More free-will paradox

Yesterday’s post on counting votes is related to a paradox I’ve been thinking about for a couple weeks. This is a from-memory paraphrase of the description in Martin Gardner’s Aha! Gotcha: Paradoxes to Puzzle and Delight:

Dr. Omega, brain specialist extraordinaire, is conducting a study. First, he scans your brain, has you fill out lots of personality tests, measures the bumps on your head, and looks deeply into your soul. Then he brings out two boxes, one opaque (A) and the other transparent (B). In the transparent box is a $100 bill. He then gives you two choices:

  • Choice 1: Take the contents of only the opaque box (box A).
  • Choice 2: Take the contents of BOTH box A and box B.

Dr. Omega explains that you that if, based on his tests, he expects that you will take Choice 1, he has placed $1000 in box A. If he expects you will take Choice 2, he has placed nothing in box A.

With that, he leaves for a vacation in Vegas.

Which choice do you take?

Assuming you believe Dr. Omega is good at what he does, I’m pretty sure this is equivalent to the Prisoner’s Dilemma with you playing against your future self.

Here’s the payoff matrix:

Your Choice
A A & B
Omega’s Prediction A $1000 $1100
A & B $0 $100

Regardless of what prediction Dr. Omega has made, your payoff is always one hundred dollars higher if you take choice #2 (A & B). The only way that can’t be true is if your choice now in some way affects the prediction that Dr. Omega has already made. And yet, if Dr. Omega’s prediction is correct then by choosing A & B you only make $100 rather than $1000 or $1100.

The paradox is even more clear if instead of a brain specialist, Dr. Omega is a time traveler. He jumps a few moments into the future, watches which choice you make, then pops back to the present and sets the boxes as before. If you believe in a single-timeline worldview (it’s already happened, so that’s how it’ll happen, ala The Hitchhiker’s Guide to the Galaxy) then you’re better off choosing A (insomuch as you can “choose” at all in this worldview). If you believe in splitting timelines (you can change the future, ala Terminator) then you’re better off choosing A & B.