Newcomb's Paradox

[noAxioms]

@felipec

It’s the fastest way to find the correct answer for you, it’s not the fastest way to demonstrate it’s the correct answer for others.

Point taken

@felipec

That’s not the full code, I omitted the constants: n=1e6;p=0.99, they don’t change the conclusion.

I clicked on the link to run it. Saw those 2 lines. That didn’t let me know where an iteration is invoked.

One of the advantages of R is that you can do vector operations

Nice. Natural language for something like a Cray (which are all gone likely before R was invented). I actually saw one.

@felipec

Interesting. I presume you meant backward causation, I didn’t have to resort to that.

No, forward. The description is somewhere above. Really hard to search for it.

@felipec

For example if I’m the moderator of a forum there’s a strong temptation of banning people who say something I don’t like, but I force myself to not do that because I have a commitment to freedom of speech, no matter what

Hear ya. I am a moderator on a science forum (small pond), and yes, I run into that. The extent of moderation is mostly confined to keeping off-consensus opinions confined to a speculation section. This mostly involves moving posts/topics, not banning.

@felipec

So basically I have to be a deontological one-boxer.

Yes. The 1K box is poison. Leave it alone even if you see the other box empty. That attitude reduces any temptation, and yea, without eliminating it. The tension is part of the fun.

[Suny]

@felipec

Correlation does not prove causation

So you understand it. The funny thing is that, in Newcomb, we know the causality. I don’t know if your stance changed, but now you are telling me that yes, we want to look for the causality; that’s why we use the correlation. But one-boxers don’t do that; they go against the established causality and use the correlation as proof. We know how one-boxers get their money and it’s not by one-boxing, and when I asked you, you said it was not important.

@felipec

I create a device that displays heads or tails randomly when you press a button. However, of the past 50,000 people who have chosen heads, 99.992% have won, and of the past 50,000 people who have chosen tails, 0.012% have won

I am not sure I understand; there is only one button? What do you mean by “choose” heads? Isn’t it random?

If I can’t choose heads or tails, why would I bet that it will display tails? I am not sure what correlation you are using. There is correlation between the display and “winning,” but if I can’t even decide the display…

[felipec]

@Suny

The funny thing is that, in Newcomb, we know the causality

No. You pretend you know the relevant causality, you don’t.

@Suny

I am not sure I understand; there is only one button? What do you mean by “choose” heads? Isn’t it random?

Stop pretending you don’t understand the problem.

1. You place a wager with me (heads or tails)
2. You press the button
3. The wager is resolved

Everyone knows this is the setup of the problem given what I explained to you. Pretty much all bets in the real world happen in this fashion.

[Suny]

Oh I see. I guess I will bet on heads.

[felipec]

@Suny

Oh I see. I guess I will bet on heads.

Why? You have spent countless replies stating that causation is the only thing that matters, and in this problem there is no causation.

You placing a wager on heads is not going to “cause” you winning the wager. There is no direct causal link.

In Newcomb’s problem you completely ignore the correlation between your choice and winning $1M and in this problem you don’t ignore the correlation between your bet and winning?

Why do you ignore the correlation in one problem, but not in the other? This isn’t rationally consistent.

[Suny]

Well, you said there is a probability p that it lands on heads. It seems p is high given the past results. So I maximize my chances by picking heads. Betting on heads will lead to me winning more money than if I bet on tails (if it’s true that p is high). But yes, I don’t cause the thing to land on heads with my choice. That’s not the problem. In fact, in Newcomb too, I don’t cause one of the prediction outcomes with my choice; that’s why one should two-box.

Just like in Newcomb, there is a certain probability that there is a million in the box. And here the good thing is that, whatever that probability is, two-boxing always leads to more money.

[felipec]

@Suny

Well, you said there is a probability p that it lands on heads. It seems p is high given the past results. So I maximize my chances by picking heads. Betting on heads will lead to me winning more money than if I bet on tails (if it’s true that p is high).

Which is exactly the same in Newcomb’s problem. p is not defined precisely, but it’s stated as “almost certain”.

You also maximize your utility by choosing one-box:

	C1	C2
Simple	1M \cdot p	1M \cdot (1-p)
Newcomb	1M \cdot p	1M \cdot (1-p) + 1K

Note: to make my simple problem symmetrical I switched the bet: you earn $1M by betting on the correct outcome.

But yes, I don’t cause the thing to land on heads with my choice. That’s not the problem. In fact, in Newcomb too, I don’t cause one of the prediction outcomes with my choice; that’s why one should two-box.

You are not explaining anything.

• In both problems your choice is not directly causally linked with $1M
• In both problems your choice is correlated with $1M

Why are you ignoring the correlation in one, but not in the other?

[Suny]

@felipec

In both problems your choice is not directly causally linked with $1M

It is in the bet thing. My choice doesn’t influence the display “heads” or “tails” but my choice influence whether I win the bet or not.

[felipec]

@Suny

My choice doesn’t influence the display “heads” or “tails” but my choice influence whether I win the bet or not.

Your choice influences the outcome in both problems.

The only reason your have to believe that your choice influences the outcome is the correlation, and it’s exactly the same thing in Newcomb’s problem, where the correlation tells you: if you choose one-box you get $1M, if you choose two-box you get $1K.

There is no difference whatsoever. In one problem you get $1M if you choose heads, in the other you get $1M if you choose one-box. It’s that simple.

[hypericin]

Missed this.

@noAxioms

Game 3 has no agency (button ignored), and I think it important to give that back ASAP, so game 4 should be post-press guesser, not infallible.

“Button ignored” was poor wording, not sure if you got it anyway. The button still determines which slots open. But unlike 2, the button no longer determines the contents.

It’s strange to me that you equate the change from 100% to 99% reliability as giving the player “agency”.

@noAxioms

I don’t think it absurd to switch to TB once 99% accuracy is introduced. You say “a one percent difference should not affect preference”, but I disagree with that dismissal.

Why should it affect preference?

[Philosophiliac]

@felipec

But that’s what I’m saying: it’s Newcomb’s problem. The fact that some people call it “Newcomb’s paradox” doesn’t mean that it is a paradox.

I don’t disagree with you here we’re both on the same page I was just speaking colloquially to begin with but now that I understand your stance better we’re both in agreement.

@felipec

Just because a problem is self-referential that doesn’t necessarily mean that the reasoning to solve it ends on a loop. I actually wrote an essay on recursivity and Newcomb’s problem precisely to explore this difficulty some humans have. I gave examples to show that just because something is self-referential that doesn’t mean it’s a paradox.

I also agree that self reference on its own doesn’t necessarily mean something is paradoxical or reasoned poorly. For example Rene Descartes “Cogito ergo sum.” is fundamentally self referential and yet it holds fast as a nearly indubitable conclusion specifically because of that self referential nature. You can’t doubt something if you don’t exist is circular yet true. It self justifies itself.

I also think that certain things which are otherwise self negating can also produce unique truths that aren’t always apparent for example Gödel’s incompleteness theorems use self referential negation about the probability of a claim to show that the claim itself may be true but unprovable.

Even the example question you gave of

@felipec

If you choose an answer to this question at random, what is the chance that you will be correct? A) 25% B) 0% C) 50% D) 25%

Does seem to check out under a paraconsistent line of reasoning that the answer is B despite B resulting in contradiction regardless, the contradiction itself only continues to confirm B as the answer no matter how you try to spin it.

So again I don’t disagree with your stance here however I do think not every self referential system is valid or reliable just as the same goes for systems that aren’t self referential.

For example like I brought up previously the Halting problem where some machine-X hypothetically is perfectly designed to complete the task of determining if any machine will loop or halt its programming when running a given task answering the question “does this machine halt?” with an answer of YES or NO. We can then add an extension off of machine-X which when fed an input of YES proceeds to loop but when fed an input of NO will proceed to halt and we’ll call this extended system machine-Y.

If we then proceed to feed machine-Y into itself if the first part of machine-Y being machine-X evaluates the halting state of machine-Y as YES then machine-Y will not halt and will loop instead. Likewise if machine-X evaluates the halting state of machine-Y as NO machine-Y will halt instead.

This contradiction shows that such a machine is not possible in reality based on the self referential nature of machine-Y.

Sure we could circumvent this by simply choosing not to build such an extension off of machine-X to produce machine-Y but the specific scenario of machine-Y’s existence is the Achilles heel so to speak of the entire idea of machine-X

My point is that while yes you can circumvent the circularity of trying to predict the predictor by just picking the one box and not playing the predictive game this doesn’t eliminate the scenario where the predictions exist on both sides each dependent on one another and are going around endlessly without a solid resolution.

Yes I will grant the human in question is at a disadvantage in this game of predictions but this still does not outright exclude the possibility of such a loop.

Perhaps a more ideal hypothetical to demonstrate this at the extremes is to entertain the possibility of a near perfect predictor and its own exact duplicate. Predictor-1 is tasked to decide the contents of the boxes based on what it predicts predictor-2 will choose as in the original scenario. meanwhile predictor-2 must choose whichever option (one box or two) it predicts will contain the most money or otherwise pick the option that will guarantee the most money is present to possibly gain. In other words if predictor-2 predicts both boxes are full it will pick both boxes but since it knows the second box will be empty if it picks both boxes it will be forced to pick one box until it is sure that one box is full in which case it reverts to two boxing. Each predictor runs its prediction simultaneously and neither has an inherent edge over the other.

If we think of these as near perfect predictive programs simulating one another there’s no issue about free will hiding anywhere in these calculations and no need to assume conscious decision making is involved in any capacity just a series of logic states which immediately sway the decision of the program, no lie or false belief about what box would be picked just the result of an ongoing calculation and following where it leads.

In this scenario does the predictor-2 pick one box or both?

If predictor-2 predicts predictor-1 would leave the second box empty if it picks 2 boxes it will pick one box.

Once predictor-2 predicts predictor-1 will fill the second box if it picks one box it will pick two boxes

Once predictor-2 predicts predictor-1 will leave the second box empty if it picks two boxes it will pick one box
…
Ad infinitum…

Likewise if we ask weather or not predictor-1 would fill the second box or not,

If predictor-1 predicts predictor-2 would pick just the second box if predictor-2 knows picking both boxes would leave the second empty, it will fill the second box

Once predictor-1 predicts predictor-2 will pick both boxes if it knows that by picking only the second box both boxes will be filled, it will not fill the second box.

Once predictor-1 predicts predictor-2 would pick just the second box if predictor-2 knows picking both boxes would leave the second empty, it will fill the second box
…
Ad infinitum…

The way this loops over and over again no matter which predictor you start with the 99.99% reliability of both predictors is constantly being applied to the next iteration of predictions with a 0.01% margin of error each time. Eventually this margin of error compounds enough that the percent chance that one predictor or the other will achieve their desired outcome is about 50% for each meaning it’d be just as effective to flip one coin to decide if the second box is filled or not and another coin to determine if one or both boxes should be taken.

This itself demonstrates that in reality such a predictor while it may be effective in some scenarios is still prone to exploitation through self referential processes.

(sorry for the late reply I have been sick for the past several days. This said I am really enjoying this conversation and I want to reiterate that I really do agree with you on a lot of this involving self referential systems so I don’t want that to get lost here)

[noAxioms]

@hypericin

It’s strange to me that you equate the change from 100% to 99% reliability as giving the player “agency”.

I think that’s the point in the original scenario as described by Newcomb. There was a valid contradiction if the predictor is perfect. I’d have to look over the literature on the subject, but yes, agency comes up.

@hypericin

@noAxioms

I don’t think it absurd to switch to TB once 99% accuracy is introduced. You say “a one percent difference should not affect preference”, but I disagree with that dismissal.

Why should it affect preference?

It introduces CDT validity. I don’t think CDT is applicable at all given a perfect predictor. Of course the whole scenario is designed to demonstrate fault in CDT reasoning.

@felipec

I create a device that displays heads or tails randomly when you press a button. However, of the past 50,000 people who have chosen heads, 99.992% have won, and of the past 50,000 people who have chosen tails, 0.012% have won

That doesn’t sound very random. By ‘won’ I presume a prize if you guess the same as the device. That contradicts the device selection being random.
It’s consistent with participants randomly selecting heads or tails in equal quantities, and the device ‘randomly’ selecting heads 49996 times and tails only 4 times, which contradicts the suggestion of it being random.

Not sure what this example illustrates.

[felipec]

@noAxioms

That doesn’t sound very random.

You misunderstand what random means. Randomness describes how an outcome is generated – not what the outcomes are. This is random because each run is independent and the outcome is unpredictable.

Getting 6 on a die is random, despite the fact that it doesn’t happen 50% of the time. Getting 0 on a roulette is also very unlikely, yet random. Winning the lottery is random, even tough it’s one in a million.

Not sure what this example illustrates.

It shows you don’t need causality to guess what the outcome will likely be.

Notice how Suny is unable to explain why he expects the outcome to be heads even when there’s no causality, but he doesn’t expect his prediction to be correctly predicted in Newcomb’s, even though it’s much more likely. He cannot explain the inconsistency in his reasoning.

[noAxioms]

@felipec

This is random because each run is independent and the outcome is unpredictable.

…

If you shuffle 99 black cards and 1 red card, the color of the next card you get is random, despite the fact that it will be 99% likely to be black.

Winning the lottery is random, even tough it’s one in a million.

Both those examples are easily predictable, despite the randomness. It will probably be black, and I’ll probably not win the lottery. The roulette wheel probably won’t hit green, but the house kind of relies on it happening now and then.

When the outcome is mostly predictable, ‘winning’ is often billed as hitting the unlikely case, from the ‘player’ POV anyway. With two parties, one of the parties always wins, so in that sense, winning can be assigned to the predictable case. So say with the 100 cards, I win $1 if it’s black, and lose 20 if it’s red. Yea, I bet on black. Not sexy, but it’s rational.

[felipec]

@noAxioms

Both those examples are easily predictable

The likely outcome is predictable, but you cannot predict which specific runs will be the exception.

It’s still random.

When the outcome is mostly predictable, ‘winning’ is often billed as hitting the unlikely case, from the ‘player’ POV anyway.

Not in Newcomb’s problem. If you choose one-box you are almost certainly going to win $1,000,000, just like in my problem if you choose heads.

It’s only rational to bet what’s going to be the likely outcome almost certainly.

Two-boxers are betting on them being the exception, they are betting on them being the 0.01% that gets tails in my problem.

That’s why even Suny would choose heads, even if there’s no causation.

[Wistly]

Two boxers aren’t really betting on anything, they don’t see this problem as a bet. They know that the contents of the mystery box have been determined before they decided whether to pick one or two boxes. Which means that whether they pick one or two boxes at this point in time will have no effect on the presence or absence of a million dollars. The only choice they are capable of making in that moment is whether to take the free 1000$ or leave it.

The predictor is basically testing for a participant’s tendency to not factor in causality, and choosing to reward them for it. Which means in retrospect the supertitious strategy for this game appears to be logically coherent, if only because the game is designed to reward an incoherent strategy.

Yet in every case where someone wins a million dollars by picking one box, we know they left 1000$ on the table when they made their choice, in other words they chose poorly. It isn’t their choice to pick one box that caused them to win, it’s their natural propensity towards picking one box, since their choice cannot have affected the past and we know that the contents of the mystery box were determined before they chose.

If we conflate “the pre-existing preference of a participant towards picking one box” with “the actual decision the participant makes during the game”, then we can say that one boxing is the correct choice. But this isn’t really answering the question of what the correct choice is, it’s answering the question of what kind of belief system leads to the most desirable outcome in this game.

The heads and tails example is very different, because unlike in the Newcomb’s game where two boxing systematically rewards you with more money than one boxing given any starting set of conditions, choosing tails doesn’t have any advantage over choosing heads.

Your example is like if in Newcomb’s game, instead of choosing between one box that might contain a million dollars and one box that might contain a million dollars + a box that is guaranteed to contain 1000$, you were choosing between one box that might contain a million dollars or JUST one box that is guaranteed to contain 1000$. It’s a completely different type of problem because here the second choice isn’t systematically better than the first.

[felipec]

@Wistly

Two boxers aren’t really betting on anything, they don’t see this problem as a bet.

But it is a bet, regardless of your inability to see it.

The probability of an event doesn’t stop existing just because you decided to close your eyes.

Which means that whether they pick one or two boxes at this point in time will have no effect on the presence or absence of a million dollars.

Two-boxers keep repeating this fact as if repeating it would make it magically matter, but it doesn’t.

Betting on heads in my example also doesn’t have an “effect” on the device displaying heads, but that does not matter: the correlation still remains.

The predictor is basically testing for a participant’s tendency to not factor in causality,

No, it’s testing for a participant’s belief that correlation doesn’t matter.

Yet in every case where someone wins a million dollars by picking one box, we know they left 1000$ on the table when they made their choice, in other words they chose poorly.

Only if you ignore the fact that if the participant were to go for that “extra” $1,000, the predictor would have predicted that almost certainly.

Let’s do a counterfactual. Let’s assume you gather 100 two-boxers, how much money would your team gather? Well, it would be 100 * $1,000, or $100,000 (almost certainly). On the other hand my team of 100 one-boxers would gather $100,000,000.

In order for your team to win more money than my team, you would need the predictor to be wrong 100 times, that has a probability of happening of 10^{-400} (assuming the predictor has an accuracy of 99.99%).

So it is pretty much a certainty that your team cannot win against my team. But you argue your team is the “rational” one.

Of course, in order to argue that you have to redefine rationality, not as what maximizes utility, but as what maximizes utility ignoring probability.

This move (ignoring probability) is not warranted. The probability will remain regardless of the unwillingness of your team to see it.

The heads and tails example is very different, because unlike in the Newcomb’s game where two boxing systematically rewards you with more money than one boxing given any starting set of conditions, choosing tails doesn’t have any advantage over choosing heads.

That doesn’t matter. Let’s modify my example: if you choose tails you get an extra $1,000 no matter what. How does that change the analysis?

That extra $1,000 is just a distraction.

Just because your choice doesn’t affect the probability doesn’t mean that the probability doesn’t affect the outcome.

[noAxioms]

@Wistly

Two boxers aren’t really betting on anything, they don’t see this problem as a bet.

Welcome to the forum and to the topic. Your post was spot on I think.

@felipec

But it is a bet, regardless of your inability to see it.

It isn’t. The only thing that is probable is the contents of the opaque box, and you get that regardless of your choice. So your only actual choice of the moment is to take the K or not, and that part isn’t a bet. Hence it’s not only not seen as a bet, it actually isn’t a bet at all.

@Wistly

It isn’t their choice to pick one box that caused them to win, it’s their natural propensity towards picking one box

As has been said in earlier posts. My propensity is to one-box, and that propensity makes me leave 1000 on the table, even if both boxes are transparent. In a way the problem, worded this way, is an illustration of lack of free will. My propensity compels me to make what you label as an incoherent choice. And yet, had I free will, I’d fare no better, so in that sense, it isn’t illustrative of lack of free will.

[felipec]

@noAxioms

So your only actual choice of the moment is to take the K or not, and that part isn’t a bet.

Just like in my example your only choice is to pick head or tails. But there’s a probability involved, which makes it a bet.

Here’s a definition of “bet”:

• an amount of money that you risk on the result of an event or a competition

If you choose two-box you are risking losing $1,000,000 whether you acknowledge it or not.

[noAxioms]

@felipec

But there’s a probability involved, which makes it a bet.

Here’s a definition of “bet”:

• an amount of money that you risk on the result of an event or a competition

The definition you give doesn’t mention ‘probability’, but I’ll accept it. So I suppose you need to demonstrate that there’s a probability involved, especially in the transparent box scenario. You said you’d still OB in that situation (as would I), but it harder to spin it as a bet when the TB guy grabs both boxes.