Granularity of wagers in games and the (im)possibility of savings


In a casino where arbitrarily small bets are admissible, any betting strategy M can be modified into a savings strategy that not only is successful on each casino sequence where M is (thus accumulating unbounded wealth inside the casino) but also saves an unbounded capital, by permanently and gradually withdrawing it from the game. Teutsch showed that this is no longer the case when a minimum wager is imposed by the casino, thus exemplifying a savings paradox where a player can win unbounded wealth inside the casino, but upon withdrawing a sufficiently large amount out of the game, he is forced into bankruptcy. We characterize the rate at which a variable minimum wager should shrink in order for saving strategies to succeed, subject to successful betting: if the minimum wager at stage s shrinks faster than 1/s, then savings are possible; otherwise Teutsch’ savings paradox occurs.