But it's still possible that some M could dominate all semicomputable SWBIs in the weaker sense of losing less than epsilon per step, for all epsilon > 0.
You may already know that, but for the benefit of onlookers: multiplicative weights over F and -F is computable and kinda sorta dominates both F and -F in the weaker sense that you specified. You can make epsilon arbitrarily close to 0 by choosing a multiplier close to 1, at the cost of also incurring an additive constant loss that goes to infinity as epsilon->0. Multiplicative weights over all lower-semicomputable SWBIs dominates them all in the same sense, but it doesn't seem to be lower-semicomputable itself, only approximable (or maybe I just failed to find a proof).
This will be unparseable to anyone except maybe ten people here. What the hell, I'm posting it anyway.
Consider game 3 from my earlier post: a possibly uncomputable sequence of bits comes in, you're trying to guess the next bit, the payoff for each correct guess is $1. Obviously any deterministic strategy can be humiliated by a sequence of bits that's chosen to make it always guess wrong. User paulfchristiano suggested that multiplicative weights could yield a randomized strategy that does at most a constant worse than any human in expectation. I noted that, unlike the Solomonoff mixture, multiplicative weights over lower-semicomputable randomized strategies is not itself lower-semicomputable (several pages of algebra omitted), so we don't have a strategy that's optimal within its own class. Then I found some slides by Hutter that include a handy table showing how unusual it is for a class do be dominated by a measure within that same class.
So here's the question: is there an analog of "dominant lower-semicomputable semimeasure" for some class of games other than the log-score game? Desiderata: I guess it must correspond to randomized strategies (because deterministic ones are not enough), it must play at most an additive constant worse than any human in expectation (or maybe a multiplicative constant that can be made arbitrarily close to 1 by parameter choice, as with multiplicative weights), and it must be good within its own class of computability (not just superior to all members of some lower class). As far as I can tell, the literature has no answer to this question, and I already spent about a week on it with no success. Anyone here have more understanding than me?