A first scribbling: Newcomb's Paradox

(Seeing a storm on the move, and having no where else to turn you stumble into a cave.  Having lit your torch you notice strange markings on the walls of the cave.  Most of these are rubbish; however you come across a section which appears mostly legible...)

I heard of Newcomb's paradox before I came to my present fate, a puzzling game in which the cruel Predictor decides the fate of a hapless Player.

There are two boxes, labelled A and B.  The Predictor, through means unknown, is purported to be able to predict with almost absolute certainty what the Player chooses to do.  In his cruelty, the Predictor presents the Player with the boxes.  Box A is transparent, and has a visible $1,000 inside.  Box B is opaque, and it is unknown to the Player what the contents are. He then offers the Player the choice to take just the contents of box B, or to take both the contents of A and B.

The Predictor's cruelty is revealed by how it fills the boxes.  Using its predictive power, the Predictor decides to place nothing in Box A if the player were to choose to take both boxes' contents, or $1,000,000 if the player were to choose to take only the contents of Box B.

Upon thinking about the Player's fate, I considered what I would do in his situation.  Many told me that, no matter what the Predictor determines, taking both boxes' contents always gives me more money.  This is in fact true; however, still others insisted that, since the Predictor is almost never wrong, taking just the contents of box B is better.

Given my current plite, I have had plenty of time to think about many curiosities such as this, and I have reached a resolution.

It befits the Player to consider likelihoods; there is some chance that the Predictor could yield the wrong prediction, but the odds of this change what the best decision would be.

Let us assume that the Predictor has the same accuracy of prediction no matter what the Player chooses, call it P.  We may now compute the expected amount the Player will receive based on his choices:

Expected gains	A and B	Just B
Predictor Correct	P * 1,000	P * 1,000,000
Predictor Incorrect	(1 - P) * 1,001,000	(1 - P) * 0

The Player's expected gains can be computed from this; I managed to find some colored berries to make a plot of these outcomes verses the Predictor's accuracy.

This plot shows the correct choice given any possible accuracy the Predictor may have, and since the predictor's accuracy is supposed to be close to 1, the best choice would be to choose only the contents of B.