Five minutes' thought allowed me to prove the following stupid "theorem":
Consider any game (haha). The only restriction is that the game must never make total wins go below zero, as in my game 2. Then there's a general-purpose winning agent: choose one strategy at the outset, sampled from the space of all computable strategies according to some distribution, and then follow it for all eternity. Obviously, this agent's expected accumulated utilities at all times cannot be worse than any individual strategy by more than a multiplicative constant, which is equal to that strategy's weight in the initial distribution.
Perhaps this result is easy in retrospect. Now I'd like to know what happens if utility can become negative (taking the exponent doesn't seem to work), and also how to improve the agent because it looks kinda stupid (even though it solves game 2 about as well as Solomonoff does). Sorry if this all sounds obvious, I've only been studying the topic for several days.
Can a computable human beat a Solomonoff hyperintelligence at making predictions about an incoming sequence of bits? If the sequence is computable, he probably can't. I'm interested in what happens when the sequence can be uncomputable. The answer depends on what you mean by "beat".
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. (This setup is designed to incentivize players to report their true beliefs, see Eliezer's technical explanation.)
Game 2: you both start with a given sum of money. On each round you're allowed to bet some of it on 0 or 1 at 1:1 odds. You cannot go below zero. (This is the "martingale game", for motivation see the section on "constructive martingales" in the Wikipedia article on Martin-Löf randomness.)
Game 3: on each round you call out 0 or 1 for the next bit. If you guess right, you win 1 dollar, otherwise you lose 1 dollar. Going below zero is allowed. (This simple game was suggested by Wei Dai in this thread on one-logic.)
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 have linked before to Shane Legg's text that can help you rederive this result.) Game 2 is a reformulation of game 1 in disguise, and you cannot beat Omega by more than a multiplicative constant. In game 3 you can beat Omega. More precisely, you can sometimes stay afloat while Omega sinks below zero at a linear rate.
Here's how. 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. As for the human, all he has to do is either always say 0 or always say 1. Intuitively it seems likely that at least one of those strategies will stay afloat, because whenever one of them sinks, the other rises.
So is Solomonoff induction really the shining jewel at the end of all science and progress, or does that depend on the payoff setup? It's not clear to me whether our own universe is computable. In the thread linked above Eliezer argued that we should be trying to approximate Solomonoff inference anyway:
Eliezer's argument sounds convincing, but to actually work it must rely on some prior over all of math, including uncomputable universes, to justify the rarity of such "devilish" or "adversarial" situations. I don't know of any prior over all of math, and Wei's restating of Berry's paradox (also linked above) seems to show that inventing such a prior is futile. So we seem to lack any formal justification for adopting the Solomonoff distribution in reasoning about physics etc., unless I'm being stupid and missing something really obvious again.
(posting this to discussion because I'm no longer convinced that my mathy posts belong in the toplevel)