How to Solve Newcomb’s “Paradox”



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How to Solve Newcomb’s “Paradox”

There is only one solution

Photo by Jan Genge on Unsplash

A godlike being — let’s call her Omega — presents you with two boxes. Box A is open and contains €1,000. Box B, however, is closed; Omega tells you it either contains €1,000,000 or nothing at all. You have a choice: take both boxes, or only take box B.

Easy, right? You don’t know what box B contains, but whatever it contains, taking both boxes (“two-boxing”) will get you €1,000 more than taking only box B (“one-boxing”).

But wait, there’s a catch: Omega has predicted what you would do. Of course, you don’t know her prediction; Omega, however, does tell you that she put the €1,000,000 in box B if and only if she predicted that you would one-box.

Let’s assume Omega can perfectly predict your choice in this dilemma, and has played this game 1000 times before and always predicted accurately: all one-boxers found €1,000,000 in box B, while each two-boxer only earned €1,000. In order to win as much money as possible, should you one-box or two-box?

The “paradox”

The supposed paradox in this problem — called Newcomb’s problem — comes from the fact that there are two arguments that both seem reasonable, but lead to opposite conclusions.

The “expected utility” argument

Historically, one-boxers have earned more than two-boxers: because Omega always predicts a player’s choice accurately, all one-boxers have made €1,000,000 while each two-boxer earned only €1,000. Based on this, you’re more likely to get a big payoff if you one-box; this argument therefore suggests one-boxing.

The “strategic dominance” argument

Strategic dominance might sound complicated, but we already saw this argument in the introduction of the problem. It goes as follows: when you make your choice, Omega has already either put €1,000,000 or nothing in box B. The contents of this box are now fixed. Whatever box B contains, getting both box A and B will always get you €1,000 more than just getting box B. This argument therefore says you should two-box.

The solution

The strategic dominance argument fails to incorporate the link between the player and Omega: Omega predicts what the player will do. This essentially means that a situation where you two-box and find €1,000,000 in box B is impossible: Omega will have predicted you would two-box and kept box B empty. One-boxing and not getting the €1,000,000 is impossible as well, because of that same link.

So it’s the expected utility argument that wins? No, that argument just “got lucky”: it has the right conclusion, but the wrong reasoning. The argument rests on a statistical relation — a correlation — between one-boxers and getting €1,000,000, but as you might now, a correlation doesn’t necessarily imply causality. The advent of cold weather might cause both increased glove sales and higher frostbite rates, but from this you shouldn’t conclude not to buy gloves — even though glove buying is correlated with frostbite. Similarly, while there is a correlation between one-boxing and earning €1,000,000, that alone is not enough reason to one-box.

So what is the correct reasoning then? The crucial point in Newcomb’s problem is the predictive power of Omega. This power isn’t magic: you have it when you predict what a calculator will answer when you feed it “2 + 2”. And that’s the point: you know how to add numbers, and therefore know — on a functional level, not necessarily on a technical level — how the calculator comes to its answer “4”. Similarly, Omega can predict your decisions, and therefore seems to have a functional model of how you make decisions.

If you and your calculator are both perfect at adding numbers, there’s no way for your calculator to give a different answer than you predicted on any given addition problem. Likewise, Omega is perfect at predicting your answer; therefore, you can’t possibly decide to two-box while she predicted you’d one-box or vice versa. What you decide is what Omega predicted you would decide, in a way; two-boxing means Omega predicted you would two-box, and therefore means there’s nothing in box B, while one-boxing means there’s €1,000,000 in box B. Therefore, the only correct choice is to one-box.

If you liked this analysis, consider visiting my publication How to Build an ASI. For now, thanks for reading!

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