In this article I present my solution to the well known Necomb's paradox. I argue it arises because of incorrect ideas about free will and determinism. The paradox is solved by a correct understanding of free will.

This is a description of Newcomb's paradox:

A superior being, who has perfect prediction abilities, presents you with two boxes, one open and one closed. The open box contains $1000. The closed box may or may not contain $1,000,000. You are given the choice of taking only the closed box, or both boxes. Assume you are greedy and prefer as much money as possible. But here's the catch. If the being predicted you would take only the closed box, he has put $1,000,000 in it. If he has predicted you would take both boxes, he has put nothing in the closed box. What should you choose?

According to the dominance argument, given the fact that the prediction has already been made and the contents of the boxes are set, you should take both boxes. The closed box either does or does not contain $1,000,000 and your choice cannot change that, because your choice cannot change the past. Whether or not the closed box contains $1,000,000, you are better off by $1000 if you take both boxes instead of only one. According to the expected utility argument you should choose only the closed box, since the being will have predicted that so you get $1,000,000 while if you take both boxes, the being will have predicted that and you get only $1000.

Note that the assumption of the paradox is that the world is deterministic (or at least you are) so that all our future actions can in principle be calculated in advance. Whether the world is deterministic and wether our actions could be calculated in advance, is not important. The paradox still describes a hypothetically possible situation of a hypothetically possible world, so the logical paradox posed is a genuine philosophical problem.

Here is an added illustration of the argument for taking both boxes from Franz Kiekeben's article (due originally to Robert Nozick):

One can make the argument for taking both boxes even more vivid by changing the setup a bit. For instance, suppose that the closed box is open on the face opposing you, so that you can't see its contents but an experiment moderator can. The moderator is watching you decide between one box and both boxes, and the money is there in front of his eyes. Wouldn't he think you are a fool for not taking both boxes?

For Newcomb's problem to be a paradox, you have to assume there is no causality from your choice to the box contents. Also, for Newcomb's problem to be a paradox, you have to assume recursive causal links from the box contents to your choice are not possible. For example, if the being's prediction is revealed to me before I make my choice (or if both boxes are open) I could be such a person who would deliberately choose so as to invalidate the being's prediction. Under these conditions a being that makes accurate predictions cannot exist, so the paradox disappears because it contains an impossible assumption (of the being being able to predict perfectly). But what of the paradox otherwise? Given no backwards causality and no recursive forward causality?

First let me state, I agree with Franz Kiekeben's solution to the paradox:

If the being predicts in the manner of a scientist, that means that there is a certain state of affairs, A, which holds at some point in time prior to your decision and the prediction, and which causes both. This connection between the prediction and the decision is what prevents your actions from being probabilistically independent of the states of the box. And it is realizing this that makes it rational to take the closed box only (i.e., it is what invalidates the dominance principle).

I want to expand on this idea and argue that the paradox arises because of incorrect notions of free will and determinism. The question creating the paradox is: what is the right (best) choice? One possible answer is that given the assumption your choice doesn't cause the prediction, the right choice is to take both boxes. But because this choice has been predicted you end up with less than had you made the "wrong" choice and taken one box, which also would have been predicted. So it seems if you make the right choice, you get less than if you make the wrong choice. This leads to the contradiction that the right choice is the wrong choice and vice versa. Alternatively we can say the paradox evolves from the fact that our choice does seem to cause the prediction, even though we are assuming that backwards causality is not possible.

Let's first see what happens if we substitute a machine for the human making the choice. Our goal is to make a machine that will make such a choice so as to maximise its gain. In that case it's simple: we program the machine to take only one box. We know the being will predict how we created our machine, therefore we deliberately create our machine so the being predicts it takes only one box and the only way to do that is to make the machine such that it actually does choose only one box. But, at the moment the machine actually chooses only one box, it is in some sense the "wrong" decision, given our assumption that taking an extra box cannot cause what was already predicted. So how does that square with what seems obvious in the case of a machine, namely that if we're smart we would program the machine to take only one box? I think the solution is that in a strict sense the choice itself doesn't cause the prediction, but rather the fact that we created the machine to make a certain choice caused the prediction. This eliminates the problem of causality backwards in time. The actual act of choosing one box doesn't cause the contents of the box, because there is no backwards causality. But the machine's propensity, or will, determined in advance, to make such choices has caused the contents of the box. This does not create a problem of backwards causality, since the machine's will was fixed before the being made it's prediction.

We could actually test Newcomb's paradox this way. A programmer writes a program to make the choice. Another programmer (the "perfect being") reads the program (or does a trial run) and predicts what it will choose and fills the closed box accordingly. Then the program is run and we'll see indeed if it chooses one box it will find $1,000,000 under the closed box, and if it chooses two boxes, it will find the closed box empty. What is the right choice? Well, the programmers who write their program to choose one box will earn $1,000,000 and the programmers who write their program to choose two boxes will earn $1000. Given that the programmers are greedy, the programs choosing only one box are making the right choice.

What now if some magic ingredient is added so that the machine has consciousness and has at least the illusion of free will, but the machine operates otherwise in exactly the same predictable manner? Would this create a paradox? I think not. It doesn't change the mechanics of what's going on, nor does it change the conclusion of causality working in a normal forward fashion rather than backwards.

This, I think, solves the paradox in the case of a human as well, given the fact that a human is also a very complex machine. The right choice is to take one box for the human as well. But somehow this still doesn't seem to sit well with intuition. We can understand why we must build a machine so that it chooses to take only one box. But somehow it feels different with a human. With a machine we understand that he must choose one box, because that choice is programmed in advance so the being will predict it and puts the $1,000,000 in the closed box. But if a human is making the choice we perceive the choice as being made at the time of choosing by an act of free will rather than as something programmed in advance. And that's what creates the paradox of how our choice can seem to cause something that already happened. If Newcomb's paradox had been phrased in terms of a machine, as described above, it probably wouldn't have been seen as a paradox at all. So the paradox arises not because the logical situation described is paradoxical, but because the concept of free will is part of the story.

What happens if we are allowed to play this game, say, 100 times in a row, instead of just once. All 100 box sets are set up in advance according to the being's predictions. We might do some testing and would find that each time we choose both boxes, we get only $1000 while each time we choose one we get $1,000,000. So perhaps after several times we would learn and choose just one box each time. There appears to be a causal effect from our choice to the contents of the boxes. How can that be?

I think that while in a strict sense our choice itself doesn't cause the contents of the box, our propensity to make the choice does cause the contents of the box. But isn't that also a causal effect backwards in time? No. Our propensity to make a certain choice is determined before the being makes his prediction, just as with the machine. In order to predict what we will choose in the future, it must be fixed at that time what we are going to choose in the future. So the fact that we will want to make a certain choice in the future causes the contents of the box, not the act of the choice itself at the time of choosing. Our will, which is set in advance, causes the contents, not the actual act which happens later. Just as the actual choosing act of the machine doesn't cause the contents, but the fact that it was programmed to make that choice does. Even though our choice does not cause the contents of the box backwards in time, we should act as if it does. Because our will, which it has been assumed is knowable in advance, to make a certain choice does cause the contents of the box. And since it's impossible to make any other choice than the one we want, we could say that our choice really is the same thing as our will to make that choice. Since our will to make a certain choice does cause the contents of the box, because that will is fixed before the being makes his prediction, we might as well say that it is in fact our choice that causes the contents of the box. The causal paradox is thus solved by realizing that the choice is fixed before the being makes his prediction. Even though subjectively we feel we make the choice later, deteministically the choice was fixed, made if you will, beforehand. So I believe the answer to the question what we should rationally choose when presented with Newcomb's paradox is: choose only one box.

One who is presented with Newcomb's paradox and is inclined to take only one box, but has reservations because he cannot comprehend how his current choice can influence the past, might be advised to realize that since free will is subjective, it isn't really related to a certain time. I'm not saying free will is an illusion, I don't think it is (or maybe I'm saying there's no difference between al illusory free will and a real free will), but it's not a phenomenon one could measure in addition to all the actual physical phenomena. Therefore I think that though we can correlate our subjective timings of choices with objective times of physical events, I don't think that consciousness thoughts themselves as we experience them subjectively can be said to have any physical timing of their own, apart from that. So if I feel I choose the one box that doesn't mean that the subjective choosing happens at the time of the physical act of the choice. The objective choosing act as implemented in the mechanics of the physics happens at that time, but not necessarily the subjective choosing. We can just as well state that as the choice was already known by the being at the time of prediction, our subjective choosing happens at or before that instant. So although we feel like we are choosing now, we could say we are actually choosing at a time before the boxes are filled. Our subjective feelings would not be any different if all our subjective experiences happened 1000 years before or after the physical events. There is no test that could establish the actual timing of our subjective experiences, therefore the idea that they actually have a timing on their own is wrong.

This assumption is another way to eliminate in our minds the causal time paradox, so that we can feel alright making the choice for only one box from the view that the choice can and does actually cause the contents of the box, because we are allowed to assume the choice is really happening at the earlier time than the prediction. This doesn't mean the choice actually does happen at a specific time, at some point before the being makes his prediction. As said, I would rather say that subjective events don't have any time instant associated with them, except that they do correlate with the times of certain physical events, which I think is not entirely the same. But if one wants to assume a time instant of the actual act of free will anyway, then causality indicates one should pick a time instant which is at least not later than the moment at which the being makes its prediction.

I think the paradox is caused by a misunderstanding of free will, in particular a refusal to accept free will can coincide with determinism, a subject I have written about in another article. Let me illustrate this by first phrasing the paradox in other terms. The paradox can be seen to arise from the problem of how our fee will act now can apparently cause something that has already been determined (the contents of the box). Since the contents of the box correlate with our choice, this is equivalent to saying the problem is how our free will act now can apparently cause something that has already been determined (our choice). So the problem becomes: how can our choice at time X to do A cause our choice to do A, which has already been determined long before? Phrased in this self-referential way, perhaps the paradox seems less of a paradox.

Some people think Newcomb's paradox shows that free will cannot coexist with determinism. The paradox is solved in that view by saying the paradox cannot exist. The assumption of both free will and determinism leads to this paradox containing the contradiction that one can argue for the correctness of both choices, therefore the assumed combination of free will and determinism simply cannot exist. Though this is a possible view to take on the paradox, I do not think it is correct. Free will and determinism can coexist, because free will means only that I have the ability to choose what I think is best. By definition what I choose is what I think is the best choice. I don't have the ability to choose that which I think is not the best choice. The freedom part of free will applies to the fact that in a logical sense you are free to make another choice. The alternative choice is not illogical. You could have chosen it if you wanted to. But since you didn't want it you didn't choose it. And since your thinking processes and values could all be determined in a physical sense, the choices you make are determined. Bu they still are determined by you. Free will means that you determine your choices and not an external agent. But you are exactly the physical processes going on in your brain. For you those are free decision processes, and for an outsider they are deterministic physical processes. But whether or not you accept free will and determinism can coexist, I think in both cases the paradox is solved. If they cannot coexist, the paradox is solved simply by the realization that the assumed situation is impossible. And if they can coexist, I hope to argue in this post that there really is no contradiction and no paradox.

Here is the paradox in another form. Suppose the world is deterministic, so everything we will ever choose is fixed. Suppose a perfect superior being has predicted everything I will do in my life. He has written a biography with the complete story of my life, and the book is in my house in a vault which I cannot open. One day before I die the being will present me with the key of the vault so I can read the book and check that it is an accurate description of how I have lived my life. The question is: why should I do anything at all in my life? And there are two possible answers. The expected utility argument is: I should do the things I want to do, because that maximises my chances of getting what I want. The dominance argument is: how much I'm going to get is determined anyway as described in the book, and what I choose now cannot have any influence on that, since the book has already been written. So given the fact that my succes in life and hapiness are already set, I'm better off doing nothing at all, since at least then I don't get tired. To make a fuss and do all sorts of things is a stupid thing to do, as it has no influence whatsoever on what my life will be like, as already described in the book. So the man who thinks that just spends his life sitting in a chair, doing practically nothing except eating and drinking and going to the bathroom to stay alive, because it's too much against his instincts not to do at least that. He has a terrible boring life, then one day gets the key and reads his biography which says: you spent your whole life in a chair doing almost nothing. And then he gets out of his chair, goes to the movies, takes a nice walk, has some romance and sex, and the next day he dies.

On the other hand the man who argues according to the expected utility argument lives a life of fun and action and the day before his death he reads an accurate description of the full life he lead. Could the first man have made any other choice? He could have in a logical sense. He could have chosen otherwise. But not in a mechanical sense. In a physical mechanical sense it was determined he would have the belief he had and make the choices he did. But the contingency of free will is still valid. If you choose to go out and have a good life, you will have a better life than if you just sit in a chair. The first man made the wrong choice. And he chose it freely, based on his own ideas and beliefs, even if they were physically determined.

Now let's take away the being and the book. The paradox is still the same. The knowlegde someone might have that the world is deterministic still creates the same problem. According to the expected utitily argument one should live as full a life as possible. According to the dominance argument one might as well sit in a chair and do nothing, because all the things I choose to do are determined in advance anyway, and what I choose to do now can't possibly influence that. The fact that in this case there isn't a book with the story of one's life doesn't change the fact that one's life is already completely determined. Just as in Newcomb's paradox, here too the sensible choice is to get up and do fun things, even though in a strict physical sense what I choose now cannot influence what has already been set. But again the point is that the contingency still holds: if I do fun things I'll be better off, even though in some technical sense what I do now cannot cause what has been set in the past. This in Newcomb's paradox is equivalent to saying: if I choose one box I'll be better off. Here too the contingency still holds: if I choose one box I'll be better off, even though in some technical sense what I choose now cannot cause what has been set in the past.

Bringing the story about the book and free will closer to the original paradox, let's make a change to the situation with the boxes. Take away the being who can predict perfectly. Let's just replace him with a simple human agent. There is still $1000 in the open box, but there is definitely nothing in the closed box. However, after you have made your choice, the agent will or will not place $1,000,000 in the closed box, according to the same rules as before. If you choose both boxes, you will get the $1000 and nothing will be put into the closed box. But if you choose only the closed box, then the agent will place $1,000,000 in the box that time. So the contingency is still the same: if you choose both boxes, you will get $1000 and if you choose one box you will get $1,000,000. It would seem that even though the logic of that is the same, the paradox seems to disappear. In this case there is a clear causal link between your choice and the contents of the closed box. Since the box will be filled after you have made a choice, it seems the dominance argument that your choice cannot change the contents of the closed box doesn't apply. So the right choice seems to be to take only one box.

But is this situation really different from the situation with the perfect being and the closed box filled or not filled in advance? If we still assume determinism, I don't think there is a fundamental difference. In this second story, the dominance argument could still be used with equal validity. One could reason: my choice has already been determined before I make it. Therefore whether or not the agent will place the $1,000,000 in the box after my choice, has also been determined before I make my choice. Hence, whatever I choose, I cannot change whether the $1,000,000 will be placed in the box, because what I do now cannot change what has been determined before. So I should choose both boxes, and whether or not I will get the $1,000,000, I am better off by $1000.

Let's look a bit closer to see more clearly how this story (story 2) is not fundamentally different from the original Newcomb's paradox (story 1). In story 1 at some time 0 the being must collect very detailed information about the world and me. Perhaps he has a record of all particles in the world outside him and all their locations and other attributes. Call all that information A. And from that he is able to calculate exactly what will happen and whether I will choose one box or both boxes. Call that calculation X. Based on that prediction the being either places, or does not place, the $1,000,000 in the closed box. In story 2 there is no being who does calculation X in advance. But still the equivalent of calculation X must be done by nature. All particles in the world and in me do evolve in a deterministric way from time 0 to time t when I make my choice for one or two boxes. And the evolution of those particles can also be said to be equal to calculation X. At least in the sense that from information A this process arrives at the same conclusion as to what choice I will make. Just as two different ways to calculate 4 X 48 can also said to be equivalent if they yield the same outcome. Now, both in story 1 and in story 2 the fact whether there will be $1,000,000 under the closed box is determined by calculation X done on information A. The only difference between the two situations is that in story 2 the calculation is done once and the calculation starts at time 0 and ends at time t. While in story 1 the same calculation is done a second time as well, starting at time 0 and ending some time before t. And in story 1 the placing of the $1,000,000 is triggered by the end of the calculation X at some time before t (when the being does or does not place $1,000,000 in the closed box), while in story 2 the placing of the $1,000,000 is triggered by the end of the calculation X at time t (when I make my choice and the human agent gives or does not give me the $1,000,000 accordingly). But why should that make a difference?

Is there a rule that a paradox arises in case the money is placed before I make my choice? In that case the being could simply write down his prediction in advance and then wait with placing or not placing the $1,000,000 under the closed box. After I have made the choice, the being is called into the room, is not informed as to what I actually chose, looks on his note to see what he had predicted, and places or does not place the $1,000,000 in the closed box. This doesn't solve the paradox, because the placing of the $1,000,000 still depends on something that already was fixed before I made my choice, namely what was written on the being's note. So the paradox doesn't depend on the timing of the actual placement of the money.

Or is there perhaps a rule that a paradox arises in case the calculation X is finished before time t? In that case, let's replace the being with a machine M that is fed information A and which can do calculation X. The first box is filled with $1000. The closed box is filled with machine M and information A. Then I make my choice. I either open one box or I open two boxes. As soon as the closed box is opened, automatically M starts doing calculation X. After it has finished its calculation it indicates the result by displaying a green light or a red light. Green if it has calculated that I have chosen one box and red if it has calculated that I have chosen two boxes. If the light is green a human agents hands me $1,000,000. If the light is red I don't get anything (except the $1000 I might already have). Does this solve the paradox? I think not. I can still argue that whether M will go green or red is entirely set in advance by the A and X already present in the machine. So the closed hat already contains the fact of whether I will get $1,000,000 in advance and my choice cannot influence that. So I might as well decide to take both boxes and be better off by $1000. So the paradox doesn't depend on the timing of the calculation determing whether I get the $1,000.000. In fact, story 1 and 2 are essentially equivalent, because in both cases A and X determine both what I choose and whether the closed box will contain $1,000,000 and they are both knowable before I make my choice.

So what does the paradox depend on? The answer remains that there is no essential difference between story 1 and 2, and the fact that we tend to see a paradox in case of story 1 and not in case of story 2 is unjustified. There is no paradox in both cases.

Although some people, such as Robert Nozick and Martin Gardner have said they would take both boxes because of the dominance argument, I think they would probably change their mind if they were allowed to make the choice 100 times in a row as described above. Out of curiosity, they might want to see what happens if they "stupidly" choose only one box just a few times. And they would note that in those cases they made $999,000 more. Even though they don't understand the rationale for making the "wrong" choice, I think they could not resist the tempation to continue to make the "wrong" choice given the fact that they would observe it make them much richer.

So, if presented with Newcomb's paradox, I would advise everybody to take only one box. In fact my advice might be the cause of the both the reader's choice for one box, and the perfect beings prediction of that choice, in accordance with normal foward time causality (if you read this article after the being has already filled the box, the being will still have been able to predict you would read this article and base his prediction on that). You then open the closed box and find the $1,000,000. But you also can clearly see the $1000 which has been visible in the open box. So you may conclude now with certainly that both boxes were filled. So, even though you now have $1,000,000 can you not say it would have been better after all to choose both boxes? It now seems certain that had you done that, you would have had $1,001,000 instead of just $1,000,000. No. The mistake here is to draw conclusions given one course of events about a situation given another course of events. It is not valid to imagine an alternative course of events with the only difference that you took both boxes instead of one, with nothing else changed. The fact is that physically your choice of two boxes would only have been possible if the closed box did not contain the $1,000,000. It is invalid to conclude from the fact that the closed box contains $1,000,000 given the course of events that the being predicted you would take only one box and that you chose one box, that the closed box would therefore also have contained the $1,000,000 given the course of events that the being predicted you would take both boxes and that you chose both boxes. Had you chosen both boxes, the closed box would have been empty. But the being knew you would choose only one box, and therefore he filled it with $1,000,000.

Concluding, I believe the dominance argument is incorrect. It is incorrect by falsely assuming that the only way our choice could be causally connected to the contents of the box, is backwards in time, which is ruled out as impossible. The choice is in fact causally connected to the contents of the box in a manner that does not require backwards information flow. Our choice is knowable in advance, as it is the assumption of the paradox that the being can know our choice in advance. And so the fact of what we will choose "happens", if not the act of our choice, before the being prepares the box, in accordance with standard causality.

As I wrote, I think the paradox is still a valid philosophical problem even if the real world doesn't allow it. Having said that, it's still interesting to see whether the paradox can arise under the Multiple World Interpretation of Quantum Mechanics, or MWI in short. (The fact that I am considering this issue does not imply that I either agree or disagree with MWI.) I think in theory it can. In the paradox, the being would have to fill the closed box with the $1,000,000 only in those universes where you choose one box, and leave it empty in the paths where you choose both. But that is impossible, because at the time of prediction the being can make only one choice: to fill the box or not. This choice is then set for all proceding decohering alternative paths, both the paths where you choose one box and the paths where you choose both boxes.

But I think the problem can be solved if we rephrase the paradox. The being cannot predict whether you will choose one box or both, because you may make a different choice in some paths. But the being can predict in which fraction of paths following its prediction you will choose only one box. Then he will multiply that fraction with $1,000,000 and put that in the closed box. So, if he predicts you will choose one box in 75% of paths, he will put $750,000 in the closed box. The second change required in the paradox is that we assume you are not concerned with what you earn in your particular path. Assume you are a utilitarian who is not concerned with the wealth or your particular copy of yourself, but rather with maximizing the average wealth of all your copies as they diverged from the moment of the being's prediction.

Now the basic problem of the paradox is equivalent to before. The being makes his prediction and fills the closed box accordingly. What must you choose? First realize that your copy's choice is not necessarily the same as all your other copy's choices. Under MWI what your choice does only is increase or decrease the fraction of universes (as divered from the time of the prediction) where you choose one or the other.

According to the expected utility argument, the best choice is to choose only one box. For say this choice increases the fraction of universes where you choose one box by 1/n. Then the value of the closed box will increase by $1,000,000 X 1/n. And the fraction of universes where you don't get the value of the open box will will also increase with 1/n. So the average wealth lost by your copy not taking the open box is $1000 X 1/n. So, increasing the fraction of universes where you choose only one box increases average wealth (by an amount $999,000 X 1/n). So, the best choice is to choose only one box.

According to the dominance argument, the contents of the closed box is already set. Whatever you choose (however you influence the fraction of universes with a certain choice) cannot change this. So, if you can increase the fraction of universes where you choose both boxes with 1/n, on average you and your copies will be better off by $1000 X 1/n. So you should choose both boxes.

So, basically the paradox is still the same.

What, however, if you believe in Copenhagen rather than MWI, and so believe you are the only copy of yourself? In that case you should choose only one box if you estimate that choice will increase the fraction of universes where you make that choice by at least 1/1000. If you think the fraction of universes you are influencing is less than 1/1000, you should choose both boxes. (1/1000 is the cutoff point because above that fraction your choice increases the closed box by more that $1000, which is more than you can gain from taking the open box as well.)

Email: henry@sturman.net