1
Dominance looks at the state
The boxes are already filled. Whatever is in the second box, taking both leaves you 1,000 better off than taking one. An act that is better in every state dominates.
Lesson · Decision and Rationality · Puzzles of expected value
Newcomb's Problem
A predictor has already filled or emptied a box according to what it forecast you would do. Two-boxing beats one-boxing in every column of the table, while one-boxing has the higher expected value once the predictor's record is counted. The accuracy at which the evidential verdict turns is computed.
box A holds 1,000 for certain
box B holds 1M if one-boxing was foreseen
accuracy a: predictor right with chance a
dominance: two-boxing ahead in every state
one-boxing wins iff a > 1001/2000
Dominance against the record
Core ideas
What this lesson establishes
Two arguments, both valid and both from reasoning the course has already taught, reach opposite conclusions about the same table.
2
Evidential value looks at the act as news
What you choose is evidence about what the predictor foresaw. If it is usually right, one-boxers usually find a full box and two-boxers an empty one, and the average over those cases favours one-boxing.
3
Accuracy is a dial
How strongly the act is evidence depends on how reliable the predictor is. At accuracy one half the act tells you nothing, and above a threshold slightly over one half the evidential verdict flips.
The material
The table, the two arguments and the accuracy at which they part
The setup, then the two arguments at full strength, then the arithmetic that locates the switch.
Robert Nozick's version of William Newcomb's problem runs like this. Two boxes sit in front of you. Box A is open and holds 1,000 dollars. Box B is closed and holds either 1,000,000 dollars or nothing. You may take both boxes or take box B alone. A predictor, who has been right in the past, has already decided what to put in B: the million if it forecast that you would take only B, nothing if it forecast that you would take both.
Write the table with two states, box full and box empty, and two acts. Dollars are scored at face value here, so the threshold below is a dollar threshold; a concave utility would move it and leave the structure alone.
act box full box empty
one-box 1,000,000 0
two-box 1,001,000 1,000
The first argument is dominance. The boxes are filled and what you do now cannot change them. Whichever state you are in, the two-box row is 1,000 higher than the one-box row, so two-boxing is the better act in every state. This is the same argument as in “The Decision Matrix and Dominance”, and it is valid.
The second argument uses the predictor's record. Let a be the probability that the predictor is right about whoever stands before it. Then a one-boxer finds a full box with probability a, and a two-boxer finds a full box with probability 1 − a. Score each act with those conditional probabilities.
EU(one-box) = a·1,000,000
EU(two-box) = (1 − a)·1,001,000 + a·1,000
Theorem · The evidential threshold
One-boxing has the higher evidential expected value exactly when a > 1001/2000. At a = 1001/2000 the two tie, and below it two-boxing is ahead.
At accuracy 9/10 the lab scores one-boxing at 900,000 and two-boxing at 101,000. The evidential rule recommends taking one box, while the dominance rule, switched on in the same lab, says the second row wins. At accuracy 1/2, a coin flip, one-boxing scores 500,000 and two-boxing 501,000, so the evidential rule also recommends both boxes. The switch lies just above one half, because the extra 1,000 in box A is small beside the million.
The two rules agree below the threshold and part above it. A predictor right fewer than 1001 times in 2000 gives both rules the same answer, two boxes, and at exactly that accuracy the evidential scores tie. For every accuracy above it the rules part, and which is correct depends on what one thinks an act is for.
The one-boxer says: look at who ends up rich. The two-boxer says: the money is there or not already, and a rational chooser takes what she can. Each says a valid thing. The dominance argument takes the states as fixed; the evidential argument treats the act as information about the state. Whether the act can be treated as independent of the contents of the box is the premise the problem exists to test.
Two quick answers fail. One says the predictor's record is irrelevant because the boxes are already filled. That confuses what the act causes with what it is evidence for, and the lab shows the record changing the evidential figure from 101,000 to 900,000. The other says a high accuracy settles the question. It settles only the evidential rule, which is the rule in dispute.
Interactive
A decision table under every rule
the same payoffs, eight ways of choosing, each computed exactly
Method
Scoring both arguments and finding the switch
Five steps; the first four are mechanical.
1
Write the table
Two acts, two states (box full, box empty), and the dollar payoffs. Check that two-boxing is 1,000 higher in each column.
2
Test dominance
An act dominates when it is at least as good in every state and better in one. Here two-boxing is better in both.
3
Write the conditional rows
For each act, the probability of each state given that you choose it. With accuracy a the one-boxer's row is a and 1 − a, the two-boxer's is 1 − a and a.
4
Score each act with its own row
Weight the payoffs by the act's conditional probabilities and add. Compare the two scores, and set them equal to find the accuracy where they tie.
5
Say which premise you reject
Dominance needs the states to be fixed whatever you do; the evidential score treats the act as evidence. Decide which you will give up before you choose.
Two arguments, one table
Score 0 / 0Choose the answer you can justify.
Both verdicts at accuracy 9/10, and the switch
One-boxing pays 1,000,000 or 0; two-boxing pays 1,001,000 or 1,000. The predictor is right with chance a.
a = 9/10
EU(one) = 9/10·1,000,000 = 900,000
EU(two) = 1/10·1,001,000 + 9/10·1,000 = 101,000
dominance: 1,001,000 > 1,000,000 and 1,000 > 0
tie: a·1,000,000 = 1,001,000 - a·1,000,000
a = 1001/2000
At 9/10 the evidential rule says one box, by 900,000 to 101,000, and dominance says two, row by row. Both can be checked in the lab. They are different rules applied to the same table, and the accuracy at which the first gives way is just above a half.
Common mistakes
1
Treating the predictor's record as irrelevant, or as decisiveThe record is irrelevant to dominance and fully relevant to the evidential score: at 9/10 it takes one-boxing from 101,000 for two-boxing up to 900,000. It is not decisive either, because it matters only if the act is to be treated as evidence about the box, which is what the two-boxer denies. The threshold, 1001/2000, shows where the evidential rule turns.
2
Thinking one of the two arguments is invalidBoth are valid. Dominance takes the states as fixed whatever you do; the evidential argument takes the act as information about the state. The disagreement is over a premise, not a step, and the lab computes both without choosing.
3
Expecting a high accuracy to be neededThe evidential switch is at 1001/2000, a hair above one half. A predictor right 51 times in 100 already favours one-boxing, because the million is so much larger than the thousand. The problem does not need a near-perfect predictor.
Check your understanding
✓
Finish when you can compute both verdicts and the accuracy at which they part.
Given the box payoffs and an accuracy, you should be able to show that two-boxing dominates, compute each act's evidential expected value from the conditional rows, find the accuracy at which the evidential scores tie, and name the premise each argument relies on.
Note: The causal alternative to the evidential rule, which scores acts by what they bring about and not by what they indicate, is not built here: the table can show dominance and the evidential figure and no more. Dollars are scored at face value, which a concave utility would change in the figures and not in the shape.