Newcomb’s Paradox: A Simple Thought Experiment

A Version of this article is published in my book, ‘TWISTED LOGIC: PUZZLES, PARADOXES, AND BIG QUESTIONS’. CRC Press/Chapman & Hall, 2024. https://www.amazon.co.uk/Twisted-Logic-Puzzles-Paradoxes-Questions/dp/1032513349

A Thought Experiment

Newcomb’s Paradox, also known as Newcomb’s Problem, is a thought experiment involving a choice between two boxes: one transparent containing $1,000 and one opaque that may contain nothing or $1 million. The twist? A highly accurate Predictor has already decided what’s in the opaque box based on what it thinks you will choose. It is a dilemma first proposed by William Newcomb, a theoretical physicist, and popularised by philosopher Robert Nozick.

The Setting of Newcomb’s Paradox

In this setting, the simple choice is being taking both boxes or just taking the opaque box. The Predictor, which is well known for its accuracy, determines the content of the opaque box based on a prediction about your decision. If the Predictor forecasts you will take both boxes, it places nothing in the opaque box. On the other hand, if it predicts that you will only take the one box, a sum of $1 million will be deposited inside. By the time you make your decision, the Predictor’s choice about the opaque box’s content has already been made. So, should we take both boxes or just the one opaque box? You could also change the amounts to update to modern day prices or in some other way and ask yourself whether it changes anything.

Two-Boxers vs. One-Boxers: The Great Debate

Essentially, Newcomb’s Paradox has divided people into two distinct camps, each adhering to a different way of looking at the problem. These factions, known as ‘two-boxers’ and ‘one-boxers’, represent different facets of decision-making theory and reflect different approaches to rational choice.

Two-Boxers: The Dominance Principle and Causal Decision Theory

Two-boxers advocate that the most rational decision is to take both boxes. This perspective aligns with the principle of dominance in decision theory, which states that if one action produces a better outcome than another in every possible scenario, then that action should be chosen. In the case of Newcomb’s Paradox, two-boxers argue that the Predictor’s decision—having already determined the content of the opaque box before you choose—cannot be influenced by your subsequent choice. This means that choosing both boxes can’t make you worse off. In the worst-case scenario, you have the guaranteed $1,000 from the transparent box, and in the best-case scenario, you could walk away with an additional $1 million if the Predictor failed in its prediction.

Two-boxers also fundamentally subscribe to causal decision theory. They reason that since your decision doesn’t cause a change in the already-decided contents of the opaque box, it’s only rational to maximise the guaranteed gains, which means taking both boxes. This standpoint portrays the logic of irreversibility, where past events (the Predictor’s decision) cannot be influenced by future actions (your choice).

One-Boxers: Evidential Decision Theory and Trusting the Predictor

Conversely, one-boxers argue for a different interpretation of rationality. They propose that the sensible decision, given the Predictor’s uncanny accuracy, is to take only the opaque box. They reason that, although the contents of the box have been decided, the Predictor’s ability to forecast accurately makes it likely that the opaque box contains the $1 million if you choose it alone.

One-boxers point to the track record: almost every participant who opted for two boxes found the opaque box empty, while the opposite was true for those who took only the opaque box. Hence, they argue that it’s not about changing the past, but about leveraging the evidence that shows a strong correlation between the decision to pick one box and winning the million dollars.

In essence, one-boxers align with evidential decision theory, which suggests that we should make decisions based on the best available evidence for the desired outcome. In the context of Newcomb’s Paradox, taking only the opaque box is based on what has happened in the past to those who took one box and two boxes, respectively.

In this way, the paradox challenges our notions of causality and rational decision-making. Can our current choice affect a decision that’s already been made? Or does the Predictor’s accuracy mean it’s better to trust the pattern of past outcomes?

Split Decision

The paradox thus splits decision-makers into two groups: ‘two-boxers’ and ‘one-boxers’, each advocating for a different decision based on their own logic.

Two-boxers argue that the rational decision is to take both boxes. As the Predictor’s decision about the content of the opaque box is already determined before you choose, your choice can’t change the contents. This implies that no matter what, you won’t be worse off taking both boxes. The least you can get is the $1,000 from the transparent box, and at best, you could get an additional $1 million if the Predictor predicted incorrectly.

On the other hand, one-boxers argue that the sensible decision, considering the Predictor’s near-perfect track record, is to take only the opaque box. They point out that almost everyone who has taken two boxes has found the opaque one empty, while those who took only the opaque box won the million dollars. Thus, based on the evidence, it seems sensible to become a one-boxer.

The decision-making here presents a fascinating conflict between reason (which seemingly lacks supporting evidence) and evidence (which seemingly lacks rational explanation). It essentially raises the question: should we trust the evidence of a well-documented pattern or rely on the rational logic of decision-making?

Causality: The Predictor and the Future

The first crucial point to clarify in Newcomb’s Paradox is the nature of causality at play. The scenario eliminates any notion of backward causality or retro-causality; your choice does not affect the Predictor’s prior decision nor alter the content of the opaque box. This stipulation aligns with our typical understanding of time’s arrow: the future does not influence the past.

The Predictor’s decision is a genuine prediction and doesn’t involve any time-travelling. It infers your choice before you make it, but it doesn’t ‘react’ to your decision.

Predictability: Unravelling the Accuracy of the Predictor

The Predictor’s high accuracy complicates the decision-making process. If you tend to be a two-boxer, you might think that it’s likely the Predictor has foreseen this and left the opaque box empty. Conversely, if you lean towards one-boxing, you might believe that the Predictor has probably predicted this and filled the opaque box with the million dollars.  The paradox then becomes less about the boxes you choose and more of a high-stakes mind game where you try to leverage the Predictor’s uncanny accuracy for your gain.

Identity: The Person You Choose to Be

This leads to another fascinating dimension: the intersection of predictability and identity. If the Predictor can predict your decision based on your inherent nature, then maybe the real strategy lies in manipulating your own disposition to game the system. The question then evolves from ‘which box should you choose?’ to ‘who should you choose to be?’

In essence, if you aspire to secure the million dollars, the optimal strategy might be to become the type of person who would always choose one box. The paradox suggests that by firmly committing to this decision, you make it likely for the Predictor to foresee this choice and fill the opaque box accordingly.

The role of the Predictor, therefore, not only tests our understanding of causality and predictability, but it also nudges us to introspect on the role our identity plays in decision-making. It prompts us to consider the potential power of a self-fulfilling prophecy, where our decision to be a certain ‘type’ of person may indeed lead to the desired outcome. Thus, Newcomb’s Paradox elegantly encapsulates the intricate interplay between causality, predictability, and personal identity in shaping our choices and their consequences.

Conclusion: One Box or Two?

The question remains: why leave the sure $1,000 in the transparent box when the content of the opaque box is already decided? Why not take both? This question is at the heart of Newcomb’s Paradox. The paradox doesn’t necessarily dictate a ‘correct’ decision. Instead, it presents a problem that forces you to rethink rationality, predictability, and decision-making. It also highlights the complexity and paradoxical nature of decision-making when dealing with highly reliable predictors. Ultimately, though, the decision rests with you. Would you take one box or two?