Newcomb's paradox without Newcomb
William Newcomb was a physicist at the Lawrence Livermore Laboratory and never published the puzzle. The philosopher Robert Nozick printed it in 1969, but named it after its author.
Two boxes are on the table. In the transparent box A you can see $1,000. The closed box B is either empty or holds $1,000,000. What is in B was decided yesterday by a predictor program that has called the choice of 990 people out of 1,000: if it foresaw you taking only B, it put the million in; if it foresaw you taking both boxes, it left B empty. Nothing in the boxes will change now. Do you take only B, or both? Lock in your choice, then play against the predictor.
Those who take only B leave with $990,000 on average, those who take both with just $11,000: the predictor rarely errs. But the money on the table is either there or not, and whatever is in B, box A adds another $1,000. Both calculations are correct. The first asks what your choice says about the box, the second asks what your choice changes in it. That is why people split almost in half: 53.5% against 46.5% in a big Guardian poll.
The physicist William Newcomb came up with this puzzle, and the philosopher Robert Nozick was the first to analyse it. Ever since, it has split philosophers, economists and ordinary readers into two camps, and each camp is sure the answer is obvious.
It is a program that has already played this game with 1,000 people and correctly predicted the choice of 990 of them, 99% of the time. How it does it doesn't matter: it simply knows people very well.
Box A is transparent and holds $1,000. Box B is closed, and you can't see what is inside.
Yesterday, before you arrived, the predictor predicted your choice. If it decided you would take only B, it put $1,000,000 in it. If it decided you would take both, it left B empty.
Nothing in the boxes will change now. You can take either box B alone or both boxes, A and B. What will you take?
"Surely it's obvious"
Most people know the answer at once and wonder how anyone could think otherwise. Notice which answer feels obvious to you, and why.
Robert Nozick noticed something strange: nearly everyone finds the answer obvious, yet people split almost in half, and each half thinks the other simply hasn't thought it through. In fact both "obvious" answers rest on a correct calculation, just a different one.
There is no trick in the setup. Just decide what you would do standing in front of these boxes, and lock in your choice before reading on.
Your answer counts as soon as you click. You can change your mind, but the first choice goes into the stats.
How other people choose, and what your choice says, will appear here.
Hard to decide? Open the hints one at a time: the easy one first, then stronger ones.
Look at the people who have already played: how much do those who took only B leave with on average, and how much do those who took both?
Now look at the table: the money in box B is either there or not. Can your choice put anything in, or take anything out?
Work out both ways for a predictor that is right 99% of the time, and ask which question each one answers.
The next rounds show why each camp has a point: a game against a real predictor, 100,000 games and the explanation. Make your choice in round 1 first, or go on?
You chose to always show all rounds.
This predictor is not all-knowing, but it doesn't guess at random either. Before each game it looks at your earlier choices: what you most often chose after the same last four moves, and bets on that. The prediction is sealed in box B before you click. Try to be unpredictable, or the opposite, consistent, and see when you take home more.
Each square is a game: blue for "only B", rust for "both". Filled if the predictor got it right, hollow if it missed.
The computer plays 100,000 times. Each player takes only B or both at random, the predictor calls the choice with accuracy p and fills box B. Move the accuracy slider. On the left we compare the players with each other, on the right we compare the two possible choices for the same boxes.
Both pictures come from the same games. Left: the more accurate the predictor, the more those who take only B leave with, and the less those who take both. The lines cross at p = 50.05%: if the predictor is right even a little more often than a coin, "only B" pays more on average. Right: take any of these games and, without touching the boxes, switch the player's choice to "both", and they would get exactly $1,000 more every time. The dots on the left are simulations of 20,000 games for each accuracy, the circles are the 100,000 games at the chosen one.
Once you answer, the simulation code appears here so you can run it yourself.
The same simulation, short and with no libraries. Run the JavaScript in your browser console (F12) or in Node; the Python needs only the standard library.
// Newcomb's paradox: 100,000 games against a predictor that is right with probability p
const N = 100000, A = 1000, M = 1000000;
for (const p of [0.9, 0.5]) {
const sum = { one: 0, two: 0 }, cnt = { one: 0, two: 0 };
let plus = 0;
for (let i = 0; i < N; i++) {
const choice = Math.random() < 0.5 ? 'one' : 'two'; // the player picks at random
const right = Math.random() < p; // did the predictor guess right?
const said = right ? choice : (choice === 'one' ? 'two' : 'one');
const B = said === 'one' ? M : 0; // yesterday: a million in B only for "only B"
const win = choice === 'one' ? B : A + B;
sum[choice] += win; cnt[choice]++;
const gain = (A + B) - B; // same boxes: "both" minus "only B"
if (gain === A) plus++;
}
console.log(`p = ${p * 100}%`);
console.log(` only B: ${Math.round(sum.one / cnt.one)} exact ${Math.round(p * M)}`);
console.log(` both: ${Math.round(sum.two / cnt.two)} exact ${Math.round(A + (1 - p) * M)}`);
console.log(` games where "both" adds exactly +1000 to the same boxes: ${(100 * plus / N).toFixed(1)}%`);
}
# Newcomb's paradox: 100,000 games against a predictor that is right with probability p
import random
N, A, M = 100_000, 1000, 1_000_000
for p in (0.9, 0.5):
total = {"one": 0, "two": 0}
count = {"one": 0, "two": 0}
plus = 0
for _ in range(N):
choice = random.choice(("one", "two")) # the player picks at random
right = random.random() < p # did the predictor guess right?
said = choice if right else ("two" if choice == "one" else "one")
B = M if said == "one" else 0 # yesterday: a million in B only for "only B"
win = B if choice == "one" else A + B
total[choice] += win
count[choice] += 1
if (A + B) - B == A: # same boxes: "both" minus "only B"
plus += 1
print(f"p = {p:.0%}")
print(f" only B: {total['one'] / count['one']:.0f} exact {p * M:.0f}")
print(f" both: {total['two'] / count['two']:.0f} exact {A + (1 - p) * M:.0f}")
print(f" games where \"both\" adds exactly +1000 to the same boxes: {100 * plus / N:.1f}%")
In most puzzles on this site intuition is wrong and a calculation puts it right. Here there are two calculations, both free of errors, and they give different answers. That is why the argument has run for more than half a century.
“In plain words” is hidden on all puzzles.
Those who took only B nearly always went home with a million. Those who took both nearly always went home with a thousand: the predictor saw it coming and left B empty. If you want to leave with a million, do what the first group did.
The money is already in B or it isn't. If it is, both boxes give $1,001,000 instead of $1,000,000. If not, $1,000 instead of nothing. Either way, box A adds $1,000.
Both lines of reasoning are correct, because they answer different questions: "what do people like me usually go home with?" and "what changes if I take A as well, right now?". That is the short version. The cards below give each side in its strongest form.
Imagine the back wall of box B is transparent and your friend is standing behind it. They can see whether the million is there, but can't tell you, only shout advice. If the million is there, your friend shouts "take both!": that makes $1,001,000. If it is empty, they also shout "take both!": $1,000 beats nothing.
A friend who sees everything gives the same advice whatever is inside. Should you go against advice that someone who knows more than you would give? That is the case for two boxes at its strongest: your choice puts nothing into B and takes nothing out.
Your choice doesn't move any money. But yesterday the predictor was looking at you in particular: at how you think and what you choose. If you feel drawn to take both right now, that is most likely just what it saw, and B is empty. If you are the kind who takes only B, it most likely saw that too, and the million is already there.
So your choice works as evidence: it doesn't change what is in the box, but it predicts it very well. With a predictor that is right 99% of the time, those who take only B find a million there 99 times out of 100. The friend behind the box sees one particular game, but you have to choose what kind of player to be. That is the case for one box at its strongest.
“Rigorous” is hidden on all puzzles.
| The predictor foresaw | You take only B | You take both |
|---|---|---|
| "only B": a million in B | $1,000,000 | $1,001,000 |
| "both": B is empty | $0 | $1,000 |
Look along the rows: in each one the right-hand cell is $1,000 bigger. That is the case for "both". Now look at the shaded diagonal: the predictor nearly always lands on it, and on the diagonal the top-left cell is a thousand times the bottom-right one. That is the case for "only B".
If your choice is evidence. The predictor is right with probability p. Those who take only B find a million in a share p of games: p · 1,000,000 on average. Those who take both get 1,000, plus the million only when the predictor got it wrong: 1,000 + (1 − p) · 1,000,000. At p = 99% that is 990,000 against 11,000.
If the contents are already fixed. Say the million is in B with some probability q, and your choice doesn't change it. Then only B gives q · 1,000,000 and both give q · 1,000,000 + 1,000. Both wins for every q.
The first approach is called evidential decision theory: a choice is judged by what it says about the world. The second is causal decision theory: a choice is judged by what it changes in the world. Newcomb's problem is the best-known place where they part ways.
A is the sum in the transparent box, M the million in the closed one, p the probability that the predictor is right. The expected win if the choice counts as evidence:
"Only B" pays more when the first exceeds the second:
If the choice has no effect on the contents and the million is in B with probability q, then for every q:
The two calculations differ in one detail. In the first, the probability of the million depends on your choice: p if you take only B, 1 − p if you take both. In the second it is the same q for both choices. Both are valid; the dispute is over which probability you should plug in when you choose.
“Deeper” is hidden on all puzzles.
The physicist William Newcomb of the Lawrence Livermore Laboratory devised the puzzle around 1960 but never published anything on it. In 1969 the philosopher Robert Nozick printed the first analysis, "Newcomb's Problem and Two Principles of Choice", in a volume of essays in honour of Carl Hempel. The two principles in the title are exactly our two calculations.
Martin Gardner made it widely known: it appeared in his Scientific American column in July 1973, and in March 1974 he printed Nozick's review of the letters readers had sent in.
In 1979 David Lewis showed in Philosophy & Public Affairs that the prisoner's dilemma is a Newcomb problem. Imagine you play not against a stranger but against an exact copy of yourself, who thinks just as you do. Defecting pays more whatever the copy does, just as taking both boxes pays more whatever is in B. But if you cooperate, the copy most likely cooperates too, like the predictor that sees "only B" in you.
How people and programs play the repeated prisoner's dilemma is shown in Axelrod's tournament.
In 2017 Eliezer Yudkowsky and Nate Soares proposed functional decision theory. The predictor foresees you by, in effect, running your way of reasoning itself. So choose as if you were choosing not one act but that way of reasoning, and with it what the predictor saw. Such an agent takes only B, while agreeing that in this particular game A would have added $1,000.
In the classic puzzle A holds a thousand and B a million, so "only B" needs a predictor only slightly better than a coin. Change the sums and the accuracy and see where the line falls. Then from the numbers to the idea behind them, and to where it shows up in life.
Only B gives p · M on average, both give A + (1 − p) · M. The first is bigger when p > (1 + A/M)/2. The more the transparent box holds compared with the closed one, the more accurate the predictor must be for "only B" to pay.
With the classic $1,000 against a million the threshold is 50.05%. If A holds $500,000, it is already 75%. If A holds as much as B, it reaches 100%: even a flawless predictor gives only a tie, and both boxes are no worse. The case for "both" doesn't depend on the sums at all: for the same boxes A always adds exactly A.
In this puzzle you are paid not for the act but for the character the predictor saw in you. Being a person who takes only B pays, even though at the moment of choice the second box would add $1,000. That is the idea of pre-commitment: binding yourself to a decision in advance, while it can still make a difference.
Odysseus had himself tied to the mast to sail past the Sirens. He knew that at the crucial moment he would want to turn aside, so he took that choice away from himself in advance.
In "Reasons and Persons" (1984) Derek Parfit described a hitchhiker stranded in the desert. A driver who reads people well will take him to town only if he believes the hitchhiker will pay him there. Once in town, paying no longer pays: he is saved, and the money changes nothing. But whoever thinks "I won't pay in town" stays in the desert.
It is the same Newcomb problem: the driver is the predictor, the rescue is the million, and the money you could keep is box A.
A Newcomb problem is easy to spot: someone decides how to treat you from a forecast of your behaviour, and the forecast is good. Then what matters is not the single act but the rule you act by. In "The Strategy of Conflict" (1960) Thomas Schelling described commitment as a strategic move: binding yourself in advance can be stronger than keeping your freedom of choice. You can test how predictable you are in round 2: the same kind of oracle by Scott Aaronson called 70–80% of the choices of people trying to outwit it.
| Situation | Who is the predictor | The "only B" policy |
|---|---|---|
| A state and hostage-takers | Kidnappers weighing whether a ransom will be paid | "We don't negotiate with terrorists". It works only if believed in advance: then kidnapping pays less, even though every single hostage is someone you badly want to save. |
| Parents and a child | A child who quickly learns whether the parents do what they say | Doing what was promised even when giving in is easier today. Giving in wins today's thousand and loses the million of trust. |
| A shop with a "money back, no questions asked" guarantee | Shoppers deciding whether to trust the guarantee | Refunding even when a particular case loses money: otherwise the guarantee is worth nothing, and shoppers will see it coming. |
| A news feed, shop recommendations | An algorithm forecasting your clicks and purchases from past behaviour | Remember that you have already been predicted: you are shown what you are most likely to bite on. The more predictable your habits, the sharper the forecast. |
Other puzzles where the calculation hinges on which question is being asked: Sleeping Beauty, Bertrand's paradox, the two envelopes. And on how rationality in a game can cost both players money, the traveler's dilemma.
William Newcomb was a physicist at the Lawrence Livermore Laboratory and never published the puzzle. The philosopher Robert Nozick printed it in 1969, but named it after its author.
Martin Gardner's Scientific American column of July 1973 drew so many letters that in March 1974 he ran a second one, in which Nozick reviewed the readers' replies.
In 2016 Alex Bellos put the puzzle to Guardian readers. 31,854 people voted, and 53.5% of them took only B.
In the 2020 PhilPapers survey, philosophers who knew the problem leaned towards two boxes: 39% against 31%. Among decision theorists the lead is bigger still: 73% against 21%.
Scott Aaronson wrote a simple program that guesses which of two keys a person will press next from their earlier presses. It called 70–80% of the presses of people trying their hardest to be random.
David Lewis showed that the prisoner's dilemma, played against your own copy, is a Newcomb problem, so anyone who takes both boxes must also defect against their twin.
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