by John Allen Paulos
Despite the fact that Newcomb’s paradox was discovered in 1960, I’ve been prompted to discuss it now for three reasons, the first being its inherent interest and counterintuitive conclusions. The two other factors are topical. One is a scheme put forth by Elon Musk in which he offered a small prize to people who publicly approved of the free speech and gun rights clauses in the Constitution. Doing so, he announced, would register them and make them eligible for a daily giveaway of a million dollars provided by him (an almost homeopathic fraction of his 400 billion dollar fortune). The other topic is the rapid rise in AI’s abilities, especially in AGI (Artificial General Intelligence). Soon enough it will be able, somewhat reliably, to predict our behaviors, at least in some contexts.
With this prologue, let me get to Newcomb’s paradox, which is a puzzle that suggests that the rational thing to do in some situations results in an outcome much worse than doing what doesn’t make sense.
As mentioned, it was first reported in 1960 by William Newcomb, a physicist at the University of California, but it was developed and popularized by the philosopher Robert Nozick in 1969.
The puzzle involves an assumed entity of some sort – a visitor from an advanced civilization, a robot with access to lightning fast computers, an all-knowing network of AI enhanced neural agents, whatever – that has the financial backing of a multi-billionaire. This billionaire claims that his ultra-sapient agent can predict with good accuracy which of two specific alternatives presented to a person he or she will choose. The billionaire further announces a sort of online lottery to demonstrate the agent’s abilities.
He explains that the agent’s assessment of people will utilize two types of boxes. Boxes of type A are transparent and all contain $1,000, whereas boxes of type B are opaque and contain either $0 or $1,000,000, the cash prizes provided by the billionaire, of course.
For each registered person, there is provided a box of both types. The person is told that he or she can choose to take the contents of box B alone or the contents of both box A and box B. However, and this is crucial, the billionaire states that the agent has already used its awesome power (enhanced with AI and a thorough perusal of his or her online presence and financial situation) to assess and analyze the psychology of the contestant. If it believes that the person will take the contents of both boxes, it has left his or her box B empty. On the other hand if it believes that the person will take only the contents of box B, it has placed $1,000,000 in his or her box B.
People registered for this contest will see for themselves that when a person chooses to take the contents of both boxes, most of the time box B is empty and the person gets only the $1,000 in box A. They also note that when a person chooses to take only the contents of box B alone, most of the time it contains $1,000,000, making the person an instant millionaire.
After watching the boxes placed before the people ahead of you in the online queue and seeing their choices and their consequences, you are finally presented with your pair of boxes. The punchline: what would you do?
Perhaps despite the evidence you’ve seen, you see no reason not to take the contents of both boxes. After all, your personal boxes were prepared well before you were asked to make your choice, so why not take them both box A and box B and possibly get $1,001,000. Or perhaps you were persuaded that the agent is almost always right, say 95% of the time, and that you should simply try to maximize your winnings by taking only box B. In the jargon of decision theory, do you subscribe to the principle of strategic dominance or to that of expected utility.
The paradox is, of course, not Musk’s million dollar scheme, but only loosely suggested by it. Furthermore, the mysterious predicting agent in Newcomb’s original story is made a little more realistic by the invocation of AI and perhaps the snooping of Musk’s minions’ searches. Other more prosaic changes might involve changing the payoffs to non-monetary outcomes, say involving doses of poison, for example, or the conferring of titles and the awarding of jobs, as some have done.
Of course if the agent is infallible, Newcomb’s exercise becomes more abstract and philosophically profound, raising questions about free will and determinism and their various interpretations. Since the boxes are prepared before people makes their choices, the issue of time travel might also arise since the agent could go back in time to make its prediction accord with people’s choices.
For the record, I’d take only box B, but I’d have misgivings about doing so. You?
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John Allen Paulos is an emeritus Professor of Mathematics at Temple University and the author of Innumeracy and A Mathematician Reads the Newspaper. These and his other books are available here.
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