Interesting Mathematical Paradox?

[dukemagic]

This is my first time posting in this subforum, so i apologize if I'm repeating an earlier problem, or if this belongs in probability.

My friend described this scenario to me the other day and I found it fascinating:

I have two envelopes, both of which have money inside. I tell you that one envelope has twice as much money as the other. I RANDOMLY choose one of the two envelopes and give it to you.

I then offer you the choice of changing envelopes before you open it. Is it in your advantage (EV wise) to do so? My initial though is no, because I randomly chose the envelope, and so you gain nothing by switching.

However, once you open the envelope and see how much money is inside (let's say its $20) I then offer you the chance to keep that $20 or change envelopes and take what is in the unopened one.

Do you then want to switch? The answer I arrived at is, yes, you do want to switch envelopes every time! I currently have $20. The other envelope muse have either twice as much or half as much- in other words, it must have either $40 or $10, so your EV by switching is $25 ($50 / 2).

So it seems to me that this is a paradox. There is no reason to change envelopes before you open them, but it is in your best interest to switch envelopes, after you open the first one, EVERY single time.

Anyone want to expand on this? There must be something I'm missing.

[MidGe]

You are making a mistake in your evaluation. Hint: in all cases you are guaranteed the minimum amount.

[dukemagic]

The one thing I wasn't sure about was whether or not you want to switch before you open up the envelope. You could use the same logic that you apply to the open envelope:

Let's say the envelope I offer you first is Envelope 1, containing A amount. Envelope 2 would then contain either A/2 or 2A. So the expected value of switching would be (A/2 + 2A) / 2, or 5/4 A (B = 5/4 A). So it seems you do want to accept the offer to switch.

But if you do accept this offer to switch and open envelope 2, you're still going to change back to Envelope 1 EVERY single time since now you expect Envelope 1 to have more than Envelope 2, on average. In fact (if my math is correct) you expected Envelope 1 to have (5/4) B, or 25/16 A which is greater than the 5/4 A you expected Envelope 2 to have. Is this a paradox? What am I missing?

[dukemagic]

[ QUOTE ]
You are making a mistake in your evaluation. Hint: in all cases you are guaranteed the minimum amount.

[/ QUOTE ]
Do you mean that in all cases I expect the other envelope to have more money, no matter what I do? Is that all there is to it? So what is the correct plan of action in this case? I'm guessing it's irrelevant whether or not you switch pre-opening, but you always want to switch post-opening. Is that correct?

[repulse]

This problem has definitely come up on 2+2 before (Probability forum I think?), it's where I was first introduced to it. The wikipedia entry for it details a number of good explanations for it, I like it: link

[MidGe]

[ QUOTE ]
Is that correct?

[/ QUOTE ] No. The switching does not change the EV.

[dukemagic]

[ QUOTE ]
This problem has definitely come up on 2+2 before (Probability forum I think?), it's where I was first introduced to it. The wikipedia entry for it details a number of good explanations for it, I like it: link

[/ QUOTE ]
Ah this is perfect. I'll read it now. Thanks!

[PairTheBoard]

[ QUOTE ]
Do you then want to switch? The answer I arrived at is, yes, you do want to switch envelopes every time! I currently have $20. The other envelope muse have either twice as much or half as much- in other words, it must have either $40 or $10, ...


[/ QUOTE ]

Yes. However you Don't know that the probability is 50% for the $40 and 50% for the $10. Therefore you cannot draw the conclusion,

[ QUOTE ]
... so your EV by switching is $25 ($50 / 2).


[/ QUOTE ]

Just because there are two possibilties doesn't make them equally likely.

PairTheBoard

[IronUnkind]

tldr. Reminded me of Monty Hall Problem. Here's a wikipedia article on a two envelope problem. Not sure if the formulation is the same.

[MidGe]

Not exactly the same formulation as the Monty Hall problem.

In the OP's case you are betting as to whether or not you got the small or the big amount, in the first envelope. In any case, you are guaranteed the small amount!