We show that if strong pseudo-random number generators exist, then betting games are equivalent to martingales, for measure on E and EXP. However, we construct betting games that succeed on certain classes whose Lutz measures are important open problems: the class of polynomial-time Turing-complete languages in EXP, and its superclass of polynomial-time Turing-autoreducible languages. If an EXP-martingale succeeds on either of these classes, or if betting games have the "finite union property" possessed by Lutz's measure, one obtains the non-relativizable consequence BPP <> EXP. We also show that if EXP <> MA, then the polynomial-time truth-table-autoreducible languages have Lutz measure zero, whereas if EXP = BPP, they have measure one.