Newcomb's Problem

Why philosophers are evenly split. Surveys of professional philosophers show a rough three-way division: one-boxing, two-boxing, and undecided. This is unusual. Most philosophical thought experiments eventually settle, or at least develop a consensus. Newcomb's Problem, more than fifty years old, has not. The division tracks a genuine disagreement about what decision theory is for.

The tickle defence of CDT. A popular response to the smoking lesion says that a rational agent who is considering smoking must not have the lesion (or the lesion, if present, cannot be producing the desire to smoke rationally). So the correlation between the decision and the lesion is broken for a genuine reasoner. Applied to Newcomb: maybe the Predictor is predicting the cause of your choice (your dispositions), not the choice itself. If so, a truly reflective two-boxer would be predicted as a two-boxer, and the correlation with Box B being empty would hold — CDT loses nothing.

The transparency variant. Suppose the Predictor tells you what it predicted before you choose. If it says "I predicted one box," two-boxing is clearly better now (Box B has the money regardless, and you take both). If it says "I predicted two boxes," two-boxing is still better (Box B is empty regardless, take what is there). Yet the Predictor is rarely wrong, so it must have predicted you would two-box when told your prediction — which means it mostly predicted two boxes and you mostly get only £1,000 by taking both. Deciding after learning the prediction differs from deciding in advance.

Game theory and commitment. Some argue the real lesson is about commitment devices. An agent who can credibly pre-commit to taking only one box — perhaps by wiring themselves to do so, perhaps by being constitutionally incapable of two-boxing — gets £1,000,000. An agent who optimises after the prediction is already locked out. This reframes the problem: maybe rational pre-commitment is better than causal reasoning applied after the fact, and the paradox is pointing to a gap in standard decision theory, not a paradox in the ordinary sense.

Robert Nozick introduced the problem to academic philosophy in a 1969 paper. He attributed it to William Newcomb, a physicist at Lawrence Livermore who had described it in conversation around 1960. Newcomb's original motivations are not recorded, and he never published on it.

Nozick's framing was deliberately neutral: he presented both arguments and said he found it genuinely hard to know which was right. The paper generated immediate responses from philosophers and decision theorists. Martin Gardner publicised the problem in his Scientific American column in 1973, bringing it to a much wider audience and generating reader correspondence that revealed a sharp split of opinion even among mathematically trained non-philosophers.

The problem became the central test case for the emerging field of causal decision theory in the 1970s and 1980s. David Lewis, Brian Skyrms, and others developed CDT partly in response to Newcomb, arguing that EDT's willingness to recommend choices purely on correlational grounds was a fatal flaw. EDT proponents, including Frank Jackson and Robert Pargetter, argued back. The debate has continued without convergence. Recent decades have seen functional decision theory proposed as a third option.

It forces a precise question about what rationality means. Before Newcomb, decision theory treated maximising expected utility as unambiguously correct. The problem shows that two formulations of "maximise expected utility" — one evidential, one causal — can diverge, and there is no theory-internal way to settle which one is right.

It separates correlation from causation in decision-making. In medicine, economics, and public policy, decisions often have to be made in the presence of correlations that may or may not be causal. The problem makes vivid why the distinction matters and why it is not always easy to determine which kind of correlation you face.

It reveals a structural limit of backwards-looking optimisation. If the predictor's accuracy comes from reading your type of decision-making rather than your specific future choice, then optimising act-by-act in isolation misses the possibility that your decision procedure can be evaluated globally.

AI alignment. A sufficiently capable AI system may be modelled as facing Newcomb-like situations: its decisions may be correlated with the dispositions its designers instilled in it, and the consequences of those decisions depend on what other agents expect of it. Understanding which decision theory to implement in an AI is not a purely academic question.

Mechanism design. A principal who can predict agents' strategies can structure rewards to elicit desired behaviour. The Newcomb structure is an extreme idealisation of this: what decision procedure should an agent adopt if it knows that accurate observers will design environments around that procedure?

Commitment and contracts. The problem illuminates the value of pre-commitment — acting as if your decisions were fixed in advance, so that observers who correctly model you can place the money in the box. Contracts, constitutions, and credible threats all operate on the same logic.

Formal setup

Let $p$ be the Predictor's accuracy: the probability that its prediction matches the player's actual choice. Assume $p$ is close to 1. Let $M$ denote the million.

Define the outcomes:

	Predictor predicted 1 box	Predictor predicted 2 boxes
Player takes 1 box	$M$	$0$
Player takes 2 boxes	$M + 1{,}000$	$1{,}000$

The prior probability that the Predictor predicted "1 box" given that you take 1 box is $p$; given that you take 2 boxes it is $1 - p$.

EDT's expected value calculation:

$$\mathrm{EU}_{\mathrm{EDT}}(1\text{-box}) = p \cdot M + (1-p)\cdot 0 = pM$$

$$\mathrm{EU}_{\mathrm{EDT}}(2\text{-box}) = (1-p)(M + 1{,}000) + p \cdot 1{,}000$$

For $p = 0.99$ and $M = 1{,}000{,}000$:

$$\mathrm{EU}{\mathrm{EDT}}(1\text{-box}) \approx 990{,}000$$ $$\mathrm{EU}{\mathrm{EDT}}(2\text{-box}) \approx 10{,}990$$

EDT recommends one box by a factor of roughly 90.

CDT's dominance argument:

For CDT, the contents of the boxes are fixed before the choice. Given any fixed state of Box B, taking 2 boxes yields £1,000 more. The act of choosing cannot causally affect what was placed in the box before the choice was made. Dominance reasoning is valid when actions cannot affect the states being compared; CDT holds that this is exactly the present case.

What the disagreement is really about

The two theories disagree about which conditional to use. EDT uses the evidential conditional: what is the expected outcome given that I choose this action? CDT uses the causal conditional: what is the expected outcome caused by this action?

In most decisions these two conditionals give the same answer. Newcomb's Problem is specifically constructed to pull them apart by introducing a predictor whose accuracy creates a correlation between choice and outcome that is not mediated by causation running from choice to outcome.

Causal decision theorists respond that EDT would advise a patient to avoid effective but unpleasant surgery because surgery correlates with bad health. EDT theorists respond that Newcomb's case is not symmetric to this: the Predictor is not a common cause of choice and outcome. It is an accurate observer of the choice's cause, and accurate observation of a cause is not the same thing as a spurious correlation.

Functional decision theory

A more recent proposal, functional decision theory (FDT), asks: what is the expected outcome of my decision procedure across all similar situations? FDT recommends one-boxing, reasoning that one-boxing decision procedures are selected for £1,000,000 by accurate predictors, so adopting a one-boxing procedure is causally better in the long run — even though any individual one-boxing choice does not cause the money to appear.

FDT dissolves the argument for two-boxing by locating the causal power not in the individual act but in the type of decision process. Critics argue this slides the question, since it is still true that the money is placed before the decision, and the player cannot retroactively change it.