Newcomb’s paradox solution and teaching

Newcomb’s paradox, also referred to as Newcomb’s problem, is a thought experiment involving a game between two players, one of whom purports to be able to predict the future. Whether the problem is actually a paradox is disputed.

Newcomb’s paradox was created by William Newcomb of the University of California’s Lawrence Livermore Laboratory. However, it was first analyzed and was published in a philosophy paper spread to the philosophical community by Robert Nozick in 1969, and appeared in Martin Gardner’s Scientific American column in 1974. Today it is a much debated problem in the philosophical branch of decision theory but has received little attention from the mathematical side.

The Resolution of Newcomb’s Paradox

We begin by defining Newcomb’s problem. Enter Newcomb’s Demon, a rather close relative of Maxwell’s Demon (the similarity goes well beyond a shared affinity for black boxes). Newcomb’s, Demon – call him ND for short – is a paragon of intelligence, a genius’s genius whose acuity transcends the temporal boundaries of merely ephemeral human beings. Like many geniuses, he displays a certain playfulness: his “hobby” is testing his own infallibility in the prediction of human behaviour by making a generous offer to random human playthings. This offer is couched as a “choice”; though ND’s human subjects are in fact forbidden to predicate their choices on external events, they are tacitly allowed to use deterministic or nondeterministic internal strategies computed by corresponding classes of neural events (you may already see the superficiality of the distinction between inward and outward events; I include it only as a customary ingredient of the formulation).(Know more ,Click Here)

What does Newcomb’s paradox teach us?

In Newcomb’s paradox you choose to receive either the contents of a particular closed box, or the contents of both that closed box and another one. Before you choose, a prediction algorithm deduces your choice, and fills the two boxes based on that deduction. Newcomb’s paradox is that Game Theory appears to provide two conflicting recommendations for what choice you should make in this scenario. We analyze Newcomb’s paradox using a recent extension of game theory in which the players set conditional probability distributions in a Bayes net. We show that the two game theory recommendations in Newcomb’s scenario have different presumptions for what Bayes net relates your choice and the algorithm’s prediction. We resolve the paradox by proving that these two Bayes nets are incompatible. We also show that the accuracy of the algorithm’s prediction, the focus of much previous work, is irrelevant. In addition we show that Newcomb’s scenario only provides a contradiction between game theory’s expected utility and dominance principles if one is sloppy in specifying the underlying Bayes net. We also show that Newcomb’s paradox is time-reversal invariant; both the paradox and its resolution are unchanged if the algorithm makes its `prediction’ after you make your choice rather than before.