But even fully general counterarguments are sometimes right.
True, though I think it is important to note the difference between an argument that a particular decision theory is doing something stupid, and an argument that no decision theory can do better than every other decision theory in every environment.
The point of my post was that we seem to have no justification for using Solomonoff induction if the universe can be uncomputable. Or rather that Eliezer has put forward a justification, but I don’t think it’s valid. Why should game 1 be a strong argument in favor of Solomonoff, if game 3 isn’t allowed to be a strong argument against?
It seems easy to invent games that favor specific epistemologies. Or do you think game 1 means something more, that it’s a “typical” game in some sense, while game 3 isn’t? I’d love to see a result like that.
How about if we restrict attention to games where at any stage the players are allowed to choose a probability distribution over the set of available moves, rather than being forced to choose one move? Is it then possible for Solomonoff induction to lose (whatever ‘lose’ means) with non-zero probability, in the limit?
In other words, does Solomonoff induction win all variants of game 1 that use different proper scoring rules in place of log score? Nice question. I’m going to sleep, will try to solve it tomorrow unless someone else does it first.
Game 1 is designed to test epistemologies as epistemologies, without considering any decision theories that use that epistemology. Figuring out a good decision theory is harder. What game 3 shows is that even starting with an ideal epistemology, you can’t build a decision theory that outperforms all others in all environments.
True, though I think it is important to note the difference between an argument that a particular decision theory is doing something stupid, and an argument that no decision theory can do better than every other decision theory in every environment.
Sorry for editing my comment.
The point of my post was that we seem to have no justification for using Solomonoff induction if the universe can be uncomputable. Or rather that Eliezer has put forward a justification, but I don’t think it’s valid. Why should game 1 be a strong argument in favor of Solomonoff, if game 3 isn’t allowed to be a strong argument against?
Game 3 is also an argument against all competing epistemologies, but game 1 is an argument in favor of only Solomonoff induction.
It seems easy to invent games that favor specific epistemologies. Or do you think game 1 means something more, that it’s a “typical” game in some sense, while game 3 isn’t? I’d love to see a result like that.
How about if we restrict attention to games where at any stage the players are allowed to choose a probability distribution over the set of available moves, rather than being forced to choose one move? Is it then possible for Solomonoff induction to lose (whatever ‘lose’ means) with non-zero probability, in the limit?
In other words, does Solomonoff induction win all variants of game 1 that use different proper scoring rules in place of log score? Nice question. I’m going to sleep, will try to solve it tomorrow unless someone else does it first.
Game 1 is designed to test epistemologies as epistemologies, without considering any decision theories that use that epistemology. Figuring out a good decision theory is harder. What game 3 shows is that even starting with an ideal epistemology, you can’t build a decision theory that outperforms all others in all environments.