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2 boxes, you open one and find it has $100

C

Chiropteran

Diamond Member
So there are two boxes. You open one, and it has $100 in it.

1- Between the two boxes, one has double the money as the other.
2- You don't if your box is the double box or not.

Given this information, you can choose to keep the $100 box, or switch to the other box.

What is the best strategy?
 
You open the other. If you're unlucky, you have $50, if you're lucky you have $200. If you stay pat you have $100. On average if you did this with a large sample, you'd have around $125.
 
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Chiropteran said:
So there are two boxes. You open one, and it has $100 in it.

1- Between the two boxes, one has double the money as the other.
2- You don't if your box is the double box or not.

Given this information, you can choose to keep the $100 box, or switch to the other box.

What is the best strategy?
Click to expand...

At first I thought it's 50/50. 50% chance of halving or doubling what you have. Then I thought about outcomes.

If your box has $100, what's in the other box is either $50 or $200.

If you guess wrong you lose $50. If you guess right you gain $100. Since the gain is larger than the loss, the best strategy is to open the other box in case of win.
 
GagHalfrunt said:
You take the $100 and hire somebody to do your homework for you rather than using us.
Click to expand...
I had a girlfriend who I was pretty serious about. She was pretty amazing. But one day I mentioned some task I wasn't crazy about doing and she with a straight face suggested I hire some kid to do it. I didn't say anything but it made me wonder about her. 😕
 
Muse said:
You open the other. If you're unlucky, you have $50, if you're lucky you have $200. If you stay pat you have $100. On average if you did this with a large sample, you'd have around $125.
Click to expand...
Exactly.

50% chance of losing $50, 50% chance of gaining $100. I like those odds.
 
Grab both boxes and walk off like a boss.

At best you walk off with $300

At worst you walk off with $150 and maybe face assault charges; hopefully you didn't give the researchers your real name.
 
Chiropteran said:
So there are two boxes. You open one, and it has $100 in it.

1- Between the two boxes, one has double the money as the other.
2- You don't if your box is the double box or not.

Given this information, you can choose to keep the $100 box, or switch to the other box.

What is the best strategy?
Click to expand...

You should read the book "Speculation" by Edmund Jorgensen
 
Given the analysis many have done above, everyone would switch. So, no matter which box you opened first, you would choose the second one. (Or rather, you would use that reasoning to determine that switching would lead to better payoffs in the long run.)

So, before you even look inside the box, you switch boxes. Think for a moment and realize that it doesn't matter.

edit: damn you chiropteran! In my defense, I was in the middle of eating the most awesome steak sandwich, with a cheddar cheese spread pasted on to the roll instead of mayo, onions and peppers, and incredibly tender strips of steak. I had to wait til I was done eating to start typing. That's why you beat me to a solution, though I didn't know this problem had a name.
 
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Muse said:
I had a girlfriend who I was pretty serious about. She was pretty amazing. But one day I mentioned some task I wasn't crazy about doing and she with a straight face suggested I hire some kid to do it. I didn't say anything but it made me wonder about her. 😕
Click to expand...

We understand. You're not sure if she hired someone to please her because you're obviously not doing a good job of it. Incidentally, the answer is yes. :whiste:
 
Chiropteran said:
You are all wrong.

https://en.wikipedia.org/wiki/Two_envelopes_problem
Click to expand...

Your description of the problem in the OP seems different than the one described in the wiki link by one item.

In your OP, you said you open the box to see what $ you got, and then can make a decision on whether to keep it or switch.

In the wiki, it says you can pick an envelope (or box), but you can't inspect/open it. You then have a choice to keep it or switch.

Doesn't that change the game significantly?
 
Schfifty Five said:
Your description of the problem in the OP seems different than the one described in the wiki link by one item.

In your OP, you said you open the box to see what $ you got, and then can make a decision on whether to keep it or switch.

In the wiki, it says you can pick an envelope (or box), but you can't inspect/open it. You then have a choice to keep it or switch.

Doesn't that change the game significantly?
Click to expand...

Good question. I can still make arguments both ways.

Let the amount in the envelope chosen by the player be A. Since the player has opened the first envelope, he knows the value of A. By swapping, the player may gain A or lose A/2. So the potential gain is strictly greater than the potential loss.

Let the amounts in the envelopes be X and 2X. The player opened an envelope, but he doesn't know if X is $50 or $100, so the situation is equivalent to never opening an envelope and not knowing the value of X at all. Now by swapping, the player may gain X or lose X. So the potential gain is equal to the potential loss.

I think the best answer depends on the player's wealth. If the player only has $1 to his name, losing $50 would be pretty bad. If the player is rich, taking the risk to double the money is more worthwhile.
 
Schfifty Five said:
Your description of the problem in the OP seems different than the one described in the wiki link by one item.

In your OP, you said you open the box to see what $ you got, and then can make a decision on whether to keep it or switch.

In the wiki, it says you can pick an envelope (or box), but you can't inspect/open it. You then have a choice to keep it or switch.

Doesn't that change the game significantly?
Click to expand...

Why would knowing what's inside the box change the game? As DrP pointed out, why bother opening it if you're always going to switch to the second box? The fallacy is that knowing what's inside the first box tells you something about the second box. It doesn't.
 
i was going to post the same thing. it is a symmetrical choice until you open one of them. and either way, if you already opened one, open the other. its free cash either way.
 
KnightBreed said:
Why would knowing what's inside the box change the game? As DrP pointed out, why bother opening it if you're always going to switch to the second box? The fallacy is that knowing what's inside the first box tells you something about the second box. It doesn't.
Click to expand...

Exactly, this passage from Wiki describes it pretty well and why the expected value calculations are made on faulty premises:

In a 2012 paper on the subject, Bliss argues that the source of the paradox is that when one mistakenly believes in the possibility of a larger payoff that does not, in actuality, exist, one is mistaken by a larger margin than when one believes in the possibility of a smaller payoff that does not actually exist.[29] If, for example, the envelopes contained $5.00 and $10.00 respectively, a player who opened the $10.00 envelope would expect the possibility of a $20.00 payout that simply does not exist. Were that player to open the $5.00 envelope instead, he would believe in the possibility of a $2.50 payout, which constitutes a smaller deviation from the true value; this results in the paradoxical discrepancy.
Click to expand...
 
Schfifty Five said:
Your description of the problem in the OP seems different than the one described in the wiki link by one item.

In your OP, you said you open the box to see what $ you got, and then can make a decision on whether to keep it or switch.

In the wiki, it says you can pick an envelope (or box), but you can't inspect/open it. You then have a choice to keep it or switch.

Doesn't that change the game significantly?
Click to expand...

I was purposely changing details to prevent instant googling for the answer, but the change doesn't materially change the discussion.

Whether you know the rewards are X & .5x or 2x, or 1 & .5 or 2, doesn't change the math. The same logic holds.
 
Chiropteran said:
I was purposely changing details to prevent instant googling for the answer, but the change doesn't materially change the discussion.

Whether you know the rewards are X & .5x or 2x, or 1 & .5 or 2, doesn't change the math. The same logic holds.
Click to expand...

Logic doesn't apply when you change the nominal values, but rather loss aversion does. The people saying to switch are misapplying the Monty Hall Problem because the situation is superficially similar.
 
Chiropteran said:
You are all wrong.

https://en.wikipedia.org/wiki/Two_envelopes_problem
Click to expand...
No one was wrong. Switching doesn't harm, so you should switch. Not necessarily for this problem, but for all the other variants where switching is the good idea. And since then you will know the truth. Not switching means it'll eat at you for the rest of your life in wonder.

Now, if you could prove that switching was bad in the long run for this case, that would be another story.
 
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Schfifty Five said:
Your description of the problem in the OP seems different than the one described in the wiki link by one item.

In your OP, you said you open the box to see what $ you got, and then can make a decision on whether to keep it or switch.

In the wiki, it says you can pick an envelope (or box), but you can't inspect/open it. You then have a choice to keep it or switch.

Doesn't that change the game significantly?
Click to expand...

No, since you still don't know how much is in any of the envelopes either way.
 
KnightBreed said:
Why would knowing what's inside the box change the game? As DrP pointed out, why bother opening it if you're always going to switch to the second box? The fallacy is that knowing what's inside the first box tells you something about the second box. It doesn't.
Click to expand...

You stand to gain more than you lose.
 
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