People have a problem with this paradox because they reject the premise that humans (ie. themselves) could be so predictable. However, that is the premise of the problem.

To understand this, imagine you have a primitive AI program on a computer under your control. You create a virtual environment for it where you pose this problem. You run the program up to the point where it makes a choice of boxes.

Then you rebuild the VR and put the money in the box based on the choice. And you run the AI again. You will predict very accurately indeed, and the right "choice" for the AI is the closed box.

I put "choice" in quotes because people have trouble reconciling choice and a deterministic mind like the AI. They refuse to believe they might have a mind so deterministic the alien could predict what it does so well.

It need not even be deterministic. The AI could include some quantum inputs so it is not perfectly predictable. However, we could run the AI 100 times and get a very good sense of the likely result. Now the choice depends on how much the AI knows about how random is decision processes actually are. If they are highly consistent, it should select the closed box and big reward. If they are highly random then the tester must be lying about the ability to correctly predict such beings 999 times in a row.

By revealing that the alien has been right 999 times in a row, the alien tells you that human decision making on this subject is indeed highly predictable to it. In which case the your choice was made long ago, a consequence of how your brain has grown and become you. The alien already knows how you will reason this out. You either already are the sort of mind that will take both boxes, or you already are the sort of mind that will take only the closed box. That's already happened. You will just be what you are, but if you are the sort of mind that will select only the closed box, you will be richer than if you are the sort of mind that takes both boxes.

This problem is not a strictly logical problem, at least not deductively so. To make it a deductive problem, replace the alien with (hypothetically speaking) God. By definition God's predictions are infallible, so we can eliminate the track record aspect and cut to the chase: God knows with absolute certainty which you will choose. Therefore, choose the closed one, you idiot.

The paradoxical angle arises from the problem whether you can rely on the alien's predictive claims. If you trust the alien, you take the closed box. If you don't, you take them both. This removes it from the realm of deductive reasoning and introduces an inductive element.

If we may assume that the alien's track record has been recorded and reported unimpeachably, then your confidence that the alien has predicted your own choice accurately should rise proportional to the success of his track record. The mechanism by which the alien predicts may be interesting, but is pragmatically irrelevant. 999 consecutive successes is a very persuasive record. Whether knowing you've been predicted influences your decision is irrelevant. The last hundred people tested have heard similarly persuasive stories, and they've been predicted accurately too.

Again, how he does it doesn't matter. The alien's error rate stands currently at worst at 1 in 1000, but is likely to be a lot lower, since a 1:1000 error rate would have been extraordinarily likely to manifest itself sometime during the first 999 attempts. We can easily chart out the odds in play here.

If you choose both boxes, there's at best a 1:1000 chance that you've fooled the alien. Multiply that chance by the value you're risking to figure out how much this choice is really worth. In this case, the total value of choosing both boxes is at best $2000.

If you choose the closed box, there's at least a 999:1000 chance that the alien predicted this correctly. You haven't got the "safe" $1000 to shore up the bottom line anymore, but the value of this option is still at least $999,000.

Therefore, factoring out irrational human decisionmaking processes, this problem is no different than if he walked up to you and said, "one of these boxes contains $2000 and the other contains $999K. Which one do you want?" With these odds it's not even an interesting conflict. If you choose anything other than the closed box, you're not deciding rationally. Your visceral concern over how the alien might be able to know what you'll do is not rational.

Given standard human guessing, of course, you're better off taking both boxes. The values of your options are $501,000 and $500,000 respectively in that case. Given the difference in values in play, the break-even point comes with even the slightest insight into how people will decide. A guesser who is right better than 50.05% of the time makes it more to your advantage to pick the closed box. A much more interesting examination of behavior will come from these very close cases, especially since such a small margin of success is very likely to be a statistical fluke.

We did read the information provided we have nothing better to do, this or work. So what is the box made of, is it alien also maybe a new element that we can sell along with the pictures. Have to go to ebay to search for the other 999 to 1998 possible alien boxes. Always take both boxes you get 2 alien boxes, a $1000 note (any average circulated $1000 note is worth $2000 and up) if the highly superior being from another part of the galaxy is wrong this time you also get the million. And your pictures posted on boingboing.net

Edited by author 12-14-2006 03:27 PM
jesse - the chances are *not* 1 in 2. Or rather - you don't know *what* they are.

Frankly, if I flip a coin and correctly guess the result 999 times correctly in a row, I'm cheating - the chance that I've done it by luck is vanishingly small. You know it, I know it - just because you don't know how I'm cheating doesn't mean that I must not be and the next result is a 50/50 chance.

(To put it another way - this isn't a proverbial 'fair coin toss' you talk about in probability. That would be 50/50, as you suggested.)

Likewise, the alien has been right 999 times in a row. We don't know how he does it, but he's doing it. It's a given of the problem (assuming the alien is lying is not really the spirit of the paradox - the question isn't 'can you trust aliens', it's 'is the universe deterministic').

[edited for clarity]

I think it's wisest not to make the question more complicated than it really is. Read the information provided carefully. If you buy the suppositions of the problem that are given, and what's the point of working on an answer if you don't, then the answer is to take the closed box only. There's no point in trying to understand how the alien predicts. If you buy that the alien was right 999 of 999 times, then there is a very very high probability that he will be right about you. Then by deciding to take the closed box only, you are doing what the alien knew you would do (regardless of how it knew that information), and walla, a million in your pocket!

Correct, bill. We crossposted. Easy (poster 9) may be onto something also.

 So the alien already knows you are going to shoot her and go through her pockets... OK, I ask, "what are the serial numbers of the bills in the box?"
Or, "what's in the box?"
"Cider!" - Nathan Detroit.

What series is the thousand dollar bill and is it uncirculated ?

I would take the closed box, without hesitation. By being the type of person who would take the closed box, I have already guaranteed that he will put $1M in it for me :) If I were the type to try and outwit the alien, he would already know that and only give me $1000. There is no paradox, assuming the alien can judge me as well as he claims.

On the other hand, is he good enough to predict the decay of a heavy atom? My buddy over here suggested to choose randomly, based on that. But maybe the alien would know you would try something like that, and simply not present you with the boxes at all :)

Let's assume that aliens can in fact predict how humans will choose between the two offers.

What if I decide by flipping a coin? Or by running a Scroedinger's cat type experiment?

Seems to be no problem to me. Take the $1000, take photos of the alien, sell them to the press for $1,000,000!

Seems to be no problem to me.

It all comes down to what you know about the alien. He claims to know all about you - maybe he does, but what do you know about him/her/it? Did you observe the 999 times? Did you make him do some cool alien tricks to prove his awesomeness? What is the motivation of a superior, possibly time-travelling omniscient-about-human-decisions alien, and does he care about money?

For you to KNOW that he is infallibly predictive of human actions would mean that you too are infallible about this guy's motivations. I would assume he's BS'ing you and take the grand. But I'm only human.

"But you *DO* know that I've been right 999 times in a row, and the chances that I can be right again are pretty good. "

Actually, the chances of you being right are still 1:2 if you don't know why you're getting it right.

Unless the alien is guaranteed to be correct about your decision or has the ability to modify the contents of the boxes after your decision there also leaves the possibility that the alien was incorrect in assuming you'd take both boxes (leaving the closed one empty) when in fact you just take the empty one. What is the result of this scenario? Do you just get screwed with the empty box and laughed at by the alien, or do you then get a million dollars out of it's pocket and the achievement of proving a supposedly superior being wrong?

Edited by author 12-14-2006 01:46 PM
danboy - he's an advanced alien from another galaxy; how do you know he *doesn't* have a time machine?

See - that's the point. Just because we don't know *how* he does his prediction doesn't mean that he can't.

If I tell you I can predict the flip of a coin, and do so correctly 999 times in a row, you still don't know how I did it - maybe I'm magic, maybe the coin is rigged, maybe something else. Maybe I'm just *really* lucky - but the chances of that are small, maybe smaller than me having a time machine. But you *DO* know that I've been right 999 times in a row, and the chances that I can be right again are pretty good.

The alien is the same case - I don't know how he makes his predictions. But he does, and with remarkable success.

If we take the premise of the argument to be correct - that he's a "superior being" from another galaxy with a proven track record - it seems silly to assume he can't predict our choice, just because we don't know how he does it.

The people who reason that the money is already in the boxes, he can't change it, and so they can go ahead and take both without changing the outcome are implicitly assuming that the alien can't really do what he claims to do - despite some fairly overwhelming evidence to the contrary. It's hubris, really - an admirable trait for a Perl programmer, but maybe not so good a way to win a million dollars from an alien.

For me - it boils down to "Will I bet $1000 I didn't have a few minutes ago that he's not lying, to win a million". I will.