The envelope paradoxon - think of it as different regimes

[cwes]

Discussion here and here .

I say we may think of being in different regimes. We are either in regime N (having picked the envelope with N in it) or regime 2N. From our point of view being in either of these regimes is equally likely (.5).

Switching envelopes in regime N will gain us N. Switching envelopes in regime 2N will lose us N. As both regimes are equally likely the expected value of switching is .5 N - .5 N = 0. Therefore switching is ev neutral.

[SamIAm]

[ QUOTE ]
Discussion here and here .

[/ QUOTE ]I have to admit, I'm kinda baffled why you thought this needed a new thread. I'd have thought you could posted in the discussion there or there.

[ QUOTE ]
From our point of view being in either of these regimes is equally likely (.5).

[/ QUOTE ]I don't know how you can blanketly assume that, even after you look in the first envelope.
-Sam

[KenProspero]

Not posting in the other threads because ..... well, this one was here.

I've thought about this in the past few days, and come up with the following solution:

Always switch. Here's why:

First of all, I firmly believe that switching is a neutral ev situation. In other words, the amount you gain by switching is exactly equal to the amount you lose by switching. However, I have also decided that this is besides the point.

I can't prove that switching is not better for me (nor, it seems can anyone else provide a mathmatical proof), plus no one has suggested thas switching is -ev, all that can be said is that it's a break-even proposition, long term.

So, faced with the proposition that if my belief is correct, I lose nothing by switching, but if I'm wrong, I gain by this strategy, it seems that there's nothing to lose (long term) by making the switch.

[SamIAm]

[ QUOTE ]
I can't prove that switching is not better for me (nor, it seems can anyone else provide a mathmatical proof), plus no one has suggested thas switching is -ev, all that can be said is that it's a break-even proposition, long term.

[/ QUOTE ]

Let's say I'm choosing X with probability proportional to 1/X^2. One envelope has X and the other has 2X. You open an envelope and it contains $100. What do you really think the +EV move is?
-Sam

[MathEconomist]

I'm not sure why you keep belaboring this point. It's obviously the case that in the real world the amount of money you observe in the envelope you open carries some information about whether you picked the N or 2N envelope. And this is clearly the case is you assume a probability distribution for N as well. However, the case most people are interested in when they ask this is the abstract case when the amount you observe in the envelope conveys no information about which one you chose.

As an aside, I've never understood why people think this is a paradox. The envelopes are seeded beforehand. One has N and the other has 2N. With 50% chance you picked the N envelope and with 50% chance you've picked the 2N. So if you always switch you end up with 2N whenever you intially picked N and N when you initially picked 2N. So always switch gets you N with 50% probability and 2N with 50% probability. But this is exactly the same as what you accomplish by never switching. And sometimes switching accomplishes the same thing (because in this case you can't selectively switch as the amount of money observed carries no information). The actual amount of money you observe is irrelevant. Of course, this is only true in the abstract case.

[PairTheBoard]

I wonder if you could actually make money with this Prop bet.


Prop Bettor -
"I'll put two numbers in these envelopes. One is twice as large as the other. I will let you choose an envelope, open it, look at the amount, and bet me a dollar that you have the smaller number. If you win I'll pay you 3-2 odds on your bet. Would you like to take that bet?"

Mark -
"Sure. It's 50-50 and you'll give me 3-2 odds? Of course I'll take the bet. I just wish I could bet more."

Prop Bettor -
"What's the most you can bet?"

Mark -
"Hell, I'd bet up to $1000 on that."

Prop Bettor -
"Tell you what. I'll make sure the numbers are both smaller than that and I'll let you bet whatever amount you see in the envelope. How's that?"

Mark -
"Well, I'm not so sure about that"

Prop Bettor -
"You just said it was 50-50 didn't you? And I'm still paying 3-2"

Mark -
"Right. It's 50-50. Shoot, your on. Deal em"


PairTheBoard

[SamIAm]

[ QUOTE ]
However, the case most people are interested in when they ask this is the abstract case when the amount you observe in the envelope conveys no information about which one you chose.

[/ QUOTE ]I literally have no idea what this means. If you ignore the fact that your envelope contains $100, maybe it leads to your confusion as to why it's called a paradox, as you illustrated below.

[ QUOTE ]
As an aside, I've never understood why people think this is a paradox. The envelopes are seeded beforehand. One has N and the other has 2N. With 50% chance you picked the N envelope and with 50% chance you've picked the 2N. So if you always switch you end up with 2N whenever you intially picked N and N when you initially picked 2N. So always switch gets you N with 50% probability and 2N with 50% probability. But this is exactly the same as what you accomplish by never switching. And sometimes switching accomplishes the same thing (because in this case you can't selectively switch as the amount of money observed carries no information). The actual amount of money you observe is irrelevant.

[/ QUOTE ]The game begins after you open the first envelope. You open it and you do see $100. You see it. It's there. NOW's the first time you actually make a decision. People call it a paradox because, under the assumption that you still have N or 2N with equal probability, switching earns you money in EV. People who ignore the contents of the envelope (like yourself) think switching is neutral. Closing your eyes and switching is NOT the same as switching knowing the contents of the 1st envelope.

[ QUOTE ]
Of course, this is only true in the abstract case.

[/ QUOTE ]What does "abstract" mean here? Simply "you don't know how N was chosen"? It's still not chosen such that every number is equally likely. Does "abstract" mean "can't happen or even be approximated"?
-Sam

[SamIAm]

Here's my strategy that definitely has higher EV than the "not switching" camp:<ul type="square">[*]Choose a number X randomly.[*]Open your first envelope.[*]If the envelope contains more than X, keep it.[*]If the envelope contains less than X, switch.[/list]Notice that if both envelopes are less than X, this strategy is EV neutral. If both envelopes are bigger than X, this strategy is EV neutral. If one envelope is bigger and the other smaller than X, I win.

Looking in the envelope matters. "Switch every time" != "Switch after learning that the envelope contains $100".
-Sam

[cwes]

[ QUOTE ]
I don't know how you can blanketly assume that, even after you look in the first envelope.


[/ QUOTE ]

I can assume that because the only thing we learn from the number we observe is that it either equals N or 2N. As we have picked N or 2N each with a probability of .5 we are in either of these regimes with a probability of .5.

[ QUOTE ]

First of all, I firmly believe that switching is a neutral ev situation.


[/ QUOTE ]

My proof supports your believes,

[ QUOTE ]

(nor, it seems can anyone else provide a mathmatical proof)

[/ QUOTE ]

Oh, MathEconomist did below and I did above.

[ QUOTE ]

With 50% chance you picked the N envelope and with 50% chance you've picked the 2N. So if you always switch you end up with 2N whenever you intially picked N and N when you initially picked 2N. So always switch gets you N with 50% probability and 2N with 50% probability. But this is exactly the same as what you accomplish by never switching.


[/ QUOTE ]
This is the same thing as I said above only your argument seems easier to understand to me. Basically we both say that the information you gain by opening one letter does not help you (opposed to the three gates / goat problem).

[PairTheBoard]

I think my last post in the other thread gets to the heart of the matter.

[ QUOTE ]
PTB -
Look at this from a Gambler's point of view. You are told about the two envelopes and asked to choose one. You are offered these Proposition Bets.

1. Given 3-2 odds, would you be willing to bet $10 that your envelope contains the smaller amount?

2. Given 3-2 odds, would you be willing to bet the amount in your Envelope that it is the smaller amount?


1 is a good bet for you while 2 is not. Do you see why?


[/ QUOTE ]

The big problem with Prop Bet 2 is that whether or not you win depends on the amount you wager. You lose when you wager more and win when you wager less. That makes you a sucker. In Prop Bet 1 your wager is independent of whether you win so the sucker is the one giving you 3-2 odds.

Prop Bet 2 is essentially what you are doing when you switch, except in the switch you are wagering half the envelope amount and getting 2-1 odds.

PairTheBoard