Newcomb's Problem

[Gogf]

By the way, just to clarify Perfy's solution, since some people don't seem to be understanding...

Most of the arguments for box b require backwards causality, something which we generally don't believe to exist. However, by creating an infinite regress of imaginary Perfs (how does imaginary Omega decide what is in the boxes for imaginary Perf?), Perf instead sets up a system where his rule determines what is in the boxes.

Interestingly enough, although it would seem as though Perf would have to know this rule before he was put into this situation, that's not true either. If the simulation is indeed a perfect (ha!) representation of Perf, then it too would logically reach this solution, so Perf can determine this rule after the money is already in the boxes and be confident that imaginary Perf also independently created it.

 

[Perfection]

If you're a perfect-for-all-practical-purposes simulation, you presumably will care about yourself just as much as the real Perfection cares about himself, and therefore the assumption that the SimPerfs desire to be good to the Real Perf makes no sense. With Newcomb's Problem, the goal presumably is to maximize one's own reward (if we allowed for benevolent concerns, then hell, even with the original problem, we could say it's better to only get $1000 so that Omega has more money), so A&B is the right choice even if you're a simulation (assuming no reverse-causality). Granted, you say that picking just B need not be altruistic, since if you misbehave the other Perfs might retaliate (at least that's how I interpret your last few sentences), but that only makes sense if this game is iterated more than once.

Well, both aren't acting in classical self-interest (doing what always gets you more) but enlightened self-interest (doing that which is the optimal solution for all game players) where pacts and promises can be made for the benefit of all. Despite Gogf's protests, I do believe it is at its heart playing prisoner's dilemma with yourself.

He does however show that englighted self-interest is not simple altruism and does give one selfish benefits.

 

[Darth_Pugwash]

Not really. Think of it this way: I flip a coin 10,000 times. Each time before I flip it, I predict what the result will be. Miraculously (and purely by luck), I am correct EVERY SINGLE TIME. I have had a 100% success rate thus far. Next, I predict the next coin toss and write down my prediction. I then ask you if you think my prediction will be accurate. If you pick right, I will give you $100. Does the expected value of me being correct ($100 minus a tiny fraction) mean that you should always pick that, despite the fact that there is in fact a 50-50 chance for each choice?

How is it likely that he will have a greater than 99% success rate in the future? If you include the past success rate, then of course that's true (assuming that there have been 99 or more "games" before), but your statement becomes meaningless and tells us nothing if you include the old success rate. In other words: how do you know that, considering only all future "games," that the future success rate will likely be 99% or greater?

Okay, so in the question we do not know Omega's methods, however, given the success rate, is it really wise to suppose that the prediction is coming from a straight 50/50 probability split? The chance of getting a 10,000 streak on a 50/50 chance is so absurdly small that it's not really worth considering for any reasonable agent. Given the success rate of Omega, and given that we can only guess at his actual method, what is wiser - to suppose that omega is making 50/50 coin tosses to predict, or that Omega is using some superior method, say mind reading, to determine his predictions?

Whatever his method is, thus far it has proved damned reliable - and as such the chance that it comes from a 50/50 coin toss is very, very small. I'm no good with numbers, maybe it's like a thousanth of a percent or something for 10,000 succesful coin flip predictions. The chance that Omega's predictions come from a method with a good - say as low as 80% for 100 successful predictions - chance of success is much, much higher.

Imagine a man who bet on the winning horse in 100 horse races in a row. Would you suppose that he is flipping a coin to pick his bets, or do you suspect that he is getting inside tips, or fixing the races? If you had to bet on a horse, would you pick a different horse to him?

And frankly, if I watched someone predict 10,000 coin flips in a row, I would assume that they were blessed by the spaghetti monster and ask them to suggest me some lottery numbers.

If you're a perfect-for-all-practical-purposes simulation, you presumably will care about yourself just as much as the real Perfection cares about himself, and therefore the assumption that the SimPerfs desire to be good to the Real Perf makes no sense. With Newcomb's Problem, the goal presumably is to maximize one's own reward (if we allowed for benevolent concerns, then hell, even with the original problem, we could say it's better to only get $1000 so that Omega has more money), so A&B is the right choice even if you're a simulation (assuming no reverse-causality). Granted, you say that picking just B need not be altruistic, since if you misbehave the other Perfs might retaliate (at least that's how I interpret your last few sentences), but that only makes sense if this game is iterated more than once.

So it still seems like the validity of picking only Box B depends on reverse-causality.

Well, if we start by saying that if RealPerf found himself in the box picking scenario, he would suppose that it might be a simulation. By extension, he would consider that he himself may be a simulation. He has no way of knowing. If he is a simulation, then he can presume that his existence will be terminated immediately after he picks a box - therefore he has no reason to be selfish. So, he figures, either I am SimMe, and when I pick a box I will disappear, or I am RealMe, and when I pick a box I will get real money and spend it on ice cream.

SimPerf and RealPerf would reason in exactly the same way, and since both care about RealPerf more than SimPerf, they will always pick box B. Why? They would reason like this:

If I am SimPerf, then I cannot be selfish as almost by definition, as I most likely have no future beyond the box game. So, what is the next best thing to looking out for SimPerf's interests? Why, looking after RealPerf's interests of course! So if I am SimPerf, then I should act in RealPerfs interests and pick box B!
If I am RealPerf, then SimPerf picked box B, and box B has $1,000,000 in it!

The clever bit is that it doesn't matter which Perf is which, since both Perfs pick box B!

But yeah I might be way wrong on this, it's late and I'm tired.

 

[Defiant47]

Okay, here's why I think you can't apply the math in the way that you are: while his accuracy in the past does, in some abstract way, suggest that he will be accurate in the future, I don't think we can assign a probability of accuracy to that and say anything meaningful.

His accuracy in the past does suggest his accuracy in the future. My problem is how exactly the probability density function works: that's what I would really like to know.

For example: A series of coin flips happen, say 1000. A person is able to predict 852 of them. Running this through our (unavailable) function, we yield that it's 30% likely he has a 85% predictive ability (i.e. will continue to have), 2% likely that he has a 97% predictive ability, 0.001% likely that he has a 30% predictive ability, and so on*.

As such, I assume (but I'm pretty sure the correct but unavailable calculation would also yield this) that Omega is most likely to have a 99% predictive ability given that he's been right 100/100 times.

* - Of course, the chance of any one result with a probability density function is 0, but I put in values to make it easier to understand for those who haven't taken advanced statistics.

Economics and statistics work on the assumption that with enough iterations everything will work out, but you only get to choose the box once. In other words, there is no reason to believe that expected value is equal to value in real terms here.

Whether you get to choose once or more, has no effect on the utility of the expected value. Indeed, a 10% chance to win $1000 does not give you $100... but it should be valued at $100.

If we KNEW that there is a 99% chance that the alien will be right, then yes, of course we could use expected value as a reasonable metric of which box to pick if we had no other. However we do not know this.

See above. Technically, we can say we know it (with a reasonable degree of certainty... even more once we get our actual function).

No matter whether he is right or not, he will be more than 99% right in his predictions because of his previous correctness. This tells us nothing.

... this tells us that he will be more than 99% right in his predictions. We can use this, say round down to exactly 99%, and evaluate expected values for both of our choices, and thus yield a conclusive result as to which choice is optimal.

I agree that the data suggests that he is very good at predicting. However, I do not agree with you that we can take this and coherently apply it in an analytical context to get a mathematical "value" for each box.

The data suggests that he is very good at predicting. If we were very good at math, we could say exactly how much he is good at predicting. Then we could use that and derive expected values, and yield a value for each choice.

So basically what you're saying is that you deal with the problem of induction by assigning a percent chance that induction works in this case? I do not think that is a rigorous response to the problem of induction.

I do not see where this "induction" comes in. We can't show that he will be correct on the nth person, so it's useless. This is a statistical analysis instead.

Um, no we wouldn't. We know that he is a "super-intelligent" alien. As you alluded to you reply to my coin-flipping analogy, it is vastly more likely that he has some sort of prediction system than that he is randomly picking either box. Perf's explanation covers just about any prediction system I can think of.

The issue is that Perfection's explanation is only a way to get two-boxers to realize the possibility of the one-box decision. I endeavour to evaluate the entire situation given unknown prediction methods.

Also, arbitrarily assigning a percentage of likely accuracy like you are doing here does not make sense.

It's not arbitrary. See above.

 

[Gogf]

Well, both aren't acting in classical self-interest (doing what always gets you more) but enlightened self-interest (doing that which is the optimal solution for all game players) where pacts and promises can be made for the benefit of all. Despite Gogf's protests, I do believe it is at its heart playing prisoner's dilemma with yourself.

He does however show that englighted self-interest is not simple altruism and does give one selfish benefits.

No way! It's definitely very similar to the prisoner's dilemma, but the crucial difference here is that there is no second round, and no way to "strike back" at the imaginary Perf. The really important point (and the one that I think most people are missing) is that this works even though you can't respond to unhelpful behavior from other Perfs.

Maybe that does qualify as the prisoner's dilemma, and I guess that's fine. The point I care about isn't what we call it, it's that this works even though you can't hurt imaginary versions of yourself who hurt you.

 

[WillJ]

Edit: I realize I've cross-posted. I'll read the posts above now.

NO! The duality of Perfs is NOT the prisoner's dilemma! These decisions are simultaneous, based on the rules that Perf has set out. If the rule is "screw other Perfs," then all Perfs get screwed by the infinite regress of imaginary Perfs. However, if the rule is "be kind to other Perfs," then all Perfs benefit.

Yes, collectively they benefit, but for one individual Perf, it's in his best interests to pick A&B, whatever any other Perf picks. Ergo, it IS the prisoner's dilemma. Let's say you're RealPerf: if SimPerf picks B, you should pick A&B; if SimPerf picks A or A&B, then as RealPerf, you shouldn't pick B, and picking A or A&B will have the same result --- so if you're RealPerf, picking A&B is the best option, because there are possibilities when this is better than the other options, and there is no possibility in which this is worse than another option. And if you're SimPerf, it doesn't matter what you pick, since I'm assuming the simulation isn't going to get any reward. So without knowledge of which one you are, you should pick A&B.

Thus, unless I'm missing something, Perf's construction hardly does anything to point us to picking just B, unless we allow for altruism (in which case we've abandoned Newcomb's Problem), or the game is iterated more than once (and even then only assuming the Perfs have a retaliatory spirit) --- or unless we're choosing what ALL of the Perfs do, rather than just one individual, in which case we've abandoned Newcomb's Problem.

 

[Gogf]

His accuracy in the past does suggest his accuracy in the future. My problem is how exactly the probability density function works: that's what I would really like to know.

For example: A series of coin flips happen, say 1000. A person is able to predict 852 of them. Running this through our (unavailable) function, we yield that it's 30% likely he has a 85% predictive ability (i.e. will continue to have), 2% likely that he has a 97% predictive ability, 0.001% likely that he has a 30% predictive ability, and so on*.

As such, I assume (but I'm pretty sure the correct but unavailable calculation would also yield this) that Omega is most likely to have a 99% predictive ability given that he's been right 100/100 times.

* - Of course, the chance of any one result with a probability density function is 0, but I put in values to make it easier to understand for those who haven't taken advanced statistics.

I'm not just asking you to tell me what this imaginary function would tell you. I'm asking you to tell me why this function is a valid tool for evaluating the value of various options.

Also, "I'm using the number that I think the function would give me" is NOT a convincing argument.

Whether you get to choose once or more, has no effect on the utility of the expected value. Indeed, a 10% chance to win $1000 does not give you $100... but it should be valued at $100.

Why?

See above. Technically, we can say we know it (with a reasonable degree of certainty... even more once we get our actual function).

WHY?

... this tells us that he will be more than 99% right in his predictions. We can use this, say round down to exactly 99%, and evaluate expected values for both of our choices, and thus yield a conclusive result as to which choice is optimal.

You are assuming that the 101 tries that will have happened after we choose our box represent the statistical norm, and I don't see why that should necessarily be true. This is the best way that we can come up with a coherent statistical representation of all this, sure, but that doesn't mean that the best statistical representation is actually a very good representation at all.

(By the way, if you know Perf's Rule, then the expected value of box b is $1,000,000 because you know that's what it will contain!)

The data suggests that he is very good at predicting. If we were very good at math, we could say exactly how much he is good at predicting. Then we could use that and derive expected values, and yield a value for each choice.

NO WE COULD NOT! We could not say exactly how good he is at predicting, no matter how could we were at math! There is always uncertainty and there is always the possibility that the sample does not represent the statistical norm. However, even if we COULD exactly predict his accuracy that, the two logical arguments (both boxes always contain more money vs. I can ensure that there is money in box b) would trump any wishy-washy, probability-based statistical model of value.

I do not see where this "induction" comes in. We can't show that he will be correct on the nth person, so it's useless. This is a statistical analysis instead.

If you looked up the problem of induction, you would see that it states that we cannot know that past results necessarily predict future results. When you make statements like "we could say exactly how much he is good at predicting," you assume that past results exactly predict or determine future results, and we have no reason to believe that that is true.

The issue is that Perfection's explanation is only a way to get two-boxers to realize the possibility of the one-box decision. I endeavour to evaluate the entire situation given unknown prediction methods.

Perfection is giving a system that will work as long as the alien is using a reasonable and accurate prediction system. If he is randomly determining the box that you will pick, then a statistical analysis might be more valid, but as you already said, it appears as if he is not doing this.

It's not arbitrary. See above.

Yes it is. You are arbitrarily using it in lieu of the results of some function that you claim would support your argument. However, even if the function churned out the same number, you still have to reply to the plethora of other issues I've raised, such as why this number is relevant whatsoever.

 

[Gogf]

Edit: I realize I've cross-posted. I'll read the posts above now.

Yes, collectively they benefit, but for one individual Perf, it's in his best interests to pick A&B, whatever any other Perf picks. Ergo, it IS the prisoner's dilemma. Let's say you're RealPerf: if SimPerf picks B, you should pick A&B; if SimPerf picks A or A&B, then as RealPerf, you shouldn't pick B, and picking A or A&B will have the same result --- so if you're RealPerf, picking A&B is the best option, because there are possibilities when this is better than the other options, and there is no possibility in which this is worse than another option. And if you're SimPerf, it doesn't matter what you pick, since I'm assuming the simulation isn't going to get any reward. So without knowledge of which one you are, you should pick A&B.

Thus, unless I'm missing something, Perf's construction hardly does anything to point us to picking just B, unless we allow for altruism (in which case we've abandoned Newcomb's Problem), or the game is iterated more than once (and even then only assuming the Perfs have a retaliatory spirit) --- or unless we're choosing what ALL of the Perfs do, rather than just one individual, in which case we've abandoned Newcomb's Problem.

What you're missing is that all the Perfs will always make the same decision. If you (real Perf) choose to take both boxes, so will all the imaginary Perfs. That means that you get only $1,000. On the other hand, if you choose only box b, then so will all the other Perfs, and you will be rewarded with $1,000,000.

The rule and thought process that lead you to your decision are what matters, not the actual decision itself.

 

[Darth_Pugwash]

Gogf is right - the trick is that SimPerf is exactly like RealPerf, and no Perf can second guess or bluff out another Perf! I think.

 

[El_Machinae]

Even if you're better off with A&B, realistically $1000 is not much incentive to cheat.

 

[Gogf]

Gogf is right - the trick is that SimPerf is exactly like RealPerf, and no Perf can second guess or bluff out another Perf! I think.

Bingo! Every Perf does the exact same thing.

 

[WillJ]

Well, if we start by saying that if RealPerf found himself in the box picking scenario, he would suppose that it might be a simulation. By extension, he would consider that he himself may be a simulation. He has no way of knowing. If he is a simulation, then he can presume that his existence will be terminated immediately after he picks a box - therefore he has no reason to be selfish. So, he figures, either I am SimMe, and when I pick a box I will disappear, or I am RealMe, and when I pick a box I will get real money and spend it on ice cream.

SimPerf and RealPerf would reason in exactly the same way, and since both care about RealPerf more than SimPerf, they will always pick box B. Why? They would reason like this:

If I am SimPerf, then I cannot be selfish as almost by definition, as I most likely have no future beyond the box game. So, what is the next best thing to looking out for SimPerf's interests? Why, looking after RealPerf's interests of course! So if I am SimPerf, then I should act in RealPerfs interests and pick box B!
If I am RealPerf, then SimPerf picked box B, and box B has $1,000,000 in it!

The clever bit is that it doesn't matter which Perf is which, since both Perfs pick box B!

But yeah I might be way wrong on this, it's late and I'm tired.

But if SimPerf picks Box B, RealPerf should pick A&B, not just B ($1,001,000 vs. $1,000,000).

What you're missing is that all the Perfs will always make the same decision. If you (real Perf) choose to take both boxes, so will all the imaginary Perfs.

SimPerf chooses, Omega bases his prediction on that, and THEN RealPerf chooses. What you are suggesting here is reverse-causation, and the point of Perfection's construction was to avoiding needing that.

He does however show that englighted self-interest is not simple altruism and does give one selfish benefits.

But I don't see how, in this case. Heck, before, I said that that's only true with repeated iteration, but now I don't even think it's true even with repeated iteration, because the fear of retaliation comes in if you're SimPerf and pick A&B, then RealPerf is screwed, but it's not like RealPerf can do anything about it, unless next time RealPerf becomes SimPerf (and chooses first) and vice versa.

 

[WillJ]

Even if you're better off with A&B, realistically $1000 is not much incentive to cheat.

Not much of an incentive, but still an incentive. A solution isn't very elegant when it relies on saying, "Eh, well $1000 isn't that much...", now is it?

 

[Darth_Pugwash]

Say RealPerf makes a decision to pick box B, and then Omega simulates RealPerf. We now have SimPerf, and since SimPerf is exactly like RealPerf, he will choose box B. So then, Omega goes to RealPerf, and presents him with two boxes. RealPerf also picks box B!

So the causation is RealPerfs initial choice, before the simulation was run.

 

[Defiant47]

I'm not just asking you to tell me what this imaginary function would tell you. I'm asking you to tell me why this function is a valid tool for evaluating the value of various options.

The calculations would be very complex, especially because this function would provide a pdf, which is continuous (I can provide more details in this regard if you want). Why it's a valid tool? This is because it would evaluate the overall expected value of each choice. Then we could see which choice is optimal.

Also, "I'm using the number that I think the function would give me" is NOT a convincing argument.

It isn't, but showing that predictive abilities above 50.05% implies that it's better to choose box B, forgoes the need of great accuracy in my estimation. I can use 90%, if you'd like, though that would be grossly low.

Why?

Why is expected value the best way to evaluate something? This is because we need to look at the big picture.

Suppose that you have an infinitesimal (say 1/2^100) chance to win a billion dollars. However, you have to buy this chance with a million dollars. Should we look at only a few possibilities, or should we look at the whole picture? Sure, for that one time in 2^100, you'll end up 999 million dollars richer! But that doesn't mean that taking the chance is the logical thing to do. You have to evaluate, given a large number of you doing the same thing, what does each you end up with on average.

Likewise, we have to evaluate the expected value of this situation:

If a million one-boxers have 1000000*$990000 distributed amongst them on average, and a million two-boxers have 1000000*$11000 distributed amongst them on average, then it's better to be a one-boxer.

WHY?

Well, if we don't say we know it (given that it's the most likely), then we'd have to go through those complex calculations mentioned above.

You are assuming that the 101 tries that will have happened after we choose our box represent the statistical norm, and I don't see why that should necessarily be true. This is the best way that we can come up with a coherent statistical representation of all this, sure, but that doesn't mean that the best statistical representation is actually a very good representation at all.

That's why we have varying degrees of "goodness" of representation. If the alien predicted 2/2 successfully, then the best statistical representation would show that he is very likely to be often correct, but the variance and potential error would be enormous. With 100/100 data, we the error is significantly lessened.

(By the way, if you know Perf's Rule, then the expected value of box b is $1,000,000 because you know that's what it will contain!)

If you "know" that's what it will contain, then the alien's predictions are 100% correct, something that cannot be derived from the fact that he's been right 100/100 times.

NO WE COULD NOT! We could not say exactly how good he is at predicting, no matter how could we were at math! There is always uncertainty and there is always the possibility that the sample does not represent the statistical norm. However, even if we COULD exactly predict his accuracy that, the two logical arguments (both boxes always contain more money vs. I can ensure that there is money in box b) would trump any wishy-washy, probability-based statistical model of value.

That's exactly the thing. We could say exactly how good he is with an exactly level of uncertainty. I.e. there's a 30% chance that he is 99% correct (and this is the highest chance). Basically, we could evaluate the likelihood of each of these prediction rates: 1%, 2%, 3%, ..., 99%, 100% ,but with real numbers instead of integers.

If we were good at math, we could say the exact level of uncertainty for each exact level of "goodness" that he is. Then we could evaluate the entire situation.

If you looked up the problem of induction, you would see that it states that we cannot know that past results necessarily predict future results. When you make statements like "we could say exactly how much he is good at predicting," you assume that past results exactly predict or determine future results, and we have no reason to believe that that is true.

Past results do not determine future results (not in a mathematical sense). However, they do predict them. We can predict future events based on past events, with a certain level of uncertainty.

Perfection is giving a system that will work as long as the alien is using a reasonable and accurate prediction system. If he is randomly determining the box that you will pick, then a statistical analysis might be more valid, but as you already said, it appears as if he is not doing this.

Perfection is assuming an arbitrary system: the alien is using simulated versions of you to figure out your response. I'm trying to ignore any such arbitrary declarations.

 

[El_Machinae]

Not much of an incentive, but still an incentive. A solution isn't very elegant when it relies on saying, "Eh, well $1000 isn't that much...", now is it?

Sure it does! The entire game would change if the values in the boxes were switched. Or if the values were varied.

 

[Fifty]

If I were to to come across Omega I would note that Omega has demonstrated a profound accuracy rate. I would presume that he has a sort of simulated version of me. This simulation would be just as capable of making choices as me, taking into the same things into account as me. In a lot of ways, the simulation is me. In fact I can't be sure that I am not the simulated version of myself in Omega's head!

When people come up with overly cute solutions like these, Paradoxers like to ask things like this:

Consider Newcomb's Paradox Plus, which is exactly like Newcomb's Paradox except you know that Omega's prediction mechanism is such that no simulation exists.

What would be your answer to Newcomb's Paradox Plus?

 

[Defiant47]

When people come up with overly cute solutions like these, Paradoxers like to ask things like this:

Consider Newcomb's Paradox Plus, which is exactly like Newcomb's Paradox except you know that Omega's prediction mechanism is such that no simulation exists.

What would be your answer to Newcomb's Paradox Plus?

I love you Fifty!

 

[Darth_Pugwash]

Way to play the game, fifty.

 

[Gogf]

But if SimPerf picks Box B, RealPerf should pick A&B, not just B ($1,001,000 vs. $1,000,000).

Untrue. If real Perf picks both boxes, then imaginary Perf also did, because both Perfs always do the same thing (they are, after all, exactly the same). Therefore, because imaginary Perf picked both boxes, there is no money in box b, and picking both results in only $1,000 for real Perf.

SimPerf chooses, Omega bases his prediction on that, and THEN RealPerf chooses. What you are suggesting here is reverse-causation, and the point of Perfection's construction was to avoiding needing that.

In this post I explain why this does not require reverse causation .

But I don't see how, in this case. Heck, before, I said that that's only true with repeated iteration, but now I don't even think it's true even with repeated iteration, because the fear of retaliation comes in if you're SimPerf and pick A&B, then RealPerf is screwed, but it's not like RealPerf can do anything about it, unless next time RealPerf becomes SimPerf (and chooses first) and vice versa.

There has to be a way that imaginary Omega determines what is in the boxes for imaginary Perf. This necessitates imaginary imaginary Perf! Of course, this chain continues, and we have an infinite regress of Perfs. What this means is that there always exists a Perf "one level higher" than whatever imaginary Perf we are considering at the moment. Therefore, because all of the Perfs choose the same thing, taking both boxes is harmful to everyone.