[quote name='Khaiy' timestamp='1301171446' post='4790767']
You are presented with three boxes, and told that one box contains $100. The other two contain nothing. You are allowed to put your hand on one box, and then the presenter will take one of the other boxes away. The box that the presenter removes is guaranteed (and shown) to be an empty box. Then you are given a choice: You may stick with the box under your hand, and get whatever's inside, or switch to the other box. Which do you choose?

There is a rational choice to be made, but many people guess. Is rationality why you answered as you did?

I know I've heard this explained before, but I can't remember the reason. Isn't it something like the box you put your hand in has a 33% chance of having the item, but after he takes away the other box the remaining box has a 50% chance of containing the item so you should choose that one?
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It is the Monty Hall Problem, which should be declared taboo, since even worse than religion or the question on what side eggs should be broken, it has the ability to divide people about whether or not one should switch boxes, leading to flame wars which end only when all participants on one side have died of old age. Or if the side that claims it doesn't make a difference finally listens and just runs a simulation which shows that switching does indeed increase your chance of winning, but unfortunately, such a happy outcome has seldom occurred in the past... So, for the sake of the children, let us speak of it no more.

Perhaps i would take the million dollars, even if i may only be 99.9% sure that i will really have it.
I dont think it would immediately mean that im greedy, but rather that i'm some kind of risk-taking radical dreamer, whose dream at that moment would be to gain lots of $$$.


and i don't think it would have implications on how the alien will treat me in the future because he will never return in the first place.

I know I've heard this explained before, but I can't remember the reason. Isn't it something like the box you put your hand in has a 33% chance of having the item, but after he takes away the other box the remaining box has a 50% chance of containing the item so you should choose that one?

Pretty much. The remaining box (the one you didn't pick initially) has a 66% chance of having the prize, because the additional information you get is that another box definitely does not have it. You picked the first box under a condition where the odds of it being the right one are 1/3, and that can't be changed because you get more information later. Sam's wikipedia link explains it better than I ever could though.

It is the Monty Hall Problem, which should be declared taboo, since even worse than religion or the question on what side eggs should be broken, it has the ability to divide people about whether or not one should switch boxes, leading to flame wars which end only when all participants on one side have died of old age. Or if the side that claims it doesn't make a difference finally listens and just runs a simulation which shows that switching does indeed increase your chance of winning, but unfortunately, such a happy outcome has seldom occurred in the past... So, for the sake of the children, let us speak of it no more.

That's odd, since it would seem that the link you shared is hard to argue against. I think it's pretty clear, I mean the probability of winning by switching is


It's so obvious!

Theory:

T1. Rational thinking never fails.

Proof that the example problem given in the OP does not counter T1:

P1. An alien that we are certain can predict our actions tells us we are free to either, take box B, or take both boxes.

P2. Given that we are certain he can predict our actions, then we are certain he can enforce the limitation of our options to one of the two options.

P3. Given that no two events can be instantaneous the only way P2 is possible is summarised in the following three sub-points.

[color="#ffffff"]...P3a. Box A becomes inaccessible if we take box B first.

[color="#ffffff"]...P3b. Both boxes are opened automatically extremely soon after we take box A first and we are presented with the according monetary reward.

[color="#ffffff"]...P3c. We take too long to make a choice and the boxes are preprogrammed to pick an option for us after we take too long, by opening box B or opening both boxes and presenting the according monetary[color="#ffffff"].reward to us. Which option the boxes choose for us is either random or predetermined by the alien.

P4. Given we are certain that the alien can predict our actions we are certain that if we open both boxes we will get only $100, and we are certain that if we open only box B we will get $1,000,000.

P5. Given P1, P2, P3, P3a, P3b, P3c, and P4, a rational thinker will take and open box B first, gaining the $1,000,000 it contains, rendering box A inaccessible, and gaining the maximum possible reward.

P6. Rational thinking wins again.

Notes:

N1. It is interesting that our greatest rational tool in making the most rewarding choice comes from knowing that the alien can predict our actions.

N2. In the event of P3c the boxes would have to follow us around (or we would be shackled nearby) and predict when we would be about to die or if we were to become permanently beyond their reach so as to present the monetary reward to us before either of those final events.

[quote name='SamLowry' timestamp='1301213414' post='4790911']
It is the Monty Hall Problem, which should be declared taboo, since even worse than religion or the question on what side eggs should be broken, it has the ability to divide people about whether or not one should switch boxes, leading to flame wars which end only when all participants on one side have died of old age. Or if the side that claims it doesn't make a difference finally listens and just runs a simulation which shows that switching does indeed increase your chance of winning, but unfortunately, such a happy outcome has seldom occurred in the past... So, for the sake of the children, let us speak of it no more.

That's odd, since it would seem that the link you shared is hard to argue against. I think it's pretty clear, I mean the probability of winning by switching is


It's so obvious!
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Perhaps it is because of the fact that in such flame wars, the common name of the puzzle is not often known so that people don't know what to google for and hence don't have access to such a nice explanation as Wikipedia's.

[quote name='Khaiy' timestamp='1301234046' post='4790963']
[quote name='SamLowry' timestamp='1301213414' post='4790911']
It is the Monty Hall Problem, which should be declared taboo, since even worse than religion or the question on what side eggs should be broken, it has the ability to divide people about whether or not one should switch boxes, leading to flame wars which end only when all participants on one side have died of old age. Or if the side that claims it doesn't make a difference finally listens and just runs a simulation which shows that switching does indeed increase your chance of winning, but unfortunately, such a happy outcome has seldom occurred in the past... So, for the sake of the children, let us speak of it no more.

That's odd, since it would seem that the link you shared is hard to argue against. I think it's pretty clear, I mean the probability of winning by switching is


It's so obvious!
[/quote]

Perhaps it is because of the fact that in such flame wars, the common name of the puzzle is not often known so that people don't know what to google for and hence don't have access to such a nice explanation as Wikipedia's.
[/quote]

which makes the problem not taboo and people stupid? :-p

To me, this "paradox" seems trivial (in the mathematical sense of the word).

First of all, this "paradox" is impossible because it assumes the world to be deterministic, which we know it isn't. So let's assume that the scenario takes place in an alternative, deterministic universe.

Now, in a deterministic universe, the choice that you will make is already determined at the very beginning of times. Hence, you never have the problem of trying to figure out which choice should you make, because in the end your choice is already predetermined. You might choose to reason it out and think about your choice anyway, but ultimately you will end up making the choice that is already predetermined. In fact, the alien could have either only brought 100$ or 100100$ from his home planet, depending on the choice he knew you would make, well ahead of time.

So, there is no intelligent thinking to be done at all, no decision making involved. It is all a matter of in "which" universe you happen to be in. Either you live in the universe where you only open box B or you live in the universe where you open both boxes. In the sense, the "paradox" is trivial.

Of course, because the paradox is so trivial, it does not imply anything about "rational thinking".

[quote name='SamLowry' timestamp='1301143832' post='4790657']
Rationality is not the same as perfection and does not necessarily lead to optimal solutions at all times. A more interesting (and realistic) scenario is the Prisoner's dilemma, where rational behavior generally leads to suboptimal results.

I would think the OP's story is rather easy to solve: assuming the alien is speaking the truth, one choice gets you one million, the other one hundred. If you take into account probabilities, you get for B = 0.9999 * $1,000,000 + 0.0001 * $1,000,000= $1,000,000 and A+B = 0.9999 * $100 + 0.0001 * $1,000,100 = $200, hence a rational actor still picks B. The only irrational part I can see is in believing the alien can predict your future, but you assign percentages of probability to that, so that makes it moot.

Actually, the payoff is that once he has left (and cannot alter the results after he has left) you either get A+B or just B. Since A is always some positive number, a rational person would choose A+B. Remember, the boxes have already been stuffed.
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Well thats not entirely true, because the outcome of A depends on B as well as B depends on A. I don't think you can use that as a rational.

[quote name='Strange Loop' timestamp='1301164850' post='4790746']
Actually, the payoff is that once he has left (and cannot alter the results after he has left) you either get A+B or just B. Since A is always some positive number, a rational person would choose A+B. Remember, the boxes have already been stuffed.

Well thats not entirely true, because the outcome of A depends on B as well as B depends on A. I don't think you can use that as a rational.
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No it doesn't. The boxes are stuffed based off an assumption the alien has made about your guess, but as they are already stuffed their contents depend on nothing that you do but rather on the aliens assumption of what you will do.

A+B where B is a non-negative number is always equal to or greater than A, so it would be silly not to open both if your goal is to maximize your gains.

There is something to be said for opening boxes based on some condition dependent on the contents of box B, which would make a somewhat infinite loop of dependencies as your choice depends on his choice that depends on your choice etc etc.

Really the riddle could be reworded with him stuffing the boxes after you make your choice and it would be pretty much exactly the same; the result being that the alien is kind of a prick.

[quote name='D.Chhetri' timestamp='1301272036' post='4791147']
[quote name='Strange Loop' timestamp='1301164850' post='4790746']
Actually, the payoff is that once he has left (and cannot alter the results after he has left) you either get A+B or just B. Since A is always some positive number, a rational person would choose A+B. Remember, the boxes have already been stuffed.

Well thats not entirely true, because the outcome of A depends on B as well as B depends on A. I don't think you can use that as a rational.
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No it doesn't. The boxes are stuffed based off an assumption the alien has made about your guess, but as they are already stuffed their contents depend on nothing that you do but rather on the aliens assumption of what you will do.
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Are u dumb or are you drunk? The alien didn't assume shit. He predicted with 99.9% of accuracy what you would do. If you try to come out with 1000100 dollars you'll only get 100: THE ALIEN WARNED YOU ABOUT THAT. Why are you still trying to do so?