How Does Newcomb's Paradox Split Rational Choice?

TL;DR

One-boxing offers the higher expected payoff when the predictor’s demonstrated accuracy is treated as evidence, while two-boxing always adds $1,000 when only present causal influence matters. Newcomb’s paradox remains divisive because evidential decision theory and causal decision theory apply different probabilities to the same setup, producing opposite recommendations from premises that each side considers rational.

Transcript

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Key Insights

• Newcomb’s paradox is a choice between taking one sealed mystery box or taking that box together with an open box containing $1,000. The sealed box was filled before the choice and contains either $1 million or nothing.
• The predictor’s rule is that a predicted one-boxer receives $1 million in the mystery box, while a predicted two-boxer receives nothing there. Its only stated goal is to predict correctly, and it has succeeded almost every time across thousands of cases.
• One-boxing has greater expected utility when the computer’s prediction accuracy exceeds 50.05%. The calculation weighs $1 million against the strong evidence supplied by the predictor’s prior record, making the sealed box alone the favored evidential choice.
• Two-boxing is the strategically dominant action when the mystery box’s existing contents are considered fixed. If it contains nothing, taking both yields $1,000 instead of zero. If it contains $1 million, taking both yields $1,001,000 instead of $1 million.
• Evidential decision theory treats a person’s choice as evidence about the predictor’s earlier action. Because previous one-boxers are associated with receiving $1 million, choosing one box provides strong evidence that the valuable outcome is waiting inside.
• Causal decision theory considers only outcomes the present decision can influence. Since the computer made its prediction and filled the boxes earlier, the current choice cannot cause the mystery box to contain money, so taking both adds $1,000 in either state.
• The disagreement depends on two compatible facts: accurately predicted one-boxers generally leave as millionaires, while the present decision cannot change the past. Each camp emphasizes one fact and embeds a different assumption about which probabilities should guide rational action.
• Public responses are closely divided, although the exact split varies by audience. A 2016 Guardian poll of more than 31,000 people found 53.5% one-boxers and 46.5% two-boxers, while more than 24,000 surveyed viewers produced about two-thirds one-boxers.

Questions & Answers

Q: What is Newcomb’s paradox?

Newcomb’s paradox asks whether to take only a sealed mystery box or to take that box plus an open box containing $1,000. A highly accurate supercomputer previously predicted the choice. It placed $1 million in the mystery box if it predicted one-boxing, but placed nothing there if it predicted two-boxing. The boxes were filled before the participant learned the problem.

Q: Why do one-boxers choose only the mystery box?

One-boxers treat the supercomputer’s history of accurate predictions as strong evidence about the mystery box’s contents. The computer has correctly predicted thousands of people almost every time, so choosing one box is associated with the computer having placed $1 million inside. Under evidential decision theory, that relationship makes one-boxing the option with the higher expected utility when accuracy is sufficiently high.

Q: Why do two-boxers take both boxes?

Two-boxers reason that the computer made its prediction and filled the boxes before the participant knew about the choice. The present action therefore cannot alter whether the mystery box contains zero or $1 million. In either possible state, taking the open box adds another $1,000, producing either $1,000 or $1,001,000. This makes two-boxing the strategically dominant choice.

Q: At what prediction accuracy does one-boxing have higher expected utility?

One-boxing has higher expected utility when the computer’s probability of predicting correctly exceeds 50.05%. The one-box calculation gives a correct prediction a payoff of $1 million and an incorrect prediction a payoff of zero. The two-box calculation gives $1,000 when the prediction is correct and $1,001,000 when it is wrong. Equating those expressions produces the 50.05% threshold.

Q: What is evidential decision theory in Newcomb’s paradox?

Evidential decision theory evaluates a choice using what that choice indicates about the likely state of the world. In Newcomb’s paradox, choosing only the mystery box is evidence that the highly accurate computer predicted one-boxing and placed $1 million inside. The decision does not need to change the past. Its value comes from the statistical relationship between the chosen action and the box’s contents.

Q: What is causal decision theory in Newcomb’s paradox?

Causal decision theory evaluates only consequences that the present choice can cause. Because the predictor already acted and the mystery box is already filled, choosing one or two boxes cannot cause $1 million to appear or disappear. If the earlier probability that the computer expected one-boxing is represented by P, two-boxing has the same million-dollar component as one-boxing plus an additional $1,000.

Q: How can both arguments in Newcomb’s paradox seem rational?

Both arguments begin with true features of the setup but prioritize different relationships. One-boxers emphasize that an extremely accurate predictor links one-boxing with receiving $1 million. Two-boxers emphasize that a later decision cannot change contents fixed in the past. Their expected utility calculations differ because evidential decision theory uses observed predictive accuracy, while causal decision theory uses probabilities tied to present causal influence.

Q: How evenly are people divided on Newcomb’s paradox?

The division is substantial but depends on the group surveyed. A 2016 Guardian poll with more than 31,000 responses found that 53.5% chose one box and 46.5% chose both. A separate audience poll with more than 24,000 responses found that about two-thirds were one-boxers. These results support the observation that many people regard opposite solutions as obviously rational.

Summary & Key Takeaways

• Newcomb’s paradox presents an open box containing $1,000 and a sealed mystery box containing either $1 million or nothing. A highly accurate supercomputer filled the mystery box according to its earlier prediction: it placed $1 million inside for a predicted one-boxer and nothing inside for a predicted two-boxer.

• One-boxers rely on the predictor’s record as evidence about what the mystery box contains. The expected utility calculation shows that one-boxing becomes preferable when prediction accuracy exceeds 50.05%. Because the computer has correctly predicted thousands of previous participants almost every time, one-boxers expect the mystery box to contain $1 million.

• Two-boxers argue that the boxes were filled before the participant learned the problem, so the present choice cannot change their contents. Whether the mystery box holds zero or $1 million, taking both boxes adds $1,000. The disagreement therefore contrasts evidential decision theory with causal decision theory and strategic dominance.