Not so. 1 is not a probability. Even if you can read your own source code, you can’t know that no outside force will alter your choice. You cannot achieve one-boxing with probability 1. Thus, the optimal strategy in your scenario is to two-box.
But Omega is just the limiting case of a perfect predictor. Newcomblike reasoning can apply even if the predictor is imperfect. Care to try again?
Sure, let’s say Omega calculates the probability that you two-box and removes that proportion of the money from box A. Then your optimal strategy is to one-box with as much probability as you can.
I think that does follow, but you’re altering The Deal. This is a different game.
The only thing Omega is allowed to do is fill the box, or not, in advance. As established in the OP, however, Omega can reduce the expected value by predicting less accurately. But over multiple games, this tarnishes Omega’s record and makes future Choosers more likely to two-box.
Thinking in this way seems to remove the possibility of meaningful strategies. That is, suppose I am considering: “Should I one-box with 80% probability? or 90% probability?” The overall probability will have to be some combination of the two. And choosing the 90% case does not seem to be a possible strategy, because I cannot make it more likely than it is that I will use the 90% case—i.e. if there is a 50% chance I will choose to use the 90% case, then I will do that in 50% of cases, and there is nothing I can do to make that 50% more or less. The chance that I will do X, is just the chance that I will do X, until I do whatever I do.
(This is like an extension of deterministic thinking: if the world is not deterministic, then it is probabilistic. So just as the determinist says, “Whatever is going to happen, is going to happen, and you can’t change that,” in a similar way someone can argue, “All possibilities currently have certain probabilities, and they will happen or not happen, following those definite probabilities, not other probabilities.”)
The probabilities are based on Omega’s state of knowledge. The original problem assumes that Omega is near-omniscient, so that he is extremely likely to make a correct prediction. If you assume that it’s possible at all to make a random choice then you must have some “hidden” source of information that Omega can’t see. Otherwise the strategy in the original post wouldn’t even work, Omega would know how your “random” choice was going to come out so every time you two boxed you would find the box empty and vice-versa.
So when I said “probability” I meant the probability as judged by Omega based on his near total knowledge of your brain and your environment, but with no knowledge of some source of randomness that you can use to generate decisions.
Many Worlds is deterministic. What relevant information is hidden? Omega can predict with certainty that both outcomes happen in the event of a quantum coin flip, in different Everett branches. This is only “random” from a subjective point of view, after the split. Yet given the rules of The Deal, Omega can only fill the box, or not, in advance of the split.
Omega can say that the box will be empty unless you one-box with probability 1. Then one-boxing is the optimal strategy.
You’d never get the million that way, there’s a greater-than-zero chance that you’ll die before making a decision.
Okay, make that “with probability greater than 99.95%”.
Not so. 1 is not a probability. Even if you can read your own source code, you can’t know that no outside force will alter your choice. You cannot achieve one-boxing with probability 1. Thus, the optimal strategy in your scenario is to two-box.
But Omega is just the limiting case of a perfect predictor. Newcomblike reasoning can apply even if the predictor is imperfect. Care to try again?
Sure, let’s say Omega calculates the probability that you two-box and removes that proportion of the money from box A. Then your optimal strategy is to one-box with as much probability as you can.
I think that does follow, but you’re altering The Deal. This is a different game.
The only thing Omega is allowed to do is fill the box, or not, in advance. As established in the OP, however, Omega can reduce the expected value by predicting less accurately. But over multiple games, this tarnishes Omega’s record and makes future Choosers more likely to two-box.
Thinking in this way seems to remove the possibility of meaningful strategies. That is, suppose I am considering: “Should I one-box with 80% probability? or 90% probability?” The overall probability will have to be some combination of the two. And choosing the 90% case does not seem to be a possible strategy, because I cannot make it more likely than it is that I will use the 90% case—i.e. if there is a 50% chance I will choose to use the 90% case, then I will do that in 50% of cases, and there is nothing I can do to make that 50% more or less. The chance that I will do X, is just the chance that I will do X, until I do whatever I do.
(This is like an extension of deterministic thinking: if the world is not deterministic, then it is probabilistic. So just as the determinist says, “Whatever is going to happen, is going to happen, and you can’t change that,” in a similar way someone can argue, “All possibilities currently have certain probabilities, and they will happen or not happen, following those definite probabilities, not other probabilities.”)
The probabilities are based on Omega’s state of knowledge. The original problem assumes that Omega is near-omniscient, so that he is extremely likely to make a correct prediction. If you assume that it’s possible at all to make a random choice then you must have some “hidden” source of information that Omega can’t see. Otherwise the strategy in the original post wouldn’t even work, Omega would know how your “random” choice was going to come out so every time you two boxed you would find the box empty and vice-versa.
So when I said “probability” I meant the probability as judged by Omega based on his near total knowledge of your brain and your environment, but with no knowledge of some source of randomness that you can use to generate decisions.
Many Worlds is deterministic. What relevant information is hidden? Omega can predict with certainty that both outcomes happen in the event of a quantum coin flip, in different Everett branches. This is only “random” from a subjective point of view, after the split. Yet given the rules of The Deal, Omega can only fill the box, or not, in advance of the split.