Newcomb's Paradox
Years ago, the periodical _Scientific American_ had a regular column titled "Mathematical Games" authored by Martin Gardner. The topics ranged from the frivolous to the profound, and one of the columns that generated a great deal of feedback was the one on what is known as "Newcomb's Paradox". A paradox is a presentation of two or more statements or arguments which appear to be correct, yet also appear to contradict one another. Newcomb's Paradox is interesting although it is deceptively simple. Of its two obvious conclusions, one can be convincingly argued on the basis of mathematics while the other, contradicting conclusion can be convincingly argued on the basis of physics or metaphysics.
Here is the paradox:
You have been selected to participate in a simple experiment that requires you to make a choice involving two boxes. These two boxes are sitting on a table in a room and the one of the left is labeled "Box A" while the one on the right is labeled "Box B". Box A is closed and it contains either ZERO dollars (euro, pounds, etc.) or ONE MILLION dollars. Box B contains FIVE THOUSAND dollars. Incidentally, Box B is open and you can see the cash. Box A is opaque and you can't see inside. There is no trickery involved, the boxes' contents do not change, the money is real, no hypnosis, no illusions, etc.
Your goal is to maximize the amount of cash you get to take out of the room. You have two choices:
Choice 1: Take Box A
Choice 2: Take Box A and Box B
To help you make your choice, you are first properly informed as to the nature of the experimentor. The experimentor (or should I write "Experimentor" with a capital "E") is known to have run this same experiment many times in the past. Furthermore, the experimentor is known to have a remarkably good (perfect?) record in predicting how you will choose. To make things interesting, if the experimentor believes that you will choose Choice 1 (Box A only), then the experimentor will have placed ONE MILLION dollars inside Box A. Otherwise, if the experimentor believes that you will choose Choice 2 (Box A and Box B), then the experimentor will leave Box A EMPTY.
The experimentor is a fair player; once the boxes are set up, they are not changed. The contents are fixed BEFORE you make your choice.
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Argument 1: (for Choice 1) The experimentor's record is very, very good. It may be perfect. As far as you know, the experimentor has NEVER made a wrong prediction. It may not be possible for the experimentor to make a wrong prediction. The experimentor may be some super-intelligent computer. It may be some incredibly advanced alien with mind-reading ability that knows all your thoughts before you enter the room. It may be God.
1.1: If the experimentor is correct and you pick Box A (Choice 1), you will get US$1,000,000.
1.2: If the experimentor is correct and you pick Box A and Box B (Choice 2), you will get only US$5,000.
Since there is no evidence that the experimentor has ever been wrong, it is obvious that you want to pick Box A (Choice 1) and get your easy MILLION.
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Argument 2: (for Choice 2) When you enter the room and make your choice, the MILLION dollars is either in Box A or it isn't. Regardless of what you choose, nothing will change the status of Box A. It was fixed before you went into the room. It can be shown mathematically that picking both boxes (Choice 2) is superior. Proof:
2.1: If the MILLION is in Box A, you will get US$1,005,000 total. This is better than US$1,000,000 which you what you would get by picking Box A (Choice 1), although not by much.
2.2: if the MILLION is not in Box A (Box A is EMPTY), you will get US$5,000. This is better than US$0, which is what you would get by picking Box A (Choice 1), although not by much.
Therefore, regardless if the MILLION is there or not, you are better off by taking both boxes (Choice 2).
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I will state that by basic game theory, the proof for Choice 2 is mathematically correct. Choice 2 is in this case called a "dominating strategy"; you can look it up. Given that the status of Box A is absolutely fixed before you make your choice, you cannot "will" or "wish" the contents of Box A one way or the other with your decision. No matter what the experimentor's prediction, the experimentor is out of the game before you make your choice. The MILLION is there or it isn't. In both possibilities of the status of Box A, you will get US$5,000 more by picking Choice 2 (both boxes). You have the free will to make the choice. No one to my knowledge has seen a hole in this proof.
But does mathematics give the complete answer? Are you willing to gamble away a MILLION by betting against the experimentor? Everyone who has tried Choice 2 in the past has, as far as you know, would up with an empty Box A! And they all seemed so sure of themselves, too!
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The deeper questions in Newcomb's Paradox go beyond mathematics. The first questions raised here are "Could such an experiment logically take place? Is there some internal contradiction? Are the premises consistent with reality?"
If the experiment is somehow impossible, why?
Can such an experimentor exist? Is true omniscience possible? Actually, perfect omniscience is not needed, the experimentor still "wins" if he/she/it/them is/are correct a "sufficient" percentage of the time.
Can free will exist? If it can, then why do those who pick both boxes always or almost always wind up without the MILLION? If the experimentor always or almost always predicts correctly, doesn't he/she/it/them determine the subjects "choice" before it is made? If so, is it really a choice?
Does causality exist? Is it possible that the subject DOES have free will and that when making the choice, somehow transmits the choice BACKWARDS in time to the point when the experimentor makes his/her/its/their "prediction"?
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By picking only Box A, you are obligated to point out the flaw in the mathematics behind the dominating strategy of picking both boxes. No one has been able to do this.
By picking both boxes, you are obligated to explain why there cannot be a sufficiently accurate experimentor. Perhaps this is the same a saying that it is impossible for God to run such an experiment, or even that it it impossible for an omniscient God to exist.
Which choice would you make?