Envelopes

[madnak]

It does if there's an upper limit. If there's no limit and the amount could be any amount, then you have to bring infinity into play. Otherwise, if the amount in the envelope is n/2+1 or greater, where n is the upper limit, you know the envelope is the top value and must stay.

In a practical situation, it might have to do with psychology. If you get $500, then you have to consider whether your opponent is willing to give up $1000. Or is $250/$500 more likely?

[Bob Moss]

[ QUOTE ]
Some friends and I had this debate at work the other day.

There are two envelopes in front of you. One contains twice as much money as the other. You choose an envelope and it contains $10. You are now given the option to switch envelopes. Do you switch?

[/ QUOTE ]

I don't switch, because how do I know that the person making the offer won't offer a switch when I pick the larger amount, and not on the smaller amount? It doesn't appear that all the rules were explained in advance. [img]/images/graemlins/smile.gif[/img]

[soon2bepro]

madnak, bob moss, those are completely subjective, and weren´t issued in the OP. One must assume the OP failed to mention that there was no cap and that the offer to switch would be there everytime; since it's clearly not what the OP is aiming at.

Anyway, I wanted to add that the OP´s argument is quite convincing, because if you know in advance that the only 2 possible scenarios are $5-$10 and $10-$20, you increase your EV considerably by switching anytime you don´t get $20. Even if you only switch when you get the $10, which was the proposed course of action, you still increase your EV. And when you open the $10 envelop you know that the only 2 possible scenarios are $5-$10 and $10-$20. This seems convincing, but one must not forget that when you get that valued $20, by following the same logic, you should switch, and that´s where you lose your extra EV.

[Evenkeal]

Answer: Doesn't matter if you switch.

Explanation (simple): If you choose an envelope, it will contain an arbitrary amount of money. Switching wont matter because the first envelope tells you no information about the contents of the second envelope.

Explanation (better): Say the envelopes are X, 2X. Half the time, you pick X, and the EV of switching is +X. The other half of the time, the EV of switching is -X. So switching is neutral EV.

To argue against the following argument: Say you choose the first envelope and it contains X. Half the time, this is the smaller one, and you gain X. Half the time, it is the bigger one, and you lose 0.5X.
Corrected argument showing the flaw: Say you choose the first envelope containing X. Half the time, it is the smaller one, and you gain X, where X is the value of the smaller one. The other half of the time, you lose 0.5X, where X is the value of the bigger one.

[pvn]

[ QUOTE ]
Corrected argument showing the flaw: Say you choose the first envelope containing X. Half the time, it is the smaller one, and you gain X, where X is the value of the smaller one. The other half of the time, you lose 0.5X, where X is the value of the bigger one.

[/ QUOTE ]

I think I can make it even simpler than that.

The envelopes have X and 2X.

If you pick X (probability: 0.5) and switch, you gain X.

If you pick 2X (probability: 0.5) and switch, you lose X.

EV = ((0.5)*+X) + ((0.5)*-X) = 0

Edit: note this does *not* depend on any weird conditions such as there being a cap on the maximum value of 2X.

[housenuts]

[ QUOTE ]
This reminded me of this problem:

You have a black box with a slot in the front and a button. There are an unknown number of marbles in the box, one of which is white, all the rest are black. The box is magic, in that the number of balls that may fit in the box does not depend on its physical size. Each time you press the button, the box spits out a marble, chosen at random from the marbles within. You push the button N times, and on the Nth attempt, the white marble is ejected.

About how many marbles would you say are in the box, why, and how certain are you?

[/ QUOTE ]

is there a logical answer to this

[bigpooch]

First of all, if I take the OP at face value, we aren't
given enough information and it's quite clear if you have
examined similar types of questions before.

Suppose I were giving you the option to switch. What do you
think is your EV? If you think hard about it, you will know
what I mean.

Secondly, this is NOT the envelope paradox in its basic form
because of AT LEAST two important factors. If you know
what those factors are, you are either well-informed or a
genius.

Thirdly, if you are thinking about an EV calculation and
think that this calculation leads you to always switching,
you don't know something about using probability theory
that modern probabilists know to see that such a calculation
is unnecessary.

Finally, the envelope paradox is much more interesting than
the OP. Also, in general, the better problem is this:

A rich uncle has put DIFFERENT AMOUNTS on two checks and has
put each of them in an envelope. Tomorrow, he will then
give you an envelope to open and you must decide within one
minute whether to switch (take the check in the other
envelope) or not. Find an effective switching strategy that
has +EV and what factors would you consider in choosing one
switching strategy over another when both have positive
expectation? Well, you do have today to think about the
switching strategy.

[bigpooch]

This is absolutely correct and is one of the unintended
"flaws" of the OP.

[Propertarian]

Depends on whether the cheapest BJ in town is $5 or $10.

[revots33]

[ QUOTE ]
[ QUOTE ]
This reminded me of this problem:

You have a black box with a slot in the front and a button. There are an unknown number of marbles in the box, one of which is white, all the rest are black. The box is magic, in that the number of balls that may fit in the box does not depend on its physical size. Each time you press the button, the box spits out a marble, chosen at random from the marbles within. You push the button N times, and on the Nth attempt, the white marble is ejected.

About how many marbles would you say are in the box, why, and how certain are you?

[/ QUOTE ]

is there a logical answer to this

[/ QUOTE ]

Since the box could contain an infinite number of marbles, I don't see how you could guess how many are left after the white marble comes out.

But I suck at these logic puzzles so I'm probably missing something.