How to formalize predictors

To me the answer depends on what kind of predictor it is. Some options below, in the order of increasing power.

a) Mind reader: Can it predict *only* Bob's mind and not the rest of the world? Then it's 1 before the flip and 2 after, since in this case the predictor's power is the same as Bob's. That's the classic way to draw the game of rock-paper-scissors against someone who can read your mind.

b) Laplace demon: Can it accurately predict the movements of Bob's coin-flipping hand, the currents in the air, and other local classical factors that go into figuring out how the coin lands? Then it's 3. Unless Bob's coin has quantum randomness, then it's 2.

c) Demiurge: Can it accurately predict the whole of the universe because it has seen it run from the Big Bang to heat death, and now is simply replaying the tape? Then it is 3.

Also, any good links to the current state of research into " bounded versions of Counterfactual Mugging, which are confusing to everyone right now"?

Like Shminux, I say: Why not all three? You've just very helpfully pointed out that there are different ways in which someone can be predictable. So going forward, when I contemplate decision theory hypotheticals in which someone is predictable, I'll make sure to specify which of these three kinds is in effect. Usually I'll consider all three kinds, to see if it changes the results.

As I mentioned elsewhere, I don't really understand...

>I think (1) is a poor formalization, because the game tree becomes unreasonably huge

What game tree? Why represent these decision problems as any kind of trees or game trees in particular? At least some problems of this type can be represented efficiently, using various methods to represent functions on the unit simplex (including decision trees)... Also: Is this decision-theoretically relevant? That is, are you saying, a good decision theory doesn't have to deal with 1 because it is cumbersome to write out (some) problems of this type? But *why* is this decision-theoretically relevant?

>some strategies of the predictor (like "fill the box unless the probability of two-boxing is exactly 1") leave no optimal strategy for the player.

Well, there are less radical ways of addressing this. E.g., expected utility-type theories just assign a preference order to the set of available actions. We could be content with that and accept that in some cases, there is no optimal action. As long as our decision theory ranks the available options in the right order... Or we could restrict attention to problems where an optimal strategy exists despite this dependence.

>And (3) seems like a poor formalization because it makes the predictor work too hard. Now it must predict all possible sources of randomness you might use, not just your internal decision-making.

For this reason, I always assume that predictors in my Newcomb-like problems are compensated appropriately and don't work on weekends! Seriously, though: what does "too hard" mean here? Is this just the point that it is in practice easy to construct agents that cannot be realistically predicted in this way when they don't want to be predicted? If so: I find that at least somewhat convincing, though I'd still be interested in developing theory that doesn't hinge on this ability.