Newcomb’s Lottery Problem

Inspired by and variant of The Ultimate Newcomb’s Problem.

In front of you are two boxes, box A and box B. You can either take only box B (one-boxing) or both (two-boxing). Box A visibly contains $1,000. Box B contains a visible number X. X is guaranteed to be equal to or larger than 1 and smaller than or equal to 1000. Also, if X is composite, box B contains $1,000,000. If X is prime, B contains $0. You observe X = 226. Omega the superintelligence has predicted your move in this game. If it predicted you will one-box, it chose X to be composite; otherwise, it made X prime. Omega is known to be correct in her predictions 99% of the time, and completely honest.

The Having Fun With Decision Theory Lottery has randomly picked a number Y, which is guaranteed to fall in the same range as X. Y is displayed on a screen visible to you. The HFWDT Lottery is organized by Omega—but again, Y is picked at random and therefore completely separately from X. If both X and Y are prime, the HFWDT Lottery gives you $4,000,000. Otherwise it gives you $0. You observe Y = 536.

Do you one-box or two-box?

Newcomb’s Lottery Problem 2: Everything is the same as before, except the HFWDT Lottery price is now $8,000,000. Do you one-box or two-box?