Sorry for the many replies, your comment continues to intrigue me.

Game 2 is different from games 1 and 3 because the available options/payoffs in game 2 also depend on your current utility or previous betting history - you cannot bet more than you've won so far. In general, if such things are allowed, there's no predictor that can win all games up to an additive or multiplicative constant, even when payoffs are bounded. Here's a game that shows why: on round 1 the player chooses x=1 or x=-1, then the universe chooses y=1 or y=-1, then the game effectively ends and the player receives utility x*y on every round thereafter. Whatever mixture of x=1 and x=-1 is chosen by your proposed predictor on the first round, the universe can choose y so as to make you go into the negatives or 0 in expectation, yet there exists a computable human that receives linearly increasing utility in that universe.

Note that my reasoning in the previous comment solves game 2 just fine, but it doesn't work for game 1 or game 3 because they can go into negative utilities. Maybe we're facing different classes of game that call for different solutions? Right now it seems we have a general theorem for game 1 (log score - regular Solomonoff induction), another general theorem for games of type 2 (positive utilities - my other comment), and a third general theorem for games of type 3 (bounded payoffs - your claimed result). We ought to unify some of them.