Granularity of wagers in games and the possibility of saving

In a casino where arbitrarily small bets are admissible, any betting strategy M can be modified into a saving 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 fixed 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 study the potential for saving under a shrinking minimum wager rule (granularity) and its dependence on the rate of decrease (inflation) as well as timid versus bold play.