The expected value under FDT would be:
1-boxing: 0.99 * big prize + 0.01 * 0
2-boxing: 0.99 * small prize + 0.01 * (small prize + big prize)
Making a decision based on that will depend on the specifics of the problem (how much bigger is the big prize than the small prize?) and your circumstances (what is your utility function with respect to the big and small prizes?)
Hmm, does this not depend on how the Oracle is making its decision? I feel like there might be versions of this that look more like the smoking lesion problem – for instance, what if the Oracle is simply using a (highly predictive) proxy to determine whether you'll 1-box or 2-box? (Say, imagine if people from cities 1-box 99% of the time, and people from the country 2-box 99% of the time, and the Oracle is just looking at where you're from).
The exact values of the payoff depend upon the unknown details of how the Oracle achieves its 99% accuracy. Here is an inconvenient scenario that is consistent with the given description:
Suppose that 99% of the population don't even think about the decision, they just follow inherent preferences that are nearly equally prevalent. The Oracle gets almost all of these correct, but fails on 1% of inherent one-boxers giving them nothing instead of $1,000,000. In the remaining cases where people do actually think about it, the Oracle is always wrong, and everyone who thinks about the situation knows this.
Since you are actually thinking about it, then you're one of the people for whom the Oracle's prediction is always wrong. If you end up taking one box, then it will be empty. If you take both boxes, you will find $1,001,000. In this scenario CDT, EDT, and FDT all agree that you should take both boxes.
In many other scenarios, possibly even "most" in some ill-defined sense, FDT says you should take one box.
Imagine a variation of Newcomb's problem where the Oracle is correct 99% of the time (say, half of people 1-box, half 2-box, and the Oracle is correct 99% of the time on each). What would FDT imply is the correct course of action then?