A Million Dollar Puzzle: The Newcomb Paradox

You are paying a visit to Perfect Predictions, the world's most accurate clairvoyancy agency, in the hope of hearing some good news about the imminent demise of your Great Aunt Maud. However, you arrive to find that your consultation has been cancelled - due to entirely foreseen circumstances, obviously - and instead you're asked to play a game that should net you a very large sum of money.

You are told that the game has something to do with Newcomb's Paradox, though you're none the wiser for that.

Really Deep Thought

A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.
--Gottlob Frege.