

How do I see it?
Well, the punter sees a choice which isn't real, but imaginary. He sees a choice between two envelopes, which contain either £10, or £40, but this is not the case. There's only one envelope, and one of those amounts is not on offer. Sadly he doesn't know which, and that choice has already been made for him.

His choice is EITHER £40/£20 OR it's £10/£20. In either case it's 50/50 that he chooses the right envelope, because it's 50-50 as to which game he's playing.

Icy

Hi Icy. As so often I have agreed with you and do so again. The illusion is that there are three alternatives to choose from: x (first envelope contents), x/2 and 2x. The choice of the second envelope offers only one of the latter - although we don't know which it is. So the choices are twofold: x and (x/2 OR 2x).

The fact that you can't lose by any choice reflects the fact that the customary fair bet of 'double or nothing' is improved upon by 'double or halve'.

toxx

I have a dstinct feeling of déjà vu. Was this entry or one just like it submitted before, some years back? Around the time of the Monty Hall entry perhaps?

Hi Recumbentman, we talked about this paradox in conversations around that time, but no-one ever wrote an entry.

So here it is!

Zube

And by the way it seems to me that maths can't escape the constraint of the upper and lower limits. Even the grains of sand are limited.

The number of atoms in the universe -- isn't that (only) about the square root of a googol?

Good Entry. I'm convinced.

But then I was convinced for a long time that the Monty Hall solution was wrong . . .

Thanks for the reply, Zube. As you know, the difference in the Monty Hall question is that information is added before the question of swapping arises. Here, of course, there is no new factor to consider, so swapping is on the same terms as the initial choice - ie: 50 - 50.

J

Quite so. The host is forced to show an empty door, not free to choose.

I like how you've set up the paradox.

I don't agree, though, that the paradox requires finite amounts of money to go away. You have stated correctly, I think, what the hidden assumption is - namely that 'no matter what amount of money is in the envelope you choose, the other envelope is equally likely to be half or double that amount'

This line of reasoning leads to the result that you can win 1.25 as much by swapping. But the flaw is not in assuming infinite supplies of money, it is in assuming that half the envelopes contain 2x and half contain 1/2x, which gives an average of 1.25x. In fact, even when there are an infinite number of envelopes containing unlimited amounts of money, one-half of them will contain 2/3 of the total and one-half will contain 1/3 the total. Therefore, with random picking of envelopes, on average you will get 1/2 the total.

The 'infinity' I'm using here is the 'countable' type, ie the process of stuffing envelopes with money is ongoing out to infinity, so that at any given time there is an even number of envelopes - there is no limit on the amount that any given envelope could contain.

Hi pailaway, I agree that the paradox must fail even with infinite amounts of money, as the Alternative Argument shows. I think your explanation is related to the one in the Alternative Argument.

Your explanation is perfectly sound, but like the Alternative Argument I use, it doesn't identify precisely which step in the faulty argument is wrong. After all, if you are holding one envelope, it remains true that the other contains half or double, so the faulty argument seems reasonable. The flaw I am trying to illustrate is that the paradoxical argument says that these possibilities are equally likely, when in fact they cannot be.

The demonstration I am using to show a flaw in the argument only works for finite amounts, though. I like it anyhow because it doesn't get into a lot of equations.

Zube

Yes, after verrrry careful re-reading, I see that you have explained this correctly.

For me, the crux is that, given an infinite amount of money, the faulty argument leads a person to calculate that there's 2.5x infinity available, which goes unnoticed until you start keeping score, which is exactly what you've done.

Very thought provoking - well done

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Bravo!

Zube

Congratulations!

Well Deserved

Congratulations - given the amount of discussion while this was in review I'm looking forward to what happens when it hits the front page !