A Decision Theory Paradox
The Predictor's
Million-Dollar Boxes
A perfect predictor offers you two boxes. Your choice reveals what kind of decision-maker you are.
A superintelligent predictor presents you with two boxes.
Box A (transparent)
Box B (opaque)
Box A is transparent. You can see it contains $1,000.
Box B is opaque. It contains either $1,000,000 or nothing.
Here's the catch: The predictor filled Box B yesterday, based on what it predicted you would do today.
If predictor thinks you'll take only Box B:
It put $1,000,000 in Box B
If predictor thinks you'll take both boxes:
It left Box B empty
The predictor has been right 99% of the time with previous players.
What do you choose?
Make Your Choice
Play multiple rounds to see how your strategy performs.
The Case for One-Boxing
The expected value argument is straightforward:
The predictor is 99% accurate.
If I one-box:
99% chance of $1,000,000, 1% chance of $0
Expected value: $990,000
If I two-box:
99% chance of $1,000, 1% chance of $1,001,000
Expected value: $11,000
One-boxing wins by $979,010 in expected value.
This is Evidential Decision Theory (EDT): Choose the action that provides the best evidence about good outcomes.
One-boxers reason: “People like me who one-box tend to find $1,000,000. People like me who two-box tend to find nothing. I want to be the kind of person who finds $1,000,000.”
“I'm not trying to outsmart the predictor. I'm trying to be the person the predictor already predicted would be rich.”
The Case for Two-Boxing
The dominance argument is equally compelling:
The predictor made its choice yesterday.
Box B is already filled or empty. Nothing I do now changes that.
If Box B has $1,000,000:
One-boxing gets $1M. Two-boxing gets $1M + $1K = $1,001,000
Two-boxing wins by $1,000
If Box B is empty:
One-boxing gets $0. Two-boxing gets $1,000
Two-boxing wins by $1,000
Two-boxing is strictly dominant. It's better in every possible state of the world.
This is Causal Decision Theory (CDT): Choose the action that causally leads to the best outcomes.
Two-boxers reason: “The money is either in Box B or it isn't. My choice can't change the past. So I should take everything available. Leaving $1,000 on the table is just irrational.”
“I refuse to let my decision be 'caused' by a prediction. The contents are fixed. I'm taking both boxes.”
When Does Two-Boxing Win?
The paradox assumes a nearly-perfect predictor. But what if the predictor isn't so accurate?
One-Boxing Expected Value
$990,000
99% x $1M + 1% x $0
Two-Boxing Expected Value
$11,000
$1K + 1% x $1M
At 99% accuracy, one-boxing has higher expected value.
Interestingly, the breakeven point is barely above 50%. Even a slightly-better-than-random predictor makes one-boxing the better expected-value play.
Two-boxers respond: “This just shows that expected value is the wrong criterion. Dominance reasoning applies regardless of the predictor's accuracy.”
What Would You Choose?
Cast your vote and see how others have decided.
Why This Matters
Robert Nozick, who popularized this paradox in 1969, called it “a problem that everyone considers easy to solve, yet about which nearly everyone disagrees.”
AI Alignment
How should AI systems make decisions when their choices are predictable? Newcomb problems arise naturally when training AI on predicted behavior.
Free Will
If a predictor can know your choice in advance, are you really choosing? The paradox probes the boundary between determinism and agency.
Game Theory
Many real strategic situations involve agents predicting each other. The paradox illuminates when standard game theory breaks down.
Rational Agency
What does it mean to be rational? CDT and EDT give different answers. There is no consensus among philosophers.
The Deeper Question
Newcomb's paradox isn't really about money or boxes. It's about whether rationality is about what kind of agent you are or what actions you take.
One-boxers: “Be the right kind of person, and good outcomes follow.”
Two-boxers: “Take the best action given what you know.”
Both can't be right. And yet both arguments feel airtight.
That's what makes it a paradox.
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Reference: Nozick (1969), Lewis (1981)