Suppose you play this interrupted newcomb + lottery problem 1 jillion times, and the lottery outputs a suitably wide range of numbers. Which strategy wins, 1-boxing or 1-boxing except for when the numbers are the same, then 2-boxing?
Or suppose that you only get to play one game of this problem in your life, so that people only get one shot. So we take 1 jillion people and have them play the game—who wins more money during their single try, the 1-boxers or the 2-box-if-same-ers?
How can 1-boxers win more over 1 jillion games when they win less when the numbers are the same? Because seeing identical numbers is a lot less common for 1-boxers than 2-boxers.
That is, not only are you controlling arithmetic, you’re also controlling the probability that the thought experiment happens at all, and yes, you do have to keep track of that.
Which strategy wins, 1-boxing or 1-boxing except for when the numbers are the same, then 2-boxing?
The 2-boxers, because you’ve misunderstood the problem.
The thought experiment only ever occurs when the numbers coincide. Equivalently, this experiment is run such that Omega will always output the same number as the lottery, in addition to its other restrictions. That’s why it’s called the Interrupted Newcomb’s Problem: it begins in medias res, and you don’t have to worry about the low probability of the coincidence itself—you don’t have to decide your algorithm to optimize for the “more likely” case.
Or at least, that’s my argument. It seems fairly obvious, but it also says “two-box” on a Newcomb-ish problem, so I’d like to have my work checked :p.
I guess I’m not totally clear on how you’re setting up the problem, then—I thought it was the same as in Eliezer’s post.
Consider this extreme version though: let’s call it “perverse Newcomb’s problem with transparent boxes.”
The way it works is that the boxes are transparent, so that you can see whether the million dollars is there or not (and as usual you can see $1000 in the other box). And the reason it’s perverse is that Omega will only put the million dollars there if you will not take the box with the thousand dollars in it no matter what. Which means that if the million dollars for some reason isn’t there, Omega expects you to take the empty box. And let’s suppose that Omega has a one in a trillion error rate, so that there’s a chance that you’ll see the empty box even if you were honestly prepared to ignore the thousand dollars.
Note that this problem is different from the vanilla Newcomb’s problem in a very important way: it doesn’t just depend on what action you eventually take, it also depends on what actions you would take in other circumstances. It’s like how in the Unexpected Hanging paradox, the prisoner who knows your strategy won’t be surprised based on what day you hang them, but rather based on how many other days you could have hanged them.
You agree to play the perverse Newcomb’s problem with transparent boxes (PNPTB), and you get just one shot. Omega gives some disclaimer (which I would argue is pointless, but may make you feel better) like “This is considered an artificially independent experiment. Your algorithm for solving this problem will not be used in my simulations of your algorithm for my various other problems. In other words, you are allowed to two-box here but one-box Newcomb’s problem, or vice versa.” Though of course Omega will still predict you correctly.
So you walk into the next room and....
a) see the boxes, with the million dollars in one box and the thousand dollars in the other. Do you one-box or two-box?
b) see the boxes, with one box empty and a thousand dollars in the other box. Do you take the thousand dollars or not?
I’d guess you avoided the thousand dollars in both scenarios. But suppose that you walk into the room and see scenario b and are a bit more conflicted than normal.
Omega gave that nice disclaimer about how no counterfactual selves would be impacted by this experiment, after all, so you really only gets one shot to make some money. Your options: either get $1000, or get nothing. So you take the $1000 - who can it hurt, right?
And since Omega predicted your actions correctly, Omega predicted that you would take the $1000, which is why you never saw the million.
Right, which would be silly, so I wouldn’t do that.
Oh, I see what’s confusing me. The “Interrupted” version of the classic Newcomb’s Problem is this: replace Omega with DumbBot that doesn’t even try to predict your actions, it just gives you outcomes at random. So you can’t affect your counterfactual selves, and don’t even bother—just two-box.
This problem—which I should rename to the Interrupted Ultimate Newcomb’s Problem—does require Omega. It would look like this: from Omega’s end, Omega simulates a jillion people, as you put it, and finds all the people who produce primes or nonprimes (depending on the primality of 1033), and then poses this question only to those people. From your point of view, though, you know neither the primality of 1033 nor your own eventual answer, so it seems like you can ambiently control 1033 to be composite—and the versions of you that didn’t make whatever choice you make are never part of the experiment, so who cares?
Suppose you play this interrupted newcomb + lottery problem 1 jillion times, and the lottery outputs a suitably wide range of numbers. Which strategy wins, 1-boxing or 1-boxing except for when the numbers are the same, then 2-boxing?
Or suppose that you only get to play one game of this problem in your life, so that people only get one shot. So we take 1 jillion people and have them play the game—who wins more money during their single try, the 1-boxers or the 2-box-if-same-ers?
How can 1-boxers win more over 1 jillion games when they win less when the numbers are the same? Because seeing identical numbers is a lot less common for 1-boxers than 2-boxers.
That is, not only are you controlling arithmetic, you’re also controlling the probability that the thought experiment happens at all, and yes, you do have to keep track of that.
The 2-boxers, because you’ve misunderstood the problem.
The thought experiment only ever occurs when the numbers coincide. Equivalently, this experiment is run such that Omega will always output the same number as the lottery, in addition to its other restrictions. That’s why it’s called the Interrupted Newcomb’s Problem: it begins in medias res, and you don’t have to worry about the low probability of the coincidence itself—you don’t have to decide your algorithm to optimize for the “more likely” case.
Or at least, that’s my argument. It seems fairly obvious, but it also says “two-box” on a Newcomb-ish problem, so I’d like to have my work checked :p.
I guess I’m not totally clear on how you’re setting up the problem, then—I thought it was the same as in Eliezer’s post.
Consider this extreme version though: let’s call it “perverse Newcomb’s problem with transparent boxes.”
The way it works is that the boxes are transparent, so that you can see whether the million dollars is there or not (and as usual you can see $1000 in the other box). And the reason it’s perverse is that Omega will only put the million dollars there if you will not take the box with the thousand dollars in it no matter what. Which means that if the million dollars for some reason isn’t there, Omega expects you to take the empty box. And let’s suppose that Omega has a one in a trillion error rate, so that there’s a chance that you’ll see the empty box even if you were honestly prepared to ignore the thousand dollars.
Note that this problem is different from the vanilla Newcomb’s problem in a very important way: it doesn’t just depend on what action you eventually take, it also depends on what actions you would take in other circumstances. It’s like how in the Unexpected Hanging paradox, the prisoner who knows your strategy won’t be surprised based on what day you hang them, but rather based on how many other days you could have hanged them.
You agree to play the perverse Newcomb’s problem with transparent boxes (PNPTB), and you get just one shot. Omega gives some disclaimer (which I would argue is pointless, but may make you feel better) like “This is considered an artificially independent experiment. Your algorithm for solving this problem will not be used in my simulations of your algorithm for my various other problems. In other words, you are allowed to two-box here but one-box Newcomb’s problem, or vice versa.” Though of course Omega will still predict you correctly.
So you walk into the next room and....
a) see the boxes, with the million dollars in one box and the thousand dollars in the other. Do you one-box or two-box?
b) see the boxes, with one box empty and a thousand dollars in the other box. Do you take the thousand dollars or not?
I’d guess you avoided the thousand dollars in both scenarios. But suppose that you walk into the room and see scenario b and are a bit more conflicted than normal.
Omega gave that nice disclaimer about how no counterfactual selves would be impacted by this experiment, after all, so you really only gets one shot to make some money. Your options: either get $1000, or get nothing. So you take the $1000 - who can it hurt, right?
And since Omega predicted your actions correctly, Omega predicted that you would take the $1000, which is why you never saw the million.
Right, which would be silly, so I wouldn’t do that.
Oh, I see what’s confusing me. The “Interrupted” version of the classic Newcomb’s Problem is this: replace Omega with DumbBot that doesn’t even try to predict your actions, it just gives you outcomes at random. So you can’t affect your counterfactual selves, and don’t even bother—just two-box.
This problem—which I should rename to the Interrupted Ultimate Newcomb’s Problem—does require Omega. It would look like this: from Omega’s end, Omega simulates a jillion people, as you put it, and finds all the people who produce primes or nonprimes (depending on the primality of 1033), and then poses this question only to those people. From your point of view, though, you know neither the primality of 1033 nor your own eventual answer, so it seems like you can ambiently control 1033 to be composite—and the versions of you that didn’t make whatever choice you make are never part of the experiment, so who cares?