How to gamble against all odds

We compare the power of betting strategies (aka martingales) whose wagers take values in different sets of reals. A martingale whose wagers take values in a set A is called an A-martingale. A set of reals B anticipates a set A, if for every A-martingale there is a countable set of B-martingales, such that on every binary sequence on which the A-martingale gains an infinite amount at least one of the B-martingales gains an infinite amount, too.We show that for two important classes of pairs of sets A and B, B anticipates A if and only if the closure of B contains rA, for some positive r. One class is when A is bounded and B is bounded away from zero; the other class is when B is well ordered. Our results generalize several recent results in algorithmic randomness and answer a question posed by Chalcraft et al. (2012).