Newcomb's paradox
Apr. 15th, 2009 11:11 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackLinks
Newcomb's paradox on wikiepedia
Newcomb's paradox on Overcoming Bias blog
Newcomb's paradox on Scott Aaronson's blog and lectures
I first came across it via overcoming bias, and discussed it with a few people, but then recently saw it again in one of the transcriptions of scott aaronson's philosophy/quantum/computing lectures.
Newcomb's paradox
In very short, Newcomb's paradox says, suppose you're a professor and a grad student (or, in some cases, a superintelligent alien) comes to you and demonstrates this experiment. She chooses a volunteer, examines them, then takes two boxes, puts £1000 in box A and either £1000000 or nothing in the box B (see below for how she decides). She brings the boxes into the room and explains the set-up to the volunteer and says that they're allowed to either take the mystery box (when they either get lots or nothing) or take both boxes (when they get at least £1000.)
She even lets them see the £100000 beforehand so they know it exists, and lets them peek into box A to show it does have the money in, though box B remains a secret until afterwards.
"What's the catch," the volunteer asks. "Ah," begins the experimenter. "I have previously examined you, and worked out which choice you're going to make. If you were going to choose both boxes, I put nothing in box B. Only if you were going to take box B only, did I choose to put £1000000 in it.
"Hm", says the volunteer. "What do I do?"
A few caveats
"What if the volunteer would change their mind when they discovered the reasoning, or is going to choose based on a coin toss?" "Then I didn't accept them as a volunteer."
"How do I know it works?" You can't be sure, but she performs the experiment lots of times and is always right, so you are convinced. (Some examples ask you to presume as part of the conditions that she can, or take it on trust, but I think "having seen it work" makes it most convincing and concrete.)
"Ah, but I don't care about £1000, and certainly not if I've got £1000000, so I don't care," Well, ask what you would do if the numbers were a bit different. Can you pretend there's no combination where you'd risk something to get the little one, yet risk more to get the big one?
"How did they know what they'd pick when this experiment was performed the first time?" It doesn't really matter, just assume that you have seen it working with pretty-perfect prediction.
What would you do? An enumeration of the two obvious arguments.
11: "Why should you take both boxes?" Duh! Because whatever's in either of the boxes, you get all of it. And if that means I fail to get the million, then it's already too late to change that, isn't it?
22: "Why should you take box B?" Duh! Because you've just seen 50 people do the experiment, and all the ones who took both got £1000 and all the ones who took B got £1000000. Follow what works, even if you can't justify it with maths.
That's why it's a paradox, because, if you squint long enough, both answers seem perfectly reasonable.
I know this seems a little convoluted, but I tried to make it comprehensible if terse even to people without a very high opinion or training in philosophy (like me, in general). And hopefully get it to the point where at least asking the question makes sense.
Wait, if we use baysian reasoning, I bet the arguments will instantly become transparent and non-controversial. Right?
11: As above. Look at the table, and enumerate the possibilities. Choosing more boxes always gives a bigger payoff.
22: Ah, no, you're cheating. Based on the previous evidence, you must assume a priori that you are on row 2 or 3. After than, the choice is easy: row 3 gives more money. (See below for more "so, there's a 2/3 chance I'm in this universe..." type reasoning.)
Can we put this on a more rational footing? How does she predict what's going to happen
In fact, there are several ways.
1. You can do it if you postulate time travel, or determinism and a copy-teleport-machine, but those are not very realistic things to postulate, whether they would be physically possible or not.
2. A super-intelligent alien scans your brain and models it in a computer.
3. They give you a short-term-memory impairing drug, and try the experiment out several times beforehand while you remain the same person with the same experiences, but have no memory of the trials.
4. They discover that 94% of the time, all men choose one way, and all women choose the other. (But the experiment is double-blind, run by a technician who doesn't know the expected results, so the grad student peeks, then tells you that there's a 94% correlation, but not which way round it is, then invites you to participate yourself.)
Further arguments
33: Aha! In method 3, you don't know which one you are, one of the trial runs, or the final experiment. The only consistent answer which gets the big money is to assume you're more likely to be a trial run, and hence choose the money.
44: Aha! If so, then (*invokes Greg Egan*), the same reasoning applies with method 2. That suggests that you don't know if you're you, or the simulation of you!
55: Nope. Not so. What about method 4? Surely you can't claim that your consciousness might be either (a) you or (b) "the statistical correlation between gender and box choice"??
Which leaves us back where we started. (But remind me to come back to the "am I equally likely to be me, or some other human or simulation of a human".)
Free will
"What does it have to do with free will?" Well, the experiment is completely (sort-of) practical to do. In theory. And so you'd think it should be also actually possible to choose which to take. And yet it doesn't seem to be, and the answer seems to depend maybe on whether you believe in something you can call "free will".
In fact, people divide between take A&B, take B, and "problem is stupid, won't consider". In general, I think the last answer is often over-overlooked. In this case, if I'd seen it work out like that, I'd agree to take only box B, even if I couldn't explain the mathematics behind it. However, I also definitely feel I should be able to justify one case or the other.
Informally, it seems most people seem to eventually take B, but I don't know how important that is.
Apocrypha
Links to prisoner's dilemma, links to doomsday paradox, etc, etc.
Newcomb's paradox on wikiepedia
Newcomb's paradox on Overcoming Bias blog
Newcomb's paradox on Scott Aaronson's blog and lectures
I first came across it via overcoming bias, and discussed it with a few people, but then recently saw it again in one of the transcriptions of scott aaronson's philosophy/quantum/computing lectures.
Newcomb's paradox
In very short, Newcomb's paradox says, suppose you're a professor and a grad student (or, in some cases, a superintelligent alien) comes to you and demonstrates this experiment. She chooses a volunteer, examines them, then takes two boxes, puts £1000 in box A and either £1000000 or nothing in the box B (see below for how she decides). She brings the boxes into the room and explains the set-up to the volunteer and says that they're allowed to either take the mystery box (when they either get lots or nothing) or take both boxes (when they get at least £1000.)
She even lets them see the £100000 beforehand so they know it exists, and lets them peek into box A to show it does have the money in, though box B remains a secret until afterwards.
| Choice | In box A | In box B | Total obtained |
| B only | £1000 | £0 | £0 |
| Both | £1000 | £0 | £1000 |
| B only | £1000 | £1000000 | £1000000 |
| Both | £1000 | £1000000 | £1001000 |
"What's the catch," the volunteer asks. "Ah," begins the experimenter. "I have previously examined you, and worked out which choice you're going to make. If you were going to choose both boxes, I put nothing in box B. Only if you were going to take box B only, did I choose to put £1000000 in it.
"Hm", says the volunteer. "What do I do?"
A few caveats
"What if the volunteer would change their mind when they discovered the reasoning, or is going to choose based on a coin toss?" "Then I didn't accept them as a volunteer."
"How do I know it works?" You can't be sure, but she performs the experiment lots of times and is always right, so you are convinced. (Some examples ask you to presume as part of the conditions that she can, or take it on trust, but I think "having seen it work" makes it most convincing and concrete.)
"Ah, but I don't care about £1000, and certainly not if I've got £1000000, so I don't care," Well, ask what you would do if the numbers were a bit different. Can you pretend there's no combination where you'd risk something to get the little one, yet risk more to get the big one?
"How did they know what they'd pick when this experiment was performed the first time?" It doesn't really matter, just assume that you have seen it working with pretty-perfect prediction.
What would you do? An enumeration of the two obvious arguments.
11: "Why should you take both boxes?" Duh! Because whatever's in either of the boxes, you get all of it. And if that means I fail to get the million, then it's already too late to change that, isn't it?
22: "Why should you take box B?" Duh! Because you've just seen 50 people do the experiment, and all the ones who took both got £1000 and all the ones who took B got £1000000. Follow what works, even if you can't justify it with maths.
That's why it's a paradox, because, if you squint long enough, both answers seem perfectly reasonable.
I know this seems a little convoluted, but I tried to make it comprehensible if terse even to people without a very high opinion or training in philosophy (like me, in general). And hopefully get it to the point where at least asking the question makes sense.
Wait, if we use baysian reasoning, I bet the arguments will instantly become transparent and non-controversial. Right?
11: As above. Look at the table, and enumerate the possibilities. Choosing more boxes always gives a bigger payoff.
22: Ah, no, you're cheating. Based on the previous evidence, you must assume a priori that you are on row 2 or 3. After than, the choice is easy: row 3 gives more money. (See below for more "so, there's a 2/3 chance I'm in this universe..." type reasoning.)
Can we put this on a more rational footing? How does she predict what's going to happen
In fact, there are several ways.
1. You can do it if you postulate time travel, or determinism and a copy-teleport-machine, but those are not very realistic things to postulate, whether they would be physically possible or not.
2. A super-intelligent alien scans your brain and models it in a computer.
3. They give you a short-term-memory impairing drug, and try the experiment out several times beforehand while you remain the same person with the same experiences, but have no memory of the trials.
4. They discover that 94% of the time, all men choose one way, and all women choose the other. (But the experiment is double-blind, run by a technician who doesn't know the expected results, so the grad student peeks, then tells you that there's a 94% correlation, but not which way round it is, then invites you to participate yourself.)
Further arguments
33: Aha! In method 3, you don't know which one you are, one of the trial runs, or the final experiment. The only consistent answer which gets the big money is to assume you're more likely to be a trial run, and hence choose the money.
44: Aha! If so, then (*invokes Greg Egan*), the same reasoning applies with method 2. That suggests that you don't know if you're you, or the simulation of you!
55: Nope. Not so. What about method 4? Surely you can't claim that your consciousness might be either (a) you or (b) "the statistical correlation between gender and box choice"??
Which leaves us back where we started. (But remind me to come back to the "am I equally likely to be me, or some other human or simulation of a human".)
Free will
"What does it have to do with free will?" Well, the experiment is completely (sort-of) practical to do. In theory. And so you'd think it should be also actually possible to choose which to take. And yet it doesn't seem to be, and the answer seems to depend maybe on whether you believe in something you can call "free will".
In fact, people divide between take A&B, take B, and "problem is stupid, won't consider". In general, I think the last answer is often over-overlooked. In this case, if I'd seen it work out like that, I'd agree to take only box B, even if I couldn't explain the mathematics behind it. However, I also definitely feel I should be able to justify one case or the other.
Informally, it seems most people seem to eventually take B, but I don't know how important that is.
Apocrypha
Links to prisoner's dilemma, links to doomsday paradox, etc, etc.
no subject
Date: 2009-04-16 11:04 am (UTC)It's important to Newcomb's paradox that the predictor might in principle be wrong but it's very unlikely. If the predictor were 100% reliably, definitionally always right then it would have to be because there was causal contraflow, at which point one-boxing is a no-brainer. The in-principle possibility of a mistake arising from the causality constraint is what makes it tricky in the first place.
1. I don't feel right saying "100% accurate", actually, because what I want to say is "definitely never ever wrong in any conceivable situation" and I feel that "100% accurate" sounds too much like the weaker concept of "wrong with probability zero". The god we're hypothesising here isn't even wrong on an outcome set of measure zero, and I think that might actually be important!
no subject
Date: 2009-04-16 11:09 am (UTC)And indeed, might lead to all sorts of "it depends on the physics of timetravel, which we don't know yet".
no subject
Date: 2009-04-16 11:12 am (UTC)Um. What I was mostly trying to say was, 'I think people who have religious leanings like mine are more likely to go one-box immediately without hesitation or overthinking, because they already made a decision to trust something which they believe is almost certainly right in order to acquire a prize they haven't seen yet and which many people tell them there is uncertainty about, over obtaining a prize which is more certain or attempting to obtain both at once'.
no subject
Date: 2009-04-16 11:18 am (UTC)But it's interesting that your belief in God makes you immediately willing to one-box on the general Newcomb problem. You might trust God to make the right prediction, but surely that doesn't mean you would trust any other apparently accurate predictor as if it were God?