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How to Gamble Against All Odds

A decision maker observes the evolving state of the world while constantly trying to predict the next state given the history of past states. The ability to benefit from such predictions depends not only on the ability to recognize patters in history, but also on the range of actions available to the decision maker. We assume there are two possible states of the world. The decision maker is a gambler who has to bet a certain amount of money on the bits of an announced binary sequence of states. If he makes a correct prediction he wins his wager, otherwise he loses it. 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 (has no left-accumulation points). Our results generalize several recent results in algorithmic randomness and answer a question posed by Chalcraft et al. (2012).