Does Solomonoff always win?

It occurs to me that an important question is, if it turns out that our universe actually contains a halting oracle (or even more powerful oracles), can an AI programmed with the Solomonoff prior exploit it properly (i.e., do no worse than a human) to accomplish its goals? This question is not really captured by any of the three games, since they all force the player to make guesses without being able to interact with the universe.

It seems hard to think of a game that does capture this question. Can anyone think of such a game?

First let's set the input sequence to be very malevolent toward Omega: it will always say the reverse of what Omega is projected to say based on the previous bits.

I'm not sure I buy this, but let me make sure I understand correctly. There is some super-super-intelligence generating bits, which is smarter than Omega and also antagonistic to it. That intelligence basically simulates Omega, figures out what Omega will guess, and sends the opposite value.

First, can't Omega just figure out that it's being simulated, and then find a source of pure randomness, and just guess according to that?

Second, can't Omega at least make sure it remains level with the pitiful human, by figuring out what the human will guess and copying that?

Third, how do you know there even exists a stream of bits that will make Omega lose constantly? It's not at all obvious to me that such strings exist. It seems like you're asserting the existence of N-bit strings that have Kolmogorov complexity substantially larger than N.

Game 3 with a bit stream that is maximally malevolent to epistemology X seems to be a fully general counterargument against epistemology X.

Suppose we are playing any game in which we repeatedly choose an option from some potentially infinite class and then receive a bounded payoff (dependent only on my choice in this round). Then I can do as well as the best computable strategy in expectation (to within a constant) by using multiplicative weights over the space of computable strategies (I've mentioned this before, but didn't realize the difference between these set-ups at the time). Note that this strategy may be randomized, if you only allow me to bet all-or-nothing (or more generally if the space of strategies isn't convex in the suitable sense).

This lets you win probabilistically at game 3, but its not immediately clear whether it lets you win at games 1 and 2, since the payoff is not bounded. I suspect it wins at games 1 and 2, but not very strongly.

Edit: Why should we restrict attention to deterministic strategies? It is immediately clear that no deterministic strategy can outperform all other deterministic strategies, but it seems like compelling evidence in Solomonoff induction's favor if it outperforms any randomized computable strategy.

To recap my position on this, intuitively it seems obvious that we don't want to assign a zero probability to the universe being uncomputable, but there are purportedly arguments showing that even if the universe is uncomputable, an AI programmed with the Solomonoff prior will do nearly as well as any human. I think there are valid counterarguments to all such arguments so far, so it seems reasonable to stick with the intuitive position until we have better arguments one way or another.

Regarding my version of Berry's Paradox, I really don't understand what it shows myself. But it seems clearly related to the fact that mathematical truth cannot be defined using any formal language. There's a cluster of related problems that are all perhaps waiting for some single major insight.

Game 3 leads to a question: Is there some way we can meaningfully define a sequence as non-hostile?

Game 1: on each round you and Omega state your probabilities that the next bit will be 1. The logarithm of the probability you assigned to the actual outcome gets added to your score.
[...]
As it turns out, in game 1 you cannot beat Omega by more than an additive constant, even if the input sequence is uncomputable and you know its definition.

I don't get this. A trivial algorithm always assigning p(0) = p(1) = 1/2 will have its score dropping by |log(1/2)| for each bit. At the same time, with the bit sequence maximally hostile to Omega, where b_{n+1} = ~OmegaPrediction((b_1...b_n)), Omega's score will always drop by at least |log(1/2)| and typically more, sometimes far more (whenever it hyper-computes any probabilities very far from 1/2).

Unless I'm missing something (as I probably am), how can the difference between the score of the trivial algorithm and Omega remain limited to "an additive constant" in any sense of the term?

There are serious weaknesses even over computable universes. Since Solomonoff induction doesn't search for ambient dependencies, considering instead explicit dependencies of reward on a particular action-symbol, it counts the same world multiple times if the agent is instantiated multiple times in it and receives the same prefix of the sequence of observations, and considers incorrect counterfactuals in these worlds, CDT-style. So two identical AIXI agents would defect in PD (CDT-style reasoning problem). In Newcomb's problem, its decision depends on the number of various explicit dependencies that carve the problem as two-boxing or one-boxing, so at least it's unclear what it'll do, and there seems to be no reason that it should reliably one-box. Given an anthropic problem where on a toss of a coin in one branch we create very many identical AIXI agents for the purpose of making some decision, while on the other branch there's only one agent (assuming the branches have sufficiently similar K-complexity), probability (complexity) of the branch with more agents would be double-counted as many times, so the consequences of actions in one-agent world won't be taken into consideration compared to consequences in the many-agents world. In Counterfactual Mugging, AIXI would not give money to Omega, since it updates. And so on.

Someone should really write up an UDT-AIXI that doesn't have these problems, I expect it should be straightforward enough (so that the problems like those I listed don't count against the arguments AIXI is used in, as a definition of impressive capability; make it stronger, not laugh at its flaws).

First let's set the input sequence to be very malevolent toward Omega: it will always say the reverse of what Omega is projected to say based on the previous bits.

This is probably a very uninformed question, but couldn't Omega break even vs. a intended-to-be malevolent bit stream by choosing a random sequence?

Suppose the data consisted of iid Bernoulli random variables which had probability 2/3 of being 1. In game 3, does Solomonoff induction's success rate converge to 2/3 (with probability 1)?