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

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

Here is an Entry about a paradox which arises from estimating probabilites when playing a simple game.

The Entry is intended to explain the paradox and its resolution in plain English, without using a lot of mathematics.

There's a problem: late on in the entry, you say "In the game, the stranger has only a limited amount of money to give away." But you didn't state that at the beginning. That really does put a different complexion on the problem entirely.

Problem is, if the stranger DOESN'T have a limited supply of money, I can't, at the moment, see the flaw in the probability argument...



SoRB

SoRB: There's a limited amount of money in the world. Even if the stranger has all of it and is giving it all away, there is an upper limit, and the paradoxical argument is wrong.

If the stranger is giving away infinite amounts of money, bags I go first!

Zube

SoRB, haven't you done a Guide Entry on statistical reversal and probability paradoxes?

If it's not you, I'm sure someone has. Doesn't this overlap somewhat?

Obviously there's a limited amount of money in the world.

But we're not talking about the world. We're talking about what you should do in *principle*, based on the maths, not in *practice*, based on things like knowing how much money they bloke has.

See also "Rosencrantz and Guildenstern Are Dead".

If the only objection you can come up with against the probability argument is the limited supply of money, you've not refuted it. Why not replace money with grains of sand, and make the object of the game to gain the most grains?

SoRB

A857793

And no, this doesn't really overlap.

Two reasons:

1. I'm not sure what's described here is a paradox. I'm still thinking about it...

2. My entry covered areas where the stats specifically reverse on combination - this isn't that.

SoRB

Either way, the stranger is giving money away - so at least play!

The apparent paradox lies in the confusion between two outcomes that are equally probable (doubling or halving your winnings) and two outcomes of which only one is possible but unknown (having already chosen, or potentially choosing, the larger or smaller sum). Here we have the second kind of situation. My second way of describing the situation is the clearer one which, to me at least, makes it apparent that there is no inherent advantage in altering ones initial choice.

I do discuss the fact that the stranger can't have an infinite supply of money later, but I think I should say so the first time it comes up. How about I reword that paragraph, moving the bit about infinity from the later section:

'In any real game, the stranger can have only a limited amount of money to give away. In the general set-up outlined at the start of the entry, the stranger doesn't say that the game is limited, but no matter how rich, generous or plain daft he is, he cannot give away an infinite amount of money. No matter how high his limit is, it affects the game in the exactly same way, which is to change the odds so that the assumption is wrong, and the game is evenly balanced.'

Zube

SoRB, I disagree that we're talking about a pure maths problem rather than a real world one featuring just this guy.

I clearly state that the paradox is about a real game where an actual stranger offers you paper envelopes. You don't have to guess how much money the bloke has to resolve the paradox, the simple fact that he can't possibly have more than all the money in the world is enough to resolve it.

Even if the game is about grains of sand, the total number of sand grains in the world is finite, and the resolution works in the same way.

Zube

jbird, I think the reason the paradoxical argument looks reasonable is the same reason the the Monty Hall solution looks wrong to many people at first sight: given two choices, people are inclined to believe the odds must be 50-50.

That's the common sense answer here, and it happens to be right. The paradoxical argument depends on assuming that those same 50-50 odds apply always and everywhere (which they don't), and that's why it looks plausible, at least to people who can follow it.

Zube

SoRB, there's an article over on wikipedia on this subject, at http://en.wikipedia.org/wiki/Two_envelopes_problem. It goes into all the business about normalization of uniform distributions, but it seems to me (not a real mathematician) to include a lot of bollix.

At any rate, it completely fails to explain what's wrong with the paradoxical argument in plain English for the real world game, which is what I'm interested in, so I'd be loath to cite it in my Entry.

Zube

Suppose it was God showing you the two envelopes. He has an infinite amount of money to offer. So would he be demonstrating his infinite goodness by giving you a finite amount in the envelopes? I think not!

This guy God shows you two envelopes. There's Aleph Null pounds in the first one and C pounds in the second. You're given the option of changing your mind once you've counted all the pounds...

No, it doesn't really work. I think we'll have to assume there is a finite limit on the amount in the envelopes.

Can God define a probability distribution such that every real number of pounds is equally likely to be in envelope A?

Zube

I cannot see where the paradox is: it is simply a matter of not understanding probability.

Does this make 'The Monty Hall Problem' a paradox as well or just not understanding probability?

OK, so I've been thinking about it a bit, and the problem seems to be, for me, the way it's described in the Entry.

"The setup" section describes playing the game just once. It does not mention the possibility of playing the game more than once. These are two very distinct cases. Anyone who knows about Prisoner's Dilemma games will know why.

"The amount you stand to gain is double the amount you stand to lose, and since there's nothing else to decide between the envelopes, swapping is a good bet."

Now I see the problem - the huckster here is telling the mark about how much he stands to GAIN, not how much he stands to WIN.

The clear-thinking person is focussed on how much money they END UP with, NOT how much they can improve where they are in the middle of the game.

"If the stranger were prepared to play all day..."

A possibility not mentioned in the setup...

"this argument suggests that by always swapping you would gain and lose equally often, but gain twice as much as you lost on average."

Again, the focus is on gaining and losing... which is an illusory state, because you start with nothing and end up with something, regardless of what goes on in between.

"A second oddity is that if you add a second player who always chooses the other envelope, he also gets a random first pick"

This sentence contradicts itself.

If the second player always chooses the other envelope, his choice is constrained, and therefore not random at all. So this argument is irrelevant.

"Call the envelopes A and B, containing £x and £2x. If you pick A and stick, you win £x. If you pick B and stick you win £2x. If you pick A and swap you win £2x, a gain of £x. If you pick B and swap you win only £x, a loss of £x."

Whoah, hang on... If you pick B and swap, you WIN £x. This is WIN of £x, not a loss of anything.

"What's Wrong With the Probability Argument?

That probability-based argument still looks good, doesn't it?"

Only because the crafty author has focussed the mind of the reader on the idea that by winning "only" £x, they've somehow "lost" £x they didn't ever have.

Any logic you apply to this problem has to work even if the stranger has an infinite money supply.

SoRB

Researcher 179541, the argument which says you gain by switching leads to the paradox, which is that whichever envelope you pick, you gain by switching to the other, so you can never stop switching and pick one.

The argument which leads to the paradox is of course faulty, and working out the probabilities correctly resolves it, but many people (including myself) take a while to understand exactly what's wrong with the faulty argument, and I understand probability reasonably well.

Zube

SoRB, I don't see how playing it just once affects the argument at all in this case. If you get to play many times, you will get some idea about the range of amounts likely to be in the envelopes. In particular, you'll quickly spot if the stranger just has fivers and tenners. This spoils the puzzle, as it means you have a better idea of the odds, and won't fall for the faulty 50-50 assumption.

The question remains what to do when asked whether to switch the first time.

'Now I see the problem - the huckster here is telling the mark about how much he stands to GAIN, not how much he stands to WIN.'

I thought it was clear from the rules that you always win at least the amount in the smaller envelope. This doesn't affect the arguments.

'Again, the focus is on gaining and losing... which is an illusory state, because you start with nothing and end up with something, regardless of what goes on in between.'

No, you win an amount, half that amount, or double that amount. If you pick high and swap, you lose half what you would win by picking high and
sticking.

'If the second player always chooses the other envelope, his choice is constrained, and therefore not random at all.'

Not so: If I toss a coin, and I take the face that lands face up and you take the one that lands face down, we both have a random chance of getting heads or tails. In the game, if I pick a random envelope I also reject a random envelope, which therefore is an equally good starting point for the other player to start gaining by switching.

'If you pick B and swap, you WIN £x. This is WIN of £x, not a loss of anything.'

As the Entry says, you win ONLY £x, whereas if you stick, you win £2x. Swapping loses you £x compared to sticking.

The Alternative Solution works perfectly even if the stranger has an infinite supply of money, but the flaw I point out in the probability argument is removed. Resolving the paradox gets into a lot of maths about partial sums of infinite series of expected gains, which I'd have to research to present accurately.

It is, in any case, irrelevant to the set-up given here, as no stranger can have an infinity of money.

Zube

G'day all.

Before Zube put this article into peer review , there was some discussion as to the validity of the "always swap" argument. As the main proponent of the "always swap" argument (well o.k., the only one) I presented a lot of what I thought was valid argument to demonstrate the fact that since the player didn't know what the upper limit was, this could not affect the "risk against return" calculation.

I even did some random generation of values in the first envelope, random decision as to whether this envelop was high or low, then compared the "stick" outcome to the "swap" outcome. This consistently got an answer which matched my assertion that swapping was a better bet.

The flaw in my argument (which I eventually worked out myself) was that by setting a limit on the amount in the first envelope, then making it possible for the second envelope to have twice as much as the first, I was setting double that limit on the amount in the second envelope.

Although I knew that the chance of ending up with the high envelope (stick or swap) must be the same as the chance of ending up with the low one, I didn't think about this when considering the "risk against return" of swapping, or of sticking.

Zube, another way to present this might be to consider the game as two choices (an A or B pick which results in a low or high result, followed by a stick or swap choice) and show all the options, and the result


e.g
pick /choice/result/change of result by second choice/final result
Low / stick /Low / no difference / win = low amount
High/ swap /Low / loss of half of the high value / win=low amount
High/ stick /High / no difference / win=high amount
Low / swap /High / gain of half of the high value / win=high amount

This also demonstrates that a "high" pick cannot be followed by "gain" result. (If you use right or wrong pick and right or wrong choice, you get the opportunity to say "two wrongs make a wrong"


Having said all this, I agree with SORB that the "all the money in the world" argument confuses rather than clarifies. While I know it is not the intent, the reader of the article may feel that you are purposely trying to confuse them.

If you stick with two envelopes, with real money in them, explain why the swap argument appears to be a better bet than the stick argument (even money chance, possibility of a two to one return), then explain why the argument is invalid (basically because you can't swap from "high" to "higher", only from "low" to "high", or from "high" to "low") - then I think it will read better.

I don't know how to word it better, but I also don't like using "lose $x" as the result of the "high to low" swap, perhaps "win $x/2" would be better.

SoRB, regarding the second player, I'll take out the reference to him getting a "random" envelope, as it's beside the point. The argument applies whether his envelope is regarded as random or constrained.

Zube

I'll think about how to rephrase the Entry to clarify the 'win' and 'lose' bits so that it reads better.

Zube