Rigorously Defining Newcomb's "Paradox"
In order to start making this tractable, I'd start by stating in which order the two events &ndash; the player making their choice, and the machine deciding how much money to put into box B &ndash; happen. There's two options:


First, the machine deposits money into box B (based on a guess as to how the player will choose); the player chooses afterward.
First, the player chooses; the machine deposits money into box B afterward.


(Technically, there's a third option: both actions being performed at the same time. I'd consider this to be equivalent to the first, in that the machine cannot observe the player's actual choice, and the machine's choice is a given for the player.)

The first option is not interesting as a one-off game, as choosing both boxes is clearly a dominant strategy. However, it would be interesting as an iterated game where the player's goal is to maximize their total payout, while the machine's goal is to minimize it. Obviously the player's optimal strategy would depend on the strategy (programming) of the machine, and vice versa. In order to identify overall good strategies, one could perhaps come up with different strategies for both player and machine, pit them against each other over a certain number of rounds, and see which strategy is robust against a wide variety of adversary strategies, the way that Axelrod did for prisoner's dilemma strategies in the early 1980s.

For the second option, the machine's decision-making process needs to be clarified; what does it mean for the machine to be "almost always correct"? I'd implement this as follows: the machine has access to an oracle that will tell it the player's choice, but which, with probability $p &gt; 0$, will give an incorrect answer. The machine always follows the oracle and puts \$1,000,000 into box B iff the oracle says the player did not choose both boxes. This scenario could then be analyzed using standard tools, and the player's optimal strategy determined for both one-off and iterated games, depending on the value of $p$.