Newcomb's paradox

[StrugglingSceptic]

I still cannot see the paradox.

I believe there is a genuine contradiction. The player has a rational argument that they should pick B2, even though they know the sum of the money in both boxes exceeds that in just B2.

I am not sure why the predictor needs to be fallible. The paradox seems to me to be present even if we assume that the predictor always makes perfect predictions.

My interpretation of the paradox is not that prediction is impossible -- this entire set up could be formalised in terms of a theorem-proving machine as I outlined above, and the predictor would still always make perfect predictions. I suspect this is actually a paradox of self-reference.

 

[David Gould]

I believe there is a genuine contradiction. The player has a rational argument that they should pick B2, even though they know the sum of the money in both boxes exceeds that in just B2.

How do they 'know' this? The predictor places the money based on their prediction of the player behaviour. Therefore, if the player picks both boxes, they know that more than likely that there will no money in B2.

I am not sure why the predictor needs to be fallible. The paradox seems to me to be present even if we assume that the predictor always makes perfect predictions.

My interpretation of the paradox is not that prediction is impossible -- this entire set up could be formalised in terms of a theorem-proving machine as I outlined above, and the predictor would still always make perfect predictions. I suspect this is actually a paradox of self-reference.

Can you explain the paradox to me? I just cannot see one.

 

[StrugglingSceptic]

How do they 'know' this? The predictor places the money based on their prediction of the player behaviour. Therefore, if the player picks both boxes, they know that more than likely that there will no money in B2.

They know that there is more money in both boxes based on arithmetical facts alone. The sum of two positive numbers exceeds either one of those numbers.

Can you explain the paradox to me? I just cannot see one.

I'll assume the predictor is infallible, because I think it still works in this case.

If I were to decide to open B2, then B2 would contain a million pounds, and I would earn a million pounds. If I were to decide to open both boxes, then B2 contains nothing, and I would earn a thousand. It appears that a decent "should choose" logic would tell me to open B2.

However, basic arithmetic tells me that there is always more money in boxes B1 and B2 together than in B2 alone, and so it appears a decent "should choose" logic would tell me to open B1 and B2.

This is a contradiction, and not one I know how to solve because I have not really thought about what a "should choose" logic would look like, and I believe such a thing is essential to a formal analysis of the problem.

 

[David Gould]

Some things are a little clearer now.

First strategy:

1.) No matter what the predictor predicts, there will always be more money in both boxes than in B alone.
2.) Therefore, it is best to pick both boxes.

Second strategy:

1.) If you pick both boxes, chances are that the predictor predicted that, and thus you only get $1,000.
2.) If you pick box B, chances are that the predictor predictd that, and thus you will get $1,000,000.


I do not see the paradox here as the first strategy ignores important information and hence leads to a less optimal outcome.

 

[David Gould]

I am now on the same page with you, so to speak. However, does not the strategy of choosing both boxes ignore information? How is ignoring information a good strategy?

In other words, for what logical reason should I use the strategy that ignores information over and above the strategy that includes it? (This assumes I want to maximise the money I get).

A flat analysis of probability tells me that the strategy of choosing box B is better - much better in the case of the predictor always being right.

 

[StrugglingSceptic]

I am now on the same page with you, so to speak. However, does not the strategy of choosing both boxes ignore information? How is ignoring information a good strategy?

In other words, for what logical reason should I use the strategy that ignores information over and above the strategy that includes it? (This assumes I want to maximise the money I get).

Because additional information cannot change the validity of an argument. It seems to me that both strategy A and B constitute valid arguments, and since they yield contradictory conclusions, there is a paradox.

The player wants to earn as much money as possible. The total monetary value of the contents of B1 and B2 must exceed that of B2. They earn the monetary value of whatever boxes they choose to open. Therefore, they will earn more money by choosing to open both boxes, and so this is what they should do.

To avoid the paradox, you must identify a flaw in this argument (or the argument corresponding to the other strategy).

A flat analysis of probability tells me that the strategy of choosing box B is better - much better in the case of the predictor always being right.

 

[David Gould]

Because additional information cannot change the validity of an argument.

Um, yes, yes it can.

It seems to me that both strategy A and B constitute valid arguments, and since they yield contradictory conclusions, there is a paradox.

The player wants to earn as much money as possible. The total monetary value of the contents of B1 and B2 must exceed that of B2. They earn the monetary value of whatever boxes they choose to open. Therefore, they will earn more money by opening both boxes, and so this is what they should do.

To avoid the paradox, you must identify a flaw in this argument (or the argument corresponding to the other strategy).

The flaw is that the first premise - the player wants to earn as much money as possible - is set aside in the first argument. In other words, if they player truly wanted to earn the most money possible, a simply probability analysis would demonstrate to them that choosing B2 is the only strategy that would indeed meet the criteria of premise one.

In other words, when you slot numbers into the equation, B2 is the logical choose as it provides you with the most money.

If you ignore the numbers - ignore information - it should be no surprise that the other logical argument is valid.

Basically, what I think is that in order to solve this you have to know about both strategies and compare them. Once you do that, there is no problem.

 

[David Gould]

In other words, logically, if your wish is to maximise your money you should evaluate these two strategies in terms of the average return they give you.

As such, there is a logical means of choosing between them, even though they are each logically valid.

 

[Osiris]

As such, there is a logical means of choosing between them, even though they are each logically valid.

But it is illogical to say that you will get $1,001,000 if you picked both boxes when the predictor is omniscient.

 

[David Gould]

But it is illogical to say that you will get $1,001,000 if you picked both boxes when the predictor is omniscient.

But that is not what they are saying. They are saying this:

1.) No matter what the predictor predicted, the total of both boxes will be higher than the total in box B.

If he predicted that I would pick box B, there will be 1,000,000 in box B and 1,001,000 in the combined boxes.

If he predicted that I would pick both boxes, there will be nothing in box B and 1,000 in the combined boxes.

As such, it is better to pick both boxes.


As I said, this ignores information, which is why it fails as a beneficial strategy. That is also why it fails logically - logically, if you want to get the most money you have to analyse the chances of the predictor predicting correctly.

 

[Osiris]

I believe there is a genuine contradiction. The player has a rational argument that they should pick B2, even though they know the sum of the money in both boxes exceeds that in just B2.

But it is illogical to expect $1,001,000 when the predictor is infallible.

I am not sure why the predictor needs to be fallible. The paradox seems to me to be present even if we assume that the predictor always makes perfect predictions.

If the predictor is fallible, how is it different if I were the predictor?

It is nothing but probabilities... if the predictor is capable or error would poker be a paradox as well?

My interpretation of the paradox is not that prediction is impossible -- this entire set up could be formalised in terms of a theorem-proving machine as I outlined above, and the predictor would still always make perfect predictions. I suspect this is actually a paradox of self-reference.

I think the paradox may be when both the predictor and the player are omniscient.

- player would pick both.
- predictor knows this and would put only 1000 in A
- player knows this and would pick only B (empty) proving predictor wrong.
- predictor would know this and would put 1,000,000
...

there be no answer unless both come into an honest agreement.

 

[Osiris]

But that is not what they are saying. They are saying this:

1.) No matter what the predictor predicted, the total of both boxes will be higher than the total in box B.

If he predicted that I would pick box B, there will be 1,000,000 in box B and 1,001,000 in the combined boxes.

If he predicted that I would pick both boxes, there will be nothing in box B and 1,000 in the combined boxes.

As such, it is better to pick both boxes.

I know, I understood that part...

It is logical to choose anything... but expecting 1,001,000 isn't.

As I said, this ignores information, which is why it fails as a beneficial strategy. That is also why it fails logically - logically, if you want to get the most money you have to analyse the chances of the predictor predicting correctly.

I agree, and the information they leave out causes their expectations to be illogical.

and I am now with you... I don't see any paradox in this paradox as how it is.

 

[David Gould]

Bloody hell, it took me a long time to understand this thing. I must be getting very, very old.

 

[StrugglingSceptic]

Um, yes, yes it can.

It cannot. A valid argument is one in which every conclusion follows from the premises by a rule of inference that preserves truth. The argument:

P1) P=>Q
P2) P
C) Q

is valid, and it can never be made invalid by adding additional information.

The flaw is that the first premise - the player wants to earn as much money as possible - is set aside in the first argument. In other words, if they player truly wanted to earn the most money possible, a simply probability analysis would demonstrate to them that choosing B2 is the only strategy that would indeed meet the criteria of premise one.

In other words, when you slot numbers into the equation, B2 is the logical choose as it provides you with the most money.

If you ignore the numbers - ignore information - it should be no surprise that the other logical argument is valid.

Ignoring information cannot make an argument valid.

Basically, what I think is that in order to solve this you have to know about both strategies and compare them. Once you do that, there is no problem.

If the two strategies constitute valid arguments, then there is a paradox by definition, because they yield contradictory conclusions. You cannot simply ignore one and claim to have solved the paradox. You must identify an invalid step in the argument, or else show that the problem was malformed to begin with.

Bloody hell, it took me a long time to understand this thing. I must be getting very, very old.

Careful. Paradoxes and other logical problems often contain many subtleties and admit of many different analyses. I am sure there were plenty of people who thought they understood the liar's paradox in the thousand years since the Greeks named it, but our understanding massively increased in the last century when we could analyse it in formal logic.

 

[t_w]

could you answer this: how could you choose something where an omniscient being won't know your choice? if you could then as David Gould said, the predictor was not omniscient to begin with...

I think that you've entirely missed the point of the paradox. The being with omniscience is assumed. If such an assumption provides several contradictions, then that assumption is wrong. Don't bother arguing that last point - it is a fact. It is one of the most widely used scientific methods - e.g. the Bell theorem assumes several things and then is proved wrong by contradictions. The being's omniscience is disproved by the paradox.

 

[t_w]

This to me still does not look like a paradox, as the premises used in each strategy are different. If the being is 90 per cent accurate, a strict probability analysis tells you that the 2nd strategy is the best one in terms of maximising your money.

I'm glad you've seen the two logical arguments. However I don't think you've taken into account the fact that the money is already sealed. Our decision doesn't change this. So logically, we should get more money if we choose both, and if we choose only B2. Our choice doesn't change the being's prediction. This is the exact mistkae Osiris made.

 

[t_w]

As I said, this ignores information, which is why it fails as a beneficial strategy. That is also why it fails logically - logically, if you want to get the most money you have to analyse the chances of the predictor predicting correctly.

No, it doesn't ignore information. It is a perfect argument for both maximising your money and in terms of logic(as is Osiris's of course...hence the paradox). Look at it this way. The being has placed the money in the box. Now, the choice you make won't change the amount in the box, so you should take both. There is absolutely no flaw here. You nearly have this paradox understood - you are simply assuming backward-acting causailty, as most people do(including me).
I will explain it the best I can. If the being makes 10 predictions twice, one set for you and one set for me(with the same predictions in each set), and then you choose B2 every time and I choose both every time I will earn more than you, even if he is omniscient!

 

[t_w]

In other words, when you slot numbers into the equation, B2 is the logical choose as it provides you with the most money.
If you ignore the numbers - ignore information - it should be no surprise that the other logical argument is valid.
Basically, what I think is that in order to solve this you have to know about both strategies and compare them. Once you do that, there is no problem.

See my post above to understood why every you have written here is wrong.

 

[Osiris]

I think that you've entirely missed the point of the paradox.

I hope we are talking about the paradox using an omniscient being.

The being with omniscience is assumed. If such an assumption provides several contradictions, then that assumption is wrong.

what would be the contradiction?

we assume an omniscient being, he predicts your choice... where is the contradiction? that you will always pick both? he will know that, remember, he's omniscient...

it is logical to pick both boxes... but it is illogical to expect $1,001,000 by picking both...

with an omniscient being, you will never get $1,001,000 -- there is no paradox.

Don't bother arguing that last point - it is a fact.

while it is true that when two contradictory things that are right happens it is a paradox... but we don't have that here.

It is one of the most widely used scientific methods - e.g. the Bell theorem assumes several things and then is proved wrong by contradictions. The being's omniscience is disproved by the paradox.

The being's omniscience hasn't been disproved...

he knows your choices, he's omniscient....
if you pick both boxes, you will never get $1,001,000 out of it... that's a fact.

 

[levi501]

If the being has 100% predicitibility power and knows that you are aware of his power, the choice is easy... you only take box B... because you knowing of his power for perfect accuracy becomes a superstition of backwards causality. The supreme being will predict your disregard of logic and place the money in box B.

If the being is fallible... you take both.