Logic Problem for GoT

[maurile]

Before you there are two boxes, one white and one black. Each box contains money; one has twice as much as the other. You may choose either box and keep whatever money is inside.

You choose the white box and find $100 in it.

You are now given the option of switching. You may either keep the contents of the white box, or you may instead opt for the contents of the black box.

What's the EV of switching?

[GuyOnTilt]

<font color="white">The black box will have $50 in it 50% of the time, and $200 in it 50% of the time. If you choose to switch, you will have $125 on average. If you stay, you will have $100 on average. So the EV of switching is $25. </font>

GoT

[blackaces13]

I'm not GOT but I say the EV of this is +$25.

EDIT: I see the real GoT got me by 2 minutes as I was calculating it. Oh well.

[maurile]

If you'd originally picked the black box, you'd have come to the exact same conclusion -- that switching is +EV. So no matter which box you choose first, you should switch.

But that can't be right.

[maurile]

We can prove two contradictory propositions:

Proposition 1. The amount that you will gain by switching, if you do gain, is greater than the amount you will lose, if you do lose.

Proposition 2. The amounts are the same.

The proof of Proposition 1 is essentially the one GoT gave: Let n be the number of dollars in the box you are now holding. Then the other box has either 2n or n/2 dollars. If you gain on the trade, you will gain n dollars, but if you lose on the trade, you will lose n/2 dollars. Since n is greater than n/2, then the amount you gain, if you do gain — which is n — is greater than the amount you will lose, if you do lose — which is n/2. This proves Proposition 1.

Now for the proof of Proposition 2. Let d be the difference between the amounts in the two boxes, or what is the same thing, let d be the lesser of the two amounts. If you gain on the trade, you will gain d dollars, and if you lose on the trade, you will lose d dollars. And so the amounts are the same after all. This proves Proposition 2.

So which proposition is really right?

[aloiz]

This is a classic paradox, (not really a paradox, but the way most people naturally think about it) almost as famous as the monty hall problem.

aloiz

[GuyOnTilt]

But that can't be right.

It can, and it is.

Problems like this one is why people who know a decent amount of game theory get frustrated watching game shows.

GoT

[maurile]

[ QUOTE ]
It can, and it is.

[/ QUOTE ]
Actually, it's not.

Let me know if you'd like the answer.

[GuyOnTilt]

Before you there are two boxes, one white and one black. Each box contains money; one has twice as much as the other. You may choose either box and keep whatever money is inside.

You choose the white box and find $100 in it.

Okay, the first rule to solving logic puzzles is to read the intro thoroughly because there's often clues and tricks in it. I've read this through a couple times now carefully and this statement still seems to be impossible to me:

Proposition 2. The amounts [that you're able to gain or lose by swtiching] are the same.

I'll keep thinking...

GoT

[AndysDaddy]

A decent, but somewhat geeky explanation for this is the fact that the dollar amounts are a geometric progression, with a fast rising curve ahead of you, and a slow falling one behind. With the big upside, and small downside you always want to take any chance to advance up the steep slope.